Subgroups — C₇×#60168 #13237384

Order 1792.

2882 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic
2	C₇×#3428	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×#518	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#2949	C₂	1	4	yes	maximal, complemented
5	C₇×#2948	C₂	1	4	yes	maximal, complemented
6	C₇×#2946	C₂	1	8	yes	maximal, complemented
7	C₄×C₇×AGL(1,ℤ/8)	C₂	1	8	yes	maximal, complemented
8	C₇×#2911	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#2896	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₇×#418	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#2873	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₇×#41	C₂²	1	1	yes	characteristic
13	C₂³×C₇×D₄	C₂²	1	1	yes	characteristic
14	C₂×C₇×AGL(1,ℤ/8)	C₄	1	16	yes	complemented
15	C₂×C₇×AGL(1,ℤ/8)	C₂²	1	4	yes	—
16	C₂×C₇×AGL(1,ℤ/8)	C₂²	1	4	yes	—
17	C₂²×C₇×SD₁₆	C₂²	1	2	yes	—
18	C₂²×C₇×D₈	C₂²	1	2	yes	—
19	C₂²×C₇×M₁₆	C₂²	1	1	yes	characteristic
20	C₂²×C₇×M₁₆	C₂²	1	1	yes	characteristic
21	C₇×#520	—	2	2	no	—
22	C₇×#520	C₂²	1	4	yes	complemented
23	C₇×#519	C₂²	1	2	yes	complemented
24	C₇×#518	—	2	2	no	—
25	C₇×#518	—	2	2	no	—
26	C₇×#518	—	2	2	no	—
27	C₇×#518	C₂²	1	4	yes	—
28	C₇×#518	C₂²	1	4	yes	complemented
29	C₇×#518	C₂²	1	2	yes	complemented
30	C₄×C₇×#41	—	2	2	no	—
31	C₄×C₇×#41	C₂²	1	4	yes	complemented
32	C₂×C₄×C₇×D₄	—	2	2	no	—
33	C₂×C₄×C₇×D₄	C₂²	1	4	yes	complemented
34	C₂×C₇×#117	—	2	2	no	—
35	C₂×C₇×#117	C₂²	1	1	yes	characteristic
36	C₂×C₇×#117	C₂²	1	1	yes	characteristic
37	C₂×C₇×#117	C₂²	1	1	yes	characteristic
38	C₂²×C₇×C₂²⋊C₄	—	2	2	no	—
39	C₇×#442	C₂²	1	16	yes	complemented
40	C₇×#440	C₂²	1	16	yes	complemented
41	C₄×C₇×SD₁₆	C₂²	1	16	yes	complemented
42	C₄×C₇×D₈	C₂²	1	16	yes	complemented
43	C₇×#428	C₂²	1	4	yes	complemented
44	C₇×#428	C₂²	1	4	yes	complemented
45	C₇×#427	C₂²	1	4	yes	complemented
46	C₇×#419	C₂²	1	2	yes	complemented
47	C₇×#418	C₂²	1	4	yes	—
48	C₇×#418	C₂²	1	4	yes	complemented
49	C₇×#418	C₂²	1	2	yes	complemented
50	C₇×#417	C₂²	1	4	yes	complemented
51	C₇×#416	C₂²	1	4	yes	complemented
52	C₂×C₇×#102	C₂²	1	4	yes	complemented
53	C₇×#405	C₂²	1	4	yes	complemented
54	C₇×#405	C₂²	1	4	yes	complemented
55	C₄×C₇×M₁₆	C₂²	1	4	yes	complemented
56	#60168	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂⁵×C₇	—	2	1	no	—
58	C₂×C₇×#41	—	2	1	no	—
59	C₂×C₇×#41	—	2	1	no	—
60	C₂×C₇×#41	—	2	4	no	—
61	C₂×C₇×#41	C₂×C₄	1	8	yes	complemented
62	C₂×C₇×#41	C₂³	1	2	yes	—
63	C₂×C₇×#41	C₂³	1	2	yes	—
64	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
65	C₂²×C₇×D₄	—	2	1	no	—
66	C₂²×C₇×D₄	—	2	1	no	—
67	C₂²×C₇×D₄	—	2	1	no	—
68	C₂²×C₇×D₄	—	2	1	no	—
69	C₂²×C₇×D₄	—	2	4	no	—
70	C₂²×C₇×D₄	C₂×C₄	1	8	yes	complemented
71	C₂²×C₇×D₄	C₂³	1	2	yes	—
72	C₂²×C₇×D₄	C₂³	1	2	yes	—
73	C₂²×C₇×D₄	C₂³	1	2	yes	—
74	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
75	C₂³×C₄×C₇	—	2	1	no	—
76	C₂³×C₄×C₇	—	2	4	no	—
77	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
78	C₇×AGL(1,ℤ/8)	C₂×C₄	1	32	yes	complemented
79	C₇×AGL(1,ℤ/8)	C₂×C₄	1	32	yes	complemented
80	C₂×C₇×SD₁₆	C₂×C₄	1	16	yes	complemented
81	C₂×C₇×SD₁₆	C₂³	1	8	yes	—
82	C₂×C₇×D₈	C₂×C₄	1	16	yes	complemented
83	C₂×C₇×D₈	C₂³	1	8	yes	—
84	C₂×C₇×M₁₆	—	2	4	no	—
85	C₂×C₇×M₁₆	C₂×C₄	1	8	yes	complemented
86	C₂×C₇×M₁₆	C₂³	1	2	yes	—
87	C₂×C₇×M₁₆	C₂³	1	2	yes	—
88	C₂×C₇×M₁₆	C₂³	1	2	yes	—
89	C₂×C₇×M₁₆	C₂³	1	2	yes	—
90	C₂²×C₇×C₈	C₂³	1	2	yes	—
91	C₂²×C₇×C₈	C₂³	1	2	yes	—
92	C₄×C₇×Q₈	—	2	4	no	—
93	C₄×C₇×Q₈	C₂³	1	8	yes	complemented
94	C₄×C₇×D₄	—	2	8	no	—
95	C₄×C₇×D₄	—	2	8	no	—
96	C₄×C₇×D₄	—	2	8	no	—
97	C₄×C₇×D₄	—	2	4	no	—
98	C₄×C₇×D₄	—	2	8	no	—
99	C₄×C₇×D₄	—	2	8	no	—
100	C₄×C₇×D₄	C₂³	1	16	yes	—
101	C₄×C₇×D₄	C₂³	1	8	yes	complemented
102	C₇×#117	—	2	1	no	—
103	C₇×#117	—	2	1	no	—
104	C₇×#117	—	2	1	no	—
105	C₇×#117	—	2	1	no	—
106	C₇×#117	—	2	4	no	—
107	C₇×#117	—	2	4	no	—
108	C₇×#117	—	2	4	no	—
109	C₇×#117	D₄	1	2	yes	—
110	C₇×#117	D₄	1	2	yes	—
111	C₇×#117	C₂³	1	2	yes	—
112	C₇×#117	C₂³	1	2	yes	—
113	C₇×#117	C₂³	1	2	yes	—
114	C₇×#117	C₂³	1	2	yes	—
115	C₇×#117	C₂³	1	2	yes	—
116	C₇×#117	C₂³	1	2	yes	—
117	C₂×C₇×C₄⋊C₄	—	2	4	no	—
118	C₂×C₇×C₄⋊C₄	D₄	1	2	yes	—
119	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
120	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
121	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
122	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
123	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
124	C₂×C₇×C₂²⋊C₄	—	2	8	no	—
125	C₂×C₇×C₂²⋊C₄	—	2	8	no	—
126	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
127	C₂×C₇×C₂²⋊C₄	D₄	1	2	yes	—
128	C₂×C₄²×C₇	—	2	1	no	—
129	C₂×C₄²×C₇	—	2	1	no	—
130	C₂×C₄²×C₇	—	2	4	no	—
131	C₂×C₄²×C₇	C₂³	1	2	yes	—
132	C₇×#107	C₂³	1	8	yes	complemented
133	C₇×#106	C₂³	1	8	yes	complemented
134	C₇×#103	C₂³	1	8	yes	complemented
135	C₇×#102	C₂³	1	16	yes	—
136	C₇×#102	C₂³	1	8	yes	complemented
137	C₇×#97	C₂³	1	8	yes	complemented
138	C₄×C₇×C₈	C₂³	1	8	yes	complemented
139	C₂²×AGL(1,ℤ/8)	C₂×C₇	1	1	yes	characteristic
140	#3428	C₂×C₇	1	1	yes	characteristic, complemented
141	C₂×#518	C₂×C₇	1	1	yes	characteristic, complemented
142	#2949	C₂×C₇	1	4	yes	complemented
143	#2948	C₂×C₇	1	4	yes	complemented
144	#2946	C₂×C₇	1	8	yes	complemented
145	C₄×AGL(1,ℤ/8)	C₂×C₇	1	8	yes	complemented
146	#2911	C₂×C₇	1	1	yes	characteristic, complemented
147	#2896	C₂×C₇	1	1	yes	characteristic, complemented
148	C₂×#418	C₂×C₇	1	1	yes	characteristic, complemented
149	#2873	C₂×C₇	1	1	yes	characteristic, complemented
150	C₂⁴×C₇	—	2	2	no	—
151	C₂⁴×C₇	—	2	8	no	—
152	C₂⁴×C₇	—	2	1	no	—
153	C₂⁴×C₇	—	2	2	no	—
154	C₂⁴×C₇	—	2	2	no	—
155	C₂⁴×C₇	—	8	4	no	—
156	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
157	C₇×#41	—	2	8	no	—
158	C₇×#41	—	2	8	no	—
159	C₇×#41	—	2	4	no	—
160	C₇×#41	—	2	4	no	—
161	C₇×#41	C₂²×C₄	1	8	yes	—
162	C₇×#41	C₂²×C₄	1	8	yes	complemented
163	C₂×C₇×Q₈	—	2	1	no	—
164	C₂×C₇×Q₈	—	2	2	no	—
165	C₂×C₇×Q₈	C₂²×C₄	1	4	yes	complemented
166	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
167	C₂×C₇×D₄	—	2	8	no	—
168	C₂×C₇×D₄	—	2	2	no	—
169	C₂×C₇×D₄	—	2	2	no	—
170	C₂×C₇×D₄	—	2	8	no	—
171	C₂×C₇×D₄	—	2	8	no	—
172	C₂×C₇×D₄	—	2	8	no	—
173	C₂×C₇×D₄	—	2	2	no	—
174	C₂×C₇×D₄	—	2	2	no	—
175	C₂×C₇×D₄	—	2	2	no	—
176	C₂×C₇×D₄	—	2	1	no	—
177	C₂×C₇×D₄	—	2	4	no	—
178	C₂×C₇×D₄	—	2	4	no	—
179	C₂×C₇×D₄	—	2	4	no	—
180	C₂×C₇×D₄	—	2	2	no	—
181	C₂×C₇×D₄	C₂²×C₄	1	8	yes	—
182	C₂×C₇×D₄	C₂²×C₄	1	8	yes	—
183	C₂×C₇×D₄	C₂²×C₄	1	8	yes	—
184	C₂×C₇×D₄	C₂²×C₄	1	4	yes	complemented
185	C₂×C₇×D₄	C₂⁴	1	4	yes	—
186	C₂×C₇×D₄	C₂⁴	1	2	yes	—
187	C₂²×C₄×C₇	—	2	1	no	—
188	C₂²×C₄×C₇	—	2	8	no	—
189	C₂²×C₄×C₇	—	2	2	no	—
190	C₂²×C₄×C₇	—	2	2	no	—
191	C₂²×C₄×C₇	—	2	2	no	—
192	C₂²×C₄×C₇	—	2	2	no	—
193	C₂²×C₄×C₇	—	2	8	no	—
194	C₂²×C₄×C₇	—	2	16	no	—
195	C₂²×C₄×C₇	—	8	8	no	—
196	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
197	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
198	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
199	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
200	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
201	C₂²×C₄×C₇	#41	1	4	yes	—
202	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
203	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
204	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
205	C₇×SD₁₆	C₂²×C₄	1	32	yes	complemented
206	C₇×D₈	C₂²×C₄	1	32	yes	complemented
207	C₇×M₁₆	—	2	4	no	—
208	C₇×M₁₆	—	2	4	no	—
209	C₇×M₁₆	C₂²×C₄	1	8	yes	—
210	C₇×M₁₆	C₂²×C₄	1	8	yes	complemented
211	C₂×C₇×C₈	—	2	4	no	—
212	C₂×C₇×C₈	C₂²×C₄	1	8	yes	complemented
213	C₂×C₇×C₈	C₂⁴	1	4	yes	—
214	C₂×C₇×C₈	C₂⁴	1	4	yes	—
215	C₇×C₄⋊C₄	—	2	2	no	—
216	C₇×C₄⋊C₄	—	2	8	no	—
217	C₇×C₄⋊C₄	—	2	2	no	—
218	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
219	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
220	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
221	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
222	C₇×C₂²⋊C₄	—	2	2	no	—
223	C₇×C₂²⋊C₄	—	2	2	no	—
224	C₇×C₂²⋊C₄	—	2	2	no	—
225	C₇×C₂²⋊C₄	—	2	2	no	—
226	C₇×C₂²⋊C₄	—	2	16	no	—
227	C₇×C₂²⋊C₄	—	2	8	no	—
228	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
229	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
230	C₄²×C₇	—	2	2	no	—
231	C₄²×C₇	—	2	8	no	—
232	C₄²×C₇	—	2	2	no	—
233	C₄²×C₇	—	2	2	no	—
234	C₄²×C₇	—	2	2	no	—
235	C₄²×C₇	C₂×D₄	1	4	yes	—
236	C₄²×C₇	C₂⁴	1	4	yes	—
237	C₂²×#41	C₂²×C₇	1	1	yes	characteristic
238	C₂³×D₄	C₂²×C₇	1	1	yes	characteristic
239	C₂×AGL(1,ℤ/8)	C₄×C₇	1	16	yes	complemented
240	C₂×AGL(1,ℤ/8)	C₂²×C₇	1	4	yes	—
241	C₂×AGL(1,ℤ/8)	C₂²×C₇	1	4	yes	—
242	C₂²×SD₁₆	C₂²×C₇	1	2	yes	—
243	C₂²×D₈	C₂²×C₇	1	2	yes	—
244	C₂²×M₁₆	C₂²×C₇	1	1	yes	characteristic
245	C₂²×M₁₆	C₂²×C₇	1	1	yes	characteristic
246	#520	—	2	2	no	—
247	#520	C₂²×C₇	1	4	yes	complemented
248	#519	C₂²×C₇	1	2	yes	complemented
249	#518	—	2	2	no	—
250	#518	—	2	2	no	—
251	#518	—	2	2	no	—
252	#518	C₂²×C₇	1	4	yes	—
253	#518	C₂²×C₇	1	4	yes	complemented
254	#518	C₂²×C₇	1	2	yes	complemented
255	C₄×#41	—	2	2	no	—
256	C₄×#41	C₂²×C₇	1	4	yes	complemented
257	C₂×C₄×D₄	—	2	2	no	—
258	C₂×C₄×D₄	C₂²×C₇	1	4	yes	complemented
259	C₂×#117	—	2	2	no	—
260	C₂×#117	C₂²×C₇	1	1	yes	characteristic
261	C₂×#117	C₂²×C₇	1	1	yes	characteristic
262	C₂×#117	C₂²×C₇	1	1	yes	characteristic
263	C₂²×C₂²⋊C₄	—	2	2	no	—
264	#442	C₂²×C₇	1	16	yes	complemented
265	#440	C₂²×C₇	1	16	yes	complemented
266	C₄×SD₁₆	C₂²×C₇	1	16	yes	complemented
267	C₄×D₈	C₂²×C₇	1	16	yes	complemented
268	#428	C₂²×C₇	1	4	yes	complemented
269	#428	C₂²×C₇	1	4	yes	complemented
270	#427	C₂²×C₇	1	4	yes	complemented
271	#419	C₂²×C₇	1	2	yes	complemented
272	#418	C₂²×C₇	1	4	yes	—
273	#418	C₂²×C₇	1	4	yes	complemented
274	#418	C₂²×C₇	1	2	yes	complemented
275	#417	C₂²×C₇	1	4	yes	complemented
276	#416	C₂²×C₇	1	4	yes	complemented
277	C₂×#102	C₂²×C₇	1	4	yes	complemented
278	#405	C₂²×C₇	1	4	yes	complemented
279	#405	C₂²×C₇	1	4	yes	complemented
280	C₄×M₁₆	C₂²×C₇	1	4	yes	complemented
281	C₂³×C₇	—	2	8	no	—
282	C₂³×C₇	—	2	2	no	—
283	C₂³×C₇	—	2	4	no	—
284	C₂³×C₇	—	2	4	no	—
285	C₂³×C₇	—	2	8	no	—
286	C₂³×C₇	—	2	8	no	—
287	C₂³×C₇	—	4	2	no	—
288	C₂³×C₇	—	4	8	no	—
289	C₂³×C₇	—	8	2	no	—
290	C₂³×C₇	—	8	4	no	—
291	C₂³×C₇	—	8	4	no	—
292	C₂³×C₇	—	8	16	no	—
293	C₂³×C₇	C₄×D₄	1	4	yes	—
294	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
295	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
296	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
297	C₇×Q₈	—	2	4	no	—
298	C₇×Q₈	—	2	2	no	—
299	C₇×Q₈	C₂³×C₄	1	4	yes	—
300	C₇×D₄	—	2	8	no	—
301	C₇×D₄	—	2	8	no	—
302	C₇×D₄	—	2	8	no	—
303	C₇×D₄	—	2	8	no	—
304	C₇×D₄	—	2	8	no	—
305	C₇×D₄	—	2	4	no	—
306	C₇×D₄	—	2	4	no	—
307	C₇×D₄	—	2	2	no	—
308	C₇×D₄	C₂³×C₄	1	4	yes	—
309	C₇×D₄	C₂³×C₄	1	8	yes	—
310	C₂×C₄×C₇	—	2	4	no	—
311	C₂×C₄×C₇	—	2	2	no	—
312	C₂×C₄×C₇	—	2	8	no	—
313	C₂×C₄×C₇	—	2	8	no	—
314	C₂×C₄×C₇	—	2	4	no	—
315	C₂×C₄×C₇	—	2	4	no	—
316	C₂×C₄×C₇	—	4	4	no	—
317	C₂×C₄×C₇	—	4	4	no	—
318	C₂×C₄×C₇	—	8	8	no	—
319	C₂×C₄×C₇	—	8	16	no	—
320	C₂×C₄×C₇	—	8	2	no	—
321	C₂×C₄×C₇	C₄×D₄	1	4	yes	—
322	C₂×C₄×C₇	C₄×D₄	1	4	yes	—
323	C₂×C₄×C₇	C₄×D₄	1	4	yes	—
324	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
325	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
326	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
327	C₂×C₄×C₇	C₂²×D₄	1	8	yes	—
328	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
329	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
330	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
331	C₂×C₄×C₇	C₂×#41	1	2	yes	—
332	C₂×C₄×C₇	C₂×#41	1	2	yes	—
333	C₂×C₄×C₇	C₂×#41	1	2	yes	—
334	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
335	C₇×C₈	—	2	4	no	—
336	C₇×C₈	C₂³×C₄	1	8	yes	—
337	C₂⁵	—	2	1	no	—
338	C₂×#41	—	2	1	no	—
339	C₂×#41	—	2	1	no	—
340	C₂×#41	—	2	4	no	—
341	C₂×#41	C₂×C₄×C₇	1	8	yes	complemented
342	C₂×#41	C₂³×C₇	1	2	yes	—
343	C₂×#41	C₂³×C₇	1	2	yes	—
344	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
345	C₂²×D₄	—	2	1	no	—
346	C₂²×D₄	—	2	1	no	—
347	C₂²×D₄	—	2	1	no	—
348	C₂²×D₄	—	2	1	no	—
349	C₂²×D₄	—	2	4	no	—
350	C₂²×D₄	C₂×C₄×C₇	1	8	yes	complemented
351	C₂²×D₄	C₂³×C₇	1	2	yes	—
352	C₂²×D₄	C₂³×C₇	1	2	yes	—
353	C₂²×D₄	C₂³×C₇	1	2	yes	—
354	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
355	C₂³×C₄	—	2	1	no	—
356	C₂³×C₄	—	2	4	no	—
357	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
358	AGL(1,ℤ/8)	C₂×C₄×C₇	1	32	yes	complemented
359	AGL(1,ℤ/8)	C₂×C₄×C₇	1	32	yes	complemented
360	C₂×SD₁₆	C₂×C₄×C₇	1	16	yes	complemented
361	C₂×SD₁₆	C₂³×C₇	1	8	yes	—
362	C₂×D₈	C₂×C₄×C₇	1	16	yes	complemented
363	C₂×D₈	C₂³×C₇	1	8	yes	—
364	C₂×M₁₆	—	2	4	no	—
365	C₂×M₁₆	C₂×C₄×C₇	1	8	yes	complemented
366	C₂×M₁₆	C₂³×C₇	1	2	yes	—
367	C₂×M₁₆	C₂³×C₇	1	2	yes	—
368	C₂×M₁₆	C₂³×C₇	1	2	yes	—
369	C₂×M₁₆	C₂³×C₇	1	2	yes	—
370	C₂²×C₈	C₂³×C₇	1	2	yes	—
371	C₂²×C₈	C₂³×C₇	1	2	yes	—
372	C₄×Q₈	—	2	4	no	—
373	C₄×Q₈	C₂³×C₇	1	8	yes	complemented
374	C₄×D₄	—	2	8	no	—
375	C₄×D₄	—	2	8	no	—
376	C₄×D₄	—	2	8	no	—
377	C₄×D₄	—	2	4	no	—
378	C₄×D₄	—	2	8	no	—
379	C₄×D₄	—	2	8	no	—
380	C₄×D₄	C₂³×C₇	1	16	yes	—
381	C₄×D₄	C₂³×C₇	1	8	yes	complemented
382	#117	—	2	1	no	—
383	#117	—	2	1	no	—
384	#117	—	2	1	no	—
385	#117	—	2	1	no	—
386	#117	—	2	4	no	—
387	#117	—	2	4	no	—
388	#117	—	2	4	no	—
389	#117	C₇×D₄	1	2	yes	—
390	#117	C₇×D₄	1	2	yes	—
391	#117	C₂³×C₇	1	2	yes	—
392	#117	C₂³×C₇	1	2	yes	—
393	#117	C₂³×C₇	1	2	yes	—
394	#117	C₂³×C₇	1	2	yes	—
395	#117	C₂³×C₇	1	2	yes	—
396	#117	C₂³×C₇	1	2	yes	—
397	C₂×C₄⋊C₄	—	2	4	no	—
398	C₂×C₄⋊C₄	C₇×D₄	1	2	yes	—
399	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
400	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
401	C₂×C₂²⋊C₄	—	2	1	no	—
402	C₂×C₂²⋊C₄	—	2	1	no	—
403	C₂×C₂²⋊C₄	—	2	4	no	—
404	C₂×C₂²⋊C₄	—	2	8	no	—
405	C₂×C₂²⋊C₄	—	2	8	no	—
406	C₂×C₂²⋊C₄	—	2	4	no	—
407	C₂×C₂²⋊C₄	C₇×D₄	1	2	yes	—
408	C₂×C₄²	—	2	1	no	—
409	C₂×C₄²	—	2	1	no	—
410	C₂×C₄²	—	2	4	no	—
411	C₂×C₄²	C₂³×C₇	1	2	yes	—
412	#107	C₂³×C₇	1	8	yes	complemented
413	#106	C₂³×C₇	1	8	yes	complemented
414	#103	C₂³×C₇	1	8	yes	complemented
415	#102	C₂³×C₇	1	16	yes	—
416	#102	C₂³×C₇	1	8	yes	complemented
417	#97	C₂³×C₇	1	8	yes	complemented
418	C₄×C₈	C₂³×C₇	1	8	yes	complemented
419	C₂²×C₇	—	2	2	no	—
420	C₂²×C₇	—	2	2	no	—
421	C₂²×C₇	—	2	2	no	—
422	C₂²×C₇	—	2	4	no	—
423	C₂²×C₇	—	2	8	no	—
424	C₂²×C₇	—	4	4	no	—
425	C₂²×C₇	—	4	16	no	—
426	C₂²×C₇	—	4	8	no	—
427	C₂²×C₇	—	4	4	no	—
428	C₂²×C₇	—	8	4	no	—
429	C₂²×C₇	—	8	8	no	—
430	C₂²×C₇	—	8	8	no	—
431	C₂²×C₇	C₂×C₄×D₄	1	2	yes	—
432	C₂²×C₇	C₂×C₄×D₄	1	2	yes	—
433	C₂²×C₇	C₂×C₄×D₄	1	2	yes	—
434	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
435	C₄×C₇	—	2	8	no	—
436	C₄×C₇	—	2	4	no	—
437	C₄×C₇	—	4	4	no	—
438	C₄×C₇	—	4	4	no	—
439	C₄×C₇	C₂×C₄×D₄	1	2	yes	—
440	C₄×C₇	C₂×C₄×D₄	1	2	yes	—
441	C₄×C₇	C₂×C₄×D₄	1	2	yes	—
442	C₄×C₇	C₂⁴×C₄	1	1	yes	characteristic
443	C₄×C₇	C₂²×#41	1	1	yes	characteristic
444	C₂⁴	—	2	2	no	—
445	C₂⁴	—	2	8	no	—
446	C₂⁴	—	2	1	no	—
447	C₂⁴	—	2	2	no	—
448	C₂⁴	—	2	2	no	—
449	C₂⁴	—	8	4	no	—
450	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
451	#41	—	2	8	no	—
452	#41	—	2	8	no	—
453	#41	—	2	4	no	—
454	#41	—	2	4	no	—
455	#41	C₂²×C₄×C₇	1	8	yes	—
456	#41	C₂²×C₄×C₇	1	8	yes	complemented
457	C₂×Q₈	—	2	1	no	—
458	C₂×Q₈	—	2	2	no	—
459	C₂×Q₈	C₂²×C₄×C₇	1	4	yes	complemented
460	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
461	C₂×D₄	—	2	8	no	—
462	C₂×D₄	—	2	2	no	—
463	C₂×D₄	—	2	2	no	—
464	C₂×D₄	—	2	8	no	—
465	C₂×D₄	—	2	8	no	—
466	C₂×D₄	—	2	8	no	—
467	C₂×D₄	—	2	2	no	—
468	C₂×D₄	—	2	2	no	—
469	C₂×D₄	—	2	2	no	—
470	C₂×D₄	—	2	1	no	—
471	C₂×D₄	—	2	4	no	—
472	C₂×D₄	—	2	4	no	—
473	C₂×D₄	—	2	4	no	—
474	C₂×D₄	—	2	2	no	—
475	C₂×D₄	C₂²×C₄×C₇	1	8	yes	—
476	C₂×D₄	C₂²×C₄×C₇	1	8	yes	—
477	C₂×D₄	C₂²×C₄×C₇	1	8	yes	—
478	C₂×D₄	C₂²×C₄×C₇	1	4	yes	complemented
479	C₂×D₄	C₂⁴×C₇	1	4	yes	—
480	C₂×D₄	C₂⁴×C₇	1	2	yes	—
481	C₂²×C₄	—	2	1	no	—
482	C₂²×C₄	—	2	8	no	—
483	C₂²×C₄	—	2	2	no	—
484	C₂²×C₄	—	2	2	no	—
485	C₂²×C₄	—	2	2	no	—
486	C₂²×C₄	—	2	2	no	—
487	C₂²×C₄	—	2	8	no	—
488	C₂²×C₄	—	2	16	no	—
489	C₂²×C₄	—	8	8	no	—
490	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
491	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
492	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
493	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
494	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
495	C₂²×C₄	C₇×#41	1	4	yes	—
496	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
497	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
498	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
499	SD₁₆	C₂²×C₄×C₇	1	32	yes	complemented
500	D₈	C₂²×C₄×C₇	1	32	yes	complemented
501	M₁₆	—	2	4	no	—
502	M₁₆	—	2	4	no	—
503	M₁₆	C₂²×C₄×C₇	1	8	yes	—
504	M₁₆	C₂²×C₄×C₇	1	8	yes	complemented
505	C₂×C₈	—	2	4	no	—
506	C₂×C₈	C₂²×C₄×C₇	1	8	yes	complemented
507	C₂×C₈	C₂⁴×C₇	1	4	yes	—
508	C₂×C₈	C₂⁴×C₇	1	4	yes	—
509	C₄⋊C₄	—	2	2	no	—
510	C₄⋊C₄	—	2	8	no	—
511	C₄⋊C₄	—	2	2	no	—
512	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
513	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
514	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
515	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
516	C₂²⋊C₄	—	2	2	no	—
517	C₂²⋊C₄	—	2	2	no	—
518	C₂²⋊C₄	—	2	2	no	—
519	C₂²⋊C₄	—	2	2	no	—
520	C₂²⋊C₄	—	2	16	no	—
521	C₂²⋊C₄	—	2	8	no	—
522	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
523	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
524	C₄²	—	2	2	no	—
525	C₄²	—	2	8	no	—
526	C₄²	—	2	2	no	—
527	C₄²	—	2	2	no	—
528	C₄²	—	2	2	no	—
529	C₄²	C₂×C₇×D₄	1	4	yes	—
530	C₄²	C₂⁴×C₇	1	4	yes	—
531	C₂×C₇	—	2	2	no	—
532	C₂×C₇	—	2	2	no	—
533	C₂×C₇	—	2	2	no	—
534	C₂×C₇	—	4	8	no	—
535	C₂×C₇	—	4	4	no	—
536	C₂×C₇	C₂²×C₄×D₄	1	1	yes	characteristic
537	C₂×C₇	#3585	1	2	yes	—
538	C₂³	—	2	2	no	—
539	C₂³	—	2	4	no	—
540	C₂³	—	2	8	no	—
541	C₂³	—	2	4	no	—
542	C₂³	—	2	8	no	—
543	C₂³	—	2	8	no	—
544	C₂³	—	4	2	no	—
545	C₂³	—	4	8	no	—
546	C₂³	—	8	2	no	—
547	C₂³	—	8	4	no	—
548	C₂³	—	8	4	no	—
549	C₂³	—	8	16	no	—
550	C₂³	C₄×C₇×D₄	1	4	yes	—
551	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
552	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
553	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
554	Q₈	—	2	2	no	—
555	Q₈	—	2	4	no	—
556	Q₈	C₂³×C₄×C₇	1	4	yes	—
557	D₄	—	2	8	no	—
558	D₄	—	2	8	no	—
559	D₄	—	2	4	no	—
560	D₄	—	2	2	no	—
561	D₄	—	2	8	no	—
562	D₄	—	2	8	no	—
563	D₄	—	2	8	no	—
564	D₄	—	2	4	no	—
565	D₄	C₂³×C₄×C₇	1	4	yes	—
566	D₄	C₂³×C₄×C₇	1	8	yes	—
567	C₂×C₄	—	2	2	no	—
568	C₂×C₄	—	2	4	no	—
569	C₂×C₄	—	2	8	no	—
570	C₂×C₄	—	2	8	no	—
571	C₂×C₄	—	2	4	no	—
572	C₂×C₄	—	2	4	no	—
573	C₂×C₄	—	4	4	no	—
574	C₂×C₄	—	4	4	no	—
575	C₂×C₄	—	8	8	no	—
576	C₂×C₄	—	8	16	no	—
577	C₂×C₄	—	8	2	no	—
578	C₂×C₄	C₄×C₇×D₄	1	4	yes	—
579	C₂×C₄	C₄×C₇×D₄	1	4	yes	—
580	C₂×C₄	C₄×C₇×D₄	1	4	yes	—
581	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
582	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
583	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
584	C₂×C₄	C₂²×C₇×D₄	1	8	yes	—
585	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
586	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
587	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
588	C₂×C₄	C₂×C₇×#41	1	2	yes	—
589	C₂×C₄	C₂×C₇×#41	1	2	yes	—
590	C₂×C₄	C₂×C₇×#41	1	2	yes	—
591	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
592	C₈	—	2	4	no	—
593	C₈	C₂³×C₄×C₇	1	8	yes	—
594	C₇	#60168	1	1	yes	characteristic, direct product factor, Sylow
595	C₂²	—	2	2	no	—
596	C₂²	—	2	2	no	—
597	C₂²	—	2	2	no	—
598	C₂²	—	2	4	no	—
599	C₂²	—	2	8	no	—
600	C₂²	—	4	4	no	—
601	C₂²	—	4	16	no	—
602	C₂²	—	4	8	no	—
603	C₂²	—	4	4	no	—
604	C₂²	—	8	8	no	—
605	C₂²	—	8	8	no	—
606	C₂²	—	8	4	no	—
607	C₂²	C₂×C₄×C₇×D₄	1	2	yes	—
608	C₂²	C₂×C₄×C₇×D₄	1	2	yes	—
609	C₂²	C₂×C₄×C₇×D₄	1	2	yes	—
610	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
611	C₄	—	2	8	no	—
612	C₄	—	2	4	no	—
613	C₄	—	4	4	no	—
614	C₄	—	4	4	no	—
615	C₄	C₂×C₄×C₇×D₄	1	2	yes	—
616	C₄	C₂×C₄×C₇×D₄	1	2	yes	—
617	C₄	C₂×C₄×C₇×D₄	1	2	yes	—
618	C₄	C₂⁴×C₄×C₇	1	1	yes	characteristic
619	C₄	C₂²×C₇×#41	1	1	yes	characteristic
620	C₂	—	2	2	no	—
621	C₂	—	2	2	no	—
622	C₂	—	2	2	no	—
623	C₂	—	4	8	no	—
624	C₂	—	4	4	no	—
625	C₂	C₂²×C₄×C₇×D₄	1	1	yes	characteristic
626	C₂	C₇×#3585	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂³×D₄
commutator	C₄	C₂⁴×C₄×C₇
Frattini, cosocle	C₂×C₄	C₂⁵×C₇
prime radical	C₂²×C₇×AGL(1,ℤ/8)	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₇×#117 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₇×#3428	C₂	semidirect
3. C₂×C₇×#518	C₂	semidirect
4. C₇×#2949	C₂	semidirect
5. C₇×#2948	C₂	semidirect
6. C₇×#2946	C₂	semidirect
7. C₄×C₇×AGL(1,ℤ/8)	C₂	semidirect
8. C₇×#2911	C₂	semidirect
9. C₇×#2896	C₂	semidirect
10. C₂×C₇×#418	C₂	semidirect
11. C₇×#2873	C₂	semidirect
14. C₂×C₇×AGL(1,ℤ/8)	C₄	semidirect
22. C₇×#520	C₂²	semidirect
23. C₇×#519	C₂²	semidirect
28. C₇×#518	C₂²	semidirect
29. C₇×#518	C₂²	semidirect
31. C₄×C₇×#41	C₂²	semidirect
33. C₂×C₄×C₇×D₄	C₂²	semidirect
39. C₇×#442	C₂²	semidirect
40. C₇×#440	C₂²	semidirect
41. C₄×C₇×SD₁₆	C₂²	semidirect
42. C₄×C₇×D₈	C₂²	semidirect
43. C₇×#428	C₂²	semidirect
44. C₇×#428	C₂²	semidirect
45. C₇×#427	C₂²	semidirect
46. C₇×#419	C₂²	semidirect
48. C₇×#418	C₂²	semidirect
49. C₇×#418	C₂²	semidirect
50. C₇×#417	C₂²	semidirect
51. C₇×#416	C₂²	semidirect
52. C₂×C₇×#102	C₂²	semidirect
53. C₇×#405	C₂²	semidirect
54. C₇×#405	C₂²	semidirect
55. C₄×C₇×M₁₆	C₂²	semidirect
56. #60168	C₇	direct (N × H)
61. C₂×C₇×#41	C₂×C₄	semidirect
70. C₂²×C₇×D₄	C₂×C₄	semidirect
78. C₇×AGL(1,ℤ/8)	C₂×C₄	semidirect
79. C₇×AGL(1,ℤ/8)	C₂×C₄	semidirect
80. C₂×C₇×SD₁₆	C₂×C₄	semidirect
82. C₂×C₇×D₈	C₂×C₄	semidirect
85. C₂×C₇×M₁₆	C₂×C₄	semidirect
93. C₄×C₇×Q₈	C₂³	semidirect
101. C₄×C₇×D₄	C₂³	semidirect
132. C₇×#107	C₂³	semidirect
133. C₇×#106	C₂³	semidirect
134. C₇×#103	C₂³	semidirect
136. C₇×#102	C₂³	semidirect
137. C₇×#97	C₂³	semidirect
138. C₄×C₇×C₈	C₂³	semidirect
140. #3428	C₂×C₇	semidirect
141. C₂×#518	C₂×C₇	semidirect
142. #2949	C₂×C₇	semidirect
143. #2948	C₂×C₇	semidirect
144. #2946	C₂×C₇	semidirect
145. C₄×AGL(1,ℤ/8)	C₂×C₇	semidirect
146. #2911	C₂×C₇	semidirect
147. #2896	C₂×C₇	semidirect
148. C₂×#418	C₂×C₇	semidirect
149. #2873	C₂×C₇	semidirect
162. C₇×#41	C₂²×C₄	semidirect
165. C₂×C₇×Q₈	C₂²×C₄	semidirect
184. C₂×C₇×D₄	C₂²×C₄	semidirect
205. C₇×SD₁₆	C₂²×C₄	semidirect
206. C₇×D₈	C₂²×C₄	semidirect
210. C₇×M₁₆	C₂²×C₄	semidirect
212. C₂×C₇×C₈	C₂²×C₄	semidirect
239. C₂×AGL(1,ℤ/8)	C₄×C₇	semidirect
247. #520	C₂²×C₇	semidirect
248. #519	C₂²×C₇	semidirect
253. #518	C₂²×C₇	semidirect
254. #518	C₂²×C₇	semidirect
256. C₄×#41	C₂²×C₇	semidirect
258. C₂×C₄×D₄	C₂²×C₇	semidirect
264. #442	C₂²×C₇	semidirect
265. #440	C₂²×C₇	semidirect
266. C₄×SD₁₆	C₂²×C₇	semidirect
267. C₄×D₈	C₂²×C₇	semidirect
268. #428	C₂²×C₇	semidirect
269. #428	C₂²×C₇	semidirect
270. #427	C₂²×C₇	semidirect
271. #419	C₂²×C₇	semidirect
273. #418	C₂²×C₇	semidirect
274. #418	C₂²×C₇	semidirect
275. #417	C₂²×C₇	semidirect
276. #416	C₂²×C₇	semidirect
277. C₂×#102	C₂²×C₇	semidirect
278. #405	C₂²×C₇	semidirect
279. #405	C₂²×C₇	semidirect
280. C₄×M₁₆	C₂²×C₇	semidirect
341. C₂×#41	C₂×C₄×C₇	semidirect
350. C₂²×D₄	C₂×C₄×C₇	semidirect
358. AGL(1,ℤ/8)	C₂×C₄×C₇	semidirect
359. AGL(1,ℤ/8)	C₂×C₄×C₇	semidirect
360. C₂×SD₁₆	C₂×C₄×C₇	semidirect
362. C₂×D₈	C₂×C₄×C₇	semidirect
365. C₂×M₁₆	C₂×C₄×C₇	semidirect
373. C₄×Q₈	C₂³×C₇	semidirect
381. C₄×D₄	C₂³×C₇	semidirect
412. #107	C₂³×C₇	semidirect
413. #106	C₂³×C₇	semidirect
414. #103	C₂³×C₇	semidirect
416. #102	C₂³×C₇	semidirect
417. #97	C₂³×C₇	semidirect
418. C₄×C₈	C₂³×C₇	semidirect
456. #41	C₂²×C₄×C₇	semidirect
459. C₂×Q₈	C₂²×C₄×C₇	semidirect
478. C₂×D₄	C₂²×C₄×C₇	semidirect
499. SD₁₆	C₂²×C₄×C₇	semidirect
500. D₈	C₂²×C₄×C₇	semidirect
504. M₁₆	C₂²×C₄×C₇	semidirect
506. C₂×C₈	C₂²×C₄×C₇	semidirect
594. C₇	#60168	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₇×AGL(1,ℤ/8)	2
C₇×#3428	2
C₂×C₇×#518	2
C₇×#2949	2
C₇×#2948	2
C₇×#2946	2
C₄×C₇×AGL(1,ℤ/8)	2
C₇×#2911	2
C₇×#2896	2
C₂×C₇×#418	2
C₇×#2873	2
#60168	7

Subgroups — C7×#60168 #13237384

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#60168 #13237384