Subgroups — C₇×#60169 #13237385

Order 1792.

2666 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×#137	C₂	1	1	yes	maximal, characteristic
2	C₇×#3428	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×#519	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#2950	C₂	1	4	yes	maximal, complemented
5	C₇×#2949	C₂	1	4	yes	maximal, complemented
6	C₇×#2947	C₂	1	8	yes	maximal, complemented
7	C₄×C₇×#137	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
10	C₂×C₇×#419	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×Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₇×#137	C₄	1	16	yes	complemented
15	C₂×C₇×#137	C₂²	1	4	yes	—
16	C₂×C₇×#137	C₂²	1	4	yes	—
17	C₂²×C₇×Q₁₆	C₂²	1	2	yes	—
18	C₂²×C₇×SD₁₆	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	—
23	C₇×#519	—	2	2	no	—
24	C₇×#519	—	2	2	no	—
25	C₇×#519	C₂²	1	4	yes	—
26	C₇×#519	C₂²	1	2	yes	—
27	C₇×#519	C₂²	1	4	yes	complemented
28	C₇×#518	—	2	2	no	—
29	C₇×#518	C₂²	1	2	yes	—
30	C₄×C₇×#41	—	2	2	no	—
31	C₄×C₇×#41	C₂²	1	4	yes	—
32	C₂×C₄×C₇×Q₈	—	2	2	no	—
33	C₂×C₄×C₇×Q₈	C₂²	1	4	yes	complemented
34	C₂×C₇×#117	—	2	2	no	—
35	C₂×C₇×#117	—	2	2	no	—
36	C₂×C₇×#117	C₂²	1	1	yes	characteristic
37	C₂×C₇×#117	C₂²	1	1	yes	characteristic
38	C₂×C₇×#117	C₂²	1	1	yes	characteristic
39	C₇×#441	C₂²	1	16	yes	complemented
40	C₇×#440	C₂²	1	16	yes	complemented
41	C₄×C₇×Q₁₆	C₂²	1	16	yes	complemented
42	C₄×C₇×SD₁₆	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	—
46	C₇×#419	C₂²	1	4	yes	—
47	C₇×#419	C₂²	1	2	yes	—
48	C₇×#419	C₂²	1	4	yes	complemented
49	C₇×#418	C₂²	1	2	yes	—
50	C₇×#417	C₂²	1	4	yes	—
51	C₇×#416	C₂²	1	4	yes	—
52	C₂×C₇×#103	C₂²	1	4	yes	complemented
53	C₇×#405	C₂²	1	4	yes	—
54	C₇×#405	C₂²	1	4	yes	complemented
55	C₄×C₇×M₁₆	C₂²	1	4	yes	complemented
56	#60169	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂×C₇×#41	—	2	1	no	—
58	C₂×C₇×#41	—	2	1	no	—
59	C₂×C₇×#41	—	2	4	no	—
60	C₂×C₇×#41	C₂×C₄	1	8	yes	—
61	C₂×C₇×#41	C₂³	1	2	yes	—
62	C₂×C₇×#41	C₂³	1	2	yes	—
63	C₂²×C₇×Q₈	—	2	1	no	—
64	C₂²×C₇×Q₈	—	2	1	no	—
65	C₂²×C₇×Q₈	—	2	1	no	—
66	C₂²×C₇×Q₈	—	2	4	no	—
67	C₂²×C₇×Q₈	C₂×C₄	1	8	yes	complemented
68	C₂²×C₇×Q₈	C₂³	1	2	yes	—
69	C₂²×C₇×Q₈	C₂³	1	2	yes	—
70	C₂²×C₇×Q₈	C₂³	1	2	yes	—
71	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
72	C₂²×C₇×D₄	—	2	1	no	—
73	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₇	—	2	1	no	—
75	C₂³×C₄×C₇	—	2	1	no	—
76	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
77	C₇×#137	C₂×C₄	1	32	yes	complemented
78	C₇×#137	C₂×C₄	1	32	yes	complemented
79	C₂×C₇×Q₁₆	C₂×C₄	1	16	yes	complemented
80	C₂×C₇×Q₁₆	C₂³	1	8	yes	—
81	C₂×C₇×SD₁₆	C₂×C₄	1	16	yes	complemented
82	C₂×C₇×SD₁₆	C₂³	1	8	yes	—
83	C₂×C₇×M₁₆	—	2	4	no	—
84	C₂×C₇×M₁₆	C₂×C₄	1	8	yes	complemented
85	C₂×C₇×M₁₆	C₂³	1	2	yes	—
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	C₂²×C₇×C₈	C₂³	1	2	yes	—
90	C₂²×C₇×C₈	C₂³	1	2	yes	—
91	C₄×C₇×Q₈	—	2	8	no	—
92	C₄×C₇×Q₈	—	2	8	no	—
93	C₄×C₇×Q₈	—	2	4	no	—
94	C₄×C₇×Q₈	—	2	8	no	—
95	C₄×C₇×Q₈	C₂³	1	16	yes	—
96	C₄×C₇×Q₈	C₂³	1	8	yes	—
97	C₄×C₇×D₄	—	2	8	no	—
98	C₄×C₇×D₄	—	2	4	no	—
99	C₄×C₇×D₄	—	2	8	no	—
100	C₄×C₇×D₄	C₂³	1	8	yes	—
101	C₇×#117	—	2	1	no	—
102	C₇×#117	—	2	1	no	—
103	C₇×#117	—	2	1	no	—
104	C₇×#117	—	2	1	no	—
105	C₇×#117	—	2	4	no	—
106	C₇×#117	—	2	4	no	—
107	C₇×#117	—	2	8	no	—
108	C₇×#117	—	2	8	no	—
109	C₇×#117	—	2	4	no	—
110	C₇×#117	D₄	1	2	yes	—
111	C₇×#117	D₄	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	C₇×#117	C₂³	1	2	yes	—
118	C₂×C₇×C₄⋊C₄	—	2	4	no	—
119	C₂×C₇×C₄⋊C₄	—	2	4	no	—
120	C₂×C₇×C₄⋊C₄	D₄	1	2	yes	—
121	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
122	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
123	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
124	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
125	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
126	C₂×C₇×C₂²⋊C₄	D₄	1	2	yes	—
127	C₂×C₄²×C₇	—	2	4	no	—
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	—
133	C₇×#106	C₂³	1	8	yes	—
134	C₇×#103	C₂³	1	16	yes	—
135	C₇×#103	C₂³	1	8	yes	—
136	C₇×#102	C₂³	1	8	yes	—
137	C₇×#97	C₂³	1	8	yes	—
138	C₄×C₇×C₈	C₂³	1	8	yes	—
139	C₂²×#137	C₂×C₇	1	1	yes	characteristic
140	#3428	C₂×C₇	1	1	yes	characteristic
141	C₂×#519	C₂×C₇	1	1	yes	characteristic, complemented
142	#2950	C₂×C₇	1	4	yes	complemented
143	#2949	C₂×C₇	1	4	yes	complemented
144	#2947	C₂×C₇	1	8	yes	complemented
145	C₄×#137	C₂×C₇	1	8	yes	complemented
146	#2911	C₂×C₇	1	1	yes	characteristic, complemented
147	#2896	C₂×C₇	1	1	yes	characteristic
148	C₂×#419	C₂×C₇	1	1	yes	characteristic, complemented
149	#2873	C₂×C₇	1	1	yes	characteristic, complemented
150	C₂⁴×C₇	—	2	1	no	—
151	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
152	C₇×#41	—	2	8	no	—
153	C₇×#41	—	2	8	no	—
154	C₇×#41	—	2	4	no	—
155	C₇×#41	—	2	4	no	—
156	C₇×#41	C₂²×C₄	1	8	yes	—
157	C₇×#41	C₂²×C₄	1	8	yes	—
158	C₂×C₇×Q₈	—	2	8	no	—
159	C₂×C₇×Q₈	—	2	8	no	—
160	C₂×C₇×Q₈	—	2	8	no	—
161	C₂×C₇×Q₈	—	2	2	no	—
162	C₂×C₇×Q₈	—	2	2	no	—
163	C₂×C₇×Q₈	—	2	2	no	—
164	C₂×C₇×Q₈	—	2	1	no	—
165	C₂×C₇×Q₈	—	2	4	no	—
166	C₂×C₇×Q₈	—	2	4	no	—
167	C₂×C₇×Q₈	—	2	4	no	—
168	C₂×C₇×Q₈	—	2	2	no	—
169	C₂×C₇×Q₈	C₂²×C₄	1	8	yes	—
170	C₂×C₇×Q₈	C₂²×C₄	1	4	yes	—
171	C₂×C₇×Q₈	C₂²×C₄	1	8	yes	—
172	C₂×C₇×Q₈	C₂²×C₄	1	8	yes	—
173	C₂×C₇×Q₈	C₂⁴	1	4	yes	—
174	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
175	C₂×C₇×D₄	—	2	8	no	—
176	C₂×C₇×D₄	—	2	2	no	—
177	C₂×C₇×D₄	—	2	2	no	—
178	C₂×C₇×D₄	—	2	1	no	—
179	C₂×C₇×D₄	—	2	2	no	—
180	C₂×C₇×D₄	C₂²×C₄	1	4	yes	—
181	C₂×C₇×D₄	C₂⁴	1	2	yes	—
182	C₂²×C₄×C₇	—	2	8	no	—
183	C₂²×C₄×C₇	—	2	2	no	—
184	C₂²×C₄×C₇	—	2	1	no	—
185	C₂²×C₄×C₇	—	2	2	no	—
186	C₂²×C₄×C₇	—	2	2	no	—
187	C₂²×C₄×C₇	—	2	8	no	—
188	C₂²×C₄×C₇	—	2	2	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	8	no	—
193	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
194	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
195	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
196	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
197	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
198	C₂²×C₄×C₇	#41	1	4	yes	—
199	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
200	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
201	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
202	C₇×Q₁₆	C₂²×C₄	1	32	yes	—
203	C₇×SD₁₆	C₂²×C₄	1	32	yes	—
204	C₇×M₁₆	—	2	4	no	—
205	C₇×M₁₆	—	2	4	no	—
206	C₇×M₁₆	C₂²×C₄	1	8	yes	—
207	C₇×M₁₆	C₂²×C₄	1	8	yes	—
208	C₂×C₇×C₈	—	2	4	no	—
209	C₂×C₇×C₈	C₂²×C₄	1	8	yes	—
210	C₂×C₇×C₈	C₂⁴	1	4	yes	—
211	C₂×C₇×C₈	C₂⁴	1	4	yes	—
212	C₇×C₄⋊C₄	—	2	2	no	—
213	C₇×C₄⋊C₄	—	2	8	no	—
214	C₇×C₄⋊C₄	—	2	16	no	—
215	C₇×C₄⋊C₄	—	2	2	no	—
216	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
217	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
218	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
219	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
220	C₇×C₂²⋊C₄	—	2	2	no	—
221	C₇×C₂²⋊C₄	—	2	2	no	—
222	C₇×C₂²⋊C₄	—	2	2	no	—
223	C₇×C₂²⋊C₄	—	2	2	no	—
224	C₇×C₂²⋊C₄	—	2	8	no	—
225	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
226	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
227	C₄²×C₇	—	2	2	no	—
228	C₄²×C₇	—	2	8	no	—
229	C₄²×C₇	—	2	16	no	—
230	C₄²×C₇	—	2	2	no	—
231	C₄²×C₇	—	2	2	no	—
232	C₄²×C₇	—	2	2	no	—
233	C₄²×C₇	C₂×D₄	1	4	yes	—
234	C₄²×C₇	C₂⁴	1	4	yes	—
235	C₂²×#41	C₂²×C₇	1	1	yes	characteristic
236	C₂³×Q₈	C₂²×C₇	1	1	yes	characteristic
237	C₂×#137	C₄×C₇	1	16	yes	complemented
238	C₂×#137	C₂²×C₇	1	4	yes	—
239	C₂×#137	C₂²×C₇	1	4	yes	—
240	C₂²×Q₁₆	C₂²×C₇	1	2	yes	—
241	C₂²×SD₁₆	C₂²×C₇	1	2	yes	—
242	C₂²×M₁₆	C₂²×C₇	1	1	yes	characteristic
243	C₂²×M₁₆	C₂²×C₇	1	1	yes	characteristic
244	#520	—	2	2	no	—
245	#520	C₂²×C₇	1	4	yes	—
246	#519	—	2	2	no	—
247	#519	—	2	2	no	—
248	#519	C₂²×C₇	1	4	yes	—
249	#519	C₂²×C₇	1	2	yes	—
250	#519	C₂²×C₇	1	4	yes	complemented
251	#518	—	2	2	no	—
252	#518	C₂²×C₇	1	2	yes	—
253	C₄×#41	—	2	2	no	—
254	C₄×#41	C₂²×C₇	1	4	yes	—
255	C₂×C₄×Q₈	—	2	2	no	—
256	C₂×C₄×Q₈	C₂²×C₇	1	4	yes	complemented
257	C₂×#117	—	2	2	no	—
258	C₂×#117	—	2	2	no	—
259	C₂×#117	C₂²×C₇	1	1	yes	characteristic
260	C₂×#117	C₂²×C₇	1	1	yes	characteristic
261	C₂×#117	C₂²×C₇	1	1	yes	characteristic
262	#441	C₂²×C₇	1	16	yes	complemented
263	#440	C₂²×C₇	1	16	yes	complemented
264	C₄×Q₁₆	C₂²×C₇	1	16	yes	complemented
265	C₄×SD₁₆	C₂²×C₇	1	16	yes	complemented
266	#428	C₂²×C₇	1	4	yes	complemented
267	#428	C₂²×C₇	1	4	yes	complemented
268	#427	C₂²×C₇	1	4	yes	—
269	#419	C₂²×C₇	1	4	yes	—
270	#419	C₂²×C₇	1	2	yes	—
271	#419	C₂²×C₇	1	4	yes	complemented
272	#418	C₂²×C₇	1	2	yes	—
273	#417	C₂²×C₇	1	4	yes	—
274	#416	C₂²×C₇	1	4	yes	—
275	C₂×#103	C₂²×C₇	1	4	yes	complemented
276	#405	C₂²×C₇	1	4	yes	—
277	#405	C₂²×C₇	1	4	yes	complemented
278	C₄×M₁₆	C₂²×C₇	1	4	yes	complemented
279	C₂³×C₇	—	2	2	no	—
280	C₂³×C₇	—	2	4	no	—
281	C₂³×C₇	—	4	2	no	—
282	C₂³×C₇	—	8	2	no	—
283	C₂³×C₇	C₄×D₄	1	4	yes	—
284	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
285	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
286	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
287	C₇×Q₈	—	2	8	no	—
288	C₇×Q₈	—	2	8	no	—
289	C₇×Q₈	—	2	8	no	—
290	C₇×Q₈	—	2	4	no	—
291	C₇×Q₈	—	2	4	no	—
292	C₇×Q₈	—	2	2	no	—
293	C₇×Q₈	C₂³×C₄	1	8	yes	—
294	C₇×Q₈	C₂³×C₄	1	4	yes	—
295	C₇×D₄	—	2	8	no	—
296	C₇×D₄	—	2	8	no	—
297	C₇×D₄	—	2	4	no	—
298	C₇×D₄	—	2	2	no	—
299	C₇×D₄	C₂³×C₄	1	4	yes	—
300	C₂×C₄×C₇	—	2	4	no	—
301	C₂×C₄×C₇	—	2	8	no	—
302	C₂×C₄×C₇	—	2	8	no	—
303	C₂×C₄×C₇	—	2	8	no	—
304	C₂×C₄×C₇	—	2	2	no	—
305	C₂×C₄×C₇	—	2	4	no	—
306	C₂×C₄×C₇	—	2	8	no	—
307	C₂×C₄×C₇	—	2	8	no	—
308	C₂×C₄×C₇	—	2	4	no	—
309	C₂×C₄×C₇	—	2	4	no	—
310	C₂×C₄×C₇	—	4	4	no	—
311	C₂×C₄×C₇	—	4	4	no	—
312	C₂×C₄×C₇	—	8	8	no	—
313	C₂×C₄×C₇	—	8	2	no	—
314	C₂×C₄×C₇	C₄×D₄	1	4	yes	—
315	C₂×C₄×C₇	C₄×D₄	1	4	yes	—
316	C₂×C₄×C₇	C₄×D₄	1	4	yes	—
317	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
318	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
319	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
320	C₂×C₄×C₇	C₂²×D₄	1	8	yes	—
321	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
322	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
323	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
324	C₂×C₄×C₇	C₂×#41	1	2	yes	—
325	C₂×C₄×C₇	C₂×#41	1	2	yes	—
326	C₂×C₄×C₇	C₂×#41	1	2	yes	—
327	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
328	C₇×C₈	—	2	4	no	—
329	C₇×C₈	C₂³×C₄	1	8	yes	—
330	C₂×#41	—	2	1	no	—
331	C₂×#41	—	2	1	no	—
332	C₂×#41	—	2	4	no	—
333	C₂×#41	C₂×C₄×C₇	1	8	yes	—
334	C₂×#41	C₂³×C₇	1	2	yes	—
335	C₂×#41	C₂³×C₇	1	2	yes	—
336	C₂²×Q₈	—	2	1	no	—
337	C₂²×Q₈	—	2	1	no	—
338	C₂²×Q₈	—	2	1	no	—
339	C₂²×Q₈	—	2	4	no	—
340	C₂²×Q₈	C₂×C₄×C₇	1	8	yes	complemented
341	C₂²×Q₈	C₂³×C₇	1	2	yes	—
342	C₂²×Q₈	C₂³×C₇	1	2	yes	—
343	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
344	C₂²×Q₈	C₂³×C₇	1	2	yes	—
345	C₂²×D₄	—	2	1	no	—
346	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
347	C₂³×C₄	—	2	1	no	—
348	C₂³×C₄	—	2	1	no	—
349	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
350	#137	C₂×C₄×C₇	1	32	yes	complemented
351	#137	C₂×C₄×C₇	1	32	yes	complemented
352	C₂×Q₁₆	C₂×C₄×C₇	1	16	yes	complemented
353	C₂×Q₁₆	C₂³×C₇	1	8	yes	—
354	C₂×SD₁₆	C₂×C₄×C₇	1	16	yes	complemented
355	C₂×SD₁₆	C₂³×C₇	1	8	yes	—
356	C₂×M₁₆	—	2	4	no	—
357	C₂×M₁₆	C₂×C₄×C₇	1	8	yes	complemented
358	C₂×M₁₆	C₂³×C₇	1	2	yes	—
359	C₂×M₁₆	C₂³×C₇	1	2	yes	—
360	C₂×M₁₆	C₂³×C₇	1	2	yes	—
361	C₂×M₁₆	C₂³×C₇	1	2	yes	—
362	C₂²×C₈	C₂³×C₇	1	2	yes	—
363	C₂²×C₈	C₂³×C₇	1	2	yes	—
364	C₄×Q₈	—	2	8	no	—
365	C₄×Q₈	—	2	8	no	—
366	C₄×Q₈	—	2	4	no	—
367	C₄×Q₈	—	2	8	no	—
368	C₄×Q₈	C₂³×C₇	1	16	yes	—
369	C₄×Q₈	C₂³×C₇	1	8	yes	—
370	C₄×D₄	—	2	8	no	—
371	C₄×D₄	—	2	4	no	—
372	C₄×D₄	—	2	8	no	—
373	C₄×D₄	C₂³×C₇	1	8	yes	—
374	#117	—	2	1	no	—
375	#117	—	2	1	no	—
376	#117	—	2	1	no	—
377	#117	—	2	1	no	—
378	#117	—	2	4	no	—
379	#117	—	2	4	no	—
380	#117	—	2	8	no	—
381	#117	—	2	8	no	—
382	#117	—	2	4	no	—
383	#117	C₇×D₄	1	2	yes	—
384	#117	C₇×D₄	1	2	yes	—
385	#117	C₂³×C₇	1	2	yes	—
386	#117	C₂³×C₇	1	2	yes	—
387	#117	C₂³×C₇	1	2	yes	—
388	#117	C₂³×C₇	1	2	yes	—
389	#117	C₂³×C₇	1	2	yes	—
390	#117	C₂³×C₇	1	2	yes	—
391	C₂×C₄⋊C₄	—	2	4	no	—
392	C₂×C₄⋊C₄	—	2	4	no	—
393	C₂×C₄⋊C₄	C₇×D₄	1	2	yes	—
394	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
395	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
396	C₂×C₂²⋊C₄	—	2	1	no	—
397	C₂×C₂²⋊C₄	—	2	1	no	—
398	C₂×C₂²⋊C₄	—	2	4	no	—
399	C₂×C₂²⋊C₄	C₇×D₄	1	2	yes	—
400	C₂×C₄²	—	2	4	no	—
401	C₂×C₄²	—	2	1	no	—
402	C₂×C₄²	—	2	1	no	—
403	C₂×C₄²	—	2	4	no	—
404	C₂×C₄²	C₂³×C₇	1	2	yes	—
405	#107	C₂³×C₇	1	8	yes	—
406	#106	C₂³×C₇	1	8	yes	—
407	#103	C₂³×C₇	1	16	yes	—
408	#103	C₂³×C₇	1	8	yes	—
409	#102	C₂³×C₇	1	8	yes	—
410	#97	C₂³×C₇	1	8	yes	—
411	C₄×C₈	C₂³×C₇	1	8	yes	—
412	C₂²×C₇	—	2	2	no	—
413	C₂²×C₇	—	2	2	no	—
414	C₂²×C₇	—	2	2	no	—
415	C₂²×C₇	—	2	4	no	—
416	C₂²×C₇	—	4	4	no	—
417	C₂²×C₇	—	4	4	no	—
418	C₂²×C₇	—	8	4	no	—
419	C₂²×C₇	C₂×C₄×D₄	1	2	yes	—
420	C₂²×C₇	C₂×C₄×D₄	1	2	yes	—
421	C₂²×C₇	C₂×C₄×D₄	1	2	yes	—
422	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
423	C₄×C₇	—	2	8	no	—
424	C₄×C₇	—	2	4	no	—
425	C₄×C₇	—	2	8	no	—
426	C₄×C₇	—	4	4	no	—
427	C₄×C₇	—	4	4	no	—
428	C₄×C₇	C₂×C₄×D₄	1	2	yes	—
429	C₄×C₇	C₂×C₄×D₄	1	2	yes	—
430	C₄×C₇	C₂×C₄×D₄	1	2	yes	—
431	C₄×C₇	C₂⁴×C₄	1	1	yes	characteristic
432	C₄×C₇	C₂²×#41	1	1	yes	characteristic
433	C₂⁴	—	2	1	no	—
434	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
435	#41	—	2	8	no	—
436	#41	—	2	8	no	—
437	#41	—	2	4	no	—
438	#41	—	2	4	no	—
439	#41	C₂²×C₄×C₇	1	8	yes	—
440	#41	C₂²×C₄×C₇	1	8	yes	—
441	C₂×Q₈	—	2	8	no	—
442	C₂×Q₈	—	2	8	no	—
443	C₂×Q₈	—	2	8	no	—
444	C₂×Q₈	—	2	2	no	—
445	C₂×Q₈	—	2	2	no	—
446	C₂×Q₈	—	2	2	no	—
447	C₂×Q₈	—	2	1	no	—
448	C₂×Q₈	—	2	4	no	—
449	C₂×Q₈	—	2	4	no	—
450	C₂×Q₈	—	2	4	no	—
451	C₂×Q₈	—	2	2	no	—
452	C₂×Q₈	C₂²×C₄×C₇	1	8	yes	—
453	C₂×Q₈	C₂²×C₄×C₇	1	4	yes	—
454	C₂×Q₈	C₂²×C₄×C₇	1	8	yes	—
455	C₂×Q₈	C₂²×C₄×C₇	1	8	yes	—
456	C₂×Q₈	C₂⁴×C₇	1	4	yes	—
457	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
458	C₂×D₄	—	2	8	no	—
459	C₂×D₄	—	2	2	no	—
460	C₂×D₄	—	2	2	no	—
461	C₂×D₄	—	2	1	no	—
462	C₂×D₄	—	2	2	no	—
463	C₂×D₄	C₂²×C₄×C₇	1	4	yes	—
464	C₂×D₄	C₂⁴×C₇	1	2	yes	—
465	C₂²×C₄	—	2	8	no	—
466	C₂²×C₄	—	2	2	no	—
467	C₂²×C₄	—	2	1	no	—
468	C₂²×C₄	—	2	2	no	—
469	C₂²×C₄	—	2	2	no	—
470	C₂²×C₄	—	2	8	no	—
471	C₂²×C₄	—	2	2	no	—
472	C₂²×C₄	—	2	2	no	—
473	C₂²×C₄	—	2	2	no	—
474	C₂²×C₄	—	2	2	no	—
475	C₂²×C₄	—	2	8	no	—
476	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
477	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
478	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
479	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
480	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
481	C₂²×C₄	C₇×#41	1	4	yes	—
482	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
483	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
484	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
485	Q₁₆	C₂²×C₄×C₇	1	32	yes	—
486	SD₁₆	C₂²×C₄×C₇	1	32	yes	—
487	M₁₆	—	2	4	no	—
488	M₁₆	—	2	4	no	—
489	M₁₆	C₂²×C₄×C₇	1	8	yes	—
490	M₁₆	C₂²×C₄×C₇	1	8	yes	—
491	C₂×C₈	—	2	4	no	—
492	C₂×C₈	C₂²×C₄×C₇	1	8	yes	—
493	C₂×C₈	C₂⁴×C₇	1	4	yes	—
494	C₂×C₈	C₂⁴×C₇	1	4	yes	—
495	C₄⋊C₄	—	2	2	no	—
496	C₄⋊C₄	—	2	8	no	—
497	C₄⋊C₄	—	2	16	no	—
498	C₄⋊C₄	—	2	2	no	—
499	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
500	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
501	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
502	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
503	C₂²⋊C₄	—	2	2	no	—
504	C₂²⋊C₄	—	2	2	no	—
505	C₂²⋊C₄	—	2	2	no	—
506	C₂²⋊C₄	—	2	2	no	—
507	C₂²⋊C₄	—	2	8	no	—
508	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
509	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
510	C₄²	—	2	2	no	—
511	C₄²	—	2	8	no	—
512	C₄²	—	2	16	no	—
513	C₄²	—	2	2	no	—
514	C₄²	—	2	2	no	—
515	C₄²	—	2	2	no	—
516	C₄²	C₂×C₇×D₄	1	4	yes	—
517	C₄²	C₂⁴×C₇	1	4	yes	—
518	C₂×C₇	—	2	2	no	—
519	C₂×C₇	—	2	2	no	—
520	C₂×C₇	—	2	2	no	—
521	C₂×C₇	—	4	4	no	—
522	C₂×C₇	C₂²×C₄×D₄	1	1	yes	characteristic
523	C₂×C₇	#3586	1	2	yes	—
524	C₂³	—	2	2	no	—
525	C₂³	—	2	4	no	—
526	C₂³	—	4	2	no	—
527	C₂³	—	8	2	no	—
528	C₂³	C₄×C₇×D₄	1	4	yes	—
529	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
530	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
531	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
532	Q₈	—	2	4	no	—
533	Q₈	—	2	2	no	—
534	Q₈	—	2	8	no	—
535	Q₈	—	2	8	no	—
536	Q₈	—	2	8	no	—
537	Q₈	—	2	4	no	—
538	Q₈	C₂³×C₄×C₇	1	8	yes	—
539	Q₈	C₂³×C₄×C₇	1	4	yes	—
540	D₄	—	2	8	no	—
541	D₄	—	2	8	no	—
542	D₄	—	2	2	no	—
543	D₄	—	2	4	no	—
544	D₄	C₂³×C₄×C₇	1	4	yes	—
545	C₂×C₄	—	2	2	no	—
546	C₂×C₄	—	2	4	no	—
547	C₂×C₄	—	2	4	no	—
548	C₂×C₄	—	2	8	no	—
549	C₂×C₄	—	2	8	no	—
550	C₂×C₄	—	2	8	no	—
551	C₂×C₄	—	2	8	no	—
552	C₂×C₄	—	2	8	no	—
553	C₂×C₄	—	2	4	no	—
554	C₂×C₄	—	2	4	no	—
555	C₂×C₄	—	4	4	no	—
556	C₂×C₄	—	4	4	no	—
557	C₂×C₄	—	8	8	no	—
558	C₂×C₄	—	8	2	no	—
559	C₂×C₄	C₄×C₇×D₄	1	4	yes	—
560	C₂×C₄	C₄×C₇×D₄	1	4	yes	—
561	C₂×C₄	C₄×C₇×D₄	1	4	yes	—
562	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
563	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
564	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
565	C₂×C₄	C₂²×C₇×D₄	1	8	yes	—
566	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
567	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
568	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
569	C₂×C₄	C₂×C₇×#41	1	2	yes	—
570	C₂×C₄	C₂×C₇×#41	1	2	yes	—
571	C₂×C₄	C₂×C₇×#41	1	2	yes	—
572	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
573	C₈	—	2	4	no	—
574	C₈	C₂³×C₄×C₇	1	8	yes	—
575	C₇	#60169	1	1	yes	characteristic, direct product factor, Sylow
576	C₂²	—	2	2	no	—
577	C₂²	—	2	2	no	—
578	C₂²	—	2	2	no	—
579	C₂²	—	2	4	no	—
580	C₂²	—	4	4	no	—
581	C₂²	—	4	4	no	—
582	C₂²	—	8	4	no	—
583	C₂²	C₂×C₄×C₇×D₄	1	2	yes	—
584	C₂²	C₂×C₄×C₇×D₄	1	2	yes	—
585	C₂²	C₂×C₄×C₇×D₄	1	2	yes	—
586	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
587	C₄	—	2	8	no	—
588	C₄	—	2	4	no	—
589	C₄	—	2	8	no	—
590	C₄	—	4	4	no	—
591	C₄	—	4	4	no	—
592	C₄	C₂×C₄×C₇×D₄	1	2	yes	—
593	C₄	C₂×C₄×C₇×D₄	1	2	yes	—
594	C₄	C₂×C₄×C₇×D₄	1	2	yes	—
595	C₄	C₂⁴×C₄×C₇	1	1	yes	characteristic
596	C₄	C₂²×C₇×#41	1	1	yes	characteristic
597	C₂	—	2	2	no	—
598	C₂	—	2	2	no	—
599	C₂	—	2	2	no	—
600	C₂	—	4	4	no	—
601	C₂	C₂²×C₄×C₇×D₄	1	1	yes	characteristic
602	C₂	C₇×#3586	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×#41	C₂²
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
3. C₂×C₇×#519	C₂	semidirect
4. C₇×#2950	C₂	semidirect
5. C₇×#2949	C₂	semidirect
6. C₇×#2947	C₂	semidirect
7. C₄×C₇×#137	C₂	semidirect
8. C₇×#2911	C₂	semidirect
10. C₂×C₇×#419	C₂	semidirect
11. C₇×#2873	C₂	semidirect
14. C₂×C₇×#137	C₄	semidirect
27. C₇×#519	C₂²	semidirect
33. C₂×C₄×C₇×Q₈	C₂²	semidirect
39. C₇×#441	C₂²	semidirect
40. C₇×#440	C₂²	semidirect
41. C₄×C₇×Q₁₆	C₂²	semidirect
42. C₄×C₇×SD₁₆	C₂²	semidirect
43. C₇×#428	C₂²	semidirect
44. C₇×#428	C₂²	semidirect
48. C₇×#419	C₂²	semidirect
52. C₂×C₇×#103	C₂²	semidirect
54. C₇×#405	C₂²	semidirect
55. C₄×C₇×M₁₆	C₂²	semidirect
56. #60169	C₇	direct (N × H)
67. C₂²×C₇×Q₈	C₂×C₄	semidirect
77. C₇×#137	C₂×C₄	semidirect
78. C₇×#137	C₂×C₄	semidirect
79. C₂×C₇×Q₁₆	C₂×C₄	semidirect
81. C₂×C₇×SD₁₆	C₂×C₄	semidirect
84. C₂×C₇×M₁₆	C₂×C₄	semidirect
141. C₂×#519	C₂×C₇	semidirect
142. #2950	C₂×C₇	semidirect
143. #2949	C₂×C₇	semidirect
144. #2947	C₂×C₇	semidirect
145. C₄×#137	C₂×C₇	semidirect
146. #2911	C₂×C₇	semidirect
148. C₂×#419	C₂×C₇	semidirect
149. #2873	C₂×C₇	semidirect
237. C₂×#137	C₄×C₇	semidirect
250. #519	C₂²×C₇	semidirect
256. C₂×C₄×Q₈	C₂²×C₇	semidirect
262. #441	C₂²×C₇	semidirect
263. #440	C₂²×C₇	semidirect
264. C₄×Q₁₆	C₂²×C₇	semidirect
265. C₄×SD₁₆	C₂²×C₇	semidirect
266. #428	C₂²×C₇	semidirect
267. #428	C₂²×C₇	semidirect
271. #419	C₂²×C₇	semidirect
275. C₂×#103	C₂²×C₇	semidirect
277. #405	C₂²×C₇	semidirect
278. C₄×M₁₆	C₂²×C₇	semidirect
340. C₂²×Q₈	C₂×C₄×C₇	semidirect
350. #137	C₂×C₄×C₇	semidirect
351. #137	C₂×C₄×C₇	semidirect
352. C₂×Q₁₆	C₂×C₄×C₇	semidirect
354. C₂×SD₁₆	C₂×C₄×C₇	semidirect
357. C₂×M₁₆	C₂×C₄×C₇	semidirect
575. C₇	#60169	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×#137	2
C₇×#3428	2
C₂×C₇×#519	2
C₇×#2950	2
C₇×#2949	2
C₇×#2947	2
C₄×C₇×#137	2
C₇×#2911	2
C₇×#2896	2
C₂×C₇×#419	2
C₇×#2873	2
#60169	7

Subgroups — C7×#60169 #13237385

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#60169 #13237385