Subgroups — C₇×#22523 #13199739

Order 1792.

1250 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×D₄×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#446	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#2355	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#1993	C₂	1	2	yes	maximal, complemented
5	C₇×#1990	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1988	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1987	C₂	1	2	yes	maximal, complemented
8	C₇×#1804	C₂	1	2	yes	maximal
9	C₇×#1802	C₂	1	1	yes	maximal, characteristic
10	C₇×#1762	C₂	1	2	yes	maximal
11	C₄×C₇×#102	C₂	1	1	yes	maximal, characteristic
12	C₂³×C₇×Q₈	—	2	1	no	—
13	C₇×D₄×Q₈	—	2	4	no	—
14	C₇×D₄×Q₈	C₄	1	8	yes	complemented
15	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
16	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₂×C₇×C₂²⋊Q₈	—	2	1	no	—
18	C₂×C₇×C₂²⋊Q₈	—	2	2	no	—
19	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₇×D₄	C₂²	1	2	yes	—
22	C₇×#446	—	2	4	no	—
23	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
24	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₇×#102	C₂²	1	1	yes	characteristic
26	C₂×C₇×#102	C₂²	1	2	yes	—
27	C₂×C₇×#102	C₂²	1	1	yes	characteristic
28	C₂×C₇×#97	C₂²	1	2	yes	—
29	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
30	C₇×#391	—	2	1	no	—
31	C₇×#387	—	2	2	no	—
32	C₇×#384	C₂²	1	1	yes	characteristic
33	C₇×#382	C₂²	1	2	yes	—
34	C₄×C₇×C₄⋊C₄	—	2	1	no	—
35	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
36	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
37	C₇×#340	C₂²	1	1	yes	characteristic
38	C₇×#340	C₂²	1	2	yes	—
39	C₇×#340	C₂²	1	2	yes	—
40	C₇×#340	C₂²	1	2	yes	—
41	C₇×#340	C₂²	1	2	yes	—
42	C₇×#340	C₂²	1	2	yes	—
43	C₇×#340	C₂²	1	1	yes	characteristic
44	#22523	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
45	C₂²×C₇×Q₈	—	2	8	no	—
46	C₂²×C₇×Q₈	—	2	2	no	—
47	C₂²×C₇×Q₈	—	2	1	no	—
48	C₂²×C₇×Q₈	—	4	4	no	—
49	C₂²×C₇×Q₈	—	4	4	no	—
50	C₂²×C₇×Q₈	D₄	1	1	yes	characteristic
51	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
52	C₂³×C₄×C₇	—	2	2	no	—
53	C₂³×C₄×C₇	—	2	1	no	—
54	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
55	C₂²×C₇×C₈	C₂³	1	2	yes	—
56	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
57	C₇×C₄⋊Q₈	—	2	2	no	—
58	C₇×C₄⋊Q₈	—	2	4	no	—
59	C₇×C₄⋊Q₈	C₂×C₄	1	8	yes	—
60	C₇×C₄⋊Q₈	C₂×C₄	1	4	yes	—
61	C₇×C₂²⋊Q₈	—	2	16	no	—
62	C₇×C₂²⋊Q₈	—	2	8	no	—
63	C₄×C₇×Q₈	C₂×C₄	1	4	yes	—
64	C₄×C₇×Q₈	D₄	1	4	yes	—
65	C₄×C₇×D₄	—	2	2	no	—
66	C₄×C₇×D₄	—	2	4	no	—
67	C₄×C₇×D₄	C₂×C₄	1	4	yes	—
68	C₄×C₇×D₄	C₂×C₄	1	8	yes	—
69	C₂×C₇×C₄⋊C₄	—	2	2	no	—
70	C₂×C₇×C₄⋊C₄	—	2	1	no	—
71	C₂×C₇×C₄⋊C₄	—	2	1	no	—
72	C₂×C₇×C₄⋊C₄	—	2	1	no	—
73	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
74	C₂×C₇×C₄⋊C₄	D₄	1	2	yes	—
75	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
76	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
77	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
78	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
79	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
80	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
81	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
82	C₂×C₄²×C₇	—	2	1	no	—
83	C₂×C₄²×C₇	—	2	2	no	—
84	C₂×C₄²×C₇	C₂³	1	2	yes	—
85	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
86	C₇×C₄⋊C₈	—	2	2	no	—
87	C₇×C₄⋊C₈	—	2	4	no	—
88	C₇×#102	—	2	2	no	—
89	C₇×#102	—	2	4	no	—
90	C₇×#102	Q₈	1	4	yes	complemented
91	C₇×#97	—	2	4	no	—
92	C₄×C₇×C₈	—	2	2	no	—
93	C₇×#95	—	2	2	no	—
94	C₇×#95	—	2	1	no	—
95	C₇×#95	—	2	2	no	—
96	C₇×#95	—	2	2	no	—
97	C₂×D₄×Q₈	C₂×C₇	1	1	yes	characteristic
98	C₂×#446	C₂×C₇	1	1	yes	characteristic, complemented
99	#2355	C₂×C₇	1	1	yes	characteristic, complemented
100	#1993	C₂×C₇	1	2	yes	complemented
101	#1990	C₂×C₇	1	1	yes	characteristic, complemented
102	#1988	C₂×C₇	1	1	yes	characteristic, complemented
103	#1987	C₂×C₇	1	2	yes	complemented
104	#1804	C₂×C₇	1	2	yes	—
105	#1802	C₂×C₇	1	1	yes	characteristic
106	#1762	C₂×C₇	1	2	yes	—
107	C₄×#102	C₂×C₇	1	1	yes	characteristic
108	C₂⁴×C₇	—	2	1	no	—
109	C₂×C₇×Q₈	—	2	8	no	—
110	C₂×C₇×Q₈	—	2	4	no	—
111	C₂×C₇×Q₈	—	2	2	no	—
112	C₂×C₇×Q₈	—	2	2	no	—
113	C₂×C₇×Q₈	—	4	4	no	—
114	C₂×C₇×Q₈	—	4	4	no	—
115	C₂×C₇×Q₈	—	4	32	no	—
116	C₂×C₇×Q₈	—	4	2	no	—
117	C₂×C₇×Q₈	—	4	2	no	—
118	C₂×C₇×Q₈	C₂²⋊C₄	1	4	yes	—
119	C₂×C₇×D₄	—	2	1	no	—
120	C₂×C₇×D₄	—	2	1	no	—
121	C₂×C₇×D₄	—	2	1	no	—
122	C₂×C₇×D₄	C₂²×C₄	1	2	yes	—
123	C₂×C₇×D₄	C₂×Q₈	1	2	yes	—
124	C₂×C₇×D₄	#41	1	2	yes	—
125	C₂²×C₄×C₇	—	2	1	no	—
126	C₂²×C₄×C₇	—	2	2	no	—
127	C₂²×C₄×C₇	—	2	1	no	—
128	C₂²×C₄×C₇	—	2	8	no	—
129	C₂²×C₄×C₇	—	2	4	no	—
130	C₂²×C₄×C₇	—	2	1	no	—
131	C₂²×C₄×C₇	—	2	2	no	—
132	C₂²×C₄×C₇	—	4	4	no	—
133	C₂²×C₄×C₇	—	4	4	no	—
134	C₂²×C₄×C₇	—	4	2	no	—
135	C₂²×C₄×C₇	—	4	2	no	—
136	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
137	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
138	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
140	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
141	C₂×C₇×C₈	—	2	1	no	—
142	C₂×C₇×C₈	—	2	1	no	—
143	C₂×C₇×C₈	—	2	1	no	—
144	C₂×C₇×C₈	—	2	2	no	—
145	C₂×C₇×C₈	—	2	1	no	—
146	C₂×C₇×C₈	—	2	2	no	—
147	C₂×C₇×C₈	—	2	1	no	—
148	C₂×C₇×C₈	—	2	2	no	—
149	C₂×C₇×C₈	C₂×Q₈	1	2	yes	complemented
150	C₇×C₄⋊C₄	—	2	4	no	—
151	C₇×C₄⋊C₄	—	2	8	no	—
152	C₇×C₄⋊C₄	—	2	1	no	—
153	C₇×C₄⋊C₄	—	2	2	no	—
154	C₇×C₄⋊C₄	—	2	1	no	—
155	C₇×C₄⋊C₄	—	2	1	no	—
156	C₇×C₄⋊C₄	—	2	2	no	—
157	C₇×C₄⋊C₄	—	2	1	no	—
158	C₇×C₄⋊C₄	—	2	2	no	—
159	C₇×C₄⋊C₄	—	4	2	no	—
160	C₇×C₄⋊C₄	—	8	1	no	—
161	C₇×C₄⋊C₄	C₂²⋊C₄	1	8	yes	—
162	C₇×C₄⋊C₄	C₂²⋊C₄	1	4	yes	—
163	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
164	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
165	C₇×C₄⋊C₄	C₂×Q₈	1	2	yes	complemented
166	C₇×C₂²⋊C₄	—	2	4	no	—
167	C₇×C₂²⋊C₄	—	2	8	no	—
168	C₄²×C₇	—	4	2	no	—
169	C₄²×C₇	—	8	2	no	—
170	C₄²×C₇	C₂²×C₄	1	4	yes	—
171	C₄²×C₇	C₂²×C₄	1	2	yes	—
172	C₄²×C₇	C₂×D₄	1	2	yes	—
173	C₄²×C₇	C₂×D₄	1	4	yes	—
174	C₂³×Q₈	—	2	1	no	—
175	D₄×Q₈	—	2	4	no	—
176	D₄×Q₈	C₄×C₇	1	8	yes	complemented
177	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
178	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
179	C₂×C₂²⋊Q₈	—	2	1	no	—
180	C₂×C₂²⋊Q₈	—	2	2	no	—
181	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
182	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
183	C₂×C₄×D₄	C₂²×C₇	1	2	yes	—
184	#446	—	2	4	no	—
185	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
186	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
187	C₂×#102	C₂²×C₇	1	2	yes	—
188	C₂×#102	C₂²×C₇	1	1	yes	characteristic
189	C₂×#102	C₂²×C₇	1	1	yes	characteristic
190	C₂×#97	C₂²×C₇	1	2	yes	—
191	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
192	#391	—	2	1	no	—
193	#387	—	2	2	no	—
194	#384	C₂²×C₇	1	1	yes	characteristic
195	#382	C₂²×C₇	1	2	yes	—
196	C₄×C₄⋊C₄	—	2	1	no	—
197	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
198	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
199	#340	C₂²×C₇	1	2	yes	—
200	#340	C₂²×C₇	1	2	yes	—
201	#340	C₂²×C₇	1	2	yes	—
202	#340	C₂²×C₇	1	1	yes	characteristic
203	#340	C₂²×C₇	1	2	yes	—
204	#340	C₂²×C₇	1	2	yes	—
205	#340	C₂²×C₇	1	1	yes	characteristic
206	C₂³×C₇	—	2	2	no	—
207	C₂³×C₇	—	2	2	no	—
208	C₂³×C₇	—	2	2	no	—
209	C₂³×C₇	—	4	1	no	—
210	C₂³×C₇	—	4	1	no	—
211	C₂³×C₇	—	4	1	no	—
212	C₂³×C₇	—	4	1	no	—
213	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
214	C₇×Q₈	—	2	8	no	—
215	C₇×Q₈	—	4	16	no	—
216	C₇×Q₈	—	4	8	no	—
217	C₇×D₄	—	2	2	no	—
218	C₇×D₄	—	2	2	no	—
219	C₇×D₄	—	2	2	no	—
220	C₇×D₄	C₄×Q₈	1	4	yes	—
221	C₂×C₄×C₇	—	2	2	no	—
222	C₂×C₄×C₇	—	2	2	no	—
223	C₂×C₄×C₇	—	2	2	no	—
224	C₂×C₄×C₇	—	2	1	no	—
225	C₂×C₄×C₇	—	2	2	no	—
226	C₂×C₄×C₇	—	2	1	no	—
227	C₂×C₄×C₇	—	2	4	no	—
228	C₂×C₄×C₇	—	2	2	no	—
229	C₂×C₄×C₇	—	2	2	no	—
230	C₂×C₄×C₇	—	4	16	no	—
231	C₂×C₄×C₇	—	4	1	no	—
232	C₂×C₄×C₇	—	4	2	no	—
233	C₂×C₄×C₇	—	4	1	no	—
234	C₂×C₄×C₇	—	4	2	no	—
235	C₂×C₄×C₇	—	4	1	no	—
236	C₂×C₄×C₇	—	4	1	no	—
237	C₂×C₄×C₇	—	4	8	no	—
238	C₂×C₄×C₇	—	4	1	no	—
239	C₂×C₄×C₇	—	4	2	no	—
240	C₂×C₄×C₇	—	4	1	no	—
241	C₂×C₄×C₇	—	4	2	no	—
242	C₂×C₄×C₇	—	4	2	no	—
243	C₂×C₄×C₇	—	4	1	no	—
244	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	2	yes	—
245	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	4	yes	—
246	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	4	yes	—
247	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	2	yes	—
248	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
249	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
250	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
251	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
252	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
253	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
254	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
255	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
256	C₇×C₈	—	2	2	no	—
257	C₇×C₈	—	4	1	no	—
258	C₇×C₈	—	4	2	no	—
259	C₂²×Q₈	—	2	8	no	—
260	C₂²×Q₈	—	2	2	no	—
261	C₂²×Q₈	—	2	1	no	—
262	C₂²×Q₈	—	4	4	no	—
263	C₂²×Q₈	—	4	4	no	—
264	C₂²×Q₈	C₇×D₄	1	1	yes	characteristic
265	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
266	C₂³×C₄	—	2	2	no	—
267	C₂³×C₄	—	2	1	no	—
268	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
269	C₂²×C₈	C₂³×C₇	1	2	yes	—
270	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
271	C₄⋊Q₈	—	2	2	no	—
272	C₄⋊Q₈	—	2	4	no	—
273	C₄⋊Q₈	C₂×C₄×C₇	1	8	yes	—
274	C₄⋊Q₈	C₂×C₄×C₇	1	4	yes	—
275	C₂²⋊Q₈	—	2	16	no	—
276	C₂²⋊Q₈	—	2	8	no	—
277	C₄×Q₈	C₂×C₄×C₇	1	4	yes	—
278	C₄×Q₈	C₇×D₄	1	4	yes	—
279	C₄×D₄	—	2	2	no	—
280	C₄×D₄	—	2	4	no	—
281	C₄×D₄	C₂×C₄×C₇	1	4	yes	—
282	C₄×D₄	C₂×C₄×C₇	1	8	yes	—
283	C₂×C₄⋊C₄	—	2	2	no	—
284	C₂×C₄⋊C₄	—	2	1	no	—
285	C₂×C₄⋊C₄	—	2	1	no	—
286	C₂×C₄⋊C₄	—	2	1	no	—
287	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
288	C₂×C₄⋊C₄	C₇×D₄	1	2	yes	—
289	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
290	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
291	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
292	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
293	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
294	C₂×C₂²⋊C₄	—	2	2	no	—
295	C₂×C₂²⋊C₄	—	2	1	no	—
296	C₂×C₄²	—	2	1	no	—
297	C₂×C₄²	—	2	2	no	—
298	C₂×C₄²	C₂³×C₇	1	2	yes	—
299	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
300	C₄⋊C₈	—	2	2	no	—
301	C₄⋊C₈	—	2	4	no	—
302	#102	—	2	4	no	—
303	#102	—	2	2	no	—
304	#102	C₇×Q₈	1	4	yes	complemented
305	#97	—	2	4	no	—
306	C₄×C₈	—	2	2	no	—
307	#95	—	2	2	no	—
308	#95	—	2	1	no	—
309	#95	—	2	2	no	—
310	#95	—	2	2	no	—
311	C₂²×C₇	—	2	4	no	—
312	C₂²×C₇	—	4	2	no	—
313	C₂²×C₇	—	4	2	no	—
314	C₂²×C₇	—	4	2	no	—
315	C₂²×C₇	—	4	2	no	—
316	C₂²×C₇	—	4	2	no	—
317	C₂²×C₇	—	4	2	no	—
318	C₂²×C₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
319	C₂²×C₇	#548	1	1	yes	characteristic
320	C₂²×C₇	D₄×Q₈	1	1	yes	characteristic
321	C₂²×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
322	C₂²×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
323	C₂²×C₇	#577	1	1	yes	characteristic
324	C₂²×C₇	#578	1	1	yes	characteristic
325	C₄×C₇	—	2	4	no	—
326	C₄×C₇	—	2	4	no	—
327	C₄×C₇	—	2	2	no	—
328	C₄×C₇	—	4	2	no	—
329	C₄×C₇	—	4	4	no	—
330	C₄×C₇	—	4	2	no	—
331	C₄×C₇	—	8	1	no	—
332	C₄×C₇	—	8	2	no	—
333	C₄×C₇	#418	1	4	yes	—
334	C₄×C₇	C₂×C₄×Q₈	1	1	yes	characteristic
335	C₄×C₇	#519	1	1	yes	characteristic
336	C₄×C₇	#548	1	1	yes	characteristic
337	C₄×C₇	D₄×Q₈	1	1	yes	characteristic
338	C₂⁴	—	2	1	no	—
339	C₂×Q₈	—	2	8	no	—
340	C₂×Q₈	—	2	4	no	—
341	C₂×Q₈	—	2	2	no	—
342	C₂×Q₈	—	2	2	no	—
343	C₂×Q₈	—	4	4	no	—
344	C₂×Q₈	—	4	4	no	—
345	C₂×Q₈	—	4	32	no	—
346	C₂×Q₈	—	4	2	no	—
347	C₂×Q₈	—	4	2	no	—
348	C₂×Q₈	C₇×C₂²⋊C₄	1	4	yes	—
349	C₂×D₄	—	2	1	no	—
350	C₂×D₄	—	2	1	no	—
351	C₂×D₄	—	2	1	no	—
352	C₂×D₄	C₂²×C₄×C₇	1	2	yes	—
353	C₂×D₄	C₂×C₇×Q₈	1	2	yes	—
354	C₂×D₄	C₇×#41	1	2	yes	—
355	C₂²×C₄	—	2	1	no	—
356	C₂²×C₄	—	2	2	no	—
357	C₂²×C₄	—	2	1	no	—
358	C₂²×C₄	—	2	8	no	—
359	C₂²×C₄	—	2	4	no	—
360	C₂²×C₄	—	2	1	no	—
361	C₂²×C₄	—	2	2	no	—
362	C₂²×C₄	—	4	4	no	—
363	C₂²×C₄	—	4	4	no	—
364	C₂²×C₄	—	4	2	no	—
365	C₂²×C₄	—	4	2	no	—
366	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
367	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
368	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
369	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
370	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
371	C₂×C₈	—	2	1	no	—
372	C₂×C₈	—	2	1	no	—
373	C₂×C₈	—	2	1	no	—
374	C₂×C₈	—	2	2	no	—
375	C₂×C₈	—	2	1	no	—
376	C₂×C₈	—	2	2	no	—
377	C₂×C₈	—	2	1	no	—
378	C₂×C₈	—	2	2	no	—
379	C₂×C₈	C₂×C₇×Q₈	1	2	yes	complemented
380	C₄⋊C₄	—	2	4	no	—
381	C₄⋊C₄	—	2	8	no	—
382	C₄⋊C₄	—	2	1	no	—
383	C₄⋊C₄	—	2	2	no	—
384	C₄⋊C₄	—	2	1	no	—
385	C₄⋊C₄	—	2	1	no	—
386	C₄⋊C₄	—	2	2	no	—
387	C₄⋊C₄	—	2	1	no	—
388	C₄⋊C₄	—	2	2	no	—
389	C₄⋊C₄	—	4	2	no	—
390	C₄⋊C₄	—	8	1	no	—
391	C₄⋊C₄	C₇×C₂²⋊C₄	1	8	yes	—
392	C₄⋊C₄	C₇×C₂²⋊C₄	1	4	yes	—
393	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
394	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
395	C₄⋊C₄	C₂×C₇×Q₈	1	2	yes	complemented
396	C₂²⋊C₄	—	2	4	no	—
397	C₂²⋊C₄	—	2	8	no	—
398	C₄²	—	4	2	no	—
399	C₄²	—	8	2	no	—
400	C₄²	C₂²×C₄×C₇	1	4	yes	—
401	C₄²	C₂²×C₄×C₇	1	2	yes	—
402	C₄²	C₂×C₇×D₄	1	2	yes	—
403	C₄²	C₂×C₇×D₄	1	4	yes	—
404	C₂×C₇	—	4	4	no	—
405	C₂×C₇	Q₈×C₂²⋊C₄	1	1	yes	characteristic
406	C₂×C₇	C₂×#418	1	1	yes	characteristic
407	C₂×C₇	#2898	1	1	yes	characteristic
408	C₂×C₇	#3367	1	1	yes	characteristic
409	C₂×C₇	#3373	1	1	yes	characteristic
410	C₂×C₇	#3384	1	1	yes	characteristic
411	C₂×C₇	#3386	1	1	yes	characteristic
412	C₂³	—	2	2	no	—
413	C₂³	—	2	2	no	—
414	C₂³	—	2	2	no	—
415	C₂³	—	4	1	no	—
416	C₂³	—	4	1	no	—
417	C₂³	—	4	1	no	—
418	C₂³	—	4	1	no	—
419	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
420	Q₈	—	2	8	no	—
421	Q₈	—	4	16	no	—
422	Q₈	—	4	8	no	—
423	D₄	—	2	2	no	—
424	D₄	—	2	2	no	—
425	D₄	—	2	2	no	—
426	D₄	C₄×C₇×Q₈	1	4	yes	—
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	1	no	—
431	C₂×C₄	—	2	2	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	4	no	—
434	C₂×C₄	—	2	2	no	—
435	C₂×C₄	—	2	2	no	—
436	C₂×C₄	—	4	16	no	—
437	C₂×C₄	—	4	1	no	—
438	C₂×C₄	—	4	2	no	—
439	C₂×C₄	—	4	1	no	—
440	C₂×C₄	—	4	2	no	—
441	C₂×C₄	—	4	1	no	—
442	C₂×C₄	—	4	1	no	—
443	C₂×C₄	—	4	8	no	—
444	C₂×C₄	—	4	1	no	—
445	C₂×C₄	—	4	2	no	—
446	C₂×C₄	—	4	1	no	—
447	C₂×C₄	—	4	2	no	—
448	C₂×C₄	—	4	2	no	—
449	C₂×C₄	—	4	1	no	—
450	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	2	yes	—
451	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	4	yes	—
452	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	4	yes	—
453	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	2	yes	—
454	C₂×C₄	C₇×AGL(1,ℤ/8)	1	2	yes	—
455	C₂×C₄	C₇×AGL(1,ℤ/8)	1	2	yes	—
456	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
457	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
458	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
459	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
460	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
461	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
462	C₈	—	2	2	no	—
463	C₈	—	4	1	no	—
464	C₈	—	4	2	no	—
465	C₇	#22523	1	1	yes	characteristic, direct product factor, Sylow
466	C₂²	—	2	4	no	—
467	C₂²	—	4	2	no	—
468	C₂²	—	4	2	no	—
469	C₂²	—	4	2	no	—
470	C₂²	—	4	2	no	—
471	C₂²	—	4	2	no	—
472	C₂²	—	4	2	no	—
473	C₂²	C₂²×C₇×C₂²⋊C₄	1	1	yes	characteristic
474	C₂²	C₇×#548	1	1	yes	characteristic
475	C₂²	C₇×D₄×Q₈	1	1	yes	characteristic
476	C₂²	C₂×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
477	C₂²	C₂×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
478	C₂²	C₇×#577	1	1	yes	characteristic
479	C₂²	C₇×#578	1	1	yes	characteristic
480	C₄	—	2	4	no	—
481	C₄	—	2	4	no	—
482	C₄	—	2	2	no	—
483	C₄	—	4	2	no	—
484	C₄	—	4	4	no	—
485	C₄	—	4	2	no	—
486	C₄	—	8	1	no	—
487	C₄	—	8	2	no	—
488	C₄	C₇×#418	1	4	yes	—
489	C₄	C₂×C₄×C₇×Q₈	1	1	yes	characteristic
490	C₄	C₇×#519	1	1	yes	characteristic
491	C₄	C₇×#548	1	1	yes	characteristic
492	C₄	C₇×D₄×Q₈	1	1	yes	characteristic
493	C₂	—	4	4	no	—
494	C₂	C₇×Q₈×C₂²⋊C₄	1	1	yes	characteristic
495	C₂	C₂×C₇×#418	1	1	yes	characteristic
496	C₂	C₇×#2898	1	1	yes	characteristic
497	C₂	C₇×#3367	1	1	yes	characteristic
498	C₂	C₇×#3373	1	1	yes	characteristic
499	C₂	C₇×#3384	1	1	yes	characteristic
500	C₂	C₇×#3386	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
prime radical	C₂²×C₇×D₄	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄×C₇×Q₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×#446	C₂	semidirect
3. C₇×#2355	C₂	semidirect
4. C₇×#1993	C₂	semidirect
5. C₇×#1990	C₂	semidirect
6. C₇×#1988	C₂	semidirect
7. C₇×#1987	C₂	semidirect
14. C₇×D₄×Q₈	C₄	semidirect
44. #22523	C₇	direct (N × H)
90. C₇×#102	Q₈	semidirect
98. C₂×#446	C₂×C₇	semidirect
99. #2355	C₂×C₇	semidirect
100. #1993	C₂×C₇	semidirect
101. #1990	C₂×C₇	semidirect
102. #1988	C₂×C₇	semidirect
103. #1987	C₂×C₇	semidirect
149. C₂×C₇×C₈	C₂×Q₈	semidirect
165. C₇×C₄⋊C₄	C₂×Q₈	semidirect
176. D₄×Q₈	C₄×C₇	semidirect
304. #102	C₇×Q₈	semidirect
379. C₂×C₈	C₂×C₇×Q₈	semidirect
395. C₄⋊C₄	C₂×C₇×Q₈	semidirect
465. C₇	#22523	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇×D₄×Q₈	2
C₂×C₇×#446	2
C₇×#2355	2
C₇×#1993	2
C₇×#1990	2
C₇×#1988	2
C₇×#1987	2
C₇×#1804	2
C₇×#1802	2
C₇×#1762	2
C₄×C₇×#102	2
#22523	7

Subgroups — C7×#22523 #13199739

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22523 #13199739