Subgroups — C₇×#23232 #13200448

Order 1792.

822 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#489	C₂	1	1	yes	maximal, characteristic
2	C₇×#2369	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#2367	C₂	1	1	yes	maximal, characteristic
4	C₇×#1988	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1983	C₂	1	1	yes	maximal, characteristic
6	C₇×#1936	C₂	1	1	yes	maximal, characteristic
7	C₇×#1933	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#1928	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#1925	C₂	1	1	yes	maximal, characteristic
10	C₇×#1865	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#1864	C₂	1	1	yes	maximal, characteristic
12	C₇×#1776	C₂	1	1	yes	maximal, characteristic, complemented
13	C₇×#1771	C₂	1	1	yes	maximal, characteristic, complemented
14	C₇×#1763	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×C₇×#102	C₂	1	1	yes	maximal, characteristic
16	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₂×C₇×#124	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
20	C₇×#489	C₄	1	8	yes	complemented
21	C₂×C₇×#107	C₂²	1	1	yes	characteristic
22	C₂×C₇×#106	C₂²	1	1	yes	characteristic
23	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₇×#103	C₂²	1	1	yes	characteristic
25	C₂×C₇×#103	C₂²	1	1	yes	characteristic
26	C₂×C₇×#103	C₂²	1	1	yes	characteristic
27	C₂×C₇×#103	C₂²	1	1	yes	characteristic
28	C₂×C₇×#102	C₂²	1	1	yes	characteristic
29	C₂×C₇×#102	C₂²	1	1	yes	characteristic
30	C₂×C₇×#102	C₂²	1	1	yes	characteristic
31	C₂×C₇×#102	C₂²	1	1	yes	characteristic
32	C₂×C₇×#97	C₂²	1	1	yes	characteristic
33	C₂×C₇×#97	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
35	C₇×#391	—	2	1	no	—
36	C₇×#391	—	2	1	no	—
37	C₇×#391	C₂²	1	1	yes	characteristic
38	C₇×#390	C₂²	1	1	yes	characteristic
39	C₇×#389	—	2	1	no	—
40	C₇×#389	C₂²	1	1	yes	characteristic
41	C₇×#389	C₂²	1	1	yes	characteristic
42	C₇×#388	—	2	1	no	—
43	C₇×#388	—	2	1	no	—
44	C₇×#385	—	2	1	no	—
45	C₄×C₇×C₄⋊C₄	—	2	1	no	—
46	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
47	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
48	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
49	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
50	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
51	C₄×C₇×C₂²⋊C₄	—	2	1	no	—
52	C₇×#340	C₂²	1	1	yes	characteristic
53	C₇×#340	C₂²	1	1	yes	characteristic
54	C₇×#340	C₂²	1	1	yes	characteristic
55	C₇×#340	C₂²	1	1	yes	characteristic
56	C₇×#340	C₂²	1	1	yes	characteristic
57	C₇×#340	C₂²	1	1	yes	characteristic
58	C₇×#336	C₂²	1	1	yes	characteristic
59	C₇×#336	C₂²	1	1	yes	characteristic
60	#23232	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂²×C₇×Q₈	—	2	1	no	—
62	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
63	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
64	C₂³×C₄×C₇	—	2	1	no	—
65	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
66	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
67	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
68	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
69	C₇×C₄⋊Q₈	—	2	2	no	—
70	C₇×C₄⋊Q₈	C₂×C₄	1	4	yes	complemented
71	C₇×#124	—	2	2	no	—
72	C₇×#124	C₂×C₄	1	4	yes	—
73	C₄×C₇×Q₈	—	2	2	no	—
74	C₄×C₇×Q₈	—	4	1	no	—
75	C₄×C₇×D₄	—	2	2	no	—
76	C₄×C₇×D₄	—	4	1	no	—
77	C₂×C₇×C₄⋊C₄	—	2	1	no	—
78	C₂×C₇×C₄⋊C₄	—	2	1	no	—
79	C₂×C₇×C₄⋊C₄	—	2	1	no	—
80	C₂×C₇×C₄⋊C₄	—	2	1	no	—
81	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
82	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
83	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
84	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
85	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
86	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
87	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
88	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
89	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
90	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
91	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
92	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
93	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
94	C₂×C₄²×C₇	—	2	1	no	—
95	C₂×C₄²×C₇	—	2	1	no	—
96	C₂×C₄²×C₇	—	2	1	no	—
97	C₂×C₄²×C₇	—	2	1	no	—
98	C₂×C₄²×C₇	—	2	1	no	—
99	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
100	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
101	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
102	C₇×#107	—	2	2	no	—
103	C₇×#106	—	2	2	no	—
104	C₇×C₄⋊C₈	—	2	2	no	—
105	C₇×#103	—	2	2	no	—
106	C₇×#103	—	4	1	no	—
107	C₇×#103	C₂×C₄	1	4	yes	complemented
108	C₇×#103	C₂×C₄	1	4	yes	complemented
109	C₇×#102	—	2	2	no	—
110	C₇×#102	—	4	1	no	—
111	C₇×#102	C₂×C₄	1	4	yes	—
112	C₇×#102	C₂×C₄	1	4	yes	—
113	C₇×#97	—	4	1	no	—
114	C₇×#97	C₂×C₄	1	4	yes	complemented
115	C₄×C₇×C₈	—	4	1	no	—
116	C₇×#95	—	2	1	no	—
117	C₇×#95	—	2	1	no	—
118	C₇×#95	—	2	1	no	—
119	C₇×#95	—	2	1	no	—
120	C₇×#95	—	2	1	no	—
121	C₇×#95	—	2	1	no	—
122	C₇×#95	—	2	1	no	—
123	C₇×#95	—	2	1	no	—
124	C₇×#95	D₄	1	1	yes	characteristic
125	C₇×#95	D₄	1	1	yes	characteristic
126	C₂×#489	C₂×C₇	1	1	yes	characteristic
127	#2369	C₂×C₇	1	1	yes	characteristic, complemented
128	#2367	C₂×C₇	1	1	yes	characteristic
129	#1988	C₂×C₇	1	1	yes	characteristic, complemented
130	#1983	C₂×C₇	1	1	yes	characteristic
131	#1936	C₂×C₇	1	1	yes	characteristic
132	#1933	C₂×C₇	1	1	yes	characteristic, complemented
133	#1928	C₂×C₇	1	1	yes	characteristic, complemented
134	#1925	C₂×C₇	1	1	yes	characteristic
135	#1865	C₂×C₇	1	1	yes	characteristic, complemented
136	#1864	C₂×C₇	1	1	yes	characteristic
137	#1776	C₂×C₇	1	1	yes	characteristic, complemented
138	#1771	C₂×C₇	1	1	yes	characteristic, complemented
139	#1763	C₂×C₇	1	1	yes	characteristic, complemented
140	C₄×#102	C₂×C₇	1	1	yes	characteristic
141	C₂⁴×C₇	—	2	1	no	—
142	C₂×C₇×Q₈	—	2	4	no	—
143	C₂×C₇×Q₈	—	2	1	no	—
144	C₂×C₇×Q₈	—	2	1	no	—
145	C₂×C₇×Q₈	—	2	1	no	—
146	C₂×C₇×Q₈	—	2	1	no	—
147	C₂×C₇×Q₈	—	2	1	no	—
148	C₂×C₇×Q₈	—	8	1	no	—
149	C₂×C₇×Q₈	—	8	1	no	—
150	C₂×C₇×Q₈	C₂²×C₄	1	2	yes	—
151	C₂×C₇×D₄	—	2	1	no	—
152	C₂×C₇×D₄	—	2	1	no	—
153	C₂×C₇×D₄	—	2	1	no	—
154	C₂×C₇×D₄	—	2	1	no	—
155	C₂×C₇×D₄	—	2	1	no	—
156	C₂×C₇×D₄	C₂²×C₄	1	2	yes	—
157	C₂²×C₄×C₇	—	2	1	no	—
158	C₂²×C₄×C₇	—	2	1	no	—
159	C₂²×C₄×C₇	—	2	1	no	—
160	C₂²×C₄×C₇	—	2	1	no	—
161	C₂²×C₄×C₇	—	2	1	no	—
162	C₂²×C₄×C₇	—	2	1	no	—
163	C₂²×C₄×C₇	—	2	1	no	—
164	C₂²×C₄×C₇	—	4	2	no	—
165	C₂²×C₄×C₇	—	4	2	no	—
166	C₂²×C₄×C₇	—	8	1	no	—
167	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
168	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
169	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
171	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
172	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
173	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
174	C₂×C₇×C₈	—	2	1	no	—
175	C₂×C₇×C₈	—	2	1	no	—
176	C₂×C₇×C₈	—	2	1	no	—
177	C₂×C₇×C₈	—	2	1	no	—
178	C₂×C₇×C₈	—	2	1	no	—
179	C₂×C₇×C₈	—	2	1	no	—
180	C₂×C₇×C₈	—	2	1	no	—
181	C₂×C₇×C₈	—	2	1	no	—
182	C₂×C₇×C₈	—	2	1	no	—
183	C₂×C₇×C₈	—	2	1	no	—
184	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
185	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
186	C₇×C₄⋊C₄	—	2	4	no	—
187	C₇×C₄⋊C₄	—	2	2	no	—
188	C₇×C₄⋊C₄	—	2	1	no	—
189	C₇×C₄⋊C₄	—	2	1	no	—
190	C₇×C₄⋊C₄	—	2	1	no	—
191	C₇×C₄⋊C₄	—	2	1	no	—
192	C₇×C₄⋊C₄	—	2	1	no	—
193	C₇×C₄⋊C₄	—	2	1	no	—
194	C₇×C₄⋊C₄	—	2	1	no	—
195	C₇×C₄⋊C₄	—	2	1	no	—
196	C₇×C₄⋊C₄	—	4	1	no	—
197	C₇×C₄⋊C₄	—	4	2	no	—
198	C₇×C₄⋊C₄	—	8	1	no	—
199	C₇×C₄⋊C₄	—	8	1	no	—
200	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
201	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
202	C₇×C₄⋊C₄	#41	1	2	yes	—
203	C₇×C₄⋊C₄	#41	1	2	yes	—
204	C₇×C₂²⋊C₄	—	2	4	no	—
205	C₇×C₂²⋊C₄	—	2	4	no	—
206	C₇×C₂²⋊C₄	—	4	2	no	—
207	C₇×C₂²⋊C₄	—	4	2	no	—
208	C₇×C₂²⋊C₄	—	8	1	no	—
209	C₄²×C₇	—	2	1	no	—
210	C₄²×C₇	—	2	1	no	—
211	C₄²×C₇	—	2	1	no	—
212	C₄²×C₇	—	2	1	no	—
213	C₄²×C₇	—	4	2	no	—
214	C₄²×C₇	—	4	2	no	—
215	C₄²×C₇	—	8	1	no	—
216	C₄²×C₇	—	8	1	no	—
217	C₄²×C₇	—	8	1	no	—
218	C₄²×C₇	C₂²×C₄	1	2	yes	—
219	C₄²×C₇	#41	1	2	yes	—
220	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
221	C₂×#124	C₂²×C₇	1	1	yes	characteristic
222	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
223	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
224	#489	C₄×C₇	1	8	yes	complemented
225	C₂×#107	C₂²×C₇	1	1	yes	characteristic
226	C₂×#106	C₂²×C₇	1	1	yes	characteristic
227	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
228	C₂×#103	C₂²×C₇	1	1	yes	characteristic
229	C₂×#103	C₂²×C₇	1	1	yes	characteristic
230	C₂×#103	C₂²×C₇	1	1	yes	characteristic
231	C₂×#103	C₂²×C₇	1	1	yes	characteristic
232	C₂×#102	C₂²×C₇	1	1	yes	characteristic
233	C₂×#102	C₂²×C₇	1	1	yes	characteristic
234	C₂×#102	C₂²×C₇	1	1	yes	characteristic
235	C₂×#102	C₂²×C₇	1	1	yes	characteristic
236	C₂×#97	C₂²×C₇	1	1	yes	characteristic
237	C₂×#97	C₂²×C₇	1	1	yes	characteristic
238	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
239	#391	—	2	1	no	—
240	#391	—	2	1	no	—
241	#391	C₂²×C₇	1	1	yes	characteristic
242	#390	C₂²×C₇	1	1	yes	characteristic
243	#389	—	2	1	no	—
244	#389	C₂²×C₇	1	1	yes	characteristic
245	#389	C₂²×C₇	1	1	yes	characteristic
246	#388	—	2	1	no	—
247	#388	—	2	1	no	—
248	#385	—	2	1	no	—
249	C₄×C₄⋊C₄	—	2	1	no	—
250	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
251	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
252	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
253	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
254	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
255	C₄×C₂²⋊C₄	—	2	1	no	—
256	#340	C₂²×C₇	1	1	yes	characteristic
257	#340	C₂²×C₇	1	1	yes	characteristic
258	#340	C₂²×C₇	1	1	yes	characteristic
259	#340	C₂²×C₇	1	1	yes	characteristic
260	#340	C₂²×C₇	1	1	yes	characteristic
261	#340	C₂²×C₇	1	1	yes	characteristic
262	#336	C₂²×C₇	1	1	yes	characteristic
263	#336	C₂²×C₇	1	1	yes	characteristic
264	C₂³×C₇	—	2	2	no	—
265	C₂³×C₇	—	4	1	no	—
266	C₂³×C₇	—	4	1	no	—
267	C₂³×C₇	—	4	1	no	—
268	C₂³×C₇	—	4	1	no	—
269	C₂³×C₇	—	4	1	no	—
270	C₂³×C₇	—	4	1	no	—
271	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
272	C₇×Q₈	—	2	2	no	—
273	C₇×Q₈	—	2	2	no	—
274	C₇×Q₈	—	4	1	no	—
275	C₇×Q₈	—	4	1	no	—
276	C₇×Q₈	—	8	4	no	—
277	C₇×D₄	—	2	2	no	—
278	C₇×D₄	—	2	2	no	—
279	C₇×D₄	—	4	1	no	—
280	C₇×D₄	—	4	1	no	—
281	C₂×C₄×C₇	—	2	1	no	—
282	C₂×C₄×C₇	—	2	1	no	—
283	C₂×C₄×C₇	—	2	1	no	—
284	C₂×C₄×C₇	—	2	1	no	—
285	C₂×C₄×C₇	—	2	1	no	—
286	C₂×C₄×C₇	—	2	1	no	—
287	C₂×C₄×C₇	—	2	2	no	—
288	C₂×C₄×C₇	—	2	2	no	—
289	C₂×C₄×C₇	—	2	2	no	—
290	C₂×C₄×C₇	—	2	1	no	—
291	C₂×C₄×C₇	—	2	1	no	—
292	C₂×C₄×C₇	—	2	1	no	—
293	C₂×C₄×C₇	—	2	2	no	—
294	C₂×C₄×C₇	—	2	2	no	—
295	C₂×C₄×C₇	—	2	1	no	—
296	C₂×C₄×C₇	—	2	1	no	—
297	C₂×C₄×C₇	—	2	1	no	—
298	C₂×C₄×C₇	—	2	1	no	—
299	C₂×C₄×C₇	—	2	1	no	—
300	C₂×C₄×C₇	—	2	1	no	—
301	C₂×C₄×C₇	—	4	1	no	—
302	C₂×C₄×C₇	—	4	1	no	—
303	C₂×C₄×C₇	—	4	1	no	—
304	C₂×C₄×C₇	—	4	1	no	—
305	C₂×C₄×C₇	—	4	1	no	—
306	C₂×C₄×C₇	—	4	1	no	—
307	C₂×C₄×C₇	—	4	1	no	—
308	C₂×C₄×C₇	—	4	1	no	—
309	C₂×C₄×C₇	—	4	1	no	—
310	C₂×C₄×C₇	—	4	1	no	—
311	C₂×C₄×C₇	—	4	1	no	—
312	C₂×C₄×C₇	—	4	1	no	—
313	C₂×C₄×C₇	—	4	1	no	—
314	C₂×C₄×C₇	—	4	1	no	—
315	C₂×C₄×C₇	—	4	1	no	—
316	C₂×C₄×C₇	—	4	1	no	—
317	C₂×C₄×C₇	—	8	4	no	—
318	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
319	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
320	C₂×C₄×C₇	#137	1	2	yes	—
321	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
322	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
323	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
324	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
325	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
326	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
327	C₇×C₈	—	2	2	no	—
328	C₇×C₈	—	2	2	no	—
329	C₇×C₈	—	4	1	no	—
330	C₇×C₈	—	4	1	no	—
331	C₂²×Q₈	—	2	1	no	—
332	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
333	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
334	C₂³×C₄	—	2	1	no	—
335	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
336	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
337	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
338	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
339	C₄⋊Q₈	—	2	2	no	—
340	C₄⋊Q₈	C₂×C₄×C₇	1	4	yes	complemented
341	#124	—	2	2	no	—
342	#124	C₂×C₄×C₇	1	4	yes	—
343	C₄×Q₈	—	2	2	no	—
344	C₄×Q₈	—	4	1	no	—
345	C₄×D₄	—	2	2	no	—
346	C₄×D₄	—	4	1	no	—
347	C₂×C₄⋊C₄	—	2	1	no	—
348	C₂×C₄⋊C₄	—	2	1	no	—
349	C₂×C₄⋊C₄	—	2	1	no	—
350	C₂×C₄⋊C₄	—	2	1	no	—
351	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
352	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
353	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
354	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
355	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
356	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
357	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
358	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
359	C₂×C₂²⋊C₄	—	2	1	no	—
360	C₂×C₂²⋊C₄	—	2	1	no	—
361	C₂×C₂²⋊C₄	—	2	1	no	—
362	C₂×C₂²⋊C₄	—	2	1	no	—
363	C₂×C₂²⋊C₄	—	2	1	no	—
364	C₂×C₄²	—	2	1	no	—
365	C₂×C₄²	—	2	1	no	—
366	C₂×C₄²	—	2	1	no	—
367	C₂×C₄²	—	2	1	no	—
368	C₂×C₄²	—	2	1	no	—
369	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
370	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
371	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
372	#107	—	2	2	no	—
373	#106	—	2	2	no	—
374	C₄⋊C₈	—	2	2	no	—
375	#103	—	2	2	no	—
376	#103	—	4	1	no	—
377	#103	C₂×C₄×C₇	1	4	yes	complemented
378	#103	C₂×C₄×C₇	1	4	yes	complemented
379	#102	—	2	2	no	—
380	#102	—	4	1	no	—
381	#102	C₂×C₄×C₇	1	4	yes	—
382	#102	C₂×C₄×C₇	1	4	yes	—
383	#97	—	4	1	no	—
384	#97	C₂×C₄×C₇	1	4	yes	complemented
385	C₄×C₈	—	4	1	no	—
386	#95	—	2	1	no	—
387	#95	—	2	1	no	—
388	#95	—	2	1	no	—
389	#95	—	2	1	no	—
390	#95	—	2	1	no	—
391	#95	—	2	1	no	—
392	#95	—	2	1	no	—
393	#95	—	2	1	no	—
394	#95	C₇×D₄	1	1	yes	characteristic
395	#95	C₇×D₄	1	1	yes	characteristic
396	C₂²×C₇	—	4	2	no	—
397	C₂²×C₇	—	4	2	no	—
398	C₂²×C₇	—	4	2	no	—
399	C₂²×C₇	—	8	1	no	—
400	C₂²×C₇	—	8	1	no	—
401	C₂²×C₇	—	8	1	no	—
402	C₂²×C₇	—	8	1	no	—
403	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
404	C₂²×C₇	#546	1	1	yes	characteristic
405	C₂²×C₇	#548	1	1	yes	characteristic
406	C₂²×C₇	C₂×#137	1	1	yes	characteristic
407	C₂²×C₇	#575	1	1	yes	characteristic
408	C₂²×C₇	#576	1	1	yes	characteristic
409	C₂²×C₇	#577	1	1	yes	characteristic
410	C₄×C₇	—	2	2	no	—
411	C₄×C₇	—	2	2	no	—
412	C₄×C₇	—	2	2	no	—
413	C₄×C₇	—	4	2	no	—
414	C₄×C₇	—	4	2	no	—
415	C₄×C₇	—	4	2	no	—
416	C₄×C₇	—	4	2	no	—
417	C₄×C₇	—	4	1	no	—
418	C₄×C₇	—	4	2	no	—
419	C₄×C₇	—	4	1	no	—
420	C₄×C₇	—	4	1	no	—
421	C₄×C₇	—	8	1	no	—
422	C₄×C₇	—	8	1	no	—
423	C₄×C₇	C₄×#41	1	1	yes	characteristic
424	C₄×C₇	#520	1	1	yes	characteristic
425	C₄×C₇	#556	1	1	yes	characteristic
426	C₄×C₇	#559	1	1	yes	characteristic
427	C₂⁴	—	2	1	no	—
428	C₂×Q₈	—	2	4	no	—
429	C₂×Q₈	—	2	1	no	—
430	C₂×Q₈	—	2	1	no	—
431	C₂×Q₈	—	2	1	no	—
432	C₂×Q₈	—	2	1	no	—
433	C₂×Q₈	—	2	1	no	—
434	C₂×Q₈	—	8	1	no	—
435	C₂×Q₈	—	8	1	no	—
436	C₂×Q₈	C₂²×C₄×C₇	1	2	yes	—
437	C₂×D₄	—	2	1	no	—
438	C₂×D₄	—	2	1	no	—
439	C₂×D₄	—	2	1	no	—
440	C₂×D₄	—	2	1	no	—
441	C₂×D₄	—	2	1	no	—
442	C₂×D₄	C₂²×C₄×C₇	1	2	yes	—
443	C₂²×C₄	—	2	1	no	—
444	C₂²×C₄	—	2	1	no	—
445	C₂²×C₄	—	2	1	no	—
446	C₂²×C₄	—	2	1	no	—
447	C₂²×C₄	—	2	1	no	—
448	C₂²×C₄	—	2	1	no	—
449	C₂²×C₄	—	2	1	no	—
450	C₂²×C₄	—	4	2	no	—
451	C₂²×C₄	—	4	2	no	—
452	C₂²×C₄	—	8	1	no	—
453	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
454	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
455	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
456	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
457	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
458	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
459	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
460	C₂×C₈	—	2	1	no	—
461	C₂×C₈	—	2	1	no	—
462	C₂×C₈	—	2	1	no	—
463	C₂×C₈	—	2	1	no	—
464	C₂×C₈	—	2	1	no	—
465	C₂×C₈	—	2	1	no	—
466	C₂×C₈	—	2	1	no	—
467	C₂×C₈	—	2	1	no	—
468	C₂×C₈	—	2	1	no	—
469	C₂×C₈	—	2	1	no	—
470	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
471	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
472	C₄⋊C₄	—	2	4	no	—
473	C₄⋊C₄	—	2	2	no	—
474	C₄⋊C₄	—	2	1	no	—
475	C₄⋊C₄	—	2	1	no	—
476	C₄⋊C₄	—	2	1	no	—
477	C₄⋊C₄	—	2	1	no	—
478	C₄⋊C₄	—	2	1	no	—
479	C₄⋊C₄	—	2	1	no	—
480	C₄⋊C₄	—	2	1	no	—
481	C₄⋊C₄	—	2	1	no	—
482	C₄⋊C₄	—	4	1	no	—
483	C₄⋊C₄	—	4	2	no	—
484	C₄⋊C₄	—	8	1	no	—
485	C₄⋊C₄	—	8	1	no	—
486	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
487	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
488	C₄⋊C₄	C₇×#41	1	2	yes	—
489	C₄⋊C₄	C₇×#41	1	2	yes	—
490	C₂²⋊C₄	—	2	4	no	—
491	C₂²⋊C₄	—	2	4	no	—
492	C₂²⋊C₄	—	4	2	no	—
493	C₂²⋊C₄	—	4	2	no	—
494	C₂²⋊C₄	—	8	1	no	—
495	C₄²	—	2	1	no	—
496	C₄²	—	2	1	no	—
497	C₄²	—	2	1	no	—
498	C₄²	—	2	1	no	—
499	C₄²	—	4	2	no	—
500	C₄²	—	4	2	no	—
501	C₄²	—	8	1	no	—
502	C₄²	—	8	1	no	—
503	C₄²	—	8	1	no	—
504	C₄²	C₂²×C₄×C₇	1	2	yes	—
505	C₄²	C₇×#41	1	2	yes	—
506	C₂×C₇	—	8	2	no	—
507	C₂×C₇	#2367	1	1	yes	characteristic
508	C₂×C₇	C₄×#137	1	1	yes	characteristic
509	C₂×C₇	#2946	1	1	yes	characteristic
510	C₂×C₇	#3314	1	1	yes	characteristic
511	C₂×C₇	#3321	1	1	yes	characteristic
512	C₂×C₇	#3369	1	1	yes	characteristic
513	C₂×C₇	#3370	1	1	yes	characteristic
514	C₂³	—	2	2	no	—
515	C₂³	—	4	1	no	—
516	C₂³	—	4	1	no	—
517	C₂³	—	4	1	no	—
518	C₂³	—	4	1	no	—
519	C₂³	—	4	1	no	—
520	C₂³	—	4	1	no	—
521	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
522	Q₈	—	2	2	no	—
523	Q₈	—	2	2	no	—
524	Q₈	—	4	1	no	—
525	Q₈	—	4	1	no	—
526	Q₈	—	8	4	no	—
527	D₄	—	2	2	no	—
528	D₄	—	2	2	no	—
529	D₄	—	4	1	no	—
530	D₄	—	4	1	no	—
531	C₂×C₄	—	2	1	no	—
532	C₂×C₄	—	2	1	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	2	1	no	—
536	C₂×C₄	—	2	1	no	—
537	C₂×C₄	—	2	2	no	—
538	C₂×C₄	—	2	2	no	—
539	C₂×C₄	—	2	2	no	—
540	C₂×C₄	—	2	1	no	—
541	C₂×C₄	—	2	1	no	—
542	C₂×C₄	—	2	1	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	2	2	no	—
545	C₂×C₄	—	2	1	no	—
546	C₂×C₄	—	2	1	no	—
547	C₂×C₄	—	2	1	no	—
548	C₂×C₄	—	2	1	no	—
549	C₂×C₄	—	2	1	no	—
550	C₂×C₄	—	2	1	no	—
551	C₂×C₄	—	4	1	no	—
552	C₂×C₄	—	4	1	no	—
553	C₂×C₄	—	4	1	no	—
554	C₂×C₄	—	4	1	no	—
555	C₂×C₄	—	4	1	no	—
556	C₂×C₄	—	4	1	no	—
557	C₂×C₄	—	4	1	no	—
558	C₂×C₄	—	4	1	no	—
559	C₂×C₄	—	4	1	no	—
560	C₂×C₄	—	4	1	no	—
561	C₂×C₄	—	4	1	no	—
562	C₂×C₄	—	4	1	no	—
563	C₂×C₄	—	4	1	no	—
564	C₂×C₄	—	4	1	no	—
565	C₂×C₄	—	4	1	no	—
566	C₂×C₄	—	4	1	no	—
567	C₂×C₄	—	8	4	no	—
568	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
569	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
570	C₂×C₄	C₇×#137	1	2	yes	—
571	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
572	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
573	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
574	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
575	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
576	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
577	C₈	—	2	2	no	—
578	C₈	—	2	2	no	—
579	C₈	—	4	1	no	—
580	C₈	—	4	1	no	—
581	C₇	#23232	1	1	yes	characteristic, direct product factor, Sylow
582	C₂²	—	4	2	no	—
583	C₂²	—	4	2	no	—
584	C₂²	—	4	2	no	—
585	C₂²	—	8	1	no	—
586	C₂²	—	8	1	no	—
587	C₂²	—	8	1	no	—
588	C₂²	—	8	1	no	—
589	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
590	C₂²	C₇×#546	1	1	yes	characteristic
591	C₂²	C₇×#548	1	1	yes	characteristic
592	C₂²	C₂×C₇×#137	1	1	yes	characteristic
593	C₂²	C₇×#575	1	1	yes	characteristic
594	C₂²	C₇×#576	1	1	yes	characteristic
595	C₂²	C₇×#577	1	1	yes	characteristic
596	C₄	—	2	2	no	—
597	C₄	—	2	2	no	—
598	C₄	—	2	2	no	—
599	C₄	—	4	2	no	—
600	C₄	—	4	2	no	—
601	C₄	—	4	1	no	—
602	C₄	—	4	2	no	—
603	C₄	—	4	1	no	—
604	C₄	—	4	1	no	—
605	C₄	—	4	2	no	—
606	C₄	—	4	2	no	—
607	C₄	—	8	1	no	—
608	C₄	—	8	1	no	—
609	C₄	C₄×C₇×#41	1	1	yes	characteristic
610	C₄	C₇×#520	1	1	yes	characteristic
611	C₄	C₇×#556	1	1	yes	characteristic
612	C₄	C₇×#559	1	1	yes	characteristic
613	C₂	—	8	2	no	—
614	C₂	C₇×#2367	1	1	yes	characteristic
615	C₂	C₄×C₇×#137	1	1	yes	characteristic
616	C₂	C₇×#2946	1	1	yes	characteristic
617	C₂	C₇×#3314	1	1	yes	characteristic
618	C₂	C₇×#3321	1	1	yes	characteristic
619	C₂	C₇×#3369	1	1	yes	characteristic
620	C₂	C₇×#3370	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 ◁ C₄×C₇×C₄⋊C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#2369	C₂	semidirect
4. C₇×#1988	C₂	semidirect
7. C₇×#1933	C₂	semidirect
8. C₇×#1928	C₂	semidirect
10. C₇×#1865	C₂	semidirect
12. C₇×#1776	C₂	semidirect
13. C₇×#1771	C₂	semidirect
14. C₇×#1763	C₂	semidirect
20. C₇×#489	C₄	semidirect
60. #23232	C₇	direct (N × H)
70. C₇×C₄⋊Q₈	C₂×C₄	semidirect
107. C₇×#103	C₂×C₄	semidirect
108. C₇×#103	C₂×C₄	semidirect
114. C₇×#97	C₂×C₄	semidirect
127. #2369	C₂×C₇	semidirect
129. #1988	C₂×C₇	semidirect
132. #1933	C₂×C₇	semidirect
133. #1928	C₂×C₇	semidirect
135. #1865	C₂×C₇	semidirect
137. #1776	C₂×C₇	semidirect
138. #1771	C₂×C₇	semidirect
139. #1763	C₂×C₇	semidirect
224. #489	C₄×C₇	semidirect
340. C₄⋊Q₈	C₂×C₄×C₇	semidirect
377. #103	C₂×C₄×C₇	semidirect
378. #103	C₂×C₄×C₇	semidirect
384. #97	C₂×C₄×C₇	semidirect
581. C₇	#23232	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×#489	2
C₇×#2369	2
C₇×#2367	2
C₇×#1988	2
C₇×#1983	2
C₇×#1936	2
C₇×#1933	2
C₇×#1928	2
C₇×#1925	2
C₇×#1865	2
C₇×#1864	2
C₇×#1776	2
C₇×#1771	2
C₇×#1763	2
C₄×C₇×#102	2
#23232	7

Subgroups — C7×#23232 #13200448

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#23232 #13200448