Subgroups — C₇×#22979 #13200195

Order 1792.

850 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#474	C₂	1	1	yes	maximal, characteristic
2	C₇×#2369	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#2349	C₂	1	1	yes	maximal, characteristic
4	C₇×#1993	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1949	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1943	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1942	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#1921	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#1918	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#1864	C₂	1	1	yes	maximal, characteristic
11	C₇×#1862	C₂	1	1	yes	maximal, characteristic
12	C₇×#1771	C₂	1	1	yes	maximal, characteristic, complemented
13	C₇×#1762	C₂	1	1	yes	maximal, characteristic
14	C₇×#1762	C₂	1	1	yes	maximal, characteristic
15	C₄×C₇×#102	C₂	1	1	yes	maximal, characteristic
16	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
20	C₇×#474	C₄	1	8	yes	complemented
21	C₂×C₇×#107	C₂²	1	1	yes	characteristic
22	C₂×C₇×#107	C₂²	1	1	yes	characteristic
23	C₂×C₇×#107	C₂²	1	1	yes	characteristic
24	C₂×C₇×#106	C₂²	1	1	yes	characteristic
25	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₇×#102	C₂²	1	1	yes	characteristic
27	C₂×C₇×#102	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×#97	C₂²	1	1	yes	characteristic
31	C₂×C₇×#97	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
33	C₇×#391	—	2	1	no	—
34	C₇×#391	—	2	1	no	—
35	C₇×#389	—	2	1	no	—
36	C₇×#389	C₂²	1	1	yes	characteristic
37	C₇×#389	C₂²	1	1	yes	characteristic
38	C₇×#389	C₂²	1	1	yes	characteristic
39	C₇×#389	C₂²	1	1	yes	characteristic
40	C₇×#385	—	2	1	no	—
41	C₇×#385	—	2	1	no	—
42	C₇×#382	C₂²	1	1	yes	characteristic
43	C₇×#380	—	2	1	no	—
44	C₄×C₇×C₄⋊C₄	—	2	1	no	—
45	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
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	C₂×C₇×#95	—	2	1	no	—
50	C₇×#340	C₂²	1	1	yes	characteristic
51	C₇×#340	C₂²	1	1	yes	characteristic
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	#22979	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂²×C₇×Q₈	—	2	1	no	—
62	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
63	C₂³×C₄×C₇	—	2	1	no	—
64	C₂³×C₄×C₇	—	2	1	no	—
65	C₂³×C₄×C₇	—	2	1	no	—
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	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
70	C₇×C₄⋊Q₈	—	2	2	no	—
71	C₇×C₄⋊Q₈	C₂×C₄	1	4	yes	complemented
72	C₄×C₇×D₄	—	2	2	no	—
73	C₄×C₇×D₄	—	2	2	no	—
74	C₄×C₇×D₄	—	2	2	no	—
75	C₄×C₇×D₄	—	4	1	no	—
76	C₄×C₇×D₄	—	4	1	no	—
77	C₄×C₇×D₄	C₂×C₄	1	4	yes	—
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	—	2	1	no	—
82	C₂×C₇×C₄⋊C₄	—	2	1	no	—
83	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
84	C₂×C₇×C₄⋊C₄	D₄	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×C₄⋊C₄	C₂³	1	1	yes	characteristic
90	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
91	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
92	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
93	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
94	C₂×C₇×C₂²⋊C₄	—	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	C₂³	1	1	yes	characteristic
99	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
100	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
101	C₇×#107	—	4	1	no	—
102	C₇×#107	C₂×C₄	1	4	yes	complemented
103	C₇×#107	C₂×C₄	1	4	yes	complemented
104	C₇×#106	—	4	1	no	—
105	C₇×C₄⋊C₈	C₂×C₄	1	4	yes	complemented
106	C₇×#102	—	2	2	no	—
107	C₇×#102	C₂×C₄	1	4	yes	—
108	C₇×#102	C₂×C₄	1	4	yes	—
109	C₇×#102	Q₈	1	4	yes	complemented
110	C₇×#97	—	2	2	no	—
111	C₇×#97	—	4	1	no	—
112	C₄×C₇×C₈	—	4	1	no	—
113	C₇×#95	—	2	1	no	—
114	C₇×#95	—	2	1	no	—
115	C₇×#95	—	2	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	D₄	1	1	yes	characteristic
123	C₇×#95	D₄	1	1	yes	characteristic
124	C₂×#474	C₂×C₇	1	1	yes	characteristic
125	#2369	C₂×C₇	1	1	yes	characteristic, complemented
126	#2349	C₂×C₇	1	1	yes	characteristic
127	#1993	C₂×C₇	1	1	yes	characteristic, complemented
128	#1949	C₂×C₇	1	1	yes	characteristic, complemented
129	#1943	C₂×C₇	1	1	yes	characteristic, complemented
130	#1942	C₂×C₇	1	1	yes	characteristic, complemented
131	#1921	C₂×C₇	1	1	yes	characteristic, complemented
132	#1918	C₂×C₇	1	1	yes	characteristic, complemented
133	#1864	C₂×C₇	1	1	yes	characteristic
134	#1862	C₂×C₇	1	1	yes	characteristic
135	#1771	C₂×C₇	1	1	yes	characteristic, complemented
136	#1762	C₂×C₇	1	1	yes	characteristic
137	#1762	C₂×C₇	1	1	yes	characteristic
138	C₄×#102	C₂×C₇	1	1	yes	characteristic
139	C₂⁴×C₇	—	2	1	no	—
140	C₂×C₇×Q₈	—	2	4	no	—
141	C₂×C₇×Q₈	—	8	1	no	—
142	C₂×C₇×Q₈	—	8	1	no	—
143	C₂×C₇×D₄	—	2	1	no	—
144	C₂×C₇×D₄	—	2	1	no	—
145	C₂×C₇×D₄	—	2	1	no	—
146	C₂×C₇×D₄	C₂²×C₄	1	2	yes	—
147	C₂×C₇×D₄	C₂×Q₈	1	2	yes	—
148	C₂×C₇×D₄	#41	1	2	yes	—
149	C₂²×C₄×C₇	—	2	1	no	—
150	C₂²×C₄×C₇	—	2	1	no	—
151	C₂²×C₄×C₇	—	2	1	no	—
152	C₂²×C₄×C₇	—	2	1	no	—
153	C₂²×C₄×C₇	—	2	4	no	—
154	C₂²×C₄×C₇	—	2	1	no	—
155	C₂²×C₄×C₇	—	2	1	no	—
156	C₂²×C₄×C₇	—	2	1	no	—
157	C₂²×C₄×C₇	—	4	2	no	—
158	C₂²×C₄×C₇	—	4	2	no	—
159	C₂²×C₄×C₇	—	4	2	no	—
160	C₂²×C₄×C₇	—	4	2	no	—
161	C₂²×C₄×C₇	—	8	1	no	—
162	C₂²×C₄×C₇	—	8	1	no	—
163	C₂²×C₄×C₇	—	8	1	no	—
164	C₂²×C₄×C₇	—	8	1	no	—
165	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
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	C₂⁴	1	1	yes	characteristic
172	C₂×C₇×C₈	—	2	1	no	—
173	C₂×C₇×C₈	—	2	1	no	—
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	C₂²×C₄	1	2	yes	—
182	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
183	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
184	C₇×C₄⋊C₄	—	2	4	no	—
185	C₇×C₄⋊C₄	—	2	2	no	—
186	C₇×C₄⋊C₄	—	2	1	no	—
187	C₇×C₄⋊C₄	—	2	1	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	—	8	1	no	—
198	C₇×C₄⋊C₄	—	8	1	no	—
199	C₇×C₄⋊C₄	—	8	1	no	—
200	C₇×C₄⋊C₄	—	8	1	no	—
201	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
202	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
203	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
204	C₇×C₄⋊C₄	C₂×Q₈	1	2	yes	—
205	C₇×C₂²⋊C₄	—	2	4	no	—
206	C₇×C₂²⋊C₄	—	4	2	no	—
207	C₇×C₂²⋊C₄	—	4	2	no	—
208	C₄²×C₇	—	2	1	no	—
209	C₄²×C₇	—	2	1	no	—
210	C₄²×C₇	—	2	1	no	—
211	C₄²×C₇	—	2	1	no	—
212	C₄²×C₇	—	4	2	no	—
213	C₄²×C₇	—	8	1	no	—
214	C₄²×C₇	—	8	1	no	—
215	C₄²×C₇	C₂²×C₄	1	2	yes	—
216	C₄²×C₇	#41	1	2	yes	—
217	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
218	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
219	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
220	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
221	#474	C₄×C₇	1	8	yes	complemented
222	C₂×#107	C₂²×C₇	1	1	yes	characteristic
223	C₂×#107	C₂²×C₇	1	1	yes	characteristic
224	C₂×#107	C₂²×C₇	1	1	yes	characteristic
225	C₂×#106	C₂²×C₇	1	1	yes	characteristic
226	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
227	C₂×#102	C₂²×C₇	1	1	yes	characteristic
228	C₂×#102	C₂²×C₇	1	1	yes	characteristic
229	C₂×#102	C₂²×C₇	1	1	yes	characteristic
230	C₂×#102	C₂²×C₇	1	1	yes	characteristic
231	C₂×#97	C₂²×C₇	1	1	yes	characteristic
232	C₂×#97	C₂²×C₇	1	1	yes	characteristic
233	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
234	#391	—	2	1	no	—
235	#391	—	2	1	no	—
236	#389	—	2	1	no	—
237	#389	C₂²×C₇	1	1	yes	characteristic
238	#389	C₂²×C₇	1	1	yes	characteristic
239	#389	C₂²×C₇	1	1	yes	characteristic
240	#389	C₂²×C₇	1	1	yes	characteristic
241	#385	—	2	1	no	—
242	#385	—	2	1	no	—
243	#382	C₂²×C₇	1	1	yes	characteristic
244	#380	—	2	1	no	—
245	C₄×C₄⋊C₄	—	2	1	no	—
246	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
247	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
248	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
249	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
250	C₂×#95	—	2	1	no	—
251	#340	C₂²×C₇	1	1	yes	characteristic
252	#340	C₂²×C₇	1	1	yes	characteristic
253	#340	C₂²×C₇	1	1	yes	characteristic
254	#340	C₂²×C₇	1	1	yes	characteristic
255	#340	C₂²×C₇	1	1	yes	characteristic
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	#336	C₂²×C₇	1	1	yes	characteristic
260	#336	C₂²×C₇	1	1	yes	characteristic
261	C₂³×C₇	—	2	2	no	—
262	C₂³×C₇	—	2	2	no	—
263	C₂³×C₇	—	2	2	no	—
264	C₂³×C₇	—	4	1	no	—
265	C₂³×C₇	—	4	1	no	—
266	C₂³×C₇	—	4	1	no	—
267	C₂³×C₇	—	4	1	no	—
268	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
269	C₇×Q₈	—	8	4	no	—
270	C₇×D₄	—	2	2	no	—
271	C₇×D₄	—	2	2	no	—
272	C₇×D₄	—	2	2	no	—
273	C₇×D₄	C₄×Q₈	1	4	yes	—
274	C₂×C₄×C₇	—	2	1	no	—
275	C₂×C₄×C₇	—	2	1	no	—
276	C₂×C₄×C₇	—	2	1	no	—
277	C₂×C₄×C₇	—	2	1	no	—
278	C₂×C₄×C₇	—	2	1	no	—
279	C₂×C₄×C₇	—	2	1	no	—
280	C₂×C₄×C₇	—	2	2	no	—
281	C₂×C₄×C₇	—	2	2	no	—
282	C₂×C₄×C₇	—	2	2	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	2	no	—
287	C₂×C₄×C₇	—	2	1	no	—
288	C₂×C₄×C₇	—	2	1	no	—
289	C₂×C₄×C₇	—	2	1	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	—	4	1	no	—
294	C₂×C₄×C₇	—	4	1	no	—
295	C₂×C₄×C₇	—	4	1	no	—
296	C₂×C₄×C₇	—	4	1	no	—
297	C₂×C₄×C₇	—	4	1	no	—
298	C₂×C₄×C₇	—	4	1	no	—
299	C₂×C₄×C₇	—	4	1	no	—
300	C₂×C₄×C₇	—	4	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	—	8	4	no	—
311	C₂×C₄×C₇	—	8	4	no	—
312	C₂×C₄×C₇	—	8	4	no	—
313	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
314	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
315	C₂×C₄×C₇	#137	1	2	yes	—
316	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
317	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
318	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
319	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
320	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
321	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
322	C₇×C₈	—	2	2	no	—
323	C₇×C₈	—	2	2	no	—
324	C₇×C₈	—	2	2	no	—
325	C₇×C₈	—	4	1	no	—
326	C₂²×Q₈	—	2	1	no	—
327	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
328	C₂³×C₄	—	2	1	no	—
329	C₂³×C₄	—	2	1	no	—
330	C₂³×C₄	—	2	1	no	—
331	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
332	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
333	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
334	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
335	C₄⋊Q₈	—	2	2	no	—
336	C₄⋊Q₈	C₂×C₄×C₇	1	4	yes	complemented
337	C₄×D₄	—	2	2	no	—
338	C₄×D₄	—	2	2	no	—
339	C₄×D₄	—	2	2	no	—
340	C₄×D₄	—	4	1	no	—
341	C₄×D₄	—	4	1	no	—
342	C₄×D₄	C₂×C₄×C₇	1	4	yes	—
343	C₂×C₄⋊C₄	—	2	1	no	—
344	C₂×C₄⋊C₄	—	2	1	no	—
345	C₂×C₄⋊C₄	—	2	1	no	—
346	C₂×C₄⋊C₄	—	2	1	no	—
347	C₂×C₄⋊C₄	—	2	1	no	—
348	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
349	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
350	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
351	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
352	C₂×C₄⋊C₄	C₂³×C₇	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×C₂²⋊C₄	—	2	1	no	—
358	C₂×C₂²⋊C₄	—	2	1	no	—
359	C₂×C₂²⋊C₄	—	2	1	no	—
360	C₂×C₄²	—	2	1	no	—
361	C₂×C₄²	—	2	1	no	—
362	C₂×C₄²	—	2	1	no	—
363	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
364	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
365	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
366	#107	—	4	1	no	—
367	#107	C₂×C₄×C₇	1	4	yes	complemented
368	#107	C₂×C₄×C₇	1	4	yes	complemented
369	#106	—	4	1	no	—
370	C₄⋊C₈	C₂×C₄×C₇	1	4	yes	complemented
371	#102	—	2	2	no	—
372	#102	C₂×C₄×C₇	1	4	yes	—
373	#102	C₂×C₄×C₇	1	4	yes	—
374	#102	C₇×Q₈	1	4	yes	complemented
375	#97	—	2	2	no	—
376	#97	—	4	1	no	—
377	C₄×C₈	—	4	1	no	—
378	#95	—	2	1	no	—
379	#95	—	2	1	no	—
380	#95	—	2	1	no	—
381	#95	—	2	1	no	—
382	#95	—	2	1	no	—
383	#95	—	2	1	no	—
384	#95	—	2	1	no	—
385	#95	—	2	1	no	—
386	#95	—	2	1	no	—
387	#95	C₇×D₄	1	1	yes	characteristic
388	#95	C₇×D₄	1	1	yes	characteristic
389	C₂²×C₇	—	2	4	no	—
390	C₂²×C₇	—	4	2	no	—
391	C₂²×C₇	—	4	2	no	—
392	C₂²×C₇	—	4	2	no	—
393	C₂²×C₇	—	4	2	no	—
394	C₂²×C₇	—	4	2	no	—
395	C₂²×C₇	—	4	2	no	—
396	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
397	C₂²×C₇	#546	1	1	yes	characteristic
398	C₂²×C₇	D₄×Q₈	1	1	yes	characteristic
399	C₂²×C₇	C₂×#137	1	1	yes	characteristic
400	C₂²×C₇	#575	1	1	yes	characteristic
401	C₂²×C₇	#576	1	1	yes	characteristic
402	C₂²×C₇	#576	1	1	yes	characteristic
403	C₄×C₇	—	2	2	no	—
404	C₄×C₇	—	2	2	no	—
405	C₄×C₇	—	2	2	no	—
406	C₄×C₇	—	4	2	no	—
407	C₄×C₇	—	4	2	no	—
408	C₄×C₇	—	4	2	no	—
409	C₄×C₇	—	4	2	no	—
410	C₄×C₇	—	4	1	no	—
411	C₄×C₇	—	4	1	no	—
412	C₄×C₇	—	4	1	no	—
413	C₄×C₇	—	8	1	no	—
414	C₄×C₇	—	8	1	no	—
415	C₄×C₇	—	8	1	no	—
416	C₄×C₇	C₂×C₄×Q₈	1	1	yes	characteristic
417	C₄×C₇	#520	1	1	yes	characteristic
418	C₄×C₇	#552	1	1	yes	characteristic
419	C₄×C₇	#554	1	1	yes	characteristic
420	C₂⁴	—	2	1	no	—
421	C₂×Q₈	—	2	4	no	—
422	C₂×Q₈	—	8	1	no	—
423	C₂×Q₈	—	8	1	no	—
424	C₂×D₄	—	2	1	no	—
425	C₂×D₄	—	2	1	no	—
426	C₂×D₄	—	2	1	no	—
427	C₂×D₄	C₂²×C₄×C₇	1	2	yes	—
428	C₂×D₄	C₂×C₇×Q₈	1	2	yes	—
429	C₂×D₄	C₇×#41	1	2	yes	—
430	C₂²×C₄	—	2	1	no	—
431	C₂²×C₄	—	2	1	no	—
432	C₂²×C₄	—	2	1	no	—
433	C₂²×C₄	—	2	1	no	—
434	C₂²×C₄	—	2	4	no	—
435	C₂²×C₄	—	2	1	no	—
436	C₂²×C₄	—	2	1	no	—
437	C₂²×C₄	—	2	1	no	—
438	C₂²×C₄	—	4	2	no	—
439	C₂²×C₄	—	4	2	no	—
440	C₂²×C₄	—	4	2	no	—
441	C₂²×C₄	—	4	2	no	—
442	C₂²×C₄	—	8	1	no	—
443	C₂²×C₄	—	8	1	no	—
444	C₂²×C₄	—	8	1	no	—
445	C₂²×C₄	—	8	1	no	—
446	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
447	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
448	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
449	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
450	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
451	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
452	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
453	C₂×C₈	—	2	1	no	—
454	C₂×C₈	—	2	1	no	—
455	C₂×C₈	—	2	1	no	—
456	C₂×C₈	—	2	1	no	—
457	C₂×C₈	—	2	1	no	—
458	C₂×C₈	—	2	1	no	—
459	C₂×C₈	—	2	1	no	—
460	C₂×C₈	—	2	1	no	—
461	C₂×C₈	—	2	1	no	—
462	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
463	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
464	C₂×C₈	C₂×C₇×Q₈	1	2	yes	—
465	C₄⋊C₄	—	2	4	no	—
466	C₄⋊C₄	—	2	2	no	—
467	C₄⋊C₄	—	2	1	no	—
468	C₄⋊C₄	—	2	1	no	—
469	C₄⋊C₄	—	2	1	no	—
470	C₄⋊C₄	—	2	1	no	—
471	C₄⋊C₄	—	2	1	no	—
472	C₄⋊C₄	—	2	1	no	—
473	C₄⋊C₄	—	2	1	no	—
474	C₄⋊C₄	—	2	1	no	—
475	C₄⋊C₄	—	2	1	no	—
476	C₄⋊C₄	—	2	1	no	—
477	C₄⋊C₄	—	4	1	no	—
478	C₄⋊C₄	—	8	1	no	—
479	C₄⋊C₄	—	8	1	no	—
480	C₄⋊C₄	—	8	1	no	—
481	C₄⋊C₄	—	8	1	no	—
482	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
483	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
484	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
485	C₄⋊C₄	C₂×C₇×Q₈	1	2	yes	—
486	C₂²⋊C₄	—	2	4	no	—
487	C₂²⋊C₄	—	4	2	no	—
488	C₂²⋊C₄	—	4	2	no	—
489	C₄²	—	2	1	no	—
490	C₄²	—	2	1	no	—
491	C₄²	—	2	1	no	—
492	C₄²	—	2	1	no	—
493	C₄²	—	4	2	no	—
494	C₄²	—	8	1	no	—
495	C₄²	—	8	1	no	—
496	C₄²	C₂²×C₄×C₇	1	2	yes	—
497	C₄²	C₇×#41	1	2	yes	—
498	C₂×C₇	—	4	4	no	—
499	C₂×C₇	#2360	1	1	yes	characteristic
500	C₂×C₇	C₄×#137	1	1	yes	characteristic
501	C₂×C₇	#2948	1	1	yes	characteristic
502	C₂×C₇	#3314	1	1	yes	characteristic
503	C₂×C₇	#3319	1	1	yes	characteristic
504	C₂×C₇	#3385	1	1	yes	characteristic
505	C₂×C₇	#3389	1	1	yes	characteristic
506	C₂³	—	2	2	no	—
507	C₂³	—	2	2	no	—
508	C₂³	—	2	2	no	—
509	C₂³	—	4	1	no	—
510	C₂³	—	4	1	no	—
511	C₂³	—	4	1	no	—
512	C₂³	—	4	1	no	—
513	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
514	Q₈	—	8	4	no	—
515	D₄	—	2	2	no	—
516	D₄	—	2	2	no	—
517	D₄	—	2	2	no	—
518	D₄	C₄×C₇×Q₈	1	4	yes	—
519	C₂×C₄	—	2	1	no	—
520	C₂×C₄	—	2	1	no	—
521	C₂×C₄	—	2	1	no	—
522	C₂×C₄	—	2	1	no	—
523	C₂×C₄	—	2	1	no	—
524	C₂×C₄	—	2	1	no	—
525	C₂×C₄	—	2	2	no	—
526	C₂×C₄	—	2	2	no	—
527	C₂×C₄	—	2	2	no	—
528	C₂×C₄	—	2	1	no	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	2	1	no	—
531	C₂×C₄	—	2	2	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	1	no	—
538	C₂×C₄	—	4	1	no	—
539	C₂×C₄	—	4	1	no	—
540	C₂×C₄	—	4	1	no	—
541	C₂×C₄	—	4	1	no	—
542	C₂×C₄	—	4	1	no	—
543	C₂×C₄	—	4	1	no	—
544	C₂×C₄	—	4	1	no	—
545	C₂×C₄	—	4	1	no	—
546	C₂×C₄	—	4	1	no	—
547	C₂×C₄	—	4	1	no	—
548	C₂×C₄	—	4	1	no	—
549	C₂×C₄	—	4	1	no	—
550	C₂×C₄	—	4	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	—	8	4	no	—
556	C₂×C₄	—	8	4	no	—
557	C₂×C₄	—	8	4	no	—
558	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
559	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
560	C₂×C₄	C₇×#137	1	2	yes	—
561	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
562	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
563	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
564	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
565	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
566	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
567	C₈	—	2	2	no	—
568	C₈	—	2	2	no	—
569	C₈	—	2	2	no	—
570	C₈	—	4	1	no	—
571	C₇	#22979	1	1	yes	characteristic, direct product factor, Sylow
572	C₂²	—	2	4	no	—
573	C₂²	—	4	2	no	—
574	C₂²	—	4	2	no	—
575	C₂²	—	4	2	no	—
576	C₂²	—	4	2	no	—
577	C₂²	—	4	2	no	—
578	C₂²	—	4	2	no	—
579	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
580	C₂²	C₇×#546	1	1	yes	characteristic
581	C₂²	C₇×D₄×Q₈	1	1	yes	characteristic
582	C₂²	C₂×C₇×#137	1	1	yes	characteristic
583	C₂²	C₇×#575	1	1	yes	characteristic
584	C₂²	C₇×#576	1	1	yes	characteristic
585	C₂²	C₇×#576	1	1	yes	characteristic
586	C₄	—	2	2	no	—
587	C₄	—	2	2	no	—
588	C₄	—	2	2	no	—
589	C₄	—	4	2	no	—
590	C₄	—	4	2	no	—
591	C₄	—	4	2	no	—
592	C₄	—	4	1	no	—
593	C₄	—	4	1	no	—
594	C₄	—	4	1	no	—
595	C₄	—	4	2	no	—
596	C₄	—	8	1	no	—
597	C₄	—	8	1	no	—
598	C₄	—	8	1	no	—
599	C₄	C₂×C₄×C₇×Q₈	1	1	yes	characteristic
600	C₄	C₇×#520	1	1	yes	characteristic
601	C₄	C₇×#552	1	1	yes	characteristic
602	C₄	C₇×#554	1	1	yes	characteristic
603	C₂	—	4	4	no	—
604	C₂	C₇×#2360	1	1	yes	characteristic
605	C₂	C₄×C₇×#137	1	1	yes	characteristic
606	C₂	C₇×#2948	1	1	yes	characteristic
607	C₂	C₇×#3314	1	1	yes	characteristic
608	C₂	C₇×#3319	1	1	yes	characteristic
609	C₂	C₇×#3385	1	1	yes	characteristic
610	C₂	C₇×#3389	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×#1993	C₂	semidirect
5. C₇×#1949	C₂	semidirect
6. C₇×#1943	C₂	semidirect
7. C₇×#1942	C₂	semidirect
8. C₇×#1921	C₂	semidirect
9. C₇×#1918	C₂	semidirect
12. C₇×#1771	C₂	semidirect
20. C₇×#474	C₄	semidirect
60. #22979	C₇	direct (N × H)
71. C₇×C₄⋊Q₈	C₂×C₄	semidirect
102. C₇×#107	C₂×C₄	semidirect
103. C₇×#107	C₂×C₄	semidirect
105. C₇×C₄⋊C₈	C₂×C₄	semidirect
109. C₇×#102	Q₈	semidirect
125. #2369	C₂×C₇	semidirect
127. #1993	C₂×C₇	semidirect
128. #1949	C₂×C₇	semidirect
129. #1943	C₂×C₇	semidirect
130. #1942	C₂×C₇	semidirect
131. #1921	C₂×C₇	semidirect
132. #1918	C₂×C₇	semidirect
135. #1771	C₂×C₇	semidirect
221. #474	C₄×C₇	semidirect
336. C₄⋊Q₈	C₂×C₄×C₇	semidirect
367. #107	C₂×C₄×C₇	semidirect
368. #107	C₂×C₄×C₇	semidirect
370. C₄⋊C₈	C₂×C₄×C₇	semidirect
374. #102	C₇×Q₈	semidirect
571. C₇	#22979	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×#474	2
C₇×#2369	2
C₇×#2349	2
C₇×#1993	2
C₇×#1949	2
C₇×#1943	2
C₇×#1942	2
C₇×#1921	2
C₇×#1918	2
C₇×#1864	2
C₇×#1862	2
C₇×#1771	2
C₇×#1762	2
C₄×C₇×#102	2
#22979	7

Subgroups — C7×#22979 #13200195

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22979 #13200195