Subgroups — C₇×#22657 #13199873

Order 1792.

982 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×C₈×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#427	C₂	1	2	yes	maximal, complemented
3	C₇×#2354	C₂	1	2	yes	maximal
4	C₇×#1947	C₂	1	2	yes	maximal, complemented
5	C₇×#1854	C₂	1	4	yes	maximal, complemented
6	C₇×#1844	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1807	C₂	1	2	yes	maximal, complemented
8	C₄×C₇×#106	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×C₇×C₈	C₂²	1	2	yes	—
10	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
11	C₂×C₇×#117	C₂²	1	4	yes	—
12	C₇×C₈×D₄	C₄	1	8	yes	complemented
13	C₇×#427	—	2	8	no	—
14	C₂×C₇×#107	C₂²	1	4	yes	—
15	C₂×C₇×#106	C₂²	1	2	yes	—
16	C₂×C₇×#106	C₂²	1	1	yes	characteristic
17	C₂×C₇×#106	C₂²	1	2	yes	—
18	C₂×C₇×#106	C₂²	1	1	yes	characteristic
19	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₇×C₂²⋊C₈	C₂²	1	2	yes	—
21	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
22	C₇×#390	—	2	2	no	—
23	C₇×#388	—	2	4	no	—
24	C₇×#384	C₂²	1	2	yes	—
25	C₇×#380	C₂²	1	4	yes	—
26	C₄×C₇×C₄⋊C₄	—	2	2	no	—
27	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
28	C₇×#340	C₂²	1	2	yes	—
29	C₇×#340	C₂²	1	4	yes	—
30	#22657	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₂²×C₇×D₄	D₄	1	1	yes	characteristic
32	C₂³×C₄×C₇	C₂³	1	2	yes	—
33	C₂²×C₇×C₈	—	2	4	no	—
34	C₂²×C₇×C₈	—	2	4	no	—
35	C₂²×C₇×C₈	C₂×C₄	1	8	yes	—
36	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
37	C₂²×C₇×C₈	C₂³	1	2	yes	—
38	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
39	C₄×C₇×D₄	C₂×C₄	1	4	yes	—
40	C₄×C₇×D₄	Q₈	1	4	yes	—
41	C₇×#117	—	2	8	no	—
42	C₇×#117	—	4	4	no	—
43	C₂×C₇×C₄⋊C₄	—	2	2	no	—
44	C₂×C₇×C₄⋊C₄	—	2	2	no	—
45	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
46	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
47	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
48	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
49	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
50	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
51	C₂×C₇×C₂²⋊C₄	D₄	1	2	yes	—
52	C₂×C₄²×C₇	—	2	4	no	—
53	C₂×C₄²×C₇	—	2	2	no	—
54	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
55	C₇×#107	—	2	8	no	—
56	C₇×#106	—	2	4	no	—
57	C₇×#106	—	2	2	no	—
58	C₇×#106	—	4	2	no	—
59	C₇×#106	D₄	1	4	yes	complemented
60	C₇×C₄⋊C₈	C₂×C₄	1	4	yes	—
61	C₇×C₂²⋊C₈	C₂×C₄	1	8	yes	—
62	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
63	C₇×#95	—	2	4	no	—
64	C₇×#95	—	2	2	no	—
65	C₂×C₈×D₄	C₂×C₇	1	1	yes	characteristic
66	C₂×#427	C₂×C₇	1	2	yes	complemented
67	#2354	C₂×C₇	1	2	yes	—
68	#1947	C₂×C₇	1	2	yes	complemented
69	#1854	C₂×C₇	1	4	yes	complemented
70	#1844	C₂×C₇	1	1	yes	characteristic, complemented
71	#1807	C₂×C₇	1	2	yes	complemented
72	C₄×#106	C₂×C₇	1	1	yes	characteristic, complemented
73	C₂⁴×C₇	C₂×D₄	1	2	yes	—
74	C₂×C₇×D₄	—	2	4	no	—
75	C₂×C₇×D₄	C₄⋊C₄	1	4	yes	—
76	C₂²×C₄×C₇	—	2	2	no	—
77	C₂²×C₄×C₇	—	2	2	no	—
78	C₂²×C₄×C₇	—	2	2	no	—
79	C₂²×C₄×C₇	—	2	2	no	—
80	C₂²×C₄×C₇	—	2	2	no	—
81	C₂²×C₄×C₇	—	2	2	no	—
82	C₂²×C₄×C₇	—	2	4	no	—
83	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
84	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₇	C₂×Q₈	1	4	yes	—
88	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
89	C₂×C₇×C₈	—	2	2	no	—
90	C₂×C₇×C₈	—	2	2	no	—
91	C₂×C₇×C₈	—	2	1	no	—
92	C₂×C₇×C₈	—	2	1	no	—
93	C₂×C₇×C₈	—	2	16	no	—
94	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
95	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
96	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
97	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
98	C₂×C₇×C₈	#41	1	2	yes	—
99	C₇×C₄⋊C₄	—	2	2	no	—
100	C₇×C₄⋊C₄	—	2	4	no	—
101	C₇×C₄⋊C₄	—	2	2	no	—
102	C₇×C₄⋊C₄	—	2	4	no	—
103	C₇×C₄⋊C₄	—	2	2	no	—
104	C₇×C₄⋊C₄	—	4	4	no	—
105	C₇×C₄⋊C₄	—	8	2	no	—
106	C₇×C₄⋊C₄	C₄⋊C₄	1	4	yes	—
107	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
108	C₇×C₂²⋊C₄	—	4	8	no	—
109	C₇×C₂²⋊C₄	—	8	4	no	—
110	C₇×C₂²⋊C₄	C₄⋊C₄	1	8	yes	—
111	C₄²×C₇	—	4	8	no	—
112	C₄²×C₇	—	4	4	no	—
113	C₄²×C₇	C₂²×C₄	1	2	yes	—
114	C₄²×C₇	C₂×Q₈	1	2	yes	—
115	C₂³×C₈	C₂²×C₇	1	2	yes	—
116	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
117	C₂×#117	C₂²×C₇	1	4	yes	—
118	C₈×D₄	C₄×C₇	1	8	yes	complemented
119	#427	—	2	8	no	—
120	C₂×#107	C₂²×C₇	1	4	yes	—
121	C₂×#106	C₂²×C₇	1	1	yes	characteristic
122	C₂×#106	C₂²×C₇	1	2	yes	—
123	C₂×#106	C₂²×C₇	1	2	yes	—
124	C₂×#106	C₂²×C₇	1	1	yes	characteristic
125	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
126	C₂×C₂²⋊C₈	C₂²×C₇	1	2	yes	—
127	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
128	#390	—	2	2	no	—
129	#388	—	2	4	no	—
130	#384	C₂²×C₇	1	2	yes	—
131	#380	C₂²×C₇	1	4	yes	—
132	C₄×C₄⋊C₄	—	2	2	no	—
133	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
134	#340	C₂²×C₇	1	4	yes	—
135	#340	C₂²×C₇	1	2	yes	—
136	C₂³×C₇	—	2	2	no	—
137	C₂³×C₇	—	2	2	no	—
138	C₂³×C₇	—	2	4	no	—
139	C₂³×C₇	—	2	4	no	—
140	C₂³×C₇	C₂×C₄⋊C₄	1	4	yes	—
141	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
142	C₇×D₄	—	2	8	no	—
143	C₂×C₄×C₇	—	2	2	no	—
144	C₂×C₄×C₇	—	2	2	no	—
145	C₂×C₄×C₇	—	2	2	no	—
146	C₂×C₄×C₇	—	2	4	no	—
147	C₂×C₄×C₇	—	2	4	no	—
148	C₂×C₄×C₇	—	2	4	no	—
149	C₂×C₄×C₇	—	2	4	no	—
150	C₂×C₄×C₇	—	2	4	no	—
151	C₂×C₄×C₇	—	4	2	no	—
152	C₂×C₄×C₇	—	4	2	no	—
153	C₂×C₄×C₇	—	4	4	no	—
154	C₂×C₄×C₇	—	4	2	no	—
155	C₂×C₄×C₇	—	4	2	no	—
156	C₂×C₄×C₇	—	4	4	no	—
157	C₂×C₄×C₇	—	4	2	no	—
158	C₂×C₄×C₇	—	4	4	no	—
159	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
160	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
161	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
162	C₂×C₄×C₇	#135	1	4	yes	—
163	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
164	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
165	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
166	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
167	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
168	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
169	C₇×C₈	—	2	4	no	—
170	C₇×C₈	—	2	2	no	—
171	C₇×C₈	C₄×D₄	1	4	yes	—
172	C₂²×D₄	C₇×D₄	1	1	yes	characteristic
173	C₂³×C₄	C₂³×C₇	1	2	yes	—
174	C₂²×C₈	—	2	4	no	—
175	C₂²×C₈	—	2	4	no	—
176	C₂²×C₈	C₂×C₄×C₇	1	8	yes	—
177	C₂²×C₈	C₂³×C₇	1	2	yes	—
178	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
179	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
180	C₄×D₄	C₂×C₄×C₇	1	4	yes	—
181	C₄×D₄	C₇×Q₈	1	4	yes	—
182	#117	—	2	8	no	—
183	#117	—	4	4	no	—
184	C₂×C₄⋊C₄	—	2	2	no	—
185	C₂×C₄⋊C₄	—	2	2	no	—
186	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
187	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
188	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
189	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
190	C₂×C₂²⋊C₄	—	2	4	no	—
191	C₂×C₂²⋊C₄	—	2	4	no	—
192	C₂×C₂²⋊C₄	C₇×D₄	1	2	yes	—
193	C₂×C₄²	—	2	4	no	—
194	C₂×C₄²	—	2	2	no	—
195	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
196	#107	—	2	8	no	—
197	#106	—	2	4	no	—
198	#106	—	2	2	no	—
199	#106	—	4	2	no	—
200	#106	C₇×D₄	1	4	yes	complemented
201	C₄⋊C₈	C₂×C₄×C₇	1	4	yes	—
202	C₂²⋊C₈	C₂×C₄×C₇	1	8	yes	—
203	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
204	#95	—	2	4	no	—
205	#95	—	2	2	no	—
206	C₂²×C₇	—	2	4	no	—
207	C₂²×C₇	—	2	4	no	—
208	C₂²×C₇	—	2	4	no	—
209	C₂²×C₇	—	4	4	no	—
210	C₂²×C₇	—	4	4	no	—
211	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
212	C₂²×C₇	D₄²	1	1	yes	characteristic
213	C₂²×C₇	#547	1	1	yes	characteristic
214	C₂²×C₇	C₂×#135	1	2	yes	—
215	C₂²×C₇	#577	1	2	yes	—
216	C₄×C₇	—	2	4	no	—
217	C₄×C₇	—	2	2	no	—
218	C₄×C₇	—	4	4	no	—
219	C₄×C₇	—	8	2	no	—
220	C₄×C₇	—	8	4	no	—
221	C₄×C₇	#427	1	4	yes	complemented
222	C₄×C₇	C₂×C₄×D₄	1	1	yes	characteristic
223	C₄×C₇	#520	1	1	yes	characteristic
224	C₄×C₇	D₄×Q₈	1	1	yes	characteristic
225	C₄×C₇	#554	1	1	yes	characteristic
226	C₂⁴	C₂×C₇×D₄	1	2	yes	—
227	C₂×D₄	—	2	4	no	—
228	C₂×D₄	C₇×C₄⋊C₄	1	4	yes	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
237	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
238	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₇×Q₈	1	4	yes	—
241	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	2	16	no	—
247	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
248	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
249	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
250	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
251	C₂×C₈	C₇×#41	1	2	yes	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	4	4	no	—
258	C₄⋊C₄	—	8	2	no	—
259	C₄⋊C₄	C₇×C₄⋊C₄	1	4	yes	—
260	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
261	C₂²⋊C₄	—	4	8	no	—
262	C₂²⋊C₄	—	8	4	no	—
263	C₂²⋊C₄	C₇×C₄⋊C₄	1	8	yes	—
264	C₄²	—	4	8	no	—
265	C₄²	—	4	4	no	—
266	C₄²	C₂²×C₄×C₇	1	2	yes	—
267	C₄²	C₂×C₇×Q₈	1	2	yes	—
268	C₂×C₇	—	4	4	no	—
269	C₂×C₇	D₄×C₄⋊C₄	1	1	yes	characteristic
270	C₂×C₇	C₂×#427	1	1	yes	characteristic
271	C₂×C₇	#2912	1	1	yes	characteristic
272	C₂×C₇	#3284	1	2	yes	—
273	C₂×C₇	#3340	1	2	yes	—
274	C₂³	—	2	2	no	—
275	C₂³	—	2	2	no	—
276	C₂³	—	2	4	no	—
277	C₂³	—	2	4	no	—
278	C₂³	C₂×C₇×C₄⋊C₄	1	4	yes	—
279	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
280	D₄	—	2	8	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	4	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
298	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
299	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
300	C₂×C₄	C₇×#135	1	4	yes	—
301	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
302	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
303	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
304	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
305	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
306	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
307	C₈	—	2	4	no	—
308	C₈	—	2	2	no	—
309	C₈	C₄×C₇×D₄	1	4	yes	—
310	C₇	#22657	1	1	yes	characteristic, direct product factor, Sylow
311	C₂²	—	2	4	no	—
312	C₂²	—	2	4	no	—
313	C₂²	—	2	4	no	—
314	C₂²	—	4	4	no	—
315	C₂²	—	4	4	no	—
316	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
317	C₂²	C₇×D₄²	1	1	yes	characteristic
318	C₂²	C₇×#547	1	1	yes	characteristic
319	C₂²	C₂×C₇×#135	1	2	yes	—
320	C₂²	C₇×#577	1	2	yes	—
321	C₄	—	2	2	no	—
322	C₄	—	2	4	no	—
323	C₄	—	4	4	no	—
324	C₄	—	8	4	no	—
325	C₄	—	8	2	no	—
326	C₄	C₇×#427	1	4	yes	complemented
327	C₄	C₂×C₄×C₇×D₄	1	1	yes	characteristic
328	C₄	C₇×#520	1	1	yes	characteristic
329	C₄	C₇×D₄×Q₈	1	1	yes	characteristic
330	C₄	C₇×#554	1	1	yes	characteristic
331	C₂	—	4	4	no	—
332	C₂	C₇×D₄×C₄⋊C₄	1	1	yes	characteristic
333	C₂	C₂×C₇×#427	1	1	yes	characteristic
334	C₂	C₇×#2912	1	1	yes	characteristic
335	C₂	C₇×#3284	1	2	yes	—
336	C₂	C₇×#3340	1	2	yes	—

	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	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄×C₇×D₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×#427	C₂	semidirect
4. C₇×#1947	C₂	semidirect
5. C₇×#1854	C₂	semidirect
6. C₇×#1844	C₂	semidirect
7. C₇×#1807	C₂	semidirect
8. C₄×C₇×#106	C₂	semidirect
12. C₇×C₈×D₄	C₄	semidirect
30. #22657	C₇	direct (N × H)
59. C₇×#106	D₄	semidirect
66. C₂×#427	C₂×C₇	semidirect
68. #1947	C₂×C₇	semidirect
69. #1854	C₂×C₇	semidirect
70. #1844	C₂×C₇	semidirect
71. #1807	C₂×C₇	semidirect
72. C₄×#106	C₂×C₇	semidirect
118. C₈×D₄	C₄×C₇	semidirect
200. #106	C₇×D₄	semidirect
221. C₄×C₇	#427	semidirect
310. C₇	#22657	direct (N × H)
326. C₄	C₇×#427	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×C₈×D₄	2
C₂×C₇×#427	2
C₇×#2354	2
C₇×#1947	2
C₇×#1854	2
C₇×#1844	2
C₇×#1807	2
C₄×C₇×#106	2
#22657	7

Subgroups — C7×#22657 #13199873

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22657 #13199873