Subgroups — C₇×#22807 #13200023

Order 1792.

934 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇×D₈	C₂	1	1	yes	maximal, characteristic
2	C₇×#2354	C₂	1	2	yes	maximal, complemented
3	C₇×#1970	C₂	1	2	yes	maximal
4	C₇×#1925	C₂	1	4	yes	maximal, complemented
5	C₇×#1845	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1802	C₂	1	2	yes	maximal, complemented
7	C₄×C₇×#107	C₂	1	2	yes	maximal, complemented
8	C₇×C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₇×D₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₇×D₄	C₂²	1	2	yes	—
11	C₂×C₇×#117	—	2	2	no	—
12	C₄×C₇×D₈	C₄	1	8	yes	complemented
13	C₂×C₇×#107	C₂²	1	2	yes	—
14	C₂×C₇×#107	C₂²	1	2	yes	—
15	C₂×C₇×#107	C₂²	1	1	yes	characteristic
16	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₇×#102	C₂²	1	4	yes	—
18	C₂×C₇×#102	C₂²	1	2	yes	—
19	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
20	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
21	C₇×#390	C₂²	1	4	yes	—
22	C₇×#388	—	2	4	no	—
23	C₇×#384	C₂²	1	2	yes	—
24	C₇×#380	—	2	2	no	—
25	C₄×C₇×C₄⋊C₄	C₂²	1	4	yes	—
26	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
27	C₇×#340	C₂²	1	4	yes	—
28	C₇×#336	C₂²	1	2	yes	—
29	#22807	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂²×C₇×D₄	C₂³	1	2	yes	—
31	C₂³×C₄×C₇	—	2	2	no	—
32	C₂×C₇×D₈	—	2	2	no	—
33	C₂×C₇×D₈	—	2	2	no	—
34	C₂×C₇×D₈	C₂×C₄	1	4	yes	—
35	C₂²×C₇×C₈	C₂³	1	2	yes	—
36	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
37	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
38	C₄×C₇×D₄	—	2	4	no	—
39	C₄×C₇×D₄	C₂×C₄	1	8	yes	—
40	C₇×#117	—	4	8	no	—
41	C₂×C₇×C₄⋊C₄	—	2	2	no	—
42	C₂×C₇×C₄⋊C₄	—	2	2	no	—
43	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
44	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
45	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
46	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
47	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
48	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
49	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
50	C₂×C₄²×C₇	—	2	4	no	—
51	C₂×C₄²×C₇	C₂³	1	2	yes	—
52	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
53	C₇×#107	—	2	4	no	—
54	C₇×#107	—	4	2	no	—
55	C₇×#107	C₂×C₄	1	4	yes	—
56	C₇×C₄⋊C₈	—	2	2	no	—
57	C₇×#102	—	4	4	no	—
58	C₇×#102	C₂×C₄	1	8	yes	—
59	C₄×C₇×C₈	—	2	4	no	—
60	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
61	C₇×#95	—	2	4	no	—
62	C₇×#95	D₄	1	2	yes	—
63	C₂×C₄×D₈	C₂×C₇	1	1	yes	characteristic
64	#2354	C₂×C₇	1	2	yes	complemented
65	#1970	C₂×C₇	1	2	yes	—
66	#1925	C₂×C₇	1	4	yes	complemented
67	#1845	C₂×C₇	1	1	yes	characteristic, complemented
68	#1802	C₂×C₇	1	2	yes	complemented
69	C₄×#107	C₂×C₇	1	2	yes	complemented
70	C₈×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic, complemented
71	C₂⁴×C₇	—	2	2	no	—
72	C₂×C₇×D₄	—	2	2	no	—
73	C₂×C₇×D₄	—	2	2	no	—
74	C₂×C₇×D₄	—	2	2	no	—
75	C₂×C₇×D₄	—	2	2	no	—
76	C₂×C₇×D₄	—	2	2	no	—
77	C₂×C₇×D₄	C₂²×C₄	1	4	yes	—
78	C₂²×C₄×C₇	—	2	2	no	—
79	C₂²×C₄×C₇	—	2	8	no	—
80	C₂²×C₄×C₇	—	2	4	no	—
81	C₂²×C₄×C₇	—	4	4	no	—
82	C₂²×C₄×C₇	—	4	4	no	—
83	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
88	C₇×D₈	—	2	8	no	—
89	C₂×C₇×C₈	—	2	1	no	—
90	C₂×C₇×C₈	—	2	1	no	—
91	C₂×C₇×C₈	—	2	2	no	—
92	C₂×C₇×C₈	—	2	2	no	—
93	C₂×C₇×C₈	—	2	2	no	—
94	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
95	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
96	C₂×C₇×C₈	#41	1	2	yes	—
97	C₂×C₇×C₈	#41	1	2	yes	—
98	C₇×C₄⋊C₄	—	2	2	no	—
99	C₇×C₄⋊C₄	—	2	2	no	—
100	C₇×C₄⋊C₄	—	2	4	no	—
101	C₇×C₄⋊C₄	—	2	4	no	—
102	C₇×C₄⋊C₄	—	4	4	no	—
103	C₇×C₄⋊C₄	—	8	2	no	—
104	C₇×C₄⋊C₄	D₈	1	4	yes	complemented
105	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
106	C₇×C₂²⋊C₄	—	2	8	no	—
107	C₇×C₂²⋊C₄	—	4	8	no	—
108	C₇×C₂²⋊C₄	—	8	4	no	—
109	C₄²×C₇	—	2	2	no	—
110	C₄²×C₇	—	2	2	no	—
111	C₄²×C₇	—	4	8	no	—
112	C₄²×C₇	C₂²×C₄	1	2	yes	—
113	C₄²×C₇	#41	1	2	yes	—
114	C₂²×D₈	C₂²×C₇	1	1	yes	characteristic
115	C₂×C₄×D₄	C₂²×C₇	1	2	yes	—
116	C₂×#117	—	2	2	no	—
117	C₄×D₈	C₄×C₇	1	8	yes	complemented
118	C₂×#107	C₂²×C₇	1	2	yes	—
119	C₂×#107	C₂²×C₇	1	2	yes	—
120	C₂×#107	C₂²×C₇	1	1	yes	characteristic
121	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
122	C₂×#102	C₂²×C₇	1	4	yes	—
123	C₂×#102	C₂²×C₇	1	2	yes	—
124	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
125	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
126	#390	C₂²×C₇	1	4	yes	—
127	#388	—	2	4	no	—
128	#384	C₂²×C₇	1	2	yes	—
129	#380	—	2	2	no	—
130	C₄×C₄⋊C₄	C₂²×C₇	1	4	yes	—
131	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
132	#340	C₂²×C₇	1	4	yes	—
133	#336	C₂²×C₇	1	2	yes	—
134	C₂³×C₇	—	2	4	no	—
135	C₂³×C₇	—	4	2	no	—
136	C₂³×C₇	—	4	2	no	—
137	C₂³×C₇	—	4	2	no	—
138	C₂³×C₇	—	4	2	no	—
139	C₂³×C₇	—	4	2	no	—
140	C₂³×C₇	—	4	2	no	—
141	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
142	C₇×D₄	—	2	4	no	—
143	C₇×D₄	—	2	4	no	—
144	C₇×D₄	—	4	2	no	—
145	C₇×D₄	—	4	2	no	—
146	C₂×C₄×C₇	—	2	2	no	—
147	C₂×C₄×C₇	—	2	1	no	—
148	C₂×C₄×C₇	—	2	2	no	—
149	C₂×C₄×C₇	—	2	1	no	—
150	C₂×C₄×C₇	—	2	4	no	—
151	C₂×C₄×C₇	—	2	2	no	—
152	C₂×C₄×C₇	—	2	2	no	—
153	C₂×C₄×C₇	—	2	2	no	—
154	C₂×C₄×C₇	—	4	2	no	—
155	C₂×C₄×C₇	—	4	2	no	—
156	C₂×C₄×C₇	—	4	16	no	—
157	C₂×C₄×C₇	—	4	4	no	—
158	C₂×C₄×C₇	—	4	4	no	—
159	C₂×C₄×C₇	—	4	4	no	—
160	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
161	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
162	C₂×C₄×C₇	C₂×D₈	1	4	yes	—
163	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
164	C₂×C₄×C₇	#135	1	2	yes	—
165	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
166	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
167	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
168	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
169	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
170	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
171	C₇×C₈	—	2	2	no	—
172	C₇×C₈	—	4	2	no	—
173	C₇×C₈	#117	1	4	yes	complemented
174	C₂²×D₄	C₂³×C₇	1	2	yes	—
175	C₂³×C₄	—	2	2	no	—
176	C₂×D₈	—	2	2	no	—
177	C₂×D₈	—	2	2	no	—
178	C₂×D₈	C₂×C₄×C₇	1	4	yes	—
179	C₂²×C₈	C₂³×C₇	1	2	yes	—
180	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
181	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
182	C₄×D₄	—	2	4	no	—
183	C₄×D₄	C₂×C₄×C₇	1	8	yes	—
184	#117	—	4	8	no	—
185	C₂×C₄⋊C₄	—	2	2	no	—
186	C₂×C₄⋊C₄	—	2	2	no	—
187	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
188	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
189	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
190	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
191	C₂×C₂²⋊C₄	—	2	2	no	—
192	C₂×C₂²⋊C₄	—	2	4	no	—
193	C₂×C₂²⋊C₄	—	2	4	no	—
194	C₂×C₄²	—	2	4	no	—
195	C₂×C₄²	C₂³×C₇	1	2	yes	—
196	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
197	#107	—	2	4	no	—
198	#107	—	4	2	no	—
199	#107	C₂×C₄×C₇	1	4	yes	—
200	C₄⋊C₈	—	2	2	no	—
201	#102	—	4	4	no	—
202	#102	C₂×C₄×C₇	1	8	yes	—
203	C₄×C₈	—	2	4	no	—
204	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
205	#95	—	2	4	no	—
206	#95	C₇×D₄	1	2	yes	—
207	C₂²×C₇	—	4	4	no	—
208	C₂²×C₇	—	4	4	no	—
209	C₂²×C₇	—	4	4	no	—
210	C₂²×C₇	—	8	2	no	—
211	C₂²×C₇	—	8	2	no	—
212	C₂²×C₇	—	8	2	no	—
213	C₂²×C₇	—	8	2	no	—
214	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
215	C₂²×C₇	#547	1	1	yes	characteristic
216	C₂²×C₇	#550	1	1	yes	characteristic
217	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
218	C₂²×C₇	C₂×#135	1	1	yes	characteristic
219	C₂²×C₇	#576	1	1	yes	characteristic
220	C₂²×C₇	#578	1	1	yes	characteristic
221	C₄×C₇	—	2	4	no	—
222	C₄×C₇	—	2	2	no	—
223	C₄×C₇	—	4	2	no	—
224	C₄×C₇	—	4	4	no	—
225	C₄×C₇	—	8	4	no	—
226	C₄×C₇	C₄×D₈	1	4	yes	—
227	C₄×C₇	C₂×#117	1	1	yes	characteristic
228	C₄×C₇	#520	1	1	yes	characteristic
229	C₄×C₇	#555	1	1	yes	characteristic
230	C₄×C₇	#556	1	1	yes	characteristic
231	C₂⁴	—	2	2	no	—
232	C₂×D₄	—	2	2	no	—
233	C₂×D₄	—	2	2	no	—
234	C₂×D₄	—	2	2	no	—
235	C₂×D₄	—	2	2	no	—
236	C₂×D₄	—	2	2	no	—
237	C₂×D₄	C₂²×C₄×C₇	1	4	yes	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	8	no	—
240	C₂²×C₄	—	2	4	no	—
241	C₂²×C₄	—	4	4	no	—
242	C₂²×C₄	—	4	4	no	—
243	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
244	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
245	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
246	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
247	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
248	D₈	—	2	8	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
255	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
256	C₂×C₈	C₇×#41	1	2	yes	—
257	C₂×C₈	C₇×#41	1	2	yes	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	4	4	no	—
263	C₄⋊C₄	—	8	2	no	—
264	C₄⋊C₄	C₇×D₈	1	4	yes	complemented
265	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
266	C₂²⋊C₄	—	2	8	no	—
267	C₂²⋊C₄	—	4	8	no	—
268	C₂²⋊C₄	—	8	4	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	4	8	no	—
272	C₄²	C₂²×C₄×C₇	1	2	yes	—
273	C₄²	C₇×#41	1	2	yes	—
274	C₂×C₇	—	8	4	no	—
275	C₂×C₇	#2354	1	1	yes	characteristic
276	C₂×C₇	C₂×C₄×D₈	1	1	yes	characteristic
277	C₂×C₇	#2951	1	1	yes	characteristic
278	C₂×C₇	#3334	1	1	yes	characteristic
279	C₂×C₇	#3341	1	1	yes	characteristic
280	C₂×C₇	#3391	1	1	yes	characteristic
281	C₂×C₇	#3393	1	1	yes	characteristic
282	C₂³	—	2	4	no	—
283	C₂³	—	4	2	no	—
284	C₂³	—	4	2	no	—
285	C₂³	—	4	2	no	—
286	C₂³	—	4	2	no	—
287	C₂³	—	4	2	no	—
288	C₂³	—	4	2	no	—
289	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
290	D₄	—	2	4	no	—
291	D₄	—	2	4	no	—
292	D₄	—	4	2	no	—
293	D₄	—	4	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	16	no	—
305	C₂×C₄	—	4	4	no	—
306	C₂×C₄	—	4	4	no	—
307	C₂×C₄	—	4	4	no	—
308	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
309	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
310	C₂×C₄	C₂×C₇×D₈	1	4	yes	—
311	C₂×C₄	C₂×C₇×D₈	1	2	yes	—
312	C₂×C₄	C₇×#135	1	2	yes	—
313	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
314	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
315	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
316	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
317	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
318	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
319	C₈	—	2	2	no	—
320	C₈	—	4	2	no	—
321	C₈	C₇×#117	1	4	yes	complemented
322	C₇	#22807	1	1	yes	characteristic, direct product factor, Sylow
323	C₂²	—	4	4	no	—
324	C₂²	—	4	4	no	—
325	C₂²	—	4	4	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	8	2	no	—
328	C₂²	—	8	2	no	—
329	C₂²	—	8	2	no	—
330	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
331	C₂²	C₇×#547	1	1	yes	characteristic
332	C₂²	C₇×#550	1	1	yes	characteristic
333	C₂²	C₂²×C₇×D₈	1	1	yes	characteristic
334	C₂²	C₂×C₇×#135	1	1	yes	characteristic
335	C₂²	C₇×#576	1	1	yes	characteristic
336	C₂²	C₇×#578	1	1	yes	characteristic
337	C₄	—	2	2	no	—
338	C₄	—	2	4	no	—
339	C₄	—	4	4	no	—
340	C₄	—	4	2	no	—
341	C₄	—	8	4	no	—
342	C₄	C₄×C₇×D₈	1	4	yes	—
343	C₄	C₂×C₇×#117	1	1	yes	characteristic
344	C₄	C₇×#520	1	1	yes	characteristic
345	C₄	C₇×#555	1	1	yes	characteristic
346	C₄	C₇×#556	1	1	yes	characteristic
347	C₂	—	8	4	no	—
348	C₂	C₇×#2354	1	1	yes	characteristic
349	C₂	C₂×C₄×C₇×D₈	1	1	yes	characteristic
350	C₂	C₇×#2951	1	1	yes	characteristic
351	C₂	C₇×#3334	1	1	yes	characteristic
352	C₂	C₇×#3341	1	1	yes	characteristic
353	C₂	C₇×#3391	1	1	yes	characteristic
354	C₂	C₇×#3393	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×D₈	C₂²
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×#2354	C₂	semidirect
4. C₇×#1925	C₂	semidirect
5. C₇×#1845	C₂	semidirect
6. C₇×#1802	C₂	semidirect
7. C₄×C₇×#107	C₂	semidirect
8. C₇×C₈×C₄⋊C₄	C₂	semidirect
12. C₄×C₇×D₈	C₄	semidirect
29. #22807	C₇	direct (N × H)
64. #2354	C₂×C₇	semidirect
66. #1925	C₂×C₇	semidirect
67. #1845	C₂×C₇	semidirect
68. #1802	C₂×C₇	semidirect
69. C₄×#107	C₂×C₇	semidirect
70. C₈×C₄⋊C₄	C₂×C₇	semidirect
104. C₇×C₄⋊C₄	D₈	semidirect
117. C₄×D₈	C₄×C₇	semidirect
173. C₇×C₈	#117	semidirect
264. C₄⋊C₄	C₇×D₈	semidirect
321. C₈	C₇×#117	semidirect
322. C₇	#22807	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×C₄×C₇×D₈	2
C₇×#2354	2
C₇×#1970	2
C₇×#1925	2
C₇×#1845	2
C₇×#1802	2
C₄×C₇×#107	2
C₇×C₈×C₄⋊C₄	2
#22807	7

Subgroups — C7×#22807 #13200023

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22807 #13200023