Subgroups — C₇×#22607 #13199823

Order 1792.

692 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2924	C₂	1	2	yes	maximal
2	C₂×C₇×#427	C₂	1	1	yes	maximal, characteristic
3	C₇×#1946	C₂	1	2	yes	maximal, complemented
4	C₇×#1922	C₂	1	4	yes	maximal, complemented
5	C₇×#1826	C₂	1	2	yes	maximal
6	C₇×#1810	C₂	1	1	yes	maximal, characteristic
7	C₇×#1773	C₂	1	2	yes	maximal, complemented
8	C₇×#1752	C₂	1	1	yes	maximal, characteristic
9	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₇×#117	C₂²	1	2	yes	—
11	C₇×#433	—	2	4	no	—
12	C₇×#432	—	2	2	no	—
13	C₇×#431	—	2	2	no	—
14	C₂×C₇×#108	C₂²	1	2	yes	—
15	C₇×#427	C₄	1	1	yes	characteristic, complemented
16	C₇×#427	C₄	1	1	yes	characteristic, complemented
17	C₇×#427	C₄	1	2	yes	complemented
18	C₇×#427	C₂²	1	1	yes	characteristic
19	C₇×#427	C₂²	1	2	yes	—
20	C₇×#427	C₂²	1	1	yes	characteristic
21	C₂×C₇×#107	C₄	1	1	yes	characteristic, complemented
22	C₂×C₇×#107	C₄	1	1	yes	characteristic, complemented
23	C₂×C₇×#106	C₄	1	2	yes	complemented
24	C₇×#424	C₂²	1	4	yes	—
25	C₇×#407	C₂²	1	2	yes	—
26	C₂×C₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
27	C₇×#405	C₂²	1	4	yes	—
28	C₂×C₇×#97	C₂²	1	2	yes	—
29	C₄×C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
30	C₇×#341	C₂²	1	2	yes	—
31	C₇×#339	C₂²	1	4	yes	—
32	C₇×#339	C₂²	1	4	yes	—
33	C₇×#336	C₂²	1	2	yes	—
34	#22607	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
35	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
36	C₂×C₇×M₁₆	C₂³	1	4	yes	—
37	C₂²×C₇×C₈	—	2	4	no	—
38	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
43	C₂²×C₇×C₈	C₂³	1	2	yes	—
44	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
45	C₇×#117	—	2	2	no	—
46	C₇×#117	—	2	2	no	—
47	C₇×#117	C₂×C₄	1	2	yes	—
48	C₇×#117	C₂×C₄	1	2	yes	—
49	C₇×#117	C₂³	1	2	yes	—
50	C₇×#117	C₂³	1	2	yes	—
51	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
52	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
53	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
54	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
55	C₂×C₄²×C₇	—	2	2	no	—
56	C₂×C₄²×C₇	C₂³	1	2	yes	—
57	C₇×#108	—	2	4	no	—
58	C₇×#107	C₂×C₄	1	2	yes	—
59	C₇×#107	C₂×C₄	1	2	yes	—
60	C₇×#107	C₂×C₄	1	1	yes	characteristic
61	C₇×#107	C₂×C₄	1	1	yes	characteristic
62	C₇×#107	C₂×C₄	1	1	yes	characteristic
63	C₇×#107	C₂×C₄	1	1	yes	characteristic
64	C₇×#106	C₂×C₄	1	2	yes	—
65	C₇×#106	C₂×C₄	1	2	yes	—
66	C₇×#106	C₂×C₄	1	2	yes	—
67	C₇×#106	C₂×C₄	1	2	yes	—
68	C₇×C₄⋊C₈	—	2	4	no	—
69	C₇×C₄⋊C₈	—	2	4	no	—
70	C₇×C₂²⋊C₈	—	2	2	no	—
71	C₇×C₂²⋊C₈	—	2	2	no	—
72	C₇×C₂²⋊C₈	D₄	1	2	yes	—
73	C₇×C₂²⋊C₈	D₄	1	2	yes	—
74	C₇×#97	—	2	4	no	—
75	C₇×#97	—	2	4	no	—
76	C₄×C₇×C₈	—	2	4	no	—
77	C₇×#95	—	2	1	no	—
78	#2924	C₂×C₇	1	2	yes	—
79	C₂×#427	C₂×C₇	1	1	yes	characteristic
80	#1946	C₂×C₇	1	2	yes	complemented
81	#1922	C₂×C₇	1	4	yes	complemented
82	#1826	C₂×C₇	1	2	yes	—
83	#1810	C₂×C₇	1	1	yes	characteristic
84	#1773	C₂×C₇	1	2	yes	complemented
85	#1752	C₂×C₇	1	1	yes	characteristic
86	C₂⁴×C₇	#41	1	1	yes	characteristic
87	C₂²×C₄×C₇	—	2	2	no	—
88	C₂²×C₄×C₇	—	2	2	no	—
89	C₂²×C₄×C₇	—	2	2	no	—
90	C₂²×C₄×C₇	—	2	2	no	—
91	C₂²×C₄×C₇	—	2	2	no	—
92	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₇	#41	1	1	yes	characteristic
97	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
98	C₇×M₁₆	—	2	4	no	—
99	C₇×M₁₆	—	2	4	no	—
100	C₂×C₇×C₈	—	2	2	no	—
101	C₂×C₇×C₈	—	2	2	no	—
102	C₂×C₇×C₈	—	2	2	no	—
103	C₂×C₇×C₈	—	2	2	no	—
104	C₂×C₇×C₈	—	2	2	no	—
105	C₂×C₇×C₈	—	2	2	no	—
106	C₂×C₇×C₈	—	2	2	no	—
107	C₂×C₇×C₈	—	2	4	no	—
108	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
113	C₂×C₇×C₈	#41	1	4	yes	—
114	C₇×C₄⋊C₄	—	2	2	no	—
115	C₇×C₄⋊C₄	—	2	2	no	—
116	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
117	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
118	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
119	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
120	C₇×C₂²⋊C₄	—	2	2	no	—
121	C₇×C₂²⋊C₄	—	2	2	no	—
122	C₇×C₂²⋊C₄	—	2	2	no	—
123	C₇×C₂²⋊C₄	—	2	2	no	—
124	C₇×C₂²⋊C₄	—	4	2	no	—
125	C₄²×C₇	—	2	2	no	—
126	C₄²×C₇	—	2	2	no	—
127	C₄²×C₇	—	2	2	no	—
128	C₄²×C₇	—	2	2	no	—
129	C₄²×C₇	—	2	2	no	—
130	C₄²×C₇	—	2	2	no	—
131	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
132	C₂×#117	C₂²×C₇	1	2	yes	—
133	#433	—	2	4	no	—
134	#432	—	2	2	no	—
135	#431	—	2	2	no	—
136	C₂×#108	C₂²×C₇	1	2	yes	—
137	#427	C₄×C₇	1	1	yes	characteristic, complemented
138	#427	C₄×C₇	1	1	yes	characteristic, complemented
139	#427	C₄×C₇	1	2	yes	complemented
140	#427	C₂²×C₇	1	2	yes	—
141	#427	C₂²×C₇	1	1	yes	characteristic
142	#427	C₂²×C₇	1	1	yes	characteristic
143	C₂×#107	C₄×C₇	1	1	yes	characteristic, complemented
144	C₂×#107	C₄×C₇	1	1	yes	characteristic, complemented
145	C₂×#106	C₄×C₇	1	2	yes	complemented
146	#424	C₂²×C₇	1	4	yes	—
147	#407	C₂²×C₇	1	2	yes	—
148	C₂×C₂²⋊C₈	C₂²×C₇	1	1	yes	characteristic
149	#405	C₂²×C₇	1	4	yes	—
150	C₂×#97	C₂²×C₇	1	2	yes	—
151	C₄×C₂²⋊C₄	C₂²×C₇	1	1	yes	characteristic
152	#341	C₂²×C₇	1	2	yes	—
153	#339	C₂²×C₇	1	4	yes	—
154	#339	C₂²×C₇	1	4	yes	—
155	#336	C₂²×C₇	1	2	yes	—
156	C₂³×C₇	—	2	2	no	—
157	C₂³×C₇	—	2	2	no	—
158	C₂³×C₇	—	4	1	no	—
159	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
160	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
161	C₂³×C₇	C₂×#41	1	1	yes	characteristic
162	C₂×C₄×C₇	—	2	2	no	—
163	C₂×C₄×C₇	—	2	1	no	—
164	C₂×C₄×C₇	—	2	1	no	—
165	C₂×C₄×C₇	—	2	1	no	—
166	C₂×C₄×C₇	—	2	1	no	—
167	C₂×C₄×C₇	—	2	2	no	—
168	C₂×C₄×C₇	—	2	2	no	—
169	C₂×C₄×C₇	—	2	2	no	—
170	C₂×C₄×C₇	—	2	2	no	—
171	C₂×C₄×C₇	—	2	2	no	—
172	C₂×C₄×C₇	—	2	2	no	—
173	C₂×C₄×C₇	—	4	4	no	—
174	C₂×C₄×C₇	—	4	2	no	—
175	C₂×C₄×C₇	—	4	2	no	—
176	C₂×C₄×C₇	—	4	2	no	—
177	C₂×C₄×C₇	—	8	2	no	—
178	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
179	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
180	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
181	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
182	C₂×C₄×C₇	C₂×#41	1	2	yes	—
183	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
184	C₇×C₈	—	2	1	no	—
185	C₇×C₈	—	2	1	no	—
186	C₇×C₈	—	2	4	no	—
187	C₇×C₈	—	4	4	no	—
188	C₇×C₈	#117	1	2	yes	—
189	C₇×C₈	#117	1	2	yes	—
190	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
191	C₂×M₁₆	C₂³×C₇	1	4	yes	—
192	C₂²×C₈	—	2	4	no	—
193	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
194	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
195	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
196	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
197	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
198	C₂²×C₈	C₂³×C₇	1	2	yes	—
199	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
200	#117	—	2	2	no	—
201	#117	—	2	2	no	—
202	#117	C₂×C₄×C₇	1	2	yes	—
203	#117	C₂×C₄×C₇	1	2	yes	—
204	#117	C₂³×C₇	1	2	yes	—
205	#117	C₂³×C₇	1	2	yes	—
206	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
207	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
208	C₂×C₂²⋊C₄	—	2	2	no	—
209	C₂×C₂²⋊C₄	—	2	1	no	—
210	C₂×C₄²	—	2	2	no	—
211	C₂×C₄²	C₂³×C₇	1	2	yes	—
212	#108	—	2	4	no	—
213	#107	C₂×C₄×C₇	1	2	yes	—
214	#107	C₂×C₄×C₇	1	2	yes	—
215	#107	C₂×C₄×C₇	1	1	yes	characteristic
216	#107	C₂×C₄×C₇	1	1	yes	characteristic
217	#107	C₂×C₄×C₇	1	1	yes	characteristic
218	#107	C₂×C₄×C₇	1	1	yes	characteristic
219	#106	C₂×C₄×C₇	1	2	yes	—
220	#106	C₂×C₄×C₇	1	2	yes	—
221	#106	C₂×C₄×C₇	1	2	yes	—
222	#106	C₂×C₄×C₇	1	2	yes	—
223	C₄⋊C₈	—	2	4	no	—
224	C₄⋊C₈	—	2	4	no	—
225	C₂²⋊C₈	—	2	2	no	—
226	C₂²⋊C₈	—	2	2	no	—
227	C₂²⋊C₈	C₇×D₄	1	2	yes	—
228	C₂²⋊C₈	C₇×D₄	1	2	yes	—
229	#97	—	2	4	no	—
230	#97	—	2	4	no	—
231	C₄×C₈	—	2	4	no	—
232	#95	—	2	1	no	—
233	C₂²×C₇	—	2	1	no	—
234	C₂²×C₇	—	2	1	no	—
235	C₂²×C₇	—	2	1	no	—
236	C₂²×C₇	—	2	1	no	—
237	C₂²×C₇	—	2	1	no	—
238	C₂²×C₇	—	2	1	no	—
239	C₂²×C₇	—	2	1	no	—
240	C₂²×C₇	—	4	4	no	—
241	C₂²×C₇	#444	1	2	yes	—
242	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
243	C₂²×C₇	#552	1	2	yes	—
244	C₄×C₇	—	2	1	no	—
245	C₄×C₇	—	2	1	no	—
246	C₄×C₇	—	4	2	no	—
247	C₄×C₇	—	4	2	no	—
248	C₄×C₇	—	4	2	no	—
249	C₄×C₇	—	4	2	no	—
250	C₄×C₇	—	8	2	no	—
251	C₄×C₇	C₂×#117	1	1	yes	characteristic
252	C₄×C₇	#519	1	1	yes	characteristic
253	C₄×C₇	#547	1	2	yes	—
254	C₂⁴	C₇×#41	1	1	yes	characteristic
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	—	2	2	no	—
258	C₂²×C₄	—	2	2	no	—
259	C₂²×C₄	—	2	2	no	—
260	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
261	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
262	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
263	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
264	C₂²×C₄	C₇×#41	1	1	yes	characteristic
265	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
266	M₁₆	—	2	4	no	—
267	M₁₆	—	2	4	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	2	4	no	—
276	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
277	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
278	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
279	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
280	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
281	C₂×C₈	C₇×#41	1	4	yes	—
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
285	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
286	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
287	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
288	C₂²⋊C₄	—	2	2	no	—
289	C₂²⋊C₄	—	2	2	no	—
290	C₂²⋊C₄	—	2	2	no	—
291	C₂²⋊C₄	—	2	2	no	—
292	C₂²⋊C₄	—	4	2	no	—
293	C₄²	—	2	2	no	—
294	C₄²	—	2	2	no	—
295	C₄²	—	2	2	no	—
296	C₄²	—	2	2	no	—
297	C₄²	—	2	2	no	—
298	C₄²	—	2	2	no	—
299	C₂×C₇	—	2	2	no	—
300	C₂×C₇	—	2	2	no	—
301	C₂×C₇	—	4	1	no	—
302	C₂×C₇	#2344	1	1	yes	characteristic
303	C₂×C₇	C₂×#444	1	1	yes	characteristic
304	C₂×C₇	#2957	1	1	yes	characteristic
305	C₂³	—	2	2	no	—
306	C₂³	—	2	2	no	—
307	C₂³	—	4	1	no	—
308	C₂³	C₄×C₇×D₄	1	1	yes	characteristic
309	C₂³	C₄×C₇×D₄	1	1	yes	characteristic
310	C₂³	C₂×C₇×#41	1	1	yes	characteristic
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	4	4	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	8	2	no	—
327	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
328	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
329	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
330	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
331	C₂×C₄	C₂×C₇×#41	1	2	yes	—
332	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
333	C₈	—	2	1	no	—
334	C₈	—	2	1	no	—
335	C₈	—	2	4	no	—
336	C₈	—	4	4	no	—
337	C₈	C₇×#117	1	2	yes	—
338	C₈	C₇×#117	1	2	yes	—
339	C₇	#22607	1	1	yes	characteristic, direct product factor, Sylow
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	4	4	no	—
348	C₂²	C₇×#444	1	2	yes	—
349	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
350	C₂²	C₇×#552	1	2	yes	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	4	2	no	—
354	C₄	—	4	2	no	—
355	C₄	—	4	2	no	—
356	C₄	—	4	2	no	—
357	C₄	—	8	2	no	—
358	C₄	C₂×C₇×#117	1	1	yes	characteristic
359	C₄	C₇×#519	1	1	yes	characteristic
360	C₄	C₇×#547	1	2	yes	—
361	C₂	—	2	2	no	—
362	C₂	—	2	2	no	—
363	C₂	—	4	1	no	—
364	C₂	C₇×#2344	1	1	yes	characteristic
365	C₂	C₂×C₇×#444	1	1	yes	characteristic
366	C₂	C₇×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle	C₂²×C₇	C₂×C₄×D₄
prime radical	C₂⁴×C₇	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₇×C₂²⋊C₈ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×#1946	C₂	semidirect
4. C₇×#1922	C₂	semidirect
7. C₇×#1773	C₂	semidirect
15. C₇×#427	C₄	semidirect
16. C₇×#427	C₄	semidirect
17. C₇×#427	C₄	semidirect
21. C₂×C₇×#107	C₄	semidirect
22. C₂×C₇×#107	C₄	semidirect
23. C₂×C₇×#106	C₄	semidirect
34. #22607	C₇	direct (N × H)
80. #1946	C₂×C₇	semidirect
81. #1922	C₂×C₇	semidirect
84. #1773	C₂×C₇	semidirect
137. #427	C₄×C₇	semidirect
138. #427	C₄×C₇	semidirect
139. #427	C₄×C₇	semidirect
143. C₂×#107	C₄×C₇	semidirect
144. C₂×#107	C₄×C₇	semidirect
145. C₂×#106	C₄×C₇	semidirect
339. C₇	#22607	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×#2924	2
C₂×C₇×#427	2
C₇×#1946	2
C₇×#1922	2
C₇×#1826	2
C₇×#1810	2
C₇×#1773	2
C₇×#1752	2
#22607	7

Subgroups — C7×#22607 #13199823

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22607 #13199823