Subgroups — C₇×#22695 #13199911

Order 1792.

808 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2957	C₂	1	2	yes	maximal
2	C₂×C₇×#444	C₂	1	1	yes	maximal, characteristic
3	C₇×#2916	C₂	1	2	yes	maximal, complemented
4	C₇×#1953	C₂	1	2	yes	maximal, complemented
5	C₇×#1861	C₂	1	4	yes	maximal, complemented
6	C₇×#1849	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1814	C₂	1	2	yes	maximal, complemented
8	C₇×#1773	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₇×#135	C₂²	1	1	yes	characteristic
10	C₇×#568	C₂²	1	4	yes	—
11	C₂×C₇×#131	C₂²	1	2	yes	—
12	C₇×#444	—	2	4	no	—
13	C₇×#444	C₄	1	8	yes	complemented
14	C₇×#443	—	2	4	no	—
15	C₇×#430	—	2	4	no	—
16	C₇×#430	C₂²	1	4	yes	—
17	C₂×C₇×#108	—	2	4	no	—
18	C₂×C₇×#108	C₂²	1	2	yes	—
19	C₂×C₇×#108	C₂²	1	2	yes	—
20	C₂×C₇×#108	C₂²	1	1	yes	characteristic
21	C₂×C₇×#108	C₂²	1	1	yes	characteristic
22	C₇×#427	C₂²	1	1	yes	characteristic
23	C₇×#423	C₂²	1	2	yes	—
24	C₇×#421	C₂²	1	4	yes	—
25	C₂×C₇×#104	C₂²	1	2	yes	—
26	C₇×#405	C₂²	1	2	yes	—
27	C₂×C₇×#97	C₂²	1	1	yes	characteristic
28	C₇×#344	C₂²	1	4	yes	—
29	C₇×#339	C₂²	1	2	yes	—
30	#22695	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₂×C₇×#41	C₂³	1	2	yes	—
32	C₇×#135	C₂×C₄	1	4	yes	—
33	C₇×#135	D₄	1	4	yes	—
34	C₂×C₇×Q₁₆	Q₈	1	1	yes	characteristic
35	C₂×C₇×SD₁₆	Q₈	1	2	yes	—
36	C₂×C₇×D₈	Q₈	1	1	yes	characteristic
37	C₇×#131	—	2	2	no	—
38	C₇×#131	—	2	2	no	—
39	C₇×#131	—	2	8	no	—
40	C₇×#131	—	4	4	no	—
41	C₇×#131	C₂×C₄	1	4	yes	—
42	C₇×#131	C₂×C₄	1	4	yes	—
43	C₂×C₇×M₁₆	—	2	2	no	—
44	C₂×C₇×M₁₆	—	2	8	no	—
45	C₂×C₇×M₁₆	C₂³	1	4	yes	—
46	C₂×C₇×M₁₆	C₂³	1	2	yes	—
47	C₂×C₇×M₁₆	C₂³	1	2	yes	—
48	C₂²×C₇×C₈	—	2	2	no	—
49	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
50	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
51	C₇×#117	C₂³	1	2	yes	—
52	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
53	C₇×#108	—	2	8	no	—
54	C₇×#108	—	2	2	no	—
55	C₇×#108	—	2	2	no	—
56	C₇×#108	—	2	2	no	—
57	C₇×#108	—	2	2	no	—
58	C₇×#108	C₂×C₄	1	4	yes	—
59	C₇×#108	D₄	1	4	yes	—
60	C₇×#107	—	2	1	no	—
61	C₇×#106	—	2	1	no	—
62	C₇×C₄⋊C₈	—	4	2	no	—
63	C₇×#104	—	2	8	no	—
64	C₇×#104	C₂×C₄	1	8	yes	—
65	C₇×#97	—	2	2	no	—
66	C₇×#97	C₂×C₄	1	4	yes	—
67	C₄×C₇×C₈	—	2	2	no	—
68	#2957	C₂×C₇	1	2	yes	—
69	C₂×#444	C₂×C₇	1	1	yes	characteristic
70	#2916	C₂×C₇	1	2	yes	complemented
71	#1953	C₂×C₇	1	2	yes	complemented
72	#1861	C₂×C₇	1	4	yes	complemented
73	#1849	C₂×C₇	1	1	yes	characteristic, complemented
74	#1814	C₂×C₇	1	2	yes	complemented
75	#1773	C₂×C₇	1	1	yes	characteristic, complemented
76	C₇×#41	—	2	2	no	—
77	C₇×#41	—	2	2	no	—
78	C₇×#41	C₂²×C₄	1	4	yes	—
79	C₇×#41	C₂×D₄	1	4	yes	—
80	C₂×C₇×Q₈	C₂×Q₈	1	2	yes	—
81	C₂×C₇×D₄	—	2	2	no	—
82	C₂×C₇×D₄	C₂×Q₈	1	2	yes	—
83	C₂²×C₄×C₇	—	2	2	no	—
84	C₂²×C₄×C₇	—	2	1	no	—
85	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
86	C₇×Q₁₆	C₄⋊C₄	1	4	yes	—
87	C₇×SD₁₆	C₄⋊C₄	1	8	yes	—
88	C₇×D₈	C₄⋊C₄	1	4	yes	—
89	C₇×M₁₆	—	2	2	no	—
90	C₇×M₁₆	—	2	2	no	—
91	C₇×M₁₆	—	2	4	no	—
92	C₇×M₁₆	—	2	4	no	—
93	C₇×M₁₆	—	2	4	no	—
94	C₇×M₁₆	—	2	4	no	—
95	C₇×M₁₆	—	4	8	no	—
96	C₇×M₁₆	C₂²×C₄	1	4	yes	—
97	C₇×M₁₆	C₂×D₄	1	4	yes	—
98	C₂×C₇×C₈	—	2	2	no	—
99	C₂×C₇×C₈	—	2	2	no	—
100	C₂×C₇×C₈	—	2	4	no	—
101	C₂×C₇×C₈	—	2	4	no	—
102	C₂×C₇×C₈	—	2	4	no	—
103	C₂×C₇×C₈	—	4	8	no	—
104	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
105	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
106	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
107	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
108	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
109	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
110	C₂×C₇×C₈	#41	1	1	yes	characteristic
111	C₂×C₇×C₈	#41	1	1	yes	characteristic
112	C₇×C₄⋊C₄	—	2	2	no	—
113	C₇×C₂²⋊C₄	—	2	2	no	—
114	C₄²×C₇	—	2	1	no	—
115	C₄²×C₇	—	2	2	no	—
116	C₄²×C₇	C₂²×C₄	1	2	yes	—
117	C₂×#135	C₂²×C₇	1	1	yes	characteristic
118	#568	C₂²×C₇	1	4	yes	—
119	C₂×#131	C₂²×C₇	1	2	yes	—
120	#444	—	2	4	no	—
121	#444	C₄×C₇	1	8	yes	complemented
122	#443	—	2	4	no	—
123	#430	—	2	4	no	—
124	#430	C₂²×C₇	1	4	yes	—
125	C₂×#108	—	2	4	no	—
126	C₂×#108	C₂²×C₇	1	2	yes	—
127	C₂×#108	C₂²×C₇	1	2	yes	—
128	C₂×#108	C₂²×C₇	1	1	yes	characteristic
129	C₂×#108	C₂²×C₇	1	1	yes	characteristic
130	#427	C₂²×C₇	1	1	yes	characteristic
131	#423	C₂²×C₇	1	2	yes	—
132	#421	C₂²×C₇	1	4	yes	—
133	C₂×#104	C₂²×C₇	1	2	yes	—
134	#405	C₂²×C₇	1	2	yes	—
135	C₂×#97	C₂²×C₇	1	1	yes	characteristic
136	#344	C₂²×C₇	1	4	yes	—
137	#339	C₂²×C₇	1	2	yes	—
138	C₂³×C₇	—	2	2	no	—
139	C₂³×C₇	2₋¹⁺⁴	1	1	yes	characteristic
140	C₇×Q₈	—	2	2	no	—
141	C₇×Q₈	C₂×C₄⋊C₄	1	4	yes	—
142	C₇×D₄	—	2	4	no	—
143	C₇×D₄	—	2	4	no	—
144	C₇×D₄	—	2	2	no	—
145	C₇×D₄	C₂×C₄⋊C₄	1	4	yes	—
146	C₂×C₄×C₇	—	2	2	no	—
147	C₂×C₄×C₇	—	2	4	no	—
148	C₂×C₄×C₇	—	2	4	no	—
149	C₂×C₄×C₇	—	2	2	no	—
150	C₂×C₄×C₇	—	4	2	no	—
151	C₂×C₄×C₇	—	8	1	no	—
152	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
153	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
156	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
157	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
158	C₇×C₈	—	2	4	no	—
159	C₇×C₈	—	2	4	no	—
160	C₇×C₈	—	4	4	no	—
161	C₇×C₈	C₂×C₄⋊C₄	1	2	yes	—
162	C₇×C₈	C₂×C₄⋊C₄	1	2	yes	—
163	C₇×C₈	C₄×D₄	1	2	yes	—
164	C₇×C₈	C₄×D₄	1	2	yes	—
165	C₂×#41	C₂³×C₇	1	2	yes	—
166	#135	C₂×C₄×C₇	1	4	yes	—
167	#135	C₇×D₄	1	4	yes	—
168	C₂×Q₁₆	C₇×Q₈	1	1	yes	characteristic
169	C₂×SD₁₆	C₇×Q₈	1	2	yes	—
170	C₂×D₈	C₇×Q₈	1	1	yes	characteristic
171	#131	—	2	2	no	—
172	#131	—	2	2	no	—
173	#131	—	2	8	no	—
174	#131	—	4	4	no	—
175	#131	C₂×C₄×C₇	1	4	yes	—
176	#131	C₂×C₄×C₇	1	4	yes	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	8	no	—
179	C₂×M₁₆	C₂³×C₇	1	4	yes	—
180	C₂×M₁₆	C₂³×C₇	1	2	yes	—
181	C₂×M₁₆	C₂³×C₇	1	2	yes	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
184	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
185	#117	C₂³×C₇	1	2	yes	—
186	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
187	#108	—	2	8	no	—
188	#108	—	2	2	no	—
189	#108	—	2	2	no	—
190	#108	—	2	2	no	—
191	#108	—	2	2	no	—
192	#108	C₂×C₄×C₇	1	4	yes	—
193	#108	C₇×D₄	1	4	yes	—
194	#107	—	2	1	no	—
195	#106	—	2	1	no	—
196	C₄⋊C₈	—	4	2	no	—
197	#104	—	2	8	no	—
198	#104	C₂×C₄×C₇	1	8	yes	—
199	#97	—	2	2	no	—
200	#97	C₂×C₄×C₇	1	4	yes	—
201	C₄×C₈	—	2	2	no	—
202	C₂²×C₇	—	2	4	no	—
203	C₂²×C₇	—	4	1	no	—
204	C₂²×C₇	—	8	2	no	—
205	C₂²×C₇	#520	1	1	yes	characteristic
206	C₂²×C₇	#547	1	1	yes	characteristic
207	C₂²×C₇	D₄×Q₈	1	1	yes	characteristic
208	C₄×C₇	—	2	4	no	—
209	C₄×C₇	—	4	2	no	—
210	C₄×C₇	—	8	2	no	—
211	C₄×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
212	C₄×C₇	C₂×C₄×D₄	1	1	yes	characteristic
213	C₄×C₇	D₄²	1	1	yes	characteristic
214	C₄×C₇	#554	1	1	yes	characteristic
215	#41	—	2	2	no	—
216	#41	—	2	2	no	—
217	#41	C₂²×C₄×C₇	1	4	yes	—
218	#41	C₂×C₇×D₄	1	4	yes	—
219	C₂×Q₈	C₂×C₇×Q₈	1	2	yes	—
220	C₂×D₄	—	2	2	no	—
221	C₂×D₄	C₂×C₇×Q₈	1	2	yes	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
225	Q₁₆	C₇×C₄⋊C₄	1	4	yes	—
226	SD₁₆	C₇×C₄⋊C₄	1	8	yes	—
227	D₈	C₇×C₄⋊C₄	1	4	yes	—
228	M₁₆	—	2	2	no	—
229	M₁₆	—	2	2	no	—
230	M₁₆	—	2	4	no	—
231	M₁₆	—	2	4	no	—
232	M₁₆	—	2	4	no	—
233	M₁₆	—	2	4	no	—
234	M₁₆	—	4	8	no	—
235	M₁₆	C₂²×C₄×C₇	1	4	yes	—
236	M₁₆	C₂×C₇×D₄	1	4	yes	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	4	8	no	—
243	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
244	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
245	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
246	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
247	C₂×C₈	C₂×C₇×Q₈	1	1	yes	characteristic
248	C₂×C₈	C₂×C₇×Q₈	1	1	yes	characteristic
249	C₂×C₈	C₇×#41	1	1	yes	characteristic
250	C₂×C₈	C₇×#41	1	1	yes	characteristic
251	C₄⋊C₄	—	2	2	no	—
252	C₂²⋊C₄	—	2	2	no	—
253	C₄²	—	2	1	no	—
254	C₄²	—	2	2	no	—
255	C₄²	C₂²×C₄×C₇	1	2	yes	—
256	C₂×C₇	—	2	1	no	—
257	C₂×C₇	—	2	1	no	—
258	C₂×C₇	—	2	1	no	—
259	C₂×C₇	—	4	4	no	—
260	C₂×C₇	D₄×C₄⋊C₄	1	1	yes	characteristic
261	C₂³	—	2	2	no	—
262	C₂³	C₇×2₋¹⁺⁴	1	1	yes	characteristic
263	Q₈	—	2	2	no	—
264	Q₈	C₂×C₇×C₄⋊C₄	1	4	yes	—
265	D₄	—	2	4	no	—
266	D₄	—	2	4	no	—
267	D₄	—	2	2	no	—
268	D₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	8	1	no	—
275	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
276	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
277	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
278	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
279	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
280	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
281	C₈	—	2	4	no	—
282	C₈	—	2	4	no	—
283	C₈	—	4	4	no	—
284	C₈	C₂×C₇×C₄⋊C₄	1	2	yes	—
285	C₈	C₂×C₇×C₄⋊C₄	1	2	yes	—
286	C₈	C₄×C₇×D₄	1	2	yes	—
287	C₈	C₄×C₇×D₄	1	2	yes	—
288	C₇	#22695	1	1	yes	characteristic, direct product factor, Sylow
289	C₂²	—	2	4	no	—
290	C₂²	—	4	1	no	—
291	C₂²	—	8	2	no	—
292	C₂²	C₇×#520	1	1	yes	characteristic
293	C₂²	C₇×#547	1	1	yes	characteristic
294	C₂²	C₇×D₄×Q₈	1	1	yes	characteristic
295	C₄	—	2	4	no	—
296	C₄	—	4	2	no	—
297	C₄	—	8	2	no	—
298	C₄	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
299	C₄	C₂×C₄×C₇×D₄	1	1	yes	characteristic
300	C₄	C₇×D₄²	1	1	yes	characteristic
301	C₄	C₇×#554	1	1	yes	characteristic
302	C₂	—	2	1	no	—
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	4	4	no	—
306	C₂	C₇×D₄×C₄⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₇	D₄²
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle	C₂×C₇	D₄×C₄⋊C₄
prime radical	C₂×C₇×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₂²×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×#2916	C₂	semidirect
4. C₇×#1953	C₂	semidirect
5. C₇×#1861	C₂	semidirect
6. C₇×#1849	C₂	semidirect
7. C₇×#1814	C₂	semidirect
8. C₇×#1773	C₂	semidirect
13. C₇×#444	C₄	semidirect
30. #22695	C₇	direct (N × H)
70. #2916	C₂×C₇	semidirect
71. #1953	C₂×C₇	semidirect
72. #1861	C₂×C₇	semidirect
73. #1849	C₂×C₇	semidirect
74. #1814	C₂×C₇	semidirect
75. #1773	C₂×C₇	semidirect
121. #444	C₄×C₇	semidirect
288. C₇	#22695	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×#2957	2
C₂×C₇×#444	2
C₇×#2916	2
C₇×#1953	2
C₇×#1861	2
C₇×#1849	2
C₇×#1814	2
C₇×#1773	2
#22695	7

Subgroups — C7×#22695 #13199911

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22695 #13199911