Subgroups — C₇×#18091 #13195307

Order 1792.

644 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3169	C₂	1	1	yes	maximal, characteristic
2	C₇×#2979	C₂	1	2	yes	maximal, complemented
3	C₇×#1594	C₂	1	2	yes	maximal, complemented
4	C₇×#1581	C₂	1	4	yes	maximal, complemented
5	C₇×#1580	C₂	1	2	yes	maximal, complemented
6	C₇×#1564	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1484	C₂	1	2	yes	maximal, complemented
8	C₇×#1451	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#532	C₂²	1	2	yes	—
10	C₇×#500	C₄	1	1	yes	characteristic
11	C₇×#493	C₄	1	1	yes	characteristic
12	C₇×#486	C₄	1	2	yes	—
13	C₇×#466	C₄	1	2	yes	—
14	C₇×#466	C₄	1	2	yes	—
15	C₄×C₇×D₈	C₂²	1	2	yes	—
16	C₄×C₇×D₈	C₂²	1	2	yes	—
17	C₇×#436	—	2	2	no	—
18	C₇×#436	C₂²	1	4	yes	—
19	C₇×C₈×D₄	—	2	2	no	—
20	C₇×C₈×D₄	C₂²	1	4	yes	—
21	C₇×#431	C₂²	1	2	yes	—
22	C₇×#431	C₂²	1	1	yes	characteristic
23	C₇×#408	—	2	4	no	—
24	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
26	C₇×#335	C₂²	1	2	yes	—
27	C₇×#334	C₂²	1	1	yes	characteristic
28	C₇×#334	C₂²	1	1	yes	characteristic
29	C₇×D₄⋊C₈	C₂²	1	4	yes	—
30	C₇×D₄⋊C₈	C₂²	1	4	yes	—
31	C₇×#322	C₂²	1	2	yes	—
32	C₇×#322	C₂²	1	2	yes	—
33	#18091	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂×C₇×#41	—	2	2	no	—
35	C₂×C₇×D₈	C₂×C₄	1	2	yes	—
36	C₂×C₇×D₈	C₂×C₄	1	2	yes	—
37	C₂×C₇×M₁₆	—	2	1	no	—
38	C₂×C₇×M₁₆	—	2	4	no	—
39	C₂²×C₇×C₈	—	2	4	no	—
40	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
42	C₇×C₄⋊Q₈	C₂×C₄	1	2	yes	—
43	C₇×#127	C₂×C₄	1	2	yes	—
44	C₇×#124	—	2	2	no	—
45	C₇×#121	—	2	2	no	—
46	C₇×#121	C₂×C₄	1	4	yes	—
47	C₄×C₇×D₄	—	2	2	no	—
48	C₄×C₇×D₄	C₂³	1	4	yes	—
49	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
50	C₇×#107	C₂×C₄	1	2	yes	—
51	C₇×#107	C₂×C₄	1	2	yes	—
52	C₇×C₄⋊C₈	D₄	1	2	yes	—
53	C₇×C₄⋊C₈	C₂³	1	2	yes	—
54	C₇×C₄⋊C₈	C₂³	1	2	yes	—
55	C₇×#102	C₂×C₄	1	4	yes	—
56	C₇×#102	C₂×C₄	1	4	yes	—
57	C₇×C₂²⋊C₈	—	2	2	no	—
58	C₇×C₂²⋊C₈	—	2	4	no	—
59	C₇×C₂²⋊C₈	—	2	4	no	—
60	C₇×C₂²⋊C₈	D₄	1	2	yes	—
61	C₇×#97	—	2	1	no	—
62	C₄×C₇×C₈	—	2	2	no	—
63	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
64	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
65	C₄×C₇×C₈	C₂³	1	2	yes	—
66	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
67	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
68	C₄×C₇×C₈	C₂³	1	2	yes	—
69	#3169	C₂×C₇	1	1	yes	characteristic
70	#2979	C₂×C₇	1	2	yes	complemented
71	#1594	C₂×C₇	1	2	yes	complemented
72	#1581	C₂×C₇	1	4	yes	complemented
73	#1580	C₂×C₇	1	2	yes	complemented
74	#1564	C₂×C₇	1	1	yes	characteristic, complemented
75	#1484	C₂×C₇	1	2	yes	complemented
76	#1451	C₂×C₇	1	1	yes	characteristic, complemented
77	C₇×#41	—	8	2	no	—
78	C₇×#41	—	8	2	no	—
79	C₂×C₇×Q₈	—	2	2	no	—
80	C₂×C₇×D₄	—	2	4	no	—
81	C₂×C₇×D₄	—	2	2	no	—
82	C₂×C₇×D₄	—	2	2	no	—
83	C₂×C₇×D₄	C₂²×C₄	1	4	yes	—
84	C₂²×C₄×C₇	—	2	4	no	—
85	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₇	#41	1	1	yes	characteristic
88	C₇×D₈	—	2	4	no	—
89	C₇×D₈	—	4	2	no	—
90	C₇×M₁₆	—	4	2	no	—
91	C₇×M₁₆	—	8	4	no	—
92	C₂×C₇×C₈	—	2	2	no	—
93	C₂×C₇×C₈	—	2	2	no	—
94	C₂×C₇×C₈	—	2	2	no	—
95	C₂×C₇×C₈	—	2	2	no	—
96	C₂×C₇×C₈	—	4	1	no	—
97	C₂×C₇×C₈	—	8	4	no	—
98	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
103	C₇×C₄⋊C₄	—	2	2	no	—
104	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
105	C₇×C₂²⋊C₄	—	2	4	no	—
106	C₇×C₂²⋊C₄	—	2	4	no	—
107	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
108	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₇	#41	1	1	yes	characteristic
110	C₄²×C₇	C₂⁴	1	1	yes	characteristic
111	#532	C₂²×C₇	1	2	yes	—
112	#500	C₄×C₇	1	1	yes	characteristic
113	#493	C₄×C₇	1	1	yes	characteristic
114	#486	C₄×C₇	1	2	yes	—
115	#466	C₄×C₇	1	2	yes	—
116	#466	C₄×C₇	1	2	yes	—
117	C₄×D₈	C₂²×C₇	1	2	yes	—
118	C₄×D₈	C₂²×C₇	1	2	yes	—
119	#436	—	2	2	no	—
120	#436	C₂²×C₇	1	4	yes	—
121	C₈×D₄	—	2	2	no	—
122	C₈×D₄	C₂²×C₇	1	4	yes	—
123	#431	C₂²×C₇	1	2	yes	—
124	#431	C₂²×C₇	1	1	yes	characteristic
125	#408	—	2	4	no	—
126	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
127	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
128	#335	C₂²×C₇	1	2	yes	—
129	#334	C₂²×C₇	1	1	yes	characteristic
130	#334	C₂²×C₇	1	1	yes	characteristic
131	D₄⋊C₈	C₂²×C₇	1	4	yes	—
132	D₄⋊C₈	C₂²×C₇	1	4	yes	—
133	#322	C₂²×C₇	1	2	yes	—
134	#322	C₂²×C₇	1	2	yes	—
135	C₂³×C₇	—	2	4	no	—
136	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
137	C₇×Q₈	—	8	2	no	—
138	C₇×D₄	—	2	4	no	—
139	C₇×D₄	—	2	4	no	—
140	C₇×D₄	—	4	4	no	—
141	C₇×D₄	—	8	2	no	—
142	C₇×D₄	—	8	4	no	—
143	C₂×C₄×C₇	—	2	1	no	—
144	C₂×C₄×C₇	—	2	1	no	—
145	C₂×C₄×C₇	—	2	2	no	—
146	C₂×C₄×C₇	—	2	2	no	—
147	C₂×C₄×C₇	—	2	4	no	—
148	C₂×C₄×C₇	—	4	4	no	—
149	C₂×C₄×C₇	—	4	4	no	—
150	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
151	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
152	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
153	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
154	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
156	C₇×C₈	—	2	1	no	—
157	C₇×C₈	—	2	1	no	—
158	C₇×C₈	—	2	1	no	—
159	C₇×C₈	—	2	4	no	—
160	C₇×C₈	—	4	2	no	—
161	C₇×C₈	—	4	2	no	—
162	C₇×C₈	—	4	2	no	—
163	C₇×C₈	#131	1	2	yes	—
164	C₂×#41	—	2	2	no	—
165	C₂×D₈	C₂×C₄×C₇	1	2	yes	—
166	C₂×D₈	C₂×C₄×C₇	1	2	yes	—
167	C₂×M₁₆	—	2	1	no	—
168	C₂×M₁₆	—	2	4	no	—
169	C₂²×C₈	—	2	4	no	—
170	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
171	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
172	C₄⋊Q₈	C₂×C₄×C₇	1	2	yes	—
173	#127	C₂×C₄×C₇	1	2	yes	—
174	#124	—	2	2	no	—
175	#121	—	2	2	no	—
176	#121	C₂×C₄×C₇	1	4	yes	—
177	C₄×D₄	—	2	2	no	—
178	C₄×D₄	C₂³×C₇	1	4	yes	—
179	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
180	#107	C₂×C₄×C₇	1	2	yes	—
181	#107	C₂×C₄×C₇	1	2	yes	—
182	C₄⋊C₈	C₇×D₄	1	2	yes	—
183	C₄⋊C₈	C₂³×C₇	1	2	yes	—
184	C₄⋊C₈	C₂³×C₇	1	2	yes	—
185	#102	C₂×C₄×C₇	1	4	yes	—
186	#102	C₂×C₄×C₇	1	4	yes	—
187	C₂²⋊C₈	—	2	2	no	—
188	C₂²⋊C₈	—	2	4	no	—
189	C₂²⋊C₈	—	2	4	no	—
190	C₂²⋊C₈	C₇×D₄	1	2	yes	—
191	#97	—	2	1	no	—
192	C₄×C₈	—	2	2	no	—
193	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
194	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
195	C₄×C₈	C₂³×C₇	1	2	yes	—
196	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
197	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
198	C₄×C₈	C₂³×C₇	1	2	yes	—
199	C₂²×C₇	—	2	1	no	—
200	C₂²×C₇	—	2	1	no	—
201	C₂²×C₇	—	2	1	no	—
202	C₂²×C₇	—	4	4	no	—
203	C₂²×C₇	—	4	4	no	—
204	C₂²×C₇	—	4	4	no	—
205	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
206	C₄×C₇	—	2	1	no	—
207	C₄×C₇	—	2	1	no	—
208	C₄×C₇	—	2	1	no	—
209	C₄×C₇	—	4	4	no	—
210	C₄×C₇	#444	1	2	yes	—
211	C₄×C₇	C₂×#131	1	1	yes	characteristic
212	C₄×C₇	#568	1	1	yes	characteristic
213	C₄×C₇	#576	1	2	yes	—
214	#41	—	8	2	no	—
215	#41	—	8	2	no	—
216	C₂×Q₈	—	2	2	no	—
217	C₂×D₄	—	2	4	no	—
218	C₂×D₄	—	2	2	no	—
219	C₂×D₄	—	2	2	no	—
220	C₂×D₄	C₂²×C₄×C₇	1	4	yes	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
224	C₂²×C₄	C₇×#41	1	1	yes	characteristic
225	D₈	—	2	4	no	—
226	D₈	—	4	2	no	—
227	M₁₆	—	4	2	no	—
228	M₁₆	—	8	4	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	4	1	no	—
234	C₂×C₈	—	8	4	no	—
235	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
236	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
237	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
238	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
239	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
242	C₂²⋊C₄	—	2	4	no	—
243	C₂²⋊C₄	—	2	4	no	—
244	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
245	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
246	C₄²	C₇×#41	1	1	yes	characteristic
247	C₄²	C₂⁴×C₇	1	1	yes	characteristic
248	C₂×C₇	—	4	1	no	—
249	C₂×C₇	—	8	4	no	—
250	C₂×C₇	#2935	1	1	yes	characteristic
251	C₂×C₇	#2948	1	1	yes	characteristic
252	C₂×C₇	C₂×#444	1	1	yes	characteristic
253	C₂³	—	2	4	no	—
254	C₂³	C₄×C₇×D₄	1	1	yes	characteristic
255	Q₈	—	8	2	no	—
256	D₄	—	2	4	no	—
257	D₄	—	2	4	no	—
258	D₄	—	4	4	no	—
259	D₄	—	8	2	no	—
260	D₄	—	8	4	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
269	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
270	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
271	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
272	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
273	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
274	C₈	—	2	1	no	—
275	C₈	—	2	1	no	—
276	C₈	—	2	1	no	—
277	C₈	—	2	4	no	—
278	C₈	—	4	2	no	—
279	C₈	—	4	2	no	—
280	C₈	—	4	2	no	—
281	C₈	C₇×#131	1	2	yes	—
282	C₇	#18091	1	1	yes	characteristic, direct product factor, Sylow
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	4	4	no	—
287	C₂²	—	4	4	no	—
288	C₂²	—	4	4	no	—
289	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	4	4	no	—
294	C₄	C₇×#444	1	2	yes	—
295	C₄	C₂×C₇×#131	1	1	yes	characteristic
296	C₄	C₇×#568	1	1	yes	characteristic
297	C₄	C₇×#576	1	2	yes	—
298	C₂	—	4	1	no	—
299	C₂	—	8	4	no	—
300	C₂	C₇×#2935	1	1	yes	characteristic
301	C₂	C₇×#2948	1	1	yes	characteristic
302	C₂	C₂×C₇×#444	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₄²	C₂⁴×C₇
socle	C₂²×C₇	C₂×C₄×D₄
prime radical	C₇×#3169	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₇×#431 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#2979	C₂	semidirect
3. C₇×#1594	C₂	semidirect
4. C₇×#1581	C₂	semidirect
5. C₇×#1580	C₂	semidirect
6. C₇×#1564	C₂	semidirect
7. C₇×#1484	C₂	semidirect
8. C₇×#1451	C₂	semidirect
33. #18091	C₇	direct (N × H)
70. #2979	C₂×C₇	semidirect
71. #1594	C₂×C₇	semidirect
72. #1581	C₂×C₇	semidirect
73. #1580	C₂×C₇	semidirect
74. #1564	C₂×C₇	semidirect
75. #1484	C₂×C₇	semidirect
76. #1451	C₂×C₇	semidirect
282. C₇	#18091	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×#3169	2
C₇×#2979	2
C₇×#1594	2
C₇×#1581	2
C₇×#1580	2
C₇×#1564	2
C₇×#1484	2
C₇×#1451	2
#18091	7

Subgroups — C7×#18091 #13195307

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#18091 #13195307