Subgroups — #13367687

Order 1792.

810 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3147	C₂	1	1	yes	maximal, characteristic
2	#11740721	C₂	1	2	yes	maximal, complemented
3	#11729841	C₂	1	8	yes	maximal, complemented
4	#11729570	C₂	1	2	yes	maximal, complemented
5	#11727545	C₂	1	1	yes	maximal, characteristic, complemented
6	#11727505	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₇×#135	C₂²	1	4	yes	—
8	C₇×#532	C₂²	1	2	yes	—
9	C₂×#6511	C₂²	1	4	yes	—
10	C₇×#495	C₄	1	4	yes	complemented
11	C₇×#494	C₄	1	1	yes	characteristic, complemented
12	C₇×#493	C₄	1	1	yes	characteristic, complemented
13	C₇×#492	C₄	1	2	yes	complemented
14	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
15	C₂×#6411	C₂²	1	8	yes	—
16	#90006	C₂²	1	4	yes	—
17	#89783	—	2	8	no	—
18	#89704	C₂²	1	1	yes	characteristic
19	#89696	C₂²	1	2	yes	—
20	C₂×#6377	C₂²	1	1	yes	characteristic
21	#89347	C₂²	1	4	yes	—
22	#89345	C₂²	1	4	yes	—
23	#22282	—	7	1	no	maximal, Sylow
24	C₂×C₇×#41	C₂³	1	4	yes	—
25	C₇×#135	—	4	8	no	—
26	C₂×C₇×Q₁₆	C₂×C₄	1	4	yes	—
27	C₂×C₇×SD₁₆	C₂×C₄	1	8	yes	—
28	C₂×C₇×D₈	C₂×C₄	1	4	yes	—
29	C₂²×C₇×C₈	C₂³	1	2	yes	—
30	C₇×C₄⋊Q₈	C₂×C₄	1	2	yes	—
31	C₇×#127	C₂×C₄	1	2	yes	—
32	C₇×#124	C₂×C₄	1	4	yes	—
33	C₇×#121	—	2	4	no	—
34	C₄×C₇×D₄	—	2	4	no	—
35	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
36	#6511	—	4	8	no	—
37	C₂×C₄×Dic₇	C₂³	1	2	yes	—
38	C₂×#1136	—	2	4	no	—
39	C₂×#1136	C₂³	1	4	yes	—
40	C₂²×C₇⋊C₈	—	2	4	no	—
41	C₂²×C₇⋊C₈	C₂³	1	2	yes	—
42	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
43	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
44	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
45	#6411	—	4	8	no	—
46	C₇⋊#95	—	2	2	no	—
47	C₂²⋊(C₇⋊C₈)	—	2	8	no	—
48	C₄⋊(C₇⋊C₈)	—	2	4	no	—
49	#6377	D₄	1	4	yes	—
50	#3147	D₇	1	1	yes	characteristic
51	#2932	—	7	2	no	—
52	#1963	—	7	2	no	—
53	#1876	—	7	8	no	—
54	#1837	—	7	1	no	—
55	#1746	—	7	1	no	—
56	C₇×#41	—	2	8	no	—
57	C₇×#41	—	2	8	no	—
58	C₂×C₇×Q₈	C₂²×C₄	1	4	yes	—
59	C₂×C₇×D₄	—	2	4	no	—
60	C₂×C₇×D₄	—	2	4	no	—
61	C₂×C₇×D₄	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₇	—	2	4	no	—
63	C₂²×C₄×C₇	#41	1	2	yes	—
64	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
65	C₂²×Dic₇	—	2	2	no	—
66	C₇×Q₁₆	—	4	4	no	—
67	C₇×SD₁₆	—	4	8	no	—
68	C₇×D₈	—	4	4	no	—
69	C₂×C₇×C₈	—	2	4	no	—
70	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
71	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
72	C₇×C₄⋊C₄	—	2	4	no	—
73	C₇×C₂²⋊C₄	—	2	8	no	—
74	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
75	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₇	C₂×D₄	1	2	yes	—
77	C₄×Dic₇	—	2	4	no	—
78	#1136	—	2	8	no	—
79	#1136	—	4	8	no	—
80	C₂×C₇⋊C₈	—	2	4	no	—
81	C₂×C₇⋊C₈	—	2	4	no	—
82	C₂×C₇⋊C₈	—	4	8	no	—
83	C₂×C₇⋊C₈	C₂×D₄	1	4	yes	—
84	C₂×#135	C₂×D₇	1	4	yes	—
85	C₂×#131	—	7	4	no	—
86	#532	C₂×D₇	1	2	yes	—
87	#495	Dic₇	1	4	yes	complemented
88	#494	Dic₇	1	1	yes	characteristic, complemented
89	#493	Dic₇	1	1	yes	characteristic, complemented
90	#492	Dic₇	1	2	yes	complemented
91	#435	—	14	8	no	—
92	#423	—	7	2	no	—
93	C₂×#104	—	7	8	no	—
94	#408	—	7	4	no	—
95	C₂×#97	—	7	1	no	—
96	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
97	#376	—	7	1	no	—
98	#341	—	7	4	no	—
99	#336	—	7	4	no	—
100	C₂³×C₇	—	2	4	no	—
101	C₂³×C₇	C₂×#41	1	1	yes	characteristic
102	C₇×Q₈	—	2	8	no	—
103	C₇×D₄	—	2	8	no	—
104	C₇×D₄	—	4	8	no	—
105	C₇×D₄	—	8	4	no	—
106	C₂×C₄×C₇	—	2	4	no	—
107	C₂×C₄×C₇	—	4	2	no	—
108	C₂×C₄×C₇	—	4	4	no	—
109	C₂×C₄×C₇	—	4	4	no	—
110	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₇	H(5,𝔽₂)	1	4	yes	—
115	C₂×Dic₇	—	4	4	no	—
116	C₂×Dic₇	—	4	2	no	—
117	C₇×C₈	—	2	2	no	—
118	C₇×C₈	—	2	2	no	—
119	C₇⋊C₈	—	2	4	no	—
120	C₇⋊C₈	—	4	4	no	—
121	C₂×#41	C₂²×D₇	1	4	yes	—
122	#135	—	4	8	no	—
123	C₂×Q₁₆	C₂×Dic₇	1	4	yes	—
124	C₂×SD₁₆	C₂×Dic₇	1	8	yes	—
125	C₂×D₈	C₂×Dic₇	1	4	yes	—
126	#131	—	28	8	no	—
127	C₂×M₁₆	—	7	4	no	—
128	C₂×M₁₆	—	14	4	no	—
129	C₂²×C₈	—	7	2	no	—
130	C₂²×C₈	—	14	4	no	—
131	C₂²×C₈	C₂²×D₇	1	2	yes	—
132	C₄⋊Q₈	C₂×Dic₇	1	2	yes	—
133	#127	C₂×Dic₇	1	2	yes	—
134	#124	C₂×Dic₇	1	4	yes	—
135	#121	—	2	4	no	—
136	C₄×D₄	—	2	4	no	—
137	C₂×C₄²	—	7	2	no	—
138	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
139	C₄⋊C₈	—	14	4	no	—
140	#104	—	28	8	no	—
141	C₂²⋊C₈	—	14	8	no	—
142	#97	—	7	4	no	—
143	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
144	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
145	C₄×C₈	C₂×Dic₇	1	2	yes	—
146	#95	—	14	2	no	—
147	C₂²×C₇	—	2	2	no	—
148	C₂²×C₇	—	4	4	no	—
149	C₂²×C₇	—	4	8	no	—
150	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
151	C₂²×C₇	#540	1	2	yes	—
152	C₄×C₇	—	2	2	no	—
153	C₄×C₇	—	2	2	no	—
154	C₄×C₇	—	4	4	no	—
155	C₄×C₇	#518	1	2	yes	—
156	C₄×C₇	#535	1	2	yes	—
157	Dic₇	—	8	2	no	—
158	#41	—	2	8	no	—
159	#41	—	2	8	no	—
160	C₂×Q₈	C₂²×Dic₇	1	4	yes	—
161	C₂×D₄	—	2	4	no	—
162	C₂×D₄	—	2	4	no	—
163	C₂×D₄	C₂²×Dic₇	1	4	yes	—
164	C₂²×C₄	—	2	4	no	—
165	C₂²×C₄	—	14	2	no	—
166	C₂²×C₄	#1159	1	2	yes	—
167	C₂²×C₄	C₂³×D₇	1	1	yes	characteristic
168	Q₁₆	—	4	4	no	—
169	SD₁₆	—	4	8	no	—
170	D₈	—	4	4	no	—
171	M₁₆	—	14	8	no	—
172	M₁₆	—	28	8	no	—
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	7	4	no	—
175	C₂×C₈	—	14	4	no	—
176	C₂×C₈	—	14	4	no	—
177	C₂×C₈	—	28	8	no	—
178	C₂×C₈	C₂²×Dic₇	1	2	yes	—
179	C₂×C₈	C₂²×Dic₇	1	2	yes	—
180	C₄⋊C₄	—	2	4	no	—
181	C₂²⋊C₄	—	2	8	no	—
182	C₄²	—	14	4	no	—
183	C₄²	D₄×D₇	1	2	yes	—
184	C₄²	C₂²×Dic₇	1	1	yes	characteristic
185	C₄²	C₂²×Dic₇	1	1	yes	characteristic
186	C₂×C₇	—	2	2	no	—
187	C₂×C₇	—	8	4	no	—
188	C₂×C₇	#2328	1	1	yes	characteristic
189	C₂×C₇	#2957	1	2	yes	—
190	C₂³	—	2	4	no	—
191	C₂³	C₂×#1159	1	1	yes	characteristic
192	Q₈	—	2	8	no	—
193	D₄	—	2	8	no	—
194	D₄	—	4	8	no	—
195	D₄	—	8	4	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	4	4	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	—	28	4	no	—
201	C₂×C₄	—	28	2	no	—
202	C₂×C₄	D₄×Dic₇	1	2	yes	—
203	C₂×C₄	D₄×Dic₇	1	2	yes	—
204	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
205	C₂×C₄	#6548	1	4	yes	—
206	C₂×C₄	C₂³×Dic₇	1	1	yes	characteristic
207	C₈	—	2	2	no	—
208	C₈	—	2	2	no	—
209	C₈	—	14	4	no	—
210	C₈	—	28	4	no	—
211	C₇	#22282	1	1	yes	characteristic, complemented, Sylow
212	C₂²	—	2	2	no	—
213	C₂²	—	4	8	no	—
214	C₂²	—	4	4	no	—
215	C₂²	#90410	1	2	yes	—
216	C₂²	C₂×D₄×Dic₇	1	1	yes	characteristic
217	C₄	—	2	2	no	—
218	C₄	—	2	2	no	—
219	C₄	—	4	4	no	—
220	C₄	—	56	2	no	—
221	C₄	#90406	1	2	yes	—
222	C₄	#90489	1	2	yes	—
223	C₂	—	2	2	no	—
224	C₂	—	8	4	no	—
225	C₂	#11740617	1	1	yes	characteristic
226	C₂	#11741355	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇
commutator	C₂×C₄×C₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇
Fitting, prime radical	C₇×#3147	C₂
socle	C₂²×C₇	C₂×C₄×D₄
cosocle	C₂²×C₄×C₇	C₂⁴
hypercenter	#3147	D₇
hypocenter	C₇	#22282
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3147

Complements

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

N	H ≅ G/N	type
2. #11740721	C₂	semidirect
3. #11729841	C₂	semidirect
4. #11729570	C₂	semidirect
5. #11727545	C₂	semidirect
6. #11727505	C₂	semidirect
10. C₇×#495	C₄	semidirect
11. C₇×#494	C₄	semidirect
12. C₇×#493	C₄	semidirect
13. C₇×#492	C₄	semidirect
87. #495	Dic₇	semidirect
88. #494	Dic₇	semidirect
89. #493	Dic₇	semidirect
90. #492	Dic₇	semidirect
211. C₇	#22282	semidirect

Maximal subgroups

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

subgroup	index
C₇×#3147	2
#11740721	2
#11729841	2
#11729570	2
#11727545	2
#11727505	2
#22282	7