Subgroups — #13506458

Order 1792.

716 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2875	C₂	1	1	yes	maximal, characteristic
2	#11741225	C₂	1	2	yes	maximal
3	#11727638	C₂	1	6	yes	maximal, complemented
4	C₂×#89261	C₂	1	6	yes	maximal, complemented
5	C₂×C₇×#131	C₄	1	1	yes	characteristic
6	C₂×C₇×#131	C₂²	1	1	yes	characteristic
7	C₄×C₇×#41	C₄	1	1	yes	characteristic
8	C₇×#405	C₄	1	3	yes	—
9	C₇×#405	C₂²	1	3	yes	—
10	C₄×C₇×M₁₆	C₄	1	3	yes	—
11	C₂×C₇×#97	C₂²	1	3	yes	—
12	#89912	C₄	1	16	yes	complemented
13	C₂×#6386	C₂²	1	12	yes	—
14	#89261	C₂²	1	12	yes	—
15	#89261	C₂²	1	4	yes	—
16	#33329	—	7	1	no	maximal, Sylow
17	C₂×C₇×#41	C₂×C₄	1	1	yes	characteristic
18	C₇×#131	—	2	4	no	—
19	C₇×#131	C₂×C₄	1	8	yes	—
20	C₂×C₇×M₁₆	C₂×C₄	1	3	yes	—
21	C₂×C₇×M₁₆	C₂×C₄	1	3	yes	—
22	C₂²×C₇×C₈	C₂×C₄	1	3	yes	—
23	C₂²×C₇×C₈	C₂³	1	3	yes	—
24	C₄×C₇×Q₈	—	2	1	no	—
25	C₄×C₇×D₄	—	2	3	no	—
26	C₇×#117	C₂×C₄	1	3	yes	—
27	C₂×C₄²×C₇	C₂×C₄	1	3	yes	—
28	C₇×#97	C₂×C₄	1	3	yes	—
29	C₇×#97	C₂×C₄	1	3	yes	—
30	C₇×#97	C₂³	1	1	yes	characteristic
31	C₇×#97	C₂³	1	3	yes	—
32	C₄×C₇×C₈	C₂×C₄	1	6	yes	—
33	#6386	C₂×C₄	1	24	yes	complemented
34	#6386	C₂³	1	8	yes	—
35	C₂×C₇⋊C₁₆	C₂×C₄	1	24	yes	complemented
36	#3407	—	7	2	no	—
37	#2875	D₇	1	1	yes	characteristic
38	#2110	—	7	6	no	—
39	C₂×#347	—	7	6	no	—
40	C₇×#41	C₄²	1	4	yes	—
41	C₇×#41	M₁₆	1	4	yes	—
42	C₂×C₇×Q₈	M₁₆	1	1	yes	characteristic
43	C₂×C₇×D₄	M₁₆	1	3	yes	—
44	C₂²×C₄×C₇	—	2	3	no	—
45	C₂²×C₄×C₇	C₂²×C₄	1	3	yes	—
46	C₇×M₁₆	—	2	6	no	—
47	C₇×M₁₆	C₄²	1	12	yes	—
48	C₂×C₇×C₈	—	2	6	no	—
49	C₂×C₇×C₈	C₂²×C₄	1	3	yes	—
50	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₇×C₈	C₂²×C₄	1	3	yes	—
52	C₂×C₇×C₈	C₂²×C₄	1	12	yes	—
53	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
54	C₇×C₄⋊C₄	—	2	3	no	—
55	C₇×C₂²⋊C₄	—	2	3	no	—
56	C₄²×C₇	—	2	3	no	—
57	C₄²×C₇	C₂²×C₄	1	3	yes	—
58	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
59	C₇⋊C₁₆	C₂²×C₄	1	16	yes	—
60	C₂×#131	Dic₇	1	1	yes	characteristic
61	C₂×#131	C₂×D₇	1	1	yes	characteristic
62	C₄×#41	Dic₇	1	1	yes	characteristic
63	#504	—	7	16	no	—
64	C₂×M₃₂	—	7	12	no	—
65	#405	Dic₇	1	3	yes	—
66	#405	C₂×D₇	1	3	yes	—
67	C₄×M₁₆	Dic₇	1	3	yes	—
68	C₂×#97	C₂×D₇	1	3	yes	—
69	#347	—	7	4	no	—
70	#347	—	7	12	no	—
71	C₂³×C₇	C₂×M₁₆	1	3	yes	—
72	C₇×Q₈	#97	1	4	yes	—
73	C₇×D₄	#97	1	12	yes	—
74	C₂×C₄×C₇	—	2	1	no	—
75	C₂×C₄×C₇	—	2	3	no	—
76	C₂×C₄×C₇	—	8	3	no	—
77	C₂×C₄×C₇	C₂×C₄²	1	6	yes	—
78	C₂×C₄×C₇	C₂×M₁₆	1	6	yes	—
79	C₂×C₄×C₇	C₂×M₁₆	1	3	yes	—
80	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
81	C₇×C₈	—	2	1	no	—
82	C₇×C₈	—	2	3	no	—
83	C₇×C₈	C₂×C₄²	1	6	yes	—
84	C₇×C₈	C₂³×C₄	1	2	yes	—
85	C₂×#41	C₂×Dic₇	1	1	yes	characteristic
86	#131	—	2	4	no	—
87	#131	C₄×D₇	1	8	yes	—
88	C₂×M₁₆	C₂×Dic₇	1	3	yes	—
89	C₂×M₁₆	C₂×Dic₇	1	3	yes	—
90	C₂²×C₈	C₂×Dic₇	1	3	yes	—
91	C₂²×C₈	C₂²×D₇	1	3	yes	—
92	C₄×Q₈	—	2	1	no	—
93	C₄×D₄	—	2	3	no	—
94	#117	C₂×Dic₇	1	3	yes	—
95	C₂×C₄²	C₂×Dic₇	1	3	yes	—
96	M₃₂	—	7	24	no	—
97	M₃₂	—	7	8	no	—
98	C₂×C₁₆	—	7	24	no	—
99	#97	C₂×Dic₇	1	3	yes	—
100	#97	C₂×Dic₇	1	3	yes	—
101	#97	C₂²×D₇	1	1	yes	characteristic
102	#97	C₂²×D₇	1	3	yes	—
103	C₄×C₈	C₂×Dic₇	1	6	yes	—
104	C₂²×C₇	—	4	3	no	—
105	C₂²×C₇	C₂×#97	1	6	yes	—
106	C₂²×C₇	C₂²×M₁₆	1	1	yes	characteristic
107	C₄×C₇	—	4	3	no	—
108	C₄×C₇	—	4	1	no	—
109	C₄×C₇	C₂×#97	1	6	yes	—
110	C₄×C₇	C₂²×C₄²	1	1	yes	characteristic
111	C₄×C₇	C₂²×M₁₆	1	1	yes	characteristic
112	#41	#1136	1	4	yes	—
113	#41	C₄×Dic₇	1	4	yes	—
114	C₂×Q₈	#1136	1	1	yes	characteristic
115	C₂×D₄	#1136	1	3	yes	—
116	C₂²×C₄	—	2	3	no	—
117	C₂²×C₄	C₂²×Dic₇	1	3	yes	—
118	M₁₆	—	2	6	no	—
119	M₁₆	C₄×Dic₇	1	12	yes	—
120	C₂×C₈	—	2	6	no	—
121	C₂×C₈	C₂×C₄×D₇	1	12	yes	—
122	C₂×C₈	C₂²×Dic₇	1	3	yes	—
123	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
124	C₂×C₈	C₂²×Dic₇	1	3	yes	—
125	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
126	C₄⋊C₄	—	2	3	no	—
127	C₂²⋊C₄	—	2	3	no	—
128	C₄²	—	2	3	no	—
129	C₄²	C₂²×Dic₇	1	3	yes	—
130	C₄²	C₂²×Dic₇	1	1	yes	characteristic
131	C₁₆	—	7	16	no	—
132	C₂×C₇	—	2	1	no	—
133	C₂×C₇	—	2	6	no	—
134	C₂×C₇	C₂²×#97	1	1	yes	characteristic
135	C₂³	C₂×#1136	1	3	yes	—
136	Q₈	#6377	1	4	yes	—
137	D₄	#6377	1	12	yes	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	3	no	—
140	C₂×C₄	—	8	3	no	—
141	C₂×C₄	C₂×#1136	1	3	yes	—
142	C₂×C₄	C₂×#1136	1	6	yes	—
143	C₂×C₄	C₂×C₄×Dic₇	1	6	yes	—
144	C₂×C₄	C₂³×Dic₇	1	1	yes	characteristic
145	C₈	—	2	1	no	—
146	C₈	—	2	3	no	—
147	C₈	C₂×C₄×Dic₇	1	6	yes	—
148	C₈	C₂²×C₄×D₇	1	2	yes	—
149	C₇	#33329	1	1	yes	characteristic, complemented, Sylow
150	C₂²	—	4	3	no	—
151	C₂²	C₂×#6377	1	6	yes	—
152	C₂²	C₂²×#1136	1	1	yes	characteristic
153	C₄	—	4	1	no	—
154	C₄	—	4	3	no	—
155	C₄	C₂×#6377	1	6	yes	—
156	C₄	C₂²×#1136	1	1	yes	characteristic
157	C₄	C₂²×C₄×Dic₇	1	1	yes	characteristic
158	C₂	—	2	1	no	—
159	C₂	—	2	6	no	—
160	C₂	C₂²×#6377	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×C₄×Dic₇
commutator	C₄×C₇	C₂²×C₄²
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₇×#2875	C₂
socle	C₂×C₇	C₂²×#97
prime radical	C₂×C₇×#41	C₂×C₄
cosocle	C₂×C₇×C₈	C₂⁴
hypercenter	#2875	D₇
hypocenter	C₇	#33329
LCS	G ▷ C₄×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ #2875

Complements

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

N	H ≅ G/N	type
3. #11727638	C₂	semidirect
4. C₂×#89261	C₂	semidirect
12. #89912	C₄	semidirect
33. #6386	C₂×C₄	semidirect
35. C₂×C₇⋊C₁₆	C₂×C₄	semidirect
149. C₇	#33329	semidirect

Maximal subgroups

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

subgroup	index
C₇×#2875	2
#11741225	2
#11727638	2
C₂×#89261	2
#33329	7