Subgroups — #13262797

Order 1792.

298 conjugacy classes of subgroups.

The subgroup lattice has 138 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#322	C₂	1	1	yes	maximal, characteristic
2	C₄×#6386	C₂	1	1	yes	maximal, characteristic
3	C₄×#6386	C₂	1	1	yes	maximal, characteristic
4	#11725248	C₂	1	4	yes	maximal, complemented
5	C₂×C₄×C₇×C₈	C₄	1	2	yes	—
6	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
7	C₂×#6386	C₄	1	2	yes	—
8	C₂×#6386	C₄	1	2	yes	—
9	C₇×#322	C₄	1	2	yes	—
10	C₇×#322	C₂²	1	2	yes	—
11	#89256	C₂²	1	2	yes	—
12	#89256	C₂²	1	2	yes	—
13	C₄×C₇⋊C₁₆	C₄	1	2	yes	—
14	C₄×C₇⋊C₁₆	C₄	1	2	yes	—
15	#11404	—	7	1	no	maximal, Sylow
16	C₂²×C₇×C₈	—	2	2	no	—
17	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
20	C₄×C₇×C₈	—	2	2	no	—
21	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
22	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
23	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
24	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
25	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
26	#6386	—	2	4	no	—
27	#6386	C₈	1	8	yes	complemented
28	C₂×C₇⋊C₁₆	C₂×C₄	1	4	yes	—
29	C₂×C₇⋊C₁₆	C₂×C₄	1	4	yes	—
30	C₄×M₃₂	—	7	1	no	—
31	C₄×M₃₂	—	7	1	no	—
32	C₂×#322	D₇	1	1	yes	characteristic
33	#1313	—	7	4	no	—
34	C₂²×C₄×C₇	C₄²	1	1	yes	characteristic
35	C₂²×C₄×C₇	M₁₆	1	2	yes	—
36	C₂×C₇×C₈	—	2	4	no	—
37	C₂×C₇×C₈	—	4	4	no	—
38	C₂×C₇×C₈	C₄²	1	1	yes	characteristic
39	C₂×C₇×C₈	C₄²	1	1	yes	characteristic
40	C₂×C₇×C₈	C₂×C₈	1	4	yes	—
41	C₂×C₇×C₈	M₁₆	1	4	yes	—
42	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
44	C₄²×C₇	C₄²	1	1	yes	characteristic
45	C₄²×C₇	M₁₆	1	2	yes	—
46	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
47	C₇⋊C₁₆	—	2	4	no	—
48	C₇⋊C₁₆	C₂×C₈	1	8	yes	complemented
49	C₂×M₃₂	—	7	2	no	—
50	C₂×M₃₂	—	7	2	no	—
51	C₂×C₄×C₈	Dic₇	1	2	yes	—
52	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
53	#346	—	7	2	no	—
54	#346	—	7	2	no	—
55	C₄×C₁₆	—	7	2	no	—
56	C₄×C₁₆	—	7	2	no	—
57	#322	Dic₇	1	2	yes	—
58	#322	C₂×D₇	1	2	yes	—
59	C₂³×C₇	#97	1	1	yes	characteristic
60	C₂×C₄×C₇	—	2	4	no	—
61	C₂×C₄×C₇	C₄×C₈	1	2	yes	—
62	C₂×C₄×C₇	#97	1	1	yes	characteristic
63	C₂×C₄×C₇	#97	1	2	yes	—
64	C₂×C₄×C₇	#97	1	2	yes	—
65	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
66	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
67	C₇×C₈	—	4	4	no	—
68	C₇×C₈	C₄×C₈	1	2	yes	—
69	C₇×C₈	#97	1	2	yes	—
70	C₇×C₈	C₂²×C₈	1	2	yes	—
71	C₇×C₈	C₂×M₁₆	1	2	yes	—
72	C₂²×C₈	—	2	2	no	—
73	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
74	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
75	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
76	M₃₂	—	7	8	no	—
77	M₃₂	—	14	4	no	—
78	C₂×C₁₆	—	7	4	no	—
79	C₂×C₁₆	—	7	4	no	—
80	C₄×C₈	—	2	2	no	—
81	C₄×C₈	C₄×D₇	1	1	yes	characteristic
82	C₄×C₈	C₄×D₇	1	1	yes	characteristic
83	C₄×C₈	C₂×Dic₇	1	4	yes	—
84	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
85	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
86	C₂²×C₇	—	2	2	no	—
87	C₂²×C₇	#322	1	2	yes	—
88	C₂²×C₇	C₂×#97	1	1	yes	characteristic
89	C₄×C₇	—	2	2	no	—
90	C₄×C₇	#322	1	2	yes	—
91	C₄×C₇	#347	1	4	yes	—
92	C₄×C₇	C₂×C₄×C₈	1	1	yes	characteristic
93	C₄×C₇	C₂×#97	1	1	yes	characteristic
94	C₂²×C₄	#1136	1	2	yes	—
95	C₂²×C₄	C₄×Dic₇	1	1	yes	characteristic
96	C₂×C₈	—	2	4	no	—
97	C₂×C₈	—	4	4	no	—
98	C₂×C₈	C₈×D₇	1	4	yes	—
99	C₂×C₈	#1131	1	4	yes	—
100	C₂×C₈	C₄×Dic₇	1	1	yes	characteristic
101	C₂×C₈	C₄×Dic₇	1	1	yes	characteristic
102	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
103	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
104	C₄²	#1136	1	2	yes	—
105	C₄²	C₄×Dic₇	1	1	yes	characteristic
106	C₄²	C₂²×Dic₇	1	1	yes	characteristic
107	C₁₆	—	7	8	no	—
108	C₁₆	—	14	4	no	—
109	C₂×C₇	—	2	2	no	—
110	C₂×C₇	C₂×#322	1	1	yes	characteristic
111	C₂×C₇	C₂×#347	1	2	yes	—
112	C₂³	#6377	1	1	yes	characteristic
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	#6377	1	1	yes	characteristic
115	C₂×C₄	#6377	1	2	yes	—
116	C₂×C₄	C₈×Dic₇	1	2	yes	—
117	C₂×C₄	#6389	1	2	yes	—
118	C₂×C₄	C₂×#1136	1	2	yes	—
119	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
120	C₈	—	4	4	no	—
121	C₈	C₈×Dic₇	1	2	yes	—
122	C₈	#6389	1	2	yes	—
123	C₈	C₂×C₈×D₇	1	2	yes	—
124	C₈	C₂×#1131	1	2	yes	—
125	C₇	#11404	1	1	yes	characteristic, complemented, Sylow
126	C₂²	—	2	2	no	—
127	C₂²	#89249	1	2	yes	—
128	C₂²	C₂×#6377	1	1	yes	characteristic
129	C₄	—	2	2	no	—
130	C₄	#89249	1	2	yes	—
131	C₄	#89261	1	4	yes	—
132	C₄	C₂×C₈×Dic₇	1	1	yes	characteristic
133	C₄	C₂×#6389	1	1	yes	characteristic
134	C₂	—	2	2	no	—
135	C₂	C₂×#89249	1	1	yes	characteristic
136	C₂	C₂×#89261	1	2	yes	—

	subgroup	G/H
center	C₄²	C₄×Dic₇
commutator	C₄×C₇	C₂×C₄×C₈
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₂×C₇×#322	C₂
socle	C₂²×C₇	C₂×#97
prime radical	C₂³×C₇	#97
cosocle	C₄×C₇×C₈	C₂³
hypercenter	C₂×#322	D₇
hypocenter	C₇	#11404
LCS	G ▷ C₄×C₇ ▷ C₇
UCS	C₁ ◁ C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
4. #11725248	C₂	semidirect
27. #6386	C₈	semidirect
48. C₇⋊C₁₆	C₂×C₈	semidirect
125. C₇	#11404	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#322	2
C₄×#6386	2
#11725248	2
#11404	7