Subgroups — #13241660

Order 1792.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#1713	C₂	1	1	yes	maximal, characteristic
2	#11725601	C₂	1	2	yes	maximal
3	C₇×#500	—	2	2	no	—
4	C₇×#494	C₄	1	1	yes	characteristic
5	C₇×#494	C₂²	1	1	yes	characteristic
6	C₇×C₈²	C₄	1	1	yes	characteristic
7	#89343	—	2	2	no	—
8	#7532	—	7	1	no	maximal, Sylow
9	C₂×C₇×Q₁₆	—	4	1	no	—
10	C₂×C₇×Q₁₆	—	4	1	no	—
11	C₇×C₄⋊Q₈	—	2	1	no	—
12	C₇×C₄⋊Q₈	D₄	1	2	yes	—
13	C₇×#107	—	4	4	no	—
14	C₄×C₇×C₈	—	2	1	no	—
15	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
16	#6410	—	4	2	no	—
17	#1713	D₇	1	1	yes	characteristic
18	#1411	—	7	2	no	—
19	C₂×C₇×Q₈	—	2	2	no	—
20	C₂×C₇×Q₈	—	4	1	no	—
21	C₇×Q₁₆	—	8	2	no	—
22	C₇×Q₁₆	—	8	2	no	—
23	C₂×C₇×C₈	—	2	2	no	—
24	C₂×C₇×C₈	—	2	1	no	—
25	C₇×C₄⋊C₄	—	4	2	no	—
26	C₇×C₄⋊C₄	—	4	2	no	—
27	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
28	#1136	—	8	2	no	—
29	#500	—	2	2	no	—
30	#494	Dic₇	1	1	yes	characteristic
31	#494	C₂×D₇	1	1	yes	characteristic
32	#356	—	14	2	no	—
33	C₈²	Dic₇	1	1	yes	characteristic
34	C₇×Q₈	—	8	2	no	—
35	C₇×Q₈	—	8	2	no	—
36	C₂×C₄×C₇	—	2	1	no	—
37	C₂×C₄×C₇	—	4	2	no	—
38	C₂×C₄×C₇	—	8	1	no	—
39	C₂×C₄×C₇	#99	1	1	yes	characteristic
40	C₇×C₈	—	2	4	no	—
41	C₇×C₈	—	2	2	no	—
42	C₇⋊C₈	—	16	2	no	—
43	C₂×Q₁₆	—	4	1	no	—
44	C₂×Q₁₆	—	4	1	no	—
45	C₄⋊Q₈	—	2	1	no	—
46	C₄⋊Q₈	C₂²⋊D₇	1	2	yes	—
47	#107	—	4	4	no	—
48	#101	—	28	2	no	—
49	C₄×C₈	—	2	1	no	—
50	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
51	C₂²×C₇	#353	1	1	yes	characteristic
52	C₄×C₇	—	2	2	no	—
53	C₄×C₇	—	2	1	no	—
54	C₄×C₇	—	8	2	no	—
55	C₄×C₇	—	16	1	no	—
56	C₂×Q₈	—	2	2	no	—
57	C₂×Q₈	—	4	1	no	—
58	Q₁₆	—	8	2	no	—
59	Q₁₆	—	8	2	no	—
60	M₁₆	—	56	2	no	—
61	C₂×C₈	—	2	2	no	—
62	C₂×C₈	—	2	1	no	—
63	C₄⋊C₄	—	4	2	no	—
64	C₄⋊C₄	—	4	2	no	—
65	C₄²	C₂²⋊Dic₇	1	1	yes	characteristic
66	C₂×C₇	—	2	1	no	—
67	C₂×C₇	#1408	1	1	yes	characteristic
68	Q₈	—	8	2	no	—
69	Q₈	—	8	2	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	4	2	no	—
72	C₂×C₄	—	8	1	no	—
73	C₂×C₄	#6408	1	1	yes	characteristic
74	C₈	—	2	4	no	—
75	C₈	—	2	2	no	—
76	C₈	—	112	2	no	—
77	C₇	#7532	1	1	yes	characteristic, complemented, Sylow
78	C₂²	#89340	1	1	yes	characteristic
79	C₄	—	2	2	no	—
80	C₄	—	2	1	no	—
81	C₄	—	8	2	no	—
82	C₄	—	16	1	no	—
83	C₂	—	2	1	no	—
84	C₂	#11725598	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#11725598
commutator	C₄×C₇×C₈	C₂×C₄
Frattini	#494	C₂×D₇
Fitting	C₇×#1713	C₂
socle	C₂×C₇	#1408
prime radical	C₂²×C₇	#353
cosocle	C₇×#494	C₂²
hypercenter	#1713	D₇
hypocenter	C₇	#7532
LCS	G ▷ C₄×C₇×C₈ ▷ C₄²×C₇ ▷ C₂×C₄×C₇ ▷ C₂²×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ #494 ◁ #1713

Complements

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

N	H ≅ G/N	type
77. C₇	#7532	semidirect

Maximal subgroups

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

subgroup	index
C₇×#1713	2
#11725601	2
#7532	7