Subgroups — #13268510

Order 1792.

310 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	#11729066	C₂	1	4	yes	maximal
3	#11725819	C₂	1	1	yes	maximal, characteristic
4	C₈×C₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₇×#500	—	2	2	no	—
6	C₇×#494	C₂²	1	2	yes	—
7	C₂×#6403	—	2	8	no	—
8	#89471	C₂²	1	2	yes	—
9	#89454	—	2	2	no	—
10	C₇×C₈²	C₂²	1	1	yes	characteristic
11	C₄×C₇⋊C₁₆	C₂²	1	2	yes	—
12	#12516	—	7	1	no	maximal, Sylow
13	C₂×C₇×Q₁₆	—	2	4	no	—
14	C₇×C₄⋊Q₈	—	2	2	no	—
15	C₂×Dic₂₈	—	2	4	no	—
16	#6432	—	2	2	no	—
17	C₇×#107	—	4	4	no	—
18	C₄×C₇×C₈	D₄	1	2	yes	—
19	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
20	#6403	—	4	16	no	—
21	#6392	—	4	4	no	—
22	C₂×C₇⋊C₁₆	D₄	1	4	yes	—
23	#2247	—	7	4	no	—
24	#1713	—	7	1	no	—
25	#1713	D₇	1	1	yes	characteristic
26	C₈×C₁₆	—	7	1	no	—
27	C₂×C₇×Q₈	—	4	2	no	—
28	C₂×Dic₁₄	—	4	2	no	—
29	C₇×Q₁₆	—	4	8	no	—
30	C₂×C₇×C₈	D₈	1	4	yes	—
31	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
32	C₇×C₄⋊C₄	—	4	4	no	—
33	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
34	C₄⋊Dic₇	—	4	4	no	—
35	Dic₂₈	—	4	8	no	—
36	C₇⋊C₁₆	D₈	1	8	yes	—
37	C₂×Q₃₂	—	14	8	no	—
38	#500	—	2	2	no	—
39	#500	—	14	2	no	—
40	#494	—	7	2	no	—
41	#494	C₂×D₇	1	2	yes	—
42	C₄×C₁₆	—	7	2	no	—
43	C₈²	C₂×D₇	1	1	yes	characteristic
44	C₇×Q₈	—	8	4	no	—
45	C₂×C₄×C₇	—	8	2	no	—
46	C₂×C₄×C₇	#127	1	1	yes	characteristic
47	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
48	C₂×Dic₇	—	8	2	no	—
49	Dic₁₄	—	8	4	no	—
50	C₇×C₈	Q₃₂	1	8	yes	complemented
51	C₇×C₈	C₂×D₈	1	4	yes	—
52	C₂×Q₁₆	—	2	4	no	—
53	C₂×Q₁₆	—	14	4	no	—
54	C₄⋊Q₈	—	2	2	no	—
55	C₄⋊Q₈	—	14	2	no	—
56	Q₃₂	—	28	16	no	—
57	C₂×C₁₆	—	7	4	no	—
58	#107	—	4	4	no	—
59	#107	—	28	4	no	—
60	C₄×C₈	C₂²⋊D₇	1	2	yes	—
61	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
62	C₂²×C₇	#493	1	1	yes	characteristic
63	C₄×C₇	—	16	2	no	—
64	C₄×C₇	#493	1	2	yes	—
65	C₄×C₇	C₂×Q₃₂	1	4	yes	—
66	Dic₇	—	16	2	no	—
67	C₂×Q₈	—	4	2	no	—
68	C₂×Q₈	—	28	2	no	—
69	Q₁₆	—	4	8	no	—
70	Q₁₆	—	28	8	no	—
71	C₂×C₈	#1141	1	4	yes	—
72	C₂×C₈	D₄×D₇	1	2	yes	—
73	C₄⋊C₄	—	4	4	no	—
74	C₄⋊C₄	—	28	4	no	—
75	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
76	C₁₆	—	7	8	no	—
77	C₂×C₇	#1712	1	1	yes	characteristic
78	C₂×C₇	#2247	1	2	yes	—
79	Q₈	—	8	4	no	—
80	Q₈	—	56	4	no	—
81	C₂×C₄	—	8	2	no	—
82	C₂×C₄	—	56	2	no	—
83	C₂×C₄	C₂×#1141	1	2	yes	—
84	C₂×C₄	#6503	1	1	yes	characteristic
85	C₈	#6403	1	8	yes	complemented
86	C₈	D₇×D₈	1	4	yes	—
87	C₇	#12516	1	1	yes	characteristic, complemented, Sylow
88	C₂²	#89845	1	1	yes	characteristic
89	C₄	—	16	2	no	—
90	C₄	—	112	2	no	—
91	C₄	#89923	1	2	yes	—
92	C₄	C₂×#6403	1	4	yes	—
93	C₂	#11729030	1	1	yes	characteristic
94	C₂	#11729066	1	2	yes	—

	subgroup	G/H
center	C₂²	#89845
commutator, cosocle	C₄×C₇×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₇×#1713	C₂
socle, prime radical	C₂²×C₇	#493
hypercenter	#1713	D₇
hypocenter	C₇	#12516
LCS	G ▷ C₄×C₇×C₈ ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

Complements

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

N	H ≅ G/N	type
50. C₇×C₈	Q₃₂	semidirect
85. C₈	#6403	semidirect
87. C₇	#12516	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
#11729066	2
#11725819	2
C₈×C₇⋊C₁₆	2
#12516	7