Subgroups — #11726837

Order 896.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#89471	C₂	1	1	yes	maximal, characteristic
2	#89455	C₂	1	1	yes	maximal, characteristic
3	C₇×#346	C₂	1	1	yes	maximal, characteristic
4	#89298	C₂	1	4	yes	maximal
5	C₂×Dic₂₈	—	2	1	no	—
6	C₂×Dic₂₈	C₂²	1	2	yes	—
7	#6433	—	2	1	no	—
8	#6432	—	2	1	no	—
9	C₂×C₇×C₁₆	C₂²	1	2	yes	—
10	C₄×C₇×C₈	C₂²	1	1	yes	characteristic
11	#6392	C₂²	1	2	yes	—
12	#6391	—	2	1	no	—
13	#2245	—	7	1	no	maximal, Sylow
14	C₂×Dic₁₄	—	2	2	no	—
15	C₂×C₇×C₈	D₄	1	1	yes	characteristic
16	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
17	C₄²×C₇	D₄	1	1	yes	characteristic
18	C₄⋊Dic₇	—	2	2	no	—
19	C₄⋊Dic₇	—	4	2	no	—
20	Dic₇⋊C₄	—	4	2	no	—
21	Dic₂₈	—	2	4	no	—
22	Dic₂₈	—	4	2	no	—
23	C₇×C₁₆	—	2	2	no	—
24	#499	—	7	1	no	—
25	#494	—	7	1	no	—
26	#358	—	7	4	no	—
27	#346	D₇	1	1	yes	characteristic
28	C₂×C₄×C₇	D₈	1	2	yes	—
29	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
30	C₂×Dic₇	—	4	2	no	—
31	C₂×Dic₇	—	4	2	no	—
32	Dic₁₄	—	4	2	no	—
33	Dic₁₄	—	4	2	no	—
34	C₇×C₈	SD₁₆	1	2	yes	—
35	C₇×C₈	#41	1	2	yes	—
36	C₂×Q₁₆	—	7	2	no	—
37	C₂×Q₁₆	—	14	1	no	—
38	C₄⋊Q₈	—	14	1	no	—
39	#125	—	14	1	no	—
40	C₂×C₁₆	C₂×D₇	1	2	yes	—
41	#107	—	7	2	no	—
42	#106	—	14	1	no	—
43	C₄×C₈	C₂×D₇	1	1	yes	characteristic
44	C₂²×C₇	C₂×D₈	1	1	yes	characteristic
45	C₄×C₇	—	2	2	no	—
46	C₄×C₇	#124	1	1	yes	characteristic
47	C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
48	Dic₇	—	8	2	no	—
49	Dic₇	—	8	2	no	—
50	C₂×Q₈	—	14	2	no	—
51	Q₁₆	—	14	4	no	—
52	Q₁₆	—	28	2	no	—
53	C₂×C₈	D₂₈	1	1	yes	characteristic
54	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
55	C₄⋊C₄	—	14	2	no	—
56	C₄⋊C₄	—	28	2	no	—
57	C₄⋊C₄	—	28	2	no	—
58	C₄²	D₂₈	1	1	yes	characteristic
59	C₁₆	—	2	2	no	—
60	C₂×C₇	#486	1	1	yes	characteristic
61	C₂×C₇	#510	1	2	yes	—
62	Q₈	—	28	2	no	—
63	Q₈	—	28	2	no	—
64	C₂×C₄	—	28	2	no	—
65	C₂×C₄	—	28	2	no	—
66	C₂×C₄	D₅₆	1	2	yes	—
67	C₂×C₄	C₂×D₂₈	1	1	yes	characteristic
68	C₈	#1132	1	2	yes	—
69	C₈	#1157	1	2	yes	—
70	C₇	#2245	1	1	yes	characteristic, complemented, Sylow
71	C₂²	C₂×D₅₆	1	1	yes	characteristic
72	C₄	—	2	2	no	—
73	C₄	—	56	2	no	—
74	C₄	—	56	2	no	—
75	C₄	#6438	1	1	yes	characteristic
76	C₄	C₂×#1132	1	1	yes	characteristic
77	C₂	#89466	1	1	yes	characteristic
78	C₂	#89681	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₅₆
commutator, cosocle	C₂×C₇×C₈	C₂³
Frattini	C₂×C₈	C₂²×D₇
Fitting	C₇×#346	C₂
socle, prime radical	C₂²×C₇	C₂×D₈
hypercenter	#346	D₇
hypocenter	C₇	#2245
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
70. C₇	#2245	semidirect

Maximal subgroups

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

subgroup	index
#89471	2
#89455	2
C₇×#346	2
#89298	2
#2245	7