Subgroups — C₂×#89317 #11729248

Order 896.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×Q₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₅₆	C₂	1	1	yes	maximal, characteristic
3	#89317	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₇⋊C₃₂	C₂	1	1	yes	maximal, characteristic
5	C₂×C₇×Q₁₆	—	2	1	no	—
6	C₂×Dic₂₈	—	2	1	no	—
7	C₇×Q₃₂	—	2	1	no	—
8	C₇×Q₃₂	C₂²	1	2	yes	—
9	C₂×C₇×C₁₆	C₂²	1	1	yes	characteristic
10	Dic₅₆	—	2	1	no	—
11	Dic₅₆	C₂²	1	2	yes	—
12	C₇⋊C₃₂	C₂²	1	2	yes	—
13	C₂×Q₆₄	—	7	1	no	maximal, Sylow
14	C₂×C₇×Q₈	—	4	1	no	—
15	C₂×Dic₁₄	—	4	1	no	—
16	C₇×Q₁₆	—	2	2	no	—
17	C₇×Q₁₆	—	4	1	no	—
18	C₂×C₇×C₈	D₄	1	1	yes	characteristic
19	Dic₂₈	—	2	2	no	—
20	Dic₂₈	—	4	1	no	—
21	C₇×C₁₆	D₄	1	1	yes	characteristic
22	C₇×C₁₆	C₂³	1	1	yes	characteristic
23	C₂×Q₃₂	—	7	1	no	—
24	C₂×Q₃₂	D₇	1	1	yes	characteristic
25	Q₆₄	—	7	4	no	—
26	C₂×C₃₂	—	7	1	no	—
27	C₇×Q₈	—	4	2	no	—
28	C₇×Q₈	—	8	1	no	—
29	C₂×C₄×C₇	—	8	1	no	—
30	C₂×C₄×C₇	D₈	1	1	yes	characteristic
31	C₂×Dic₇	—	8	1	no	—
32	Dic₁₄	—	4	2	no	—
33	Dic₁₄	—	8	1	no	—
34	C₇×C₈	D₈	1	1	yes	characteristic
35	C₇×C₈	C₂×D₄	1	1	yes	characteristic
36	C₂×Q₁₆	—	2	1	no	—
37	C₂×Q₁₆	—	14	1	no	—
38	Q₃₂	—	2	1	no	—
39	Q₃₂	—	7	2	no	—
40	Q₃₂	—	14	1	no	—
41	Q₃₂	C₂×D₇	1	2	yes	—
42	C₂×C₁₆	C₂×D₇	1	1	yes	characteristic
43	C₃₂	—	7	2	no	—
44	C₂²×C₇	D₁₆	1	1	yes	characteristic
45	C₄×C₇	—	8	2	no	—
46	C₄×C₇	D₁₆	1	1	yes	characteristic
47	C₄×C₇	C₂×D₈	1	1	yes	characteristic
48	Dic₇	—	8	2	no	—
49	C₂×Q₈	—	4	1	no	—
50	C₂×Q₈	—	28	1	no	—
51	Q₁₆	—	2	2	no	—
52	Q₁₆	—	4	1	no	—
53	Q₁₆	—	14	2	no	—
54	Q₁₆	—	28	1	no	—
55	C₂×C₈	C₂²⋊D₇	1	1	yes	characteristic
56	C₁₆	C₂²⋊D₇	1	1	yes	characteristic
57	C₁₆	C₂²×D₇	1	1	yes	characteristic
58	C₂×C₇	Q₆₄	1	2	yes	complemented
59	C₂×C₇	C₂×D₁₆	1	1	yes	characteristic
60	Q₈	—	4	2	no	—
61	Q₈	—	8	1	no	—
62	Q₈	—	28	2	no	—
63	Q₈	—	56	1	no	—
64	C₂×C₄	—	8	1	no	—
65	C₂×C₄	—	56	1	no	—
66	C₂×C₄	#1141	1	1	yes	characteristic
67	C₈	#1141	1	1	yes	characteristic
68	C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
69	C₇	C₂×Q₆₄	1	1	yes	characteristic, complemented, Sylow
70	C₂²	#6400	1	1	yes	characteristic
71	C₄	—	8	2	no	—
72	C₄	—	56	2	no	—
73	C₄	#6400	1	1	yes	characteristic
74	C₄	C₂×#1141	1	1	yes	characteristic
75	C₂	#89317	1	2	yes	direct product factor
76	C₂	C₂×#6400	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#6400
commutator, cosocle	C₇×C₁₆	C₂³
Frattini	C₁₆	C₂²×D₇
Fitting	C₂×C₇×Q₃₂	C₂
socle, prime radical	C₂²×C₇	D₁₆
hypercenter	C₂×Q₃₂	D₇
hypocenter	C₇	C₂×Q₆₄
LCS	G ▷ C₇×C₁₆ ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×Q₃₂

Complements

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

N	H ≅ G/N	type
3. #89317	C₂	direct (N × H)
58. C₂×C₇	Q₆₄	semidirect
69. C₇	C₂×Q₆₄	semidirect
75. C₂	#89317	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇×Q₃₂	2
C₂×Dic₅₆	2
#89317	2
C₂×C₇⋊C₃₂	2
C₂×Q₆₄	7

Subgroups — C2×#89317 #11729248

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#89317 #11729248