Subgroups — C₂³×#1144 #11744310

Order 896.

1464 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₇×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂³×Dic₁₄	C₂	1	1	yes	maximal, characteristic
3	C₂²×#1144	C₂	1	28	yes	maximal, direct product factor
4	C₂³×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₇×Q₈	—	2	7	no	—
6	C₂²×C₇×Q₈	C₂²	1	14	yes	—
7	C₂³×C₄×C₇	—	2	1	no	—
8	C₂³×C₄×C₇	C₂²	1	1	yes	characteristic
9	C₂³×Dic₇	—	2	1	no	—
10	C₂²×Dic₁₄	—	2	7	no	—
11	C₂²×Dic₁₄	C₂²	1	14	yes	—
12	C₂×#1144	C₂²	1	112	yes	direct product factor
13	C₂²×C₇⋊C₈	C₂²	1	14	yes	—
14	C₂³×Q₁₆	—	7	1	no	maximal, Sylow
15	C₂⁴×C₇	D₄	1	1	yes	characteristic
16	C₂×C₇×Q₈	—	2	42	no	—
17	C₂×C₇×Q₈	C₂³	1	28	yes	—
18	C₂²×C₄×C₇	—	2	14	no	—
19	C₂²×C₄×C₇	D₄	1	7	yes	—
20	C₂²×C₄×C₇	C₂³	1	7	yes	—
21	C₂²×Dic₇	—	2	14	no	—
22	C₂×Dic₁₄	—	2	42	no	—
23	C₂×Dic₁₄	C₂³	1	28	yes	—
24	#1144	C₂³	1	64	yes	direct product factor
25	C₂×C₇⋊C₈	C₂³	1	28	yes	—
26	C₂³×Q₈	—	7	1	no	—
27	C₂³×Q₈	D₇	1	1	yes	characteristic
28	C₂²×Q₁₆	—	7	28	no	—
29	C₂³×C₈	—	7	1	no	—
30	C₂³×C₇	Q₁₆	1	8	yes	complemented
31	C₂³×C₇	C₂×D₄	1	7	yes	—
32	C₇×Q₈	—	2	28	no	—
33	C₇×Q₈	C₂⁴	1	8	yes	—
34	C₂×C₄×C₇	—	2	28	no	—
35	C₂×C₄×C₇	C₂×D₄	1	21	yes	—
36	C₂×C₄×C₇	C₂⁴	1	7	yes	—
37	C₂×Dic₇	—	2	28	no	—
38	Dic₁₄	—	2	28	no	—
39	Dic₁₄	C₂⁴	1	8	yes	—
40	C₇⋊C₈	C₂⁴	1	8	yes	—
41	C₂²×Q₈	—	2	7	no	—
42	C₂²×Q₈	—	7	14	no	—
43	C₂²×Q₈	—	14	7	no	—
44	C₂²×Q₈	C₂×D₇	1	14	yes	—
45	C₂³×C₄	—	2	1	no	—
46	C₂³×C₄	—	14	1	no	—
47	C₂³×C₄	C₂×D₇	1	1	yes	characteristic
48	C₂×Q₁₆	—	7	112	no	—
49	C₂²×C₈	—	7	14	no	—
50	C₂²×C₇	C₂×Q₁₆	1	28	yes	complemented
51	C₂²×C₇	C₂²×D₄	1	7	yes	—
52	C₄×C₇	—	2	8	no	—
53	C₄×C₇	C₂²×D₄	1	7	yes	—
54	C₄×C₇	C₂⁵	1	1	yes	characteristic
55	Dic₇	—	2	8	no	—
56	C₂⁴	C₂²⋊D₇	1	1	yes	characteristic
57	C₂×Q₈	—	2	42	no	—
58	C₂×Q₈	—	7	28	no	—
59	C₂×Q₈	—	14	42	no	—
60	C₂×Q₈	C₂²×D₇	1	28	yes	—
61	C₂²×C₄	—	2	14	no	—
62	C₂²×C₄	—	14	14	no	—
63	C₂²×C₄	C₂²⋊D₇	1	7	yes	—
64	C₂²×C₄	C₂²×D₇	1	7	yes	—
65	Q₁₆	—	7	64	no	—
66	C₂×C₈	—	7	28	no	—
67	C₂×C₇	C₂²×Q₁₆	1	14	yes	complemented
68	C₂×C₇	C₂³×D₄	1	1	yes	characteristic
69	C₂³	#1144	1	8	yes	direct product factor
70	C₂³	C₂×C₂²⋊D₇	1	7	yes	—
71	Q₈	—	2	28	no	—
72	Q₈	—	7	8	no	—
73	Q₈	—	14	28	no	—
74	Q₈	C₂³×D₇	1	8	yes	—
75	C₂×C₄	—	2	28	no	—
76	C₂×C₄	—	14	28	no	—
77	C₂×C₄	C₂×C₂²⋊D₇	1	21	yes	—
78	C₂×C₄	C₂³×D₇	1	7	yes	—
79	C₈	—	7	8	no	—
80	C₇	C₂³×Q₁₆	1	1	yes	characteristic, complemented, Sylow
81	C₂²	C₂×#1144	1	28	yes	direct product factor
82	C₂²	C₂²×C₂²⋊D₇	1	7	yes	—
83	C₄	—	2	8	no	—
84	C₄	—	14	8	no	—
85	C₄	C₂²×C₂²⋊D₇	1	7	yes	—
86	C₄	C₂⁴×D₇	1	1	yes	characteristic
87	C₂	C₂²×#1144	1	14	yes	direct product factor
88	C₂	C₂³×C₂²⋊D₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²⋊D₇
commutator, cosocle	C₄×C₇	C₂⁵
Frattini	C₄	C₂⁴×D₇
Fitting	C₂³×C₇×Q₈	C₂
socle, prime radical	C₂⁴×C₇	D₄
hypercenter	C₂³×Q₈	D₇
hypocenter	C₇	C₂³×Q₁₆
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×Q₈

Complements

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

N	H ≅ G/N	type
3. C₂²×#1144	C₂	direct (N × H)
12. C₂×#1144	C₂²	direct (N × H)
24. #1144	C₂³	direct (N × H)
30. C₂³×C₇	Q₁₆	semidirect
50. C₂²×C₇	C₂×Q₁₆	semidirect
67. C₂×C₇	C₂²×Q₁₆	semidirect
69. C₂³	#1144	direct (N × H)
80. C₇	C₂³×Q₁₆	semidirect
81. C₂²	C₂×#1144	direct (N × H)
87. C₂	C₂²×#1144	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₇×Q₈	2
C₂³×Dic₁₄	2
C₂²×#1144	2
C₂³×C₇⋊C₈	2
C₂³×Q₁₆	7

Subgroups — C23×#1144 #11744310

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#1144 #11744310