Subgroups — C₂²×#1144 #90500

Order 448.

258 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×Dic₁₄	C₂	1	1	yes	maximal, characteristic
3	C₂×#1144	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×C₇×Q₈	—	2	3	no	—
6	C₂×C₇×Q₈	C₂²	1	6	yes	—
7	C₂²×C₄×C₇	—	2	1	no	—
8	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
9	C₂²×Dic₇	—	2	1	no	—
10	C₂×Dic₁₄	—	2	3	no	—
11	C₂×Dic₁₄	C₂²	1	6	yes	—
12	#1144	C₂²	1	16	yes	direct product factor
13	C₂×C₇⋊C₈	C₂²	1	6	yes	—
14	C₂²×Q₁₆	—	7	1	no	maximal, Sylow
15	C₂³×C₇	D₄	1	1	yes	characteristic
16	C₇×Q₈	—	2	6	no	—
17	C₇×Q₈	C₂³	1	4	yes	—
18	C₂×C₄×C₇	—	2	6	no	—
19	C₂×C₄×C₇	D₄	1	3	yes	—
20	C₂×C₄×C₇	C₂³	1	3	yes	—
21	C₂×Dic₇	—	2	6	no	—
22	Dic₁₄	—	2	6	no	—
23	Dic₁₄	C₂³	1	4	yes	—
24	C₇⋊C₈	C₂³	1	4	yes	—
25	C₂²×Q₈	—	7	1	no	—
26	C₂²×Q₈	D₇	1	1	yes	characteristic
27	C₂×Q₁₆	—	7	12	no	—
28	C₂²×C₈	—	7	1	no	—
29	C₂²×C₇	Q₁₆	1	4	yes	complemented
30	C₂²×C₇	C₂×D₄	1	3	yes	—
31	C₄×C₇	—	2	4	no	—
32	C₄×C₇	C₂×D₄	1	3	yes	—
33	C₄×C₇	C₂⁴	1	1	yes	characteristic
34	Dic₇	—	2	4	no	—
35	C₂×Q₈	—	2	3	no	—
36	C₂×Q₈	—	7	6	no	—
37	C₂×Q₈	—	14	3	no	—
38	C₂×Q₈	C₂×D₇	1	6	yes	—
39	C₂²×C₄	—	2	1	no	—
40	C₂²×C₄	—	14	1	no	—
41	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
42	Q₁₆	—	7	16	no	—
43	C₂×C₈	—	7	6	no	—
44	C₂×C₇	C₂×Q₁₆	1	6	yes	complemented
45	C₂×C₇	C₂²×D₄	1	1	yes	characteristic
46	C₂³	C₂²⋊D₇	1	1	yes	characteristic
47	Q₈	—	2	6	no	—
48	Q₈	—	7	4	no	—
49	Q₈	—	14	6	no	—
50	Q₈	C₂²×D₇	1	4	yes	—
51	C₂×C₄	—	2	6	no	—
52	C₂×C₄	—	14	6	no	—
53	C₂×C₄	C₂²⋊D₇	1	3	yes	—
54	C₂×C₄	C₂²×D₇	1	3	yes	—
55	C₈	—	7	4	no	—
56	C₇	C₂²×Q₁₆	1	1	yes	characteristic, complemented, Sylow
57	C₂²	#1144	1	4	yes	direct product factor
58	C₂²	C₂×C₂²⋊D₇	1	3	yes	—
59	C₄	—	2	4	no	—
60	C₄	—	14	4	no	—
61	C₄	C₂×C₂²⋊D₇	1	3	yes	—
62	C₄	C₂³×D₇	1	1	yes	characteristic
63	C₂	C₂×#1144	1	6	yes	direct product factor
64	C₂	C₂²×C₂²⋊D₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₂×#1144	C₂	direct (N × H)
12. #1144	C₂²	direct (N × H)
29. C₂²×C₇	Q₁₆	semidirect
44. C₂×C₇	C₂×Q₁₆	semidirect
56. C₇	C₂²×Q₁₆	semidirect
57. C₂²	#1144	direct (N × H)
63. C₂	C₂×#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
C₂²×C₇×Q₈	2
C₂²×Dic₁₄	2
C₂×#1144	2
C₂²×C₇⋊C₈	2
C₂²×Q₁₆	7

Subgroups — C22×#1144 #90500

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#1144 #90500