Subgroups — C₂²×#67275 #13093193

Order 1472.

258 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×Dic₄₆	C₂	1	1	yes	maximal, characteristic
3	C₂×#67275	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₈×C₂₃	—	2	3	no	—
6	C₂×Q₈×C₂₃	C₂²	1	6	yes	—
7	C₂²×C₄×C₂₃	—	2	1	no	—
8	C₂²×C₄×C₂₃	C₂²	1	1	yes	characteristic
9	C₂²×Dic₂₃	—	2	1	no	—
10	C₂×Dic₄₆	—	2	3	no	—
11	C₂×Dic₄₆	C₂²	1	6	yes	—
12	#67275	C₂²	1	16	yes	direct product factor
13	C₂×C₂₃⋊C₈	C₂²	1	6	yes	—
14	C₂³×C₂₃	D₄	1	1	yes	characteristic
15	Q₈×C₂₃	—	2	6	no	—
16	Q₈×C₂₃	C₂³	1	4	yes	—
17	C₂×C₄×C₂₃	—	2	6	no	—
18	C₂×C₄×C₂₃	D₄	1	3	yes	—
19	C₂×C₄×C₂₃	C₂³	1	3	yes	—
20	C₂×Dic₂₃	—	2	6	no	—
21	Dic₄₆	—	2	6	no	—
22	Dic₄₆	C₂³	1	4	yes	—
23	C₂₃⋊C₈	C₂³	1	4	yes	—
24	C₂²×C₂₃	Q₁₆	1	4	yes	complemented
25	C₂²×C₂₃	C₂×D₄	1	3	yes	—
26	C₄×C₂₃	—	2	4	no	—
27	C₄×C₂₃	C₂×D₄	1	3	yes	—
28	C₄×C₂₃	C₂⁴	1	1	yes	characteristic
29	Dic₂₃	—	2	4	no	—
30	C₂²×Q₁₆	—	23	1	no	maximal, Sylow
31	C₂×C₂₃	C₂×Q₁₆	1	6	yes	complemented
32	C₂×C₂₃	C₂²×D₄	1	1	yes	characteristic
33	C₂²×Q₈	—	23	1	no	—
34	C₂²×Q₈	D₂₃	1	1	yes	characteristic
35	C₂×Q₁₆	—	23	12	no	—
36	C₂²×C₈	—	23	1	no	—
37	C₂₃	C₂²×Q₁₆	1	1	yes	characteristic, complemented, Sylow
38	C₂×Q₈	—	2	3	no	—
39	C₂×Q₈	—	23	6	no	—
40	C₂×Q₈	—	46	3	no	—
41	C₂×Q₈	C₂×D₂₃	1	6	yes	—
42	C₂²×C₄	—	2	1	no	—
43	C₂²×C₄	—	46	1	no	—
44	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
45	Q₁₆	—	23	16	no	—
46	C₂×C₈	—	23	6	no	—
47	C₂³	C₂²⋊D₂₃	1	1	yes	characteristic
48	Q₈	—	2	6	no	—
49	Q₈	—	23	4	no	—
50	Q₈	—	46	6	no	—
51	Q₈	C₂²×D₂₃	1	4	yes	—
52	C₂×C₄	—	2	6	no	—
53	C₂×C₄	—	46	6	no	—
54	C₂×C₄	C₂²⋊D₂₃	1	3	yes	—
55	C₂×C₄	C₂²×D₂₃	1	3	yes	—
56	C₈	—	23	4	no	—
57	C₂²	#67275	1	4	yes	direct product factor
58	C₂²	C₂×C₂²⋊D₂₃	1	3	yes	—
59	C₄	—	2	4	no	—
60	C₄	—	46	4	no	—
61	C₄	C₂×C₂²⋊D₂₃	1	3	yes	—
62	C₄	C₂³×D₂₃	1	1	yes	characteristic
63	C₂	C₂×#67275	1	6	yes	direct product factor
64	C₂	C₂²×C₂²⋊D₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊D₂₃
commutator, cosocle	C₄×C₂₃	C₂⁴
Frattini	C₄	C₂³×D₂₃
Fitting	C₂²×Q₈×C₂₃	C₂
socle, prime radical	C₂³×C₂₃	D₄
hypercenter	C₂²×Q₈	D₂₃
hypocenter	C₂₃	C₂²×Q₁₆
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
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×#67275	C₂	direct (N × H)
12. #67275	C₂²	direct (N × H)
24. C₂²×C₂₃	Q₁₆	semidirect
31. C₂×C₂₃	C₂×Q₁₆	semidirect
37. C₂₃	C₂²×Q₁₆	semidirect
57. C₂²	#67275	direct (N × H)
63. C₂	C₂×#67275	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×Q₈×C₂₃	2
C₂²×Dic₄₆	2
C₂×#67275	2
C₂²×C₂₃⋊C₈	2
C₂²×Q₁₆	23

Subgroups — C22×#67275 #13093193

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#67275 #13093193