Subgroups — #13092277

Order 1472.

234 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊Dic₂₃	C₂	1	2	yes	maximal
3	C₂×C₄⋊Dic₂₃	C₂	1	2	yes	maximal, complemented
4	C₂₃⋊#95	C₂	1	2	yes	maximal, complemented
5	C₂⁴×C₂₃	C₂²	1	1	yes	characteristic
6	C₂²×C₄×C₂₃	—	2	4	no	—
7	C₂²×C₄×C₂₃	C₄	1	4	yes	complemented
8	C₂²×C₄×C₂₃	C₂²	1	1	yes	characteristic
9	C₂²×C₄×C₂₃	C₂²	1	1	yes	characteristic
10	C₂²×Dic₂₃	C₂²	1	4	yes	—
11	C₂²⋊Dic₂₃	—	2	4	no	—
12	C₄⋊Dic₂₃	—	2	4	no	—
13	C₂³×C₂₃	—	2	2	no	—
14	C₂³×C₂₃	—	2	2	no	—
15	C₂³×C₂₃	C₂×C₄	1	2	yes	—
16	C₂³×C₂₃	D₄	1	2	yes	—
17	C₂³×C₂₃	Q₈	1	2	yes	—
18	C₂³×C₂₃	C₂³	1	1	yes	characteristic
19	C₂×C₄×C₂₃	—	2	2	no	—
20	C₂×C₄×C₂₃	—	2	8	no	—
21	C₂×C₄×C₂₃	C₂×C₄	1	2	yes	—
22	C₂×C₄×C₂₃	C₂×C₄	1	2	yes	—
23	C₂×C₄×C₂₃	D₄	1	4	yes	—
24	C₂×Dic₂₃	—	2	4	no	—
25	C₂×Dic₂₃	—	2	4	no	—
26	C₂×Dic₂₃	—	2	4	no	—
27	C₂²×C₂₃	—	2	2	no	—
28	C₂²×C₂₃	—	2	2	no	—
29	C₂²×C₂₃	—	2	4	no	—
30	C₂²×C₂₃	—	2	4	no	—
31	C₂²×C₂₃	C₄⋊C₄	1	4	yes	complemented
32	C₂²×C₂₃	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₂₃	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₂₃	C₂×D₄	1	2	yes	—
35	C₂²×C₂₃	C₂×Q₈	1	1	yes	characteristic
36	C₂²×C₂₃	#41	1	2	yes	—
37	C₄×C₂₃	—	2	2	no	—
38	C₄×C₂₃	C₂²⋊C₄	1	4	yes	complemented
39	Dic₂₃	—	4	4	no	—
40	#380	—	23	1	no	maximal, Sylow
41	C₂×C₂₃	—	2	4	no	—
42	C₂×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₂₃	#117	1	1	yes	characteristic
45	C₂×C₂₃	#121	1	2	yes	—
46	C₂×C₂₃	C₂²⋊Q₈	1	2	yes	—
47	C₂³×C₄	D₂₃	1	1	yes	characteristic
48	C₂×C₄⋊C₄	—	23	2	no	—
49	C₂×C₂²⋊C₄	—	23	2	no	—
50	#95	—	23	2	no	—
51	C₂₃	#380	1	1	yes	characteristic, complemented, Sylow
52	C₂⁴	C₂×D₂₃	1	1	yes	characteristic
53	C₂²×C₄	—	2	4	no	—
54	C₂²×C₄	—	23	4	no	—
55	C₂²×C₄	Dic₂₃	1	4	yes	complemented
56	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
57	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
58	C₄⋊C₄	—	46	4	no	—
59	C₂²⋊C₄	—	46	4	no	—
60	C₂³	—	2	2	no	—
61	C₂³	—	2	2	no	—
62	C₂³	Dic₄₆	1	2	yes	—
63	C₂³	D₉₂	1	2	yes	—
64	C₂³	C₂×Dic₂₃	1	2	yes	—
65	C₂³	C₂²×D₂₃	1	1	yes	characteristic
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	—	2	8	no	—
68	C₂×C₄	—	46	4	no	—
69	C₂×C₄	—	46	4	no	—
70	C₂×C₄	—	46	4	no	—
71	C₂×C₄	C₂×Dic₂₃	1	2	yes	—
72	C₂×C₄	C₂×Dic₂₃	1	2	yes	—
73	C₂×C₄	C₂²⋊D₂₃	1	4	yes	—
74	C₂²	—	2	4	no	—
75	C₂²	—	2	2	no	—
76	C₂²	—	2	4	no	—
77	C₂²	—	2	2	no	—
78	C₂²	C₄⋊Dic₂₃	1	4	yes	complemented
79	C₂²	C₂×Dic₄₆	1	1	yes	characteristic
80	C₂²	C₂×D₉₂	1	1	yes	characteristic
81	C₂²	#67288	1	2	yes	—
82	C₂²	C₂²×Dic₂₃	1	1	yes	characteristic
83	C₂²	C₂×C₂²⋊D₂₃	1	2	yes	—
84	C₄	—	2	2	no	—
85	C₄	—	92	4	no	—
86	C₄	C₂²⋊Dic₂₃	1	4	yes	complemented
87	C₂	—	2	4	no	—
88	C₂	#10625095	1	2	yes	—
89	C₂	C₂×C₄⋊Dic₂₃	1	1	yes	characteristic
90	C₂	#10625097	1	1	yes	characteristic
91	C₂	#10625101	1	2	yes	—
92	C₂	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₂₃
commutator	C₂²×C₂₃	C₂²×C₄
Fitting	C₂³×C₄×C₂₃	C₂
socle, cosocle	C₂³×C₂₃	C₂³
prime radical	C₂⁴×C₂₃	C₂²
hypercenter	C₂³×C₄	D₂₃
hypocenter	C₂₃	#380
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
3. C₂×C₄⋊Dic₂₃	C₂	semidirect
4. C₂₃⋊#95	C₂	semidirect
7. C₂²×C₄×C₂₃	C₄	semidirect
31. C₂²×C₂₃	C₄⋊C₄	semidirect
38. C₄×C₂₃	C₂²⋊C₄	semidirect
51. C₂₃	#380	semidirect
55. C₂²×C₄	Dic₂₃	semidirect
78. C₂²	C₄⋊Dic₂₃	semidirect
86. C₄	C₂²⋊Dic₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₄×C₂₃	2
C₂×C₂²⋊Dic₂₃	2
C₂×C₄⋊Dic₂₃	2
C₂₃⋊#95	2
#380	23