Subgroups — #15671700

Order 2944.

214 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#67267	C₂	1	2	yes	maximal
2	#13092139	C₂	1	2	yes	maximal, complemented
3	#13092138	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂×M₁₆×C₂₃	C₄	1	2	yes	—
7	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×#67267	C₄	1	4	yes	—
9	C₂₃×#97	C₄	1	2	yes	—
10	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
12	#10624985	C₄	1	4	yes	—
13	C₄×C₂₃⋊C₈	C₂²	1	2	yes	—
14	C₄×C₂₃⋊C₈	C₂²	1	2	yes	—
15	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
16	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
17	M₁₆×C₂₃	—	2	4	no	—
18	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
19	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
20	C₄²×C₂₃	C₂×C₄	1	2	yes	—
21	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₂₃	C₂³	1	1	yes	characteristic
23	#67267	—	2	8	no	—
24	C₂×C₂₃⋊C₈	C₂×C₄	1	4	yes	—
25	C₂×C₂₃⋊C₈	C₂×C₄	1	4	yes	—
26	C₂³×C₂₃	C₄²	1	1	yes	characteristic
27	C₂×C₄×C₂₃	—	2	4	no	—
28	C₂×C₄×C₂₃	—	2	2	no	—
29	C₂×C₄×C₂₃	C₄²	1	1	yes	characteristic
30	C₂×C₄×C₂₃	C₄²	1	2	yes	—
31	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
32	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
33	C₈×C₂₃	—	2	2	no	—
34	C₈×C₂₃	M₁₆	1	4	yes	complemented
35	C₂₃⋊C₈	—	2	4	no	—
36	C₂₃⋊C₈	M₁₆	1	8	yes	complemented
37	#1455	—	23	1	no	maximal, Sylow
38	C₂²×C₂₃	—	2	2	no	—
39	C₂²×C₂₃	—	2	1	no	—
40	C₂²×C₂₃	C₂×C₄²	1	1	yes	characteristic
41	C₄×C₂₃	—	2	2	no	—
42	C₄×C₂₃	—	2	1	no	—
43	C₄×C₂₃	C₂×M₁₆	1	4	yes	—
44	C₄×C₂₃	C₂×M₁₆	1	2	yes	—
45	C₄×M₁₆	—	23	2	no	—
46	C₄×M₁₆	D₂₃	1	1	yes	characteristic
47	#322	—	23	1	no	—
48	#322	—	23	2	no	—
49	C₈²	—	23	1	no	—
50	C₂×C₂₃	—	4	1	no	—
51	C₂×C₂₃	C₄×M₁₆	1	2	yes	—
52	C₂×C₂₃	C₄×M₁₆	1	1	yes	characteristic
53	C₂×M₁₆	—	23	4	no	—
54	C₂×M₁₆	Dic₂₃	1	2	yes	—
55	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
56	#97	—	23	4	no	—
57	#97	Dic₂₃	1	2	yes	—
58	C₄×C₈	—	23	2	no	—
59	C₄×C₈	—	23	2	no	—
60	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
61	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
62	C₂₃	#1455	1	1	yes	characteristic, complemented, Sylow
63	C₂²×C₄	C₄×D₂₃	1	2	yes	—
64	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
65	M₁₆	—	2	4	no	—
66	M₁₆	—	46	8	no	—
67	C₂×C₈	—	23	4	no	—
68	C₂×C₈	—	23	4	no	—
69	C₂×C₈	C₂×Dic₂₃	1	2	yes	—
70	C₂×C₈	C₂×Dic₂₃	1	2	yes	—
71	C₄²	C₄×D₂₃	1	2	yes	—
72	C₄²	C₂×Dic₂₃	1	1	yes	characteristic
73	C₄²	C₂²×D₂₃	1	1	yes	characteristic
74	C₂³	C₄×Dic₂₃	1	1	yes	characteristic
75	C₂×C₄	—	2	4	no	—
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	C₄×Dic₂₃	1	1	yes	characteristic
78	C₂×C₄	C₄×Dic₂₃	1	2	yes	—
79	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
80	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
81	C₈	—	2	2	no	—
82	C₈	—	23	8	no	—
83	C₈	—	46	4	no	—
84	C₈	#67267	1	4	yes	complemented
85	C₂²	—	2	2	no	—
86	C₂²	—	2	1	no	—
87	C₂²	C₂×C₄×Dic₂₃	1	1	yes	characteristic
88	C₄	—	2	2	no	—
89	C₄	—	2	1	no	—
90	C₄	M₁₆×D₂₃	1	4	yes	—
91	C₄	C₂×#67267	1	2	yes	—
92	C₂	—	4	1	no	—
93	C₂	M₁₆×Dic₂₃	1	2	yes	—
94	C₂	C₄×#67267	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₂₃
commutator, socle	C₂²×C₂₃	C₂×C₄²
Fitting	C₄×M₁₆×C₂₃	C₂
prime radical	C₂³×C₂₃	C₄²
cosocle	C₄²×C₂₃	C₂³
hypercenter	C₄×M₁₆	D₂₃
hypocenter	C₂₃	#1455
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₄² ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #13092139	C₂	semidirect
3. #13092138	C₂	semidirect
4. C₈×C₂₃⋊C₈	C₂	semidirect
34. C₈×C₂₃	M₁₆	semidirect
36. C₂₃⋊C₈	M₁₆	semidirect
62. C₂₃	#1455	semidirect
84. C₈	#67267	semidirect

Maximal subgroups

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

subgroup	index
C₄×#67267	2
#13092139	2
#13092138	2
C₈×C₂₃⋊C₈	2
C₄×M₁₆×C₂₃	2
#1455	23