Subgroups — #15675427

Order 2944.

224 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092599	C₂	1	3	yes	maximal
2	#13092216	C₂	1	3	yes	maximal, complemented
3	Q₈²×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#10625158	C₂²	1	1	yes	characteristic
5	C₂₃×C₄⋊Q₈	—	2	3	no	—
6	C₂₃×C₄⋊Q₈	C₂²	1	3	yes	—
7	C₄×Q₈×C₂₃	—	2	3	no	—
8	#10625113	—	2	3	no	—
9	#10625018	C₂²	1	3	yes	—
10	#10624987	—	2	3	no	—
11	C₂×Q₈×C₂₃	—	2	1	no	—
12	C₂×Q₈×C₂₃	—	2	1	no	—
13	C₂×Q₈×C₂₃	D₄	1	3	yes	—
14	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
15	#67292	—	4	1	no	—
16	Q₈×D₂₃	—	4	1	no	—
17	#67288	—	2	3	no	—
18	C₄⋊C₄×C₂₃	—	2	3	no	—
19	C₄⋊C₄×C₂₃	—	2	6	no	—
20	C₄²×C₂₃	—	2	3	no	—
21	C₄²×C₂₃	D₄	1	3	yes	—
22	#67275	—	4	3	no	—
23	#67274	—	4	3	no	—
24	#67267	—	2	3	no	—
25	Q₈×C₂₃	—	2	4	no	—
26	Q₈×C₂₃	—	4	1	no	—
27	Q₈×C₂₃	—	4	3	no	—
28	Q₈×C₂₃	—	8	1	no	—
29	C₂×C₄×C₂₃	—	2	3	no	—
30	C₂×C₄×C₂₃	—	2	3	no	—
31	C₂×C₄×C₂₃	C₂×D₄	1	3	yes	—
32	C₂²⋊D₂₃	—	4	1	no	—
33	D₉₂	—	2	3	no	—
34	C₄×D₂₃	—	4	3	no	—
35	Dic₄₆	—	2	3	no	—
36	C₂₃⋊C₈	—	4	3	no	—
37	#2205	—	23	1	no	maximal, Sylow
38	C₂²×C₂₃	#120	1	1	yes	characteristic
39	C₂×D₂₃	—	4	1	no	—
40	C₄×C₂₃	—	2	6	no	—
41	C₄×C₂₃	—	2	3	no	—
42	C₄×C₂₃	—	4	3	no	—
43	Dic₂₃	—	4	1	no	—
44	Q₈²	D₂₃	1	1	yes	characteristic, complemented
45	#456	—	23	3	no	—
46	#356	—	23	3	no	—
47	C₂×C₂₃	—	2	1	no	—
48	C₂×C₂₃	B(4,𝔽₂)	1	1	yes	characteristic
49	D₂₃	—	8	1	no	—
50	2₋¹⁺⁴	—	23	1	no	—
51	#137	—	46	3	no	—
52	C₄⋊Q₈	—	2	3	no	—
53	C₄⋊Q₈	C₂×D₂₃	1	3	yes	—
54	C₄×Q₈	—	2	3	no	—
55	#104	—	46	3	no	—
56	#101	—	23	3	no	—
57	C₂₃	#2205	1	1	yes	characteristic, complemented, Sylow
58	#41	—	46	3	no	—
59	#41	—	92	1	no	—
60	C₂×Q₈	—	2	1	no	—
61	C₂×Q₈	—	2	1	no	—
62	C₂×Q₈	—	92	1	no	—
63	C₂×Q₈	C₂²⋊D₂₃	1	3	yes	—
64	C₂×Q₈	C₂²×D₂₃	1	1	yes	characteristic
65	Q₁₆	—	92	3	no	—
66	SD₁₆	—	92	3	no	—
67	M₁₆	—	46	3	no	—
68	C₄⋊C₄	—	2	3	no	—
69	C₄⋊C₄	—	2	6	no	—
70	C₄²	—	2	3	no	—
71	C₄²	C₂²⋊D₂₃	1	3	yes	—
72	Q₈	—	2	4	no	—
73	Q₈	—	4	1	no	—
74	Q₈	—	4	3	no	—
75	Q₈	—	8	1	no	—
76	Q₈	—	46	3	no	—
77	D₄	—	46	3	no	—
78	D₄	—	92	1	no	—
79	C₂×C₄	—	2	3	no	—
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	—	92	3	no	—
82	C₂×C₄	C₂×C₂²⋊D₂₃	1	3	yes	—
83	C₈	—	92	3	no	—
84	C₂²	—	92	1	no	—
85	C₂²	#10625124	1	1	yes	characteristic
86	C₄	—	2	6	no	—
87	C₄	—	2	3	no	—
88	C₄	—	4	3	no	—
89	C₄	—	92	1	no	—
90	C₂	—	2	1	no	—
91	C₂	—	184	1	no	—
92	C₂	#13092604	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#13092604
commutator, cosocle	C₂×Q₈×C₂₃	C₂³
Frattini	C₂×Q₈	C₂²×D₂₃
Fitting	Q₈²×C₂₃	C₂
socle	C₂×C₂₃	B(4,𝔽₂)
prime radical	#10625158	C₂²
hypercenter	Q₈²	D₂₃
hypocenter	C₂₃	#2205
LCS	G ▷ C₂×Q₈×C₂₃ ▷ C₂²×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ Q₈²

Complements

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

N	H ≅ G/N	type
2. #13092216	C₂	semidirect
3. Q₈²×C₂₃	C₂	semidirect
44. Q₈²	D₂₃	semidirect
57. C₂₃	#2205	semidirect

Maximal subgroups

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

subgroup	index
#13092599	2
#13092216	2
Q₈²×C₂₃	2
#2205	23