Subgroups — #37565687

Order 24256.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27131353	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₁₅₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₃₀₃₂	C₂	1	2	yes	maximal, complemented
4	#27131276	C₂	1	2	yes	maximal, complemented
5	M₃₂×C₃₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#20258606	—	2	1	no	—
7	C₂×Dic₇₅₈	—	2	1	no	—
8	Dic₁₅₁₆	—	2	1	no	—
9	Dic₁₅₁₆	C₂²	1	2	yes	—
10	Dic₁₅₁₆	C₂²	1	1	yes	characteristic
11	#20258595	—	2	1	no	—
12	D₃₀₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₈×C₃₇₉	C₂²	1	1	yes	characteristic
14	C₁₆×C₃₇₉	C₂²	1	2	yes	—
15	Dic₇₅₈	—	2	1	no	—
16	Dic₇₅₈	—	2	2	no	—
17	Dic₇₅₈	—	4	1	no	—
18	D₁₅₁₆	—	2	1	no	—
19	C₂²⋊D₃₇₉	—	4	1	no	—
20	C₂×Dic₃₇₉	—	4	1	no	—
21	C₄×D₃₇₉	—	4	1	no	—
22	C₂×C₄×C₃₇₉	D₄	1	1	yes	characteristic
23	C₈×C₃₇₉	D₄	1	1	yes	characteristic
24	C₈×C₃₇₉	C₂³	1	1	yes	characteristic
25	C₂²×C₃₇₉	D₈	1	1	yes	characteristic
26	C₂×D₃₇₉	—	4	1	no	—
27	C₄×C₃₇₉	D₈	1	1	yes	characteristic
28	C₄×C₃₇₉	C₂×D₄	1	1	yes	characteristic
29	Dic₃₇₉	—	4	1	no	—
30	Dic₃₇₉	—	4	2	no	—
31	C₂×C₃₇₉	—	2	1	no	—
32	C₂×C₃₇₉	C₂×D₈	1	1	yes	characteristic
33	D₃₇₉	—	8	1	no	—
34	C₃₇₉	#510	1	1	yes	characteristic, complemented, Sylow
35	#510	—	379	1	no	maximal, Sylow
36	#135	—	379	1	no	—
37	C₂×Q₁₆	—	379	1	no	—
38	Q₃₂	—	379	2	no	—
39	SD₃₂	—	379	2	no	—
40	M₃₂	D₃₇₉	1	1	yes	characteristic, complemented
41	#41	—	758	1	no	—
42	C₂×Q₈	—	758	1	no	—
43	Q₁₆	—	379	2	no	—
44	Q₁₆	—	379	1	no	—
45	Q₁₆	—	758	1	no	—
46	SD₁₆	—	758	1	no	—
47	D₈	—	379	1	no	—
48	C₂×C₈	C₂×D₃₇₉	1	1	yes	characteristic
49	C₁₆	C₂×D₃₇₉	1	2	yes	—
50	Q₈	—	1516	1	no	—
51	Q₈	—	758	1	no	—
52	Q₈	—	758	2	no	—
53	D₄	—	1516	1	no	—
54	D₄	—	758	1	no	—
55	C₂×C₄	—	1516	1	no	—
56	C₂×C₄	—	1516	1	no	—
57	C₂×C₄	D₁₅₁₆	1	1	yes	characteristic
58	C₈	D₁₅₁₆	1	1	yes	characteristic
59	C₈	C₂²×D₃₇₉	1	1	yes	characteristic
60	C₂²	—	1516	1	no	—
61	C₂²	D₃₀₃₂	1	1	yes	characteristic
62	C₄	—	1516	1	no	—
63	C₄	—	1516	2	no	—
64	C₄	D₃₀₃₂	1	1	yes	characteristic
65	C₄	C₂×D₁₅₁₆	1	1	yes	characteristic
66	C₂	—	2	1	no	—
67	C₂	—	3032	1	no	—
68	C₂	C₂×D₃₀₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₃₀₃₂
commutator, cosocle	C₈×C₃₇₉	C₂³
Frattini	C₈	C₂²×D₃₇₉
Fitting	M₃₂×C₃₇₉	C₂
socle	C₂×C₃₇₉	C₂×D₈
prime radical	#27131353	C₂
hypercenter	M₃₂	D₃₇₉
hypocenter	C₃₇₉	#510
LCS	G ▷ C₈×C₃₇₉ ▷ C₄×C₃₇₉ ▷ C₂×C₃₇₉ ▷ C₃₇₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₁₅₁₆	C₂	semidirect
3. Dic₃₀₃₂	C₂	semidirect
4. #27131276	C₂	semidirect
5. M₃₂×C₃₇₉	C₂	semidirect
34. C₃₇₉	#510	semidirect
40. M₃₂	D₃₇₉	semidirect

Maximal subgroups

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

subgroup	index
#27131353	2
C₂×Dic₁₅₁₆	2
Dic₃₀₃₂	2
#27131276	2
M₃₂×C₃₇₉	2
#510	379