Subgroups — #37565426

Order 24256.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27131379	C₂	1	1	yes	maximal, characteristic
2	#27131362	C₂	1	2	yes	maximal, complemented
3	#27131259	C₂	1	2	yes	maximal, complemented
4	#27131250	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄⋊Q₈×C₃₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#20258607	—	2	2	no	—
7	#20258606	—	2	2	no	—
8	#20258606	C₂²	1	2	yes	—
9	Q₈×D₃₇₉	—	2	2	no	—
10	#20258596	—	2	2	no	—
11	#20258593	—	2	2	no	—
12	#20258590	C₂²	1	2	yes	—
13	C₂×Q₈×C₃₇₉	C₂²	1	1	yes	characteristic
14	C₂×Q₈×C₃₇₉	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×C₃₇₉	—	2	2	no	—
16	C₄²×C₃₇₉	C₂²	1	1	yes	characteristic
17	Dic₇₅₈	—	2	2	no	—
18	Dic₇₅₈	D₄	1	2	yes	—
19	D₁₅₁₆	—	2	2	no	—
20	D₁₅₁₆	D₄	1	2	yes	—
21	C₂²⋊D₃₇₉	—	2	2	no	—
22	C₄×D₃₇₉	—	2	4	no	—
23	C₄×D₃₇₉	—	2	2	no	—
24	C₃₇₉⋊C₈	—	2	2	no	—
25	Q₈×C₃₇₉	—	2	2	no	—
26	Q₈×C₃₇₉	—	2	2	no	—
27	C₂×C₄×C₃₇₉	—	2	1	no	—
28	C₂×C₄×C₃₇₉	—	2	1	no	—
29	C₂×C₄×C₃₇₉	D₄	1	2	yes	—
30	C₂×C₄×C₃₇₉	C₂³	1	1	yes	characteristic
31	C₂²×C₃₇₉	C₂×D₄	1	1	yes	characteristic
32	C₂×D₃₇₉	—	2	2	no	—
33	C₄×C₃₇₉	—	2	2	no	—
34	C₄×C₃₇₉	—	2	2	no	—
35	C₄×C₃₇₉	—	4	1	no	—
36	C₄×C₃₇₉	C₂×D₄	1	2	yes	—
37	Dic₃₇₉	—	2	2	no	—
38	C₂×C₃₇₉	—	2	1	no	—
39	C₂×C₃₇₉	#120	1	1	yes	characteristic
40	D₃₇₉	—	4	2	no	—
41	C₃₇₉	#456	1	1	yes	characteristic, complemented, Sylow
42	#456	—	379	1	no	maximal, Sylow
43	2₋¹⁺⁴	—	379	1	no	—
44	#137	—	379	2	no	—
45	C₄⋊Q₈	D₃₇₉	1	1	yes	characteristic, complemented
46	#104	—	379	2	no	—
47	#101	—	379	1	no	—
48	#41	—	379	2	no	—
49	#41	—	758	2	no	—
50	#41	—	758	2	no	—
51	C₂×Q₈	—	758	2	no	—
52	C₂×Q₈	C₂×D₃₇₉	1	1	yes	characteristic
53	C₂×Q₈	C₂×D₃₇₉	1	1	yes	characteristic
54	Q₁₆	—	758	2	no	—
55	SD₁₆	—	758	2	no	—
56	M₁₆	—	379	2	no	—
57	C₄⋊C₄	—	2	2	no	—
58	C₄²	C₂×D₃₇₉	1	1	yes	characteristic
59	Q₈	—	2	2	no	—
60	Q₈	—	2	2	no	—
61	Q₈	—	379	2	no	—
62	Q₈	—	758	2	no	—
63	D₄	—	379	2	no	—
64	D₄	—	758	2	no	—
65	D₄	—	758	2	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	758	4	no	—
69	C₂×C₄	—	758	2	no	—
70	C₂×C₄	C₂²⋊D₃₇₉	1	2	yes	—
71	C₂×C₄	C₂²×D₃₇₉	1	1	yes	characteristic
72	C₈	—	758	2	no	—
73	C₂²	—	758	2	no	—
74	C₂²	C₂×C₂²⋊D₃₇₉	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	—	2	2	no	—
77	C₄	—	4	1	no	—
78	C₄	—	758	2	no	—
79	C₄	D₄×D₃₇₉	1	2	yes	—
80	C₂	—	2	1	no	—
81	C₂	—	1516	2	no	—
82	C₂	#27131299	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#27131299
commutator, cosocle	C₂×C₄×C₃₇₉	C₂³
Frattini	C₂×C₄	C₂²×D₃₇₉
Fitting	C₄⋊Q₈×C₃₇₉	C₂
socle	C₂×C₃₇₉	#120
prime radical	#27131379	C₂
hypercenter	C₄⋊Q₈	D₃₇₉
hypocenter	C₃₇₉	#456
LCS	G ▷ C₂×C₄×C₃₇₉ ▷ C₂×C₃₇₉ ▷ C₃₇₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #27131362	C₂	semidirect
3. #27131259	C₂	semidirect
4. #27131250	C₂	semidirect
5. C₄⋊Q₈×C₃₇₉	C₂	semidirect
41. C₃₇₉	#456	semidirect
45. C₄⋊Q₈	D₃₇₉	semidirect

Maximal subgroups

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

subgroup	index
#27131379	2
#27131362	2
#27131259	2
#27131250	2
C₄⋊Q₈×C₃₇₉	2
#456	379