Subgroups — #37565606

Order 24256.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27131319	C₂	1	1	yes	maximal, characteristic
2	#27131254	C₂	1	2	yes	maximal, complemented
3	#27131251	C₂	1	2	yes	maximal
4	#27131242	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄⋊Q₈×C₃₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₇₅₈	C₂²	1	1	yes	characteristic
7	C₂×D₁₅₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₃₇₉⋊C₈	C₂²	1	2	yes	—
9	(C₂×D₃₇₉)⋊C₄	—	2	1	no	—
10	(C₂×D₃₇₉)⋊C₄	—	2	1	no	—
11	C₂×Q₈×C₃₇₉	—	2	1	no	—
12	C₄⋊C₄×C₃₇₉	—	2	1	no	—
13	C₄⋊C₄×C₃₇₉	C₂²	1	2	yes	—
14	C₄²×C₃₇₉	C₂²	1	1	yes	characteristic
15	C₂²×D₃₇₉	—	2	1	no	—
16	Dic₇₅₈	—	2	2	no	—
17	D₁₅₁₆	—	2	2	no	—
18	C₂×Dic₃₇₉	—	2	1	no	—
19	C₃₇₉⋊C₈	—	2	2	no	—
20	Q₈×C₃₇₉	—	4	2	no	—
21	C₂×C₄×C₃₇₉	—	2	2	no	—
22	C₂×C₄×C₃₇₉	D₄	1	1	yes	characteristic
23	C₂×C₄×C₃₇₉	D₄	1	1	yes	characteristic
24	C₂×C₄×C₃₇₉	C₂³	1	1	yes	characteristic
25	C₂²×C₃₇₉	C₂×D₄	1	1	yes	characteristic
26	C₂×D₃₇₉	—	4	1	no	—
27	C₂×D₃₇₉	—	4	1	no	—
28	C₂×D₃₇₉	—	4	1	no	—
29	C₄×C₃₇₉	—	2	1	no	—
30	C₄×C₃₇₉	—	2	1	no	—
31	C₄×C₃₇₉	—	4	2	no	—
32	C₄×C₃₇₉	#41	1	2	yes	—
33	Dic₃₇₉	—	4	1	no	—
34	C₂×C₃₇₉	#124	1	1	yes	characteristic
35	C₂×C₃₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
36	C₂×C₃₇₉	#137	1	1	yes	characteristic
37	D₃₇₉	—	8	1	no	—
38	C₃₇₉	#489	1	1	yes	characteristic, complemented, Sylow
39	#489	—	379	1	no	maximal, Sylow
40	C₄⋊Q₈	D₃₇₉	1	1	yes	characteristic, complemented
41	#124	—	379	1	no	—
42	#103	—	379	2	no	—
43	#102	—	379	2	no	—
44	#97	—	379	1	no	—
45	C₂×Q₈	—	2	1	no	—
46	C₂×Q₈	—	379	1	no	—
47	C₂×D₄	—	379	1	no	—
48	C₂×C₈	—	379	2	no	—
49	C₄⋊C₄	—	2	1	no	—
50	C₄⋊C₄	C₂×D₃₇₉	1	2	yes	—
51	C₂²⋊C₄	—	758	1	no	—
52	C₂²⋊C₄	—	758	1	no	—
53	C₄²	C₂×D₃₇₉	1	1	yes	characteristic
54	C₂³	—	758	1	no	—
55	Q₈	—	4	2	no	—
56	Q₈	—	758	2	no	—
57	D₄	—	758	2	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	758	1	no	—
60	C₂×C₄	C₂²⋊D₃₇₉	1	1	yes	characteristic
61	C₂×C₄	C₂²⋊D₃₇₉	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₃₇₉	1	1	yes	characteristic
63	C₈	—	758	2	no	—
64	C₂²	—	1516	1	no	—
65	C₂²	—	1516	1	no	—
66	C₂²	—	1516	1	no	—
67	C₂²	C₂×C₂²⋊D₃₇₉	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	2	1	no	—
70	C₄	—	4	2	no	—
71	C₄	—	1516	1	no	—
72	C₄	#20258607	1	2	yes	—
73	C₂	—	3032	1	no	—
74	C₂	#27131320	1	1	yes	characteristic
75	C₂	#27131357	1	1	yes	characteristic
76	C₂	#27131362	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₃₇₉
commutator, cosocle	C₂×C₄×C₃₇₉	C₂³
Frattini	C₂×C₄	C₂²×D₃₇₉
Fitting	C₄⋊Q₈×C₃₇₉	C₂
socle	C₂²×C₃₇₉	C₂×D₄
prime radical	C₂×D₁₅₁₆	C₂²
hypercenter	C₄⋊Q₈	D₃₇₉
hypocenter	C₃₇₉	#489
LCS	G ▷ C₂×C₄×C₃₇₉ ▷ C₂×C₃₇₉ ▷ C₃₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #27131254	C₂	semidirect
4. #27131242	C₂	semidirect
5. C₄⋊Q₈×C₃₇₉	C₂	semidirect
38. C₃₇₉	#489	semidirect
40. 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
#27131319	2
#27131254	2
#27131251	2
#27131242	2
C₄⋊Q₈×C₃₇₉	2
#489	379