Subgroups — #37565527

Order 24256.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27131330	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₃₇₉	C₂	1	1	yes	maximal, characteristic
3	#27131263	C₂	1	1	yes	maximal, characteristic
4	#27131262	C₂	1	1	yes	maximal, characteristic
5	Dic₃₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic
6	#27131255	C₂	1	1	yes	maximal, characteristic
7	#103×C₃₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₇₅₈	—	2	1	no	—
9	C₂×C₃₇₉⋊C₈	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₃₇₉	—	2	1	no	—
11	C₄⋊Dic₃₇₉	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₃₇₉	C₂²	1	1	yes	characteristic
13	Dic₃₇₉⋊C₄	—	2	1	no	—
14	C₄×Dic₃₇₉	—	2	1	no	—
15	C₄×Dic₃₇₉	C₂²	1	1	yes	characteristic
16	C₂×Q₈×C₃₇₉	C₂²	1	1	yes	characteristic
17	C₂×C₈×C₃₇₉	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₃₇₉	C₂²	1	1	yes	characteristic
19	Dic₇₅₈	—	4	2	no	—
20	C₂×Dic₃₇₉	—	2	1	no	—
21	C₂×Dic₃₇₉	—	2	1	no	—
22	C₂×Dic₃₇₉	D₄	1	1	yes	characteristic
23	C₂×Dic₃₇₉	D₄	1	1	yes	characteristic
24	C₃₇₉⋊C₈	—	2	1	no	—
25	Q₈×C₃₇₉	—	2	1	no	—
26	Q₈×C₃₇₉	Q₈	1	2	yes	complemented
27	C₂×C₄×C₃₇₉	—	2	1	no	—
28	C₂×C₄×C₃₇₉	—	2	1	no	—
29	C₂×C₄×C₃₇₉	C₂³	1	1	yes	characteristic
30	C₈×C₃₇₉	—	2	1	no	—
31	C₂²×C₃₇₉	C₂×D₄	1	1	yes	characteristic
32	C₄×C₃₇₉	—	2	2	no	—
33	C₄×C₃₇₉	—	4	1	no	—
34	C₄×C₃₇₉	C₂×Q₈	1	1	yes	characteristic
35	C₄×C₃₇₉	#41	1	1	yes	characteristic
36	Dic₃₇₉	—	2	1	no	—
37	Dic₃₇₉	—	4	1	no	—
38	Dic₃₇₉	—	4	1	no	—
39	Dic₃₇₉	SD₁₆	1	2	yes	—
40	C₂×C₃₇₉	C₂²⋊Q₈	1	1	yes	characteristic
41	C₂×C₃₇₉	C₂×SD₁₆	1	1	yes	characteristic
42	C₂×C₃₇₉	#137	1	1	yes	characteristic
43	C₃₇₉	#475	1	1	yes	characteristic, complemented, Sylow
44	#475	—	379	1	no	maximal, Sylow
45	C₄⋊Q₈	—	379	1	no	—
46	C₄×Q₈	—	379	1	no	—
47	#106	—	379	1	no	—
48	#106	—	379	1	no	—
49	C₄⋊C₈	—	379	1	no	—
50	#103	—	379	1	no	—
51	#103	D₃₇₉	1	1	yes	characteristic
52	C₂×Q₈	—	758	1	no	—
53	C₂×Q₈	C₂×D₃₇₉	1	1	yes	characteristic
54	C₂×C₈	—	379	1	no	—
55	C₂×C₈	C₂×D₃₇₉	1	1	yes	characteristic
56	C₄⋊C₄	—	379	1	no	—
57	C₄⋊C₄	—	379	1	no	—
58	C₄⋊C₄	—	758	1	no	—
59	C₄⋊C₄	—	758	1	no	—
60	C₄⋊C₄	C₂×D₃₇₉	1	1	yes	characteristic
61	C₄²	—	379	1	no	—
62	C₄²	—	758	1	no	—
63	Q₈	—	2	1	no	—
64	Q₈	—	1516	2	no	—
65	Q₈	Dic₇₅₈	1	2	yes	complemented
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	379	1	no	—
69	C₂×C₄	—	379	1	no	—
70	C₂×C₄	—	758	1	no	—
71	C₂×C₄	—	758	1	no	—
72	C₂×C₄	C₂²×D₃₇₉	1	1	yes	characteristic
73	C₈	—	2	1	no	—
74	C₈	—	758	1	no	—
75	C₂²	D₄×D₃₇₉	1	1	yes	characteristic
76	C₄	—	2	2	no	—
77	C₄	—	4	1	no	—
78	C₄	—	379	2	no	—
79	C₄	—	1516	1	no	—
80	C₄	—	1516	1	no	—
81	C₄	—	758	1	no	—
82	C₄	C₂×Dic₇₅₈	1	1	yes	characteristic
83	C₄	#20258605	1	1	yes	characteristic
84	C₂	#27131308	1	1	yes	characteristic
85	C₂	SD₁₆×D₃₇₉	1	1	yes	characteristic
86	C₂	#27131363	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₃₇₉
commutator, cosocle	C₂×C₄×C₃₇₉	C₂³
Frattini	C₂×C₄	C₂²×D₃₇₉
Fitting	#103×C₃₇₉	C₂
socle, prime radical	C₂²×C₃₇₉	C₂×D₄
hypercenter	#103	D₃₇₉
hypocenter	C₃₇₉	#475
LCS	G ▷ C₂×C₄×C₃₇₉ ▷ C₂×C₃₇₉ ▷ C₃₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

Complements

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

N	H ≅ G/N	type
26. Q₈×C₃₇₉	Q₈	semidirect
43. C₃₇₉	#475	semidirect
65. Q₈	Dic₇₅₈	semidirect

Maximal subgroups

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

subgroup	index
#27131330	2
Q₈×Dic₃₇₉	2
#27131263	2
#27131262	2
Dic₃₇₉⋊C₈	2
#27131255	2
#103×C₃₇₉	2
#475	379