Subgroups — #38763251

Order 25408.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27163096	C₂	1	3	yes	maximal
2	#27163087	C₂	1	6	yes	maximal
3	Q₈×Dic₃₉₇	C₂	1	2	yes	maximal
4	C₄×Dic₇₉₄	C₂	1	3	yes	maximal
5	C₄×Q₈×C₃₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₇₉₄	C₂²	1	3	yes	—
7	C₄⋊Dic₃₉₇	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₃₉₇	C₂²	1	6	yes	—
9	C₄⋊Dic₃₉₇	C₂²	1	6	yes	—
10	#20263710	C₂²	1	6	yes	—
11	C₄×Dic₃₉₇	C₂²	1	6	yes	—
12	C₂×Q₈×C₃₉₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₃₉₇	C₂²	1	3	yes	—
14	C₄²×C₃₉₇	C₂²	1	3	yes	—
15	Dic₇₉₄	—	2	6	no	—
16	C₂×Dic₃₉₇	C₂³	1	2	yes	—
17	C₂×Dic₃₉₇	C₂³	1	6	yes	—
18	Q₈×C₃₉₇	Q₈	1	4	yes	complemented
19	C₂×C₄×C₃₉₇	C₂³	1	3	yes	—
20	C₂×C₄×C₃₉₇	C₂³	1	3	yes	—
21	C₂×C₄×C₃₉₇	C₂³	1	1	yes	characteristic
22	C₂²×C₃₉₇	C₂⁴	1	1	yes	characteristic
23	C₄×C₃₉₇	—	2	3	no	—
24	C₄×C₃₉₇	C₂×Q₈	1	6	yes	—
25	C₄×C₃₉₇	#41	1	2	yes	—
26	Dic₃₉₇	—	2	6	no	—
27	Dic₃₉₇	—	2	2	no	—
28	C₂×C₃₉₇	C₂²×Q₈	1	1	yes	characteristic
29	C₂×C₃₉₇	C₂×#41	1	1	yes	characteristic
30	C₂×C₃₉₇	2₋¹⁺⁴	1	1	yes	characteristic
31	C₃₉₇	#557	1	1	yes	characteristic, complemented, Sylow
32	#557	—	397	1	no	maximal, Sylow
33	C₄⋊Q₈	—	397	3	no	—
34	#125	—	397	6	no	—
35	C₄×Q₈	—	397	2	no	—
36	C₄×Q₈	—	397	3	no	—
37	C₄×Q₈	D₃₉₇	1	1	yes	characteristic
38	C₂×Q₈	—	397	3	no	—
39	C₂×Q₈	C₂×D₃₉₇	1	1	yes	characteristic
40	C₄⋊C₄	—	397	6	no	—
41	C₄⋊C₄	—	397	1	no	—
42	C₄⋊C₄	—	397	6	no	—
43	C₄⋊C₄	—	397	6	no	—
44	C₄⋊C₄	C₂×D₃₉₇	1	3	yes	—
45	C₄²	—	397	6	no	—
46	C₄²	C₂×D₃₉₇	1	3	yes	—
47	Q₈	—	794	6	no	—
48	Q₈	Dic₇₉₄	1	4	yes	complemented
49	C₂×C₄	—	397	2	no	—
50	C₂×C₄	—	397	6	no	—
51	C₂×C₄	C₂²×D₃₉₇	1	3	yes	—
52	C₂×C₄	C₂²×D₃₉₇	1	3	yes	—
53	C₂×C₄	C₂²×D₃₉₇	1	1	yes	characteristic
54	C₂²	C₂³×D₃₉₇	1	1	yes	characteristic
55	C₄	—	2	3	no	—
56	C₄	—	794	6	no	—
57	C₄	—	794	2	no	—
58	C₄	C₂×Dic₇₉₄	1	6	yes	—
59	C₄	#20263732	1	2	yes	—
60	C₂	C₂²×Dic₇₉₄	1	1	yes	characteristic
61	C₂	C₂×#20263732	1	1	yes	characteristic
62	C₂	#27163143	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₉₇
commutator, socle, prime radical, cosocle	C₂²×C₃₉₇	C₂⁴
Fitting	C₄×Q₈×C₃₉₇	C₂
hypercenter	C₄×Q₈	D₃₉₇
hypocenter	C₃₉₇	#557
LCS	G ▷ C₂²×C₃₉₇ ▷ C₃₉₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
18. Q₈×C₃₉₇	Q₈	semidirect
31. C₃₉₇	#557	semidirect
48. Q₈	Dic₇₉₄	semidirect

Maximal subgroups

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

subgroup	index
#27163096	2
#27163087	2
Q₈×Dic₃₉₇	2
C₄×Dic₇₉₄	2
C₄×Q₈×C₃₉₇	2
#557	397