Subgroups — #324×C₃₉₇ #38761978

Order 25408.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊C₄×C₃₉₇	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊C₈×C₃₉₇	C₂	1	2	yes	maximal
3	C₂²×C₄×C₃₉₇	C₄	1	1	yes	characteristic
4	C₂²×C₄×C₃₉₇	C₄	1	1	yes	characteristic
5	C₂²×C₄×C₃₉₇	C₂²	1	1	yes	characteristic
6	C₂×C₈×C₃₉₇	—	2	2	no	—
7	C₄⋊C₄×C₃₉₇	—	2	2	no	—
8	C₂³×C₃₉₇	C₂×C₄	1	1	yes	characteristic
9	C₂×C₄×C₃₉₇	—	2	1	no	—
10	C₂×C₄×C₃₉₇	—	2	1	no	—
11	C₂×C₄×C₃₉₇	—	2	1	no	—
12	C₂×C₄×C₃₉₇	—	4	1	no	—
13	C₂×C₄×C₃₉₇	—	4	1	no	—
14	C₂×C₄×C₃₉₇	C₈	1	2	yes	complemented
15	C₂×C₄×C₃₉₇	D₄	1	2	yes	—
16	C₈×C₃₉₇	—	4	2	no	—
17	C₂²×C₃₉₇	—	2	1	no	—
18	C₂²×C₃₉₇	—	2	1	no	—
19	C₂²×C₃₉₇	C₂²⋊C₄	1	1	yes	characteristic
20	C₂²×C₃₉₇	C₂×C₈	1	1	yes	characteristic
21	C₂²×C₃₉₇	M₁₆	1	1	yes	characteristic
22	C₄×C₃₉₇	—	2	2	no	—
23	C₄×C₃₉₇	—	2	2	no	—
24	C₄×C₃₉₇	—	4	1	no	—
25	C₂×C₃₉₇	—	2	1	no	—
26	C₂×C₃₉₇	—	2	1	no	—
27	C₂×C₃₉₇	C₂²⋊C₈	1	1	yes	characteristic
28	C₂×C₃₉₇	#99	1	1	yes	characteristic
29	C₂×C₃₉₇	#101	1	1	yes	characteristic
30	C₃₉₇	#324	1	1	yes	characteristic, direct product factor, Sylow
31	#324	C₃₉₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂×C₄⋊C₄	C₂×C₃₉₇	1	1	yes	characteristic
33	C₂²⋊C₈	C₂×C₃₉₇	1	2	yes	—
34	C₂²×C₄	C₄×C₃₉₇	1	1	yes	characteristic
35	C₂²×C₄	C₄×C₃₉₇	1	1	yes	characteristic
36	C₂²×C₄	C₂²×C₃₉₇	1	1	yes	characteristic
37	C₂×C₈	—	2	2	no	—
38	C₄⋊C₄	—	2	2	no	—
39	C₂³	C₂×C₄×C₃₉₇	1	1	yes	characteristic
40	C₂×C₄	—	2	1	no	—
41	C₂×C₄	—	2	1	no	—
42	C₂×C₄	—	2	1	no	—
43	C₂×C₄	—	4	1	no	—
44	C₂×C₄	—	4	1	no	—
45	C₂×C₄	C₈×C₃₉₇	1	2	yes	complemented
46	C₂×C₄	D₄×C₃₉₇	1	2	yes	—
47	C₈	—	4	2	no	—
48	C₂²	—	2	1	no	—
49	C₂²	—	2	1	no	—
50	C₂²	C₂²⋊C₄×C₃₉₇	1	1	yes	characteristic
51	C₂²	C₂×C₈×C₃₉₇	1	1	yes	characteristic
52	C₂²	M₁₆×C₃₉₇	1	1	yes	characteristic
53	C₄	—	2	2	no	—
54	C₄	—	2	2	no	—
55	C₄	—	4	1	no	—
56	C₂	—	2	1	no	—
57	C₂	—	2	1	no	—
58	C₂	C₂²⋊C₈×C₃₉₇	1	1	yes	characteristic
59	C₂	#99×C₃₉₇	1	1	yes	characteristic
60	C₂	#101×C₃₉₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₉₇	C₂²⋊C₄
commutator	C₂²	C₂×C₈×C₃₉₇
Frattini, cosocle	C₂²×C₄	C₂²×C₃₉₇
prime radical	C₂³×C₃₉₇	C₂×C₄
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₉₇ ◁ C₂²×C₄×C₃₉₇ ◁ G

Complements

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

N	H ≅ G/N	type
14. C₂×C₄×C₃₉₇	C₈	semidirect
30. C₃₉₇	#324	direct (N × H)
31. #324	C₃₉₇	direct (N × H)
45. C₂×C₄	C₈×C₃₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊C₄×C₃₉₇	2
C₂²⋊C₈×C₃₉₇	2
#324	397

Subgroups — #324×C397 #38761978

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #324×C₃₉₇ #38761978