Subgroups — C₂²×#20263741 #38763596

Order 25408.

298 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20263741	C₂	1	12	yes	maximal, direct product factor
2	C₂²×#15691794	C₂	1	2	yes	maximal, complemented
3	C₂²×C₄×D₃₉₇	C₂	1	1	yes	maximal, characteristic
4	#20263741	C₂²	1	16	yes	direct product factor
5	C₂×#15691794	C₂²	1	12	yes	complemented
6	C₂³×D₃₉₇	C₄	1	1	yes	characteristic
7	C₂²×Dic₃₉₇	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₃₉₇	C₄	1	6	yes	—
9	C₂×C₄×D₃₉₇	C₂²	1	6	yes	—
10	C₂²×C₄×C₃₉₇	C₄	1	1	yes	characteristic
11	#15691794	C₂³	1	8	yes	complemented
12	C₂²×D₃₉₇	—	2	4	no	—
13	C₂²×D₃₉₇	C₂×C₄	1	6	yes	—
14	C₂×Dic₃₉₇	C₂×C₄	1	3	yes	—
15	C₂×Dic₃₉₇	C₂³	1	3	yes	—
16	C₄×D₃₉₇	C₂×C₄	1	12	yes	—
17	C₄×D₃₉₇	C₂³	1	4	yes	—
18	C₂³×C₃₉₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₃₉₇	C₂×C₄	1	6	yes	—
20	C₂²×C₃₉₇	M₁₆	1	4	yes	complemented
21	C₂²×C₃₉₇	C₂²×C₄	1	3	yes	—
22	C₂×D₃₉₇	—	2	12	no	—
23	C₂×D₃₉₇	C₂²×C₄	1	4	yes	—
24	C₄×C₃₉₇	C₂²×C₄	1	4	yes	—
25	Dic₃₉₇	C₂²×C₄	1	3	yes	—
26	Dic₃₉₇	C₂⁴	1	1	yes	characteristic
27	C₂×C₃₉₇	C₂×M₁₆	1	6	yes	complemented
28	C₂×C₃₉₇	C₂³×C₄	1	1	yes	characteristic
29	D₃₉₇	—	2	4	no	—
30	C₃₉₇	C₂²×M₁₆	1	1	yes	characteristic, complemented, Sylow
31	C₂²×M₁₆	—	397	1	no	maximal, Sylow
32	C₂³×C₄	—	397	1	no	—
33	C₂×M₁₆	—	397	12	no	—
34	C₂²×C₈	—	397	2	no	—
35	C₂⁴	—	397	1	no	—
36	C₂²×C₄	—	397	1	no	—
37	C₂²×C₄	—	397	6	no	—
38	C₂²×C₄	—	397	6	no	—
39	C₂²×C₄	#13097544	1	1	yes	characteristic
40	M₁₆	—	397	16	no	—
41	C₂×C₈	—	397	12	no	—
42	C₂³	—	794	4	no	—
43	C₂³	—	397	6	no	—
44	C₂³	C₂×#13097544	1	1	yes	characteristic
45	C₂×C₄	—	397	3	no	—
46	C₂×C₄	—	397	3	no	—
47	C₂×C₄	—	397	12	no	—
48	C₂×C₄	—	397	4	no	—
49	C₂×C₄	C₂×#13097544	1	6	yes	—
50	C₈	—	397	8	no	—
51	C₂²	—	794	12	no	—
52	C₂²	—	397	4	no	—
53	C₂²	#20263741	1	4	yes	direct product factor
54	C₂²	C₂²×#13097544	1	3	yes	—
55	C₄	—	397	1	no	—
56	C₄	—	397	3	no	—
57	C₄	C₂²×#13097544	1	4	yes	—
58	C₂	—	794	4	no	—
59	C₂	C₂×#20263741	1	6	yes	direct product factor
60	C₂	C₂³×#13097544	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#13097544
commutator	C₂×C₃₉₇	C₂³×C₄
Frattini	C₂	C₂³×#13097544
Fitting	C₂²×C₄×C₃₉₇	C₄
socle	C₂³×C₃₉₇	C₂×C₄
prime radical	C₂³×D₃₉₇	C₄
cosocle	Dic₃₉₇	C₂⁴
hypercenter	C₂²×C₄	#13097544
hypocenter	C₃₉₇	C₂²×M₁₆
LCS	G ▷ C₂×C₃₉₇ ▷ C₃₉₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
1. C₂×#20263741	C₂	direct (N × H)
2. C₂²×#15691794	C₂	semidirect
4. #20263741	C₂²	direct (N × H)
5. C₂×#15691794	C₂²	semidirect
11. #15691794	C₂³	semidirect
20. C₂²×C₃₉₇	M₁₆	semidirect
27. C₂×C₃₉₇	C₂×M₁₆	semidirect
30. C₃₉₇	C₂²×M₁₆	semidirect
53. C₂²	#20263741	direct (N × H)
59. C₂	C₂×#20263741	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#20263741	2
C₂²×#15691794	2
C₂²×C₄×D₃₉₇	2
C₂²×M₁₆	397

Subgroups — C22×#20263741 #38763596

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#20263741 #38763596