Subgroups — C₂×C₂²⋊#13097544 #27163173

Order 12704.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13097544	C₂	1	2	yes	maximal, complemented
2	C₂²⋊#13097544	C₂	1	4	yes	maximal, direct product factor
3	C₂³×D₃₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#13097544	—	2	4	no	—
5	C₂×#13097544	C₂²	1	4	yes	complemented
6	C₂²×D₃₉₇	—	2	4	no	—
7	C₂²×D₃₉₇	C₄	1	2	yes	complemented
8	C₂²×D₃₉₇	C₄	1	1	yes	characteristic, complemented
9	C₂²×D₃₉₇	C₂²	1	2	yes	—
10	C₂²×D₃₉₇	C₂²	1	1	yes	characteristic
11	C₂³×C₃₉₇	C₄	1	1	yes	characteristic, complemented
12	C₂²×C₃₉₇	—	2	2	no	—
13	C₂²×C₃₉₇	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₃₉₇	C₂×C₄	1	2	yes	complemented
15	C₂×D₃₉₇	—	2	2	no	—
16	C₂×D₃₉₇	—	2	4	no	—
17	C₂×D₃₉₇	—	2	4	no	—
18	C₂×D₃₉₇	C₂×C₄	1	1	yes	characteristic
19	C₂×D₃₉₇	C₂×C₄	1	2	yes	complemented
20	C₂×D₃₉₇	D₄	1	4	yes	—
21	C₂×D₃₉₇	C₂³	1	1	yes	characteristic
22	#13097544	—	2	4	no	—
23	C₂×C₃₉₇	—	2	2	no	—
24	C₂×C₃₉₇	C₂²⋊C₄	1	2	yes	complemented
25	C₂×C₃₉₇	C₂²×C₄	1	1	yes	characteristic
26	D₃₉₇	—	2	2	no	—
27	D₃₉₇	C₂²⋊C₄	1	2	yes	complemented
28	D₃₉₇	C₂×D₄	1	2	yes	—
29	C₃₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
30	C₂×C₂²⋊C₄	—	397	1	no	maximal, Sylow
31	C₂⁴	—	397	1	no	—
32	C₂²×C₄	—	397	2	no	—
33	C₂²⋊C₄	—	397	4	no	—
34	C₂³	—	794	4	no	—
35	C₂³	—	397	1	no	—
36	C₂³	—	397	2	no	—
37	C₂³	—	397	2	no	—
38	C₂³	—	397	1	no	—
39	C₂³	#13097544	1	1	yes	characteristic, complemented
40	C₂×C₄	—	794	4	no	—
41	C₂×C₄	—	397	4	no	—
42	C₂²	—	2	2	no	—
43	C₂²	—	794	4	no	—
44	C₂²	—	794	4	no	—
45	C₂²	—	794	2	no	—
46	C₂²	—	397	2	no	—
47	C₂²	—	397	4	no	—
48	C₂²	—	397	1	no	—
49	C₂²	—	397	1	no	—
50	C₂²	C₂×#13097544	1	1	yes	characteristic
51	C₂²	C₂×#13097544	1	2	yes	complemented
52	C₄	—	794	4	no	—
53	C₂	—	2	2	no	—
54	C₂	—	794	2	no	—
55	C₂	—	397	2	no	—
56	C₂	—	397	2	no	—
57	C₂	C₂²⋊#13097544	1	2	yes	direct product factor
58	C₂	C₂²×#13097544	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂²×#13097544	C₂	semidirect
2. C₂²⋊#13097544	C₂	direct (N × H)
5. C₂×#13097544	C₂²	semidirect
7. C₂²×D₃₉₇	C₄	semidirect
8. C₂²×D₃₉₇	C₄	semidirect
11. C₂³×C₃₉₇	C₄	semidirect
14. C₂²×C₃₉₇	C₂×C₄	semidirect
19. C₂×D₃₉₇	C₂×C₄	semidirect
24. C₂×C₃₉₇	C₂²⋊C₄	semidirect
27. D₃₉₇	C₂²⋊C₄	semidirect
29. C₃₉₇	C₂×C₂²⋊C₄	semidirect
39. C₂³	#13097544	semidirect
51. C₂²	C₂×#13097544	semidirect
57. C₂	C₂²⋊#13097544	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#13097544	2
C₂²⋊#13097544	2
C₂³×D₃₉₇	2
C₂×C₂²⋊C₄	397

Subgroups — C2×C22⋊#13097544 #27163173

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂²⋊#13097544 #27163173