Subgroups — #38762912

Order 25408.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20263718	C₂	1	2	yes	maximal, complemented
2	C₂×#20263716	C₂	1	2	yes	maximal, complemented
3	#27163093	C₂	1	1	yes	maximal, characteristic, complemented
4	#27163007	C₂	1	1	yes	maximal, characteristic, complemented
5	#124×C₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₅₈₈	—	2	2	no	—
7	C₂×D₁₅₈₈	C₂²	1	2	yes	—
8	#20263718	—	2	4	no	—
9	#20263716	—	2	4	no	—
10	C₂×#15691787	C₂²	1	2	yes	—
11	C₂×Q₈×C₃₉₇	C₂²	1	1	yes	characteristic
12	C₂×D₄×C₃₉₇	C₂²	1	1	yes	characteristic
13	C₂²⋊C₄×C₃₉₇	—	2	2	no	—
14	C₄²×C₃₉₇	C₂²	1	1	yes	characteristic
15	C₂²×D₃₉₇	—	2	2	no	—
16	D₁₅₈₈	—	2	2	no	—
17	D₁₅₈₈	—	2	2	no	—
18	D₁₅₈₈	—	4	4	no	—
19	#15691787	D₄	1	4	yes	complemented
20	C₂³×C₃₉₇	—	2	1	no	—
21	Q₈×C₃₉₇	—	2	2	no	—
22	D₄×C₃₉₇	—	2	2	no	—
23	C₂×C₄×C₃₉₇	—	2	1	no	—
24	C₂×C₄×C₃₉₇	D₄	1	2	yes	—
25	C₂×C₄×C₃₉₇	C₂³	1	1	yes	characteristic
26	C₂²×C₃₉₇	—	4	1	no	—
27	C₂²×C₃₉₇	—	4	2	no	—
28	C₂²×C₃₉₇	C₂×D₄	1	1	yes	characteristic
29	C₂×D₃₉₇	—	4	2	no	—
30	C₂×D₃₉₇	—	4	4	no	—
31	C₄×C₃₉₇	—	2	2	no	—
32	C₄×C₃₉₇	—	4	1	no	—
33	C₄×C₃₉₇	C₂×D₄	1	2	yes	—
34	C₂×C₃₉₇	—	8	1	no	—
35	C₂×C₃₉₇	#127	1	1	yes	characteristic
36	C₂×C₃₉₇	AGL(1,ℤ/8)	1	2	yes	—
37	D₃₉₇	—	8	2	no	—
38	C₃₉₇	#496	1	1	yes	characteristic, complemented, Sylow
39	#496	—	397	1	no	maximal, Sylow
40	C₂×SD₁₆	—	397	2	no	—
41	C₂×D₈	—	397	2	no	—
42	#127	—	397	1	no	—
43	#124	D₃₉₇	1	1	yes	characteristic, complemented
44	#97	—	397	1	no	—
45	C₂×Q₈	C₂×D₃₉₇	1	1	yes	characteristic
46	C₂×D₄	—	794	2	no	—
47	C₂×D₄	—	397	2	no	—
48	C₂×D₄	C₂×D₃₉₇	1	1	yes	characteristic
49	SD₁₆	—	794	4	no	—
50	D₈	—	794	4	no	—
51	C₂×C₈	—	397	2	no	—
52	C₂²⋊C₄	—	2	2	no	—
53	C₄²	C₂×D₃₉₇	1	1	yes	characteristic
54	C₂³	—	2	1	no	—
55	C₂³	—	794	2	no	—
56	Q₈	—	2	2	no	—
57	D₄	—	2	2	no	—
58	D₄	—	794	2	no	—
59	D₄	—	794	2	no	—
60	D₄	—	1588	4	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	C₂²⋊D₃₉₇	1	2	yes	—
63	C₂×C₄	C₂²×D₃₉₇	1	1	yes	characteristic
64	C₈	—	397	4	no	—
65	C₂²	—	4	1	no	—
66	C₂²	—	4	2	no	—
67	C₂²	—	1588	2	no	—
68	C₂²	—	1588	4	no	—
69	C₂²	C₂×C₂²⋊D₃₉₇	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	4	1	no	—
72	C₄	D₄×D₃₉₇	1	2	yes	—
73	C₂	—	8	1	no	—
74	C₂	—	3176	2	no	—
75	C₂	#27163092	1	1	yes	characteristic
76	C₂	#27163121	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₃₉₇
commutator, cosocle	C₂×C₄×C₃₉₇	C₂³
Frattini	C₂×C₄	C₂²×D₃₉₇
Fitting	#124×C₃₉₇	C₂
socle	C₂²×C₃₉₇	C₂×D₄
hypercenter	#124	D₃₉₇
hypocenter	C₃₉₇	#496
LCS	G ▷ C₂×C₄×C₃₉₇ ▷ C₂×C₃₉₇ ▷ C₃₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #124

Complements

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

N	H ≅ G/N	type
1. C₂×#20263718	C₂	semidirect
2. C₂×#20263716	C₂	semidirect
3. #27163093	C₂	semidirect
4. #27163007	C₂	semidirect
5. #124×C₃₉₇	C₂	semidirect
19. #15691787	D₄	semidirect
38. C₃₉₇	#496	semidirect
43. #124	D₃₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20263718	2
C₂×#20263716	2
#27163093	2
#27163007	2
#124×C₃₉₇	2
#496	397