Subgroups — #27141888

Order 12416.

300 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#11715895	C₂	1	6	yes	maximal
2	C₄³×C₉₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11715895	C₂²	1	4	yes	—
4	C₄⋊#11715895	—	2	12	no	—
5	C₂×C₄²×C₉₇	C₄	1	3	yes	—
6	C₂×C₄²×C₉₇	C₄	1	1	yes	characteristic
7	C₂×C₄²×C₉₇	C₂²	1	3	yes	—
8	C₂×#11715895	—	2	12	no	—
9	C₂²×C₄×C₉₇	C₂×C₄	1	3	yes	—
10	C₂²×C₄×C₉₇	C₂×C₄	1	3	yes	—
11	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
12	C₄²×C₉₇	—	2	6	no	—
13	C₄²×C₉₇	C₈	1	4	yes	complemented
14	C₄²×C₉₇	D₄	1	6	yes	—
15	C₄²×C₉₇	Q₈	1	6	yes	—
16	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
17	C₂×C₄×C₉₇	—	2	6	no	—
18	C₂×C₄×C₉₇	C₄⋊C₄	1	6	yes	—
19	C₂×C₄×C₉₇	C₄⋊C₄	1	6	yes	—
20	C₂×C₄×C₉₇	C₂×C₈	1	6	yes	—
21	C₂×C₄×C₉₇	M₁₆	1	6	yes	—
22	C₂×C₄×C₉₇	C₂×D₄	1	3	yes	—
23	C₂×C₄×C₉₇	C₂×Q₈	1	3	yes	—
24	#11715895	—	4	4	no	—
25	C₂²×C₉₇	C₂×C₄⋊C₄	1	3	yes	—
26	C₂²×C₉₇	C₂²×C₈	1	1	yes	characteristic
27	C₂²×C₉₇	C₂×M₁₆	1	3	yes	—
28	C₄×C₉₇	—	2	6	no	—
29	C₄×C₉₇	C₄⋊C₈	1	12	yes	complemented
30	C₄×C₉₇	#127	1	1	yes	characteristic
31	C₄×C₉₇	C₄⋊Q₈	1	3	yes	—
32	C₂×C₉₇	#384	1	1	yes	characteristic
33	C₂×C₉₇	C₂×C₄⋊C₈	1	3	yes	—
34	C₂×C₉₇	#423	1	3	yes	—
35	#1842	—	97	1	no	maximal, Sylow
36	C₉₇	#1842	1	1	yes	characteristic, complemented, Sylow
37	C₂×C₄⋊C₈	—	97	6	no	—
38	C₄³	D₉₇	1	1	yes	characteristic
39	C₂²×C₈	—	97	4	no	—
40	C₂×C₄²	Dic₉₇	1	3	yes	—
41	C₂×C₄²	Dic₉₇	1	1	yes	characteristic
42	C₂×C₄²	C₂×D₉₇	1	3	yes	—
43	C₄⋊C₈	—	194	12	no	—
44	C₂²×C₄	C₂×Dic₉₇	1	3	yes	—
45	C₂²×C₄	C₂×Dic₉₇	1	3	yes	—
46	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
47	C₂×C₈	—	194	12	no	—
48	C₄²	—	2	6	no	—
49	C₄²	#11715895	1	4	yes	complemented
50	C₄²	Dic₁₉₄	1	6	yes	—
51	C₄²	D₃₈₈	1	6	yes	—
52	C₂³	C₂²×Dic₉₇	1	1	yes	characteristic
53	C₂×C₄	—	2	6	no	—
54	C₂×C₄	C₄⋊Dic₉₇	1	6	yes	—
55	C₂×C₄	C₄⋊Dic₉₇	1	6	yes	—
56	C₂×C₄	C₂×#11715895	1	6	yes	—
57	C₂×C₄	#13095495	1	6	yes	—
58	C₂×C₄	C₂×D₃₈₈	1	3	yes	—
59	C₂×C₄	C₂×Dic₁₉₄	1	3	yes	—
60	C₈	—	388	4	no	—
61	C₂²	C₂×C₄⋊Dic₉₇	1	3	yes	—
62	C₂²	C₂²×#11715895	1	1	yes	characteristic
63	C₂²	C₂×#13095495	1	3	yes	—
64	C₄	—	2	6	no	—
65	C₄	C₄⋊#11715895	1	12	yes	complemented
66	C₄	#15690807	1	1	yes	characteristic
67	C₄	#15690810	1	3	yes	—
68	C₂	#20260605	1	1	yes	characteristic
69	C₂	C₂×C₄⋊#11715895	1	3	yes	—
70	C₂	#20260759	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₉₇
commutator	C₂²×C₉₇	C₂²×C₈
Fitting	C₄³×C₉₇	C₂
socle, prime radical	C₂³×C₉₇	C₂²×C₄
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	C₄³	D₉₇
hypocenter	C₉₇	#1842
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³

Complements

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

N	H ≅ G/N	type
13. C₄²×C₉₇	C₈	semidirect
29. C₄×C₉₇	C₄⋊C₈	semidirect
36. C₉₇	#1842	semidirect
49. C₄²	#11715895	semidirect
65. C₄	C₄⋊#11715895	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11715895	2
C₄³×C₉₇	2
#1842	97