Subgroups — #27141867

Order 12416.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20260601	C₂	1	2	yes	maximal
2	#20260498	C₂	1	4	yes	maximal
3	C₂×C₄×C₈×C₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊Dic₉₇	C₂²	1	4	yes	—
5	#15690717	—	2	4	no	—
6	C₂²×C₈×C₉₇	C₂²	1	2	yes	—
7	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
8	C₄×C₈×C₉₇	C₄	1	4	yes	complemented
9	C₂²×Dic₉₇	—	2	4	no	—
10	C₄⋊Dic₉₇	—	2	8	no	—
11	C₂²×C₄×C₉₇	D₄	1	2	yes	—
12	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
13	C₂×C₈×C₉₇	—	2	4	no	—
14	C₂×C₈×C₉₇	C₂×C₄	1	4	yes	—
15	C₄²×C₉₇	C₂×C₄	1	2	yes	—
16	C₄²×C₉₇	Q₈	1	2	yes	—
17	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
18	C₂×C₄×C₉₇	—	2	4	no	—
19	C₂×C₄×C₉₇	C₄⋊C₄	1	4	yes	—
20	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
21	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
22	C₂×C₄×C₉₇	#41	1	4	yes	—
23	C₂×Dic₉₇	—	4	4	no	—
24	C₂×Dic₉₇	—	4	8	no	—
25	C₈×C₉₇	—	2	4	no	—
26	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
27	C₂²×C₉₇	#124	1	2	yes	—
28	C₂²×C₉₇	#135	1	4	yes	—
29	C₄×C₉₇	—	2	4	no	—
30	C₄×C₉₇	#117	1	2	yes	—
31	C₄×C₉₇	#125	1	2	yes	—
32	Dic₉₇	—	8	4	no	—
33	C₂×C₉₇	#382	1	1	yes	characteristic
34	C₂×C₉₇	#427	1	2	yes	—
35	C₂×C₉₇	#488	1	4	yes	—
36	#1835	—	97	1	no	maximal, Sylow
37	C₉₇	#1835	1	1	yes	characteristic, complemented, Sylow
38	C₂×C₄×C₈	D₉₇	1	1	yes	characteristic
39	#382	—	97	2	no	—
40	#340	—	97	4	no	—
41	C₂²×C₈	C₂×D₉₇	1	2	yes	—
42	C₂×C₄⋊C₄	—	97	4	no	—
43	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
44	C₄×C₈	Dic₉₇	1	4	yes	complemented
45	#95	—	194	4	no	—
46	C₂²×C₄	—	194	4	no	—
47	C₂²×C₄	D₃₈₈	1	2	yes	—
48	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
49	C₂×C₈	—	2	4	no	—
50	C₂×C₈	C₂×Dic₉₇	1	4	yes	—
51	C₄⋊C₄	—	194	8	no	—
52	C₄²	Dic₁₉₄	1	2	yes	—
53	C₄²	C₂×Dic₉₇	1	2	yes	—
54	C₂³	C₂×D₃₈₈	1	1	yes	characteristic
55	C₂×C₄	—	2	4	no	—
56	C₂×C₄	—	388	4	no	—
57	C₂×C₄	—	388	8	no	—
58	C₂×C₄	C₄⋊Dic₉₇	1	4	yes	—
59	C₂×C₄	C₂²×Dic₉₇	1	1	yes	characteristic
60	C₂×C₄	C₂×Dic₁₉₄	1	1	yes	characteristic
61	C₂×C₄	#13095511	1	4	yes	—
62	C₈	—	2	4	no	—
63	C₂²	C₂×C₄⋊Dic₉₇	1	1	yes	characteristic
64	C₂²	#15690798	1	2	yes	—
65	C₂²	#15690832	1	4	yes	—
66	C₄	—	2	4	no	—
67	C₄	—	776	4	no	—
68	C₄	#15690769	1	2	yes	—
69	C₄	#15690802	1	2	yes	—
70	C₂	#20260601	1	1	yes	characteristic
71	C₂	#20260773	1	2	yes	—
72	C₂	#20261090	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂×D₃₈₈
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	C₂×C₄×C₈×C₉₇	C₂
socle, prime radical	C₂³×C₉₇	C₂×D₄
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	C₂×C₄×C₈	D₉₇
hypocenter	C₉₇	#1835
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
8. C₄×C₈×C₉₇	C₄	semidirect
37. C₉₇	#1835	semidirect
44. C₄×C₈	Dic₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20260601	2
#20260498	2
C₂×C₄×C₈×C₉₇	2
#1835	97