Subgroups — #27157661

Order 12416.

188 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261740	C₂	1	2	yes	maximal
2	#20261611	C₂	1	4	yes	maximal
3	C₄×C₈×D₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#13095521	C₂²	1	4	yes	—
5	#15690877	—	2	4	no	—
6	C₂×C₈×D₉₇	C₂²	1	2	yes	—
7	C₄²×D₉₇	C₂²	1	1	yes	characteristic
8	C₄×#11715895	C₄	1	1	yes	characteristic
9	C₈×Dic₉₇	C₄	1	2	yes	—
10	C₄×C₈×C₉₇	C₄	1	1	yes	characteristic
11	#13095521	—	2	8	no	—
12	C₂×#11715897	—	2	4	no	—
13	C₂×C₄×D₉₇	Q₈	1	2	yes	—
14	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
15	C₂×#11715895	C₂×C₄	1	2	yes	—
16	C₈×D₉₇	—	2	4	no	—
17	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
18	C₄×Dic₉₇	D₄	1	2	yes	—
19	C₂×C₈×C₉₇	C₂×C₄	1	2	yes	—
20	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×D₉₇	C₂×Q₈	1	1	yes	characteristic
22	C₂×C₄×C₉₇	C₄⋊C₄	1	2	yes	—
23	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
24	C₂×Dic₉₇	C₄⋊C₄	1	2	yes	—
25	C₂×Dic₉₇	C₂×D₄	1	1	yes	characteristic
26	C₄×D₉₇	—	2	4	no	—
27	C₄×D₉₇	#41	1	4	yes	—
28	#11715897	—	4	4	no	—
29	C₈×C₉₇	—	2	2	no	—
30	#11715895	—	2	2	no	—
31	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
32	C₂×D₉₇	—	2	2	no	—
33	C₂×D₉₇	#125	1	2	yes	—
34	C₄×C₉₇	—	2	2	no	—
35	C₄×C₉₇	#117	1	2	yes	—
36	Dic₉₇	#108	1	4	yes	—
37	Dic₉₇	#124	1	2	yes	—
38	C₂×C₉₇	#382	1	1	yes	characteristic
39	C₂×C₉₇	C₂×#108	1	2	yes	—
40	D₉₇	—	2	2	no	—
41	#1837	—	97	1	no	maximal, Sylow
42	C₉₇	#1837	1	1	yes	characteristic, complemented, Sylow
43	#423	—	97	2	no	—
44	C₂×C₄×C₈	—	97	1	no	—
45	#341	—	97	4	no	—
46	C₂×M₁₆	—	97	4	no	—
47	C₂²×C₈	—	97	2	no	—
48	C₂×C₄²	—	97	1	no	—
49	C₄⋊C₈	—	194	4	no	—
50	C₄×C₈	—	97	2	no	—
51	C₄×C₈	—	97	1	no	—
52	C₄×C₈	#87604	1	1	yes	characteristic
53	C₂²×C₄	—	97	1	no	—
54	C₂²×C₄	—	97	2	no	—
55	M₁₆	—	194	8	no	—
56	C₂×C₈	—	97	2	no	—
57	C₂×C₈	—	194	4	no	—
58	C₂×C₈	—	194	4	no	—
59	C₂×C₈	C₂×#87604	1	2	yes	—
60	C₄²	—	97	2	no	—
61	C₄²	—	97	1	no	—
62	C₄²	C₂×#87604	1	1	yes	characteristic
63	C₂³	—	97	1	no	—
64	C₂×C₄	—	97	1	no	—
65	C₂×C₄	—	97	2	no	—
66	C₂×C₄	—	97	4	no	—
67	C₂×C₄	—	194	4	no	—
68	C₂×C₄	C₄⋊#87604	1	2	yes	—
69	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
70	C₈	—	2	2	no	—
71	C₈	—	388	4	no	—
72	C₈	—	194	2	no	—
73	C₂²	—	97	2	no	—
74	C₂²	—	194	2	no	—
75	C₂²	C₂×C₄⋊#87604	1	1	yes	characteristic
76	C₄	—	2	2	no	—
77	C₄	—	97	2	no	—
78	C₄	—	97	4	no	—
79	C₄	#15690889	1	2	yes	—
80	C₂	—	194	2	no	—
81	C₂	#20261679	1	1	yes	characteristic
82	C₂	D₉₇⋊#108	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₄⋊#87604
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#87604
Fitting	C₄×C₈×C₉₇	C₄
socle	C₂²×C₉₇	C₂×C₄⋊C₄
prime radical	C₂²×D₉₇	C₂×Q₈
cosocle	C₂×C₄×D₉₇	C₂³
hypercenter	C₄×C₈	#87604
hypocenter	C₉₇	#1837
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
42. C₉₇	#1837	semidirect

Maximal subgroups

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

subgroup	index
#20261740	2
#20261611	2
C₄×C₈×D₉₇	2
#1837	97