Subgroups — #27156693

Order 12416.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261621	C₂	1	1	yes	maximal, characteristic
2	#20261598	C₂	1	1	yes	maximal, characteristic
3	#20260829	C₂	1	1	yes	maximal, characteristic
4	C₂×#13095514	—	2	1	no	—
5	C₄⋊#11715897	—	2	1	no	—
6	C₂×Dic₃₈₈	C₄	1	1	yes	characteristic
7	#15690775	—	2	1	no	—
8	#15690733	—	2	1	no	—
9	C₈×Dic₉₇	C₂²	1	1	yes	characteristic
10	#107×C₉₇	C₄	1	1	yes	characteristic
11	C₂×#11715897	—	4	1	no	—
12	#13095514	—	2	2	no	—
13	C₂×Dic₁₉₄	—	2	1	no	—
14	Dic₃₈₈	—	2	1	no	—
15	Dic₃₈₈	C₈	1	2	yes	complemented
16	C₂×#11715895	D₄	1	1	yes	characteristic
17	#13095490	—	4	1	no	—
18	C₄×Dic₉₇	—	4	1	no	—
19	C₄×Dic₉₇	D₄	1	1	yes	characteristic
20	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
21	C₄⋊C₄×C₉₇	—	2	1	no	—
22	C₂×C₄×C₉₇	—	4	1	no	—
23	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
24	C₂×Dic₉₇	—	4	1	no	—
25	C₂×Dic₉₇	D₈	1	1	yes	characteristic
26	C₂×Dic₉₇	SD₁₆	1	1	yes	characteristic
27	Dic₁₉₄	—	2	2	no	—
28	Dic₁₉₄	—	4	1	no	—
29	#11715897	—	8	1	no	—
30	C₈×C₉₇	C₂×C₈	1	1	yes	characteristic
31	C₈×C₉₇	M₁₆	1	1	yes	characteristic
32	#11715895	—	2	1	no	—
33	C₂²×C₉₇	#102	1	1	yes	characteristic
34	C₄×C₉₇	—	8	1	no	—
35	C₄×C₉₇	C₂²⋊C₈	1	1	yes	characteristic
36	C₄×C₉₇	#104	1	1	yes	characteristic
37	Dic₉₇	—	2	1	no	—
38	Dic₉₇	—	4	2	no	—
39	Dic₉₇	SD₃₂	1	1	yes	characteristic
40	Dic₉₇	Q₃₂	1	1	yes	characteristic
41	C₂×C₉₇	D₄⋊C₈	1	1	yes	characteristic
42	C₂×C₉₇	#358	1	1	yes	characteristic
43	C₂×C₉₇	#359	1	1	yes	characteristic
44	#1332	—	97	1	no	maximal, Sylow
45	C₉₇	#1332	1	1	yes	characteristic, complemented, Sylow
46	C₄×Q₁₆	—	97	1	no	—
47	C₄×C₁₆	—	97	1	no	—
48	#335	—	97	1	no	—
49	C₂×Q₁₆	—	97	1	no	—
50	C₄×Q₈	—	194	1	no	—
51	C₂×C₁₆	—	194	1	no	—
52	#107	#87604	1	1	yes	characteristic
53	C₄⋊C₈	—	194	1	no	—
54	#103	—	194	1	no	—
55	C₄×C₈	—	97	1	no	—
56	C₂×Q₈	—	194	1	no	—
57	Q₁₆	—	97	2	no	—
58	Q₁₆	—	194	1	no	—
59	C₂×C₈	—	97	1	no	—
60	C₂×C₈	—	388	1	no	—
61	C₂×C₈	C₂×#87604	1	1	yes	characteristic
62	C₄⋊C₄	—	2	1	no	—
63	C₄⋊C₄	—	388	1	no	—
64	C₄²	—	97	1	no	—
65	C₄²	—	388	1	no	—
66	C₁₆	—	194	2	no	—
67	Q₈	—	388	1	no	—
68	Q₈	—	194	2	no	—
69	C₂×C₄	—	4	1	no	—
70	C₂×C₄	—	97	1	no	—
71	C₂×C₄	—	97	1	no	—
72	C₂×C₄	—	388	1	no	—
73	C₂×C₄	C₂²⋊#87604	1	1	yes	characteristic
74	C₈	—	776	1	no	—
75	C₈	—	194	1	no	—
76	C₈	#13095519	1	1	yes	characteristic
77	C₈	#13095521	1	1	yes	characteristic
78	C₂²	#15690872	1	1	yes	characteristic
79	C₄	—	8	1	no	—
80	C₄	—	97	1	no	—
81	C₄	—	97	1	no	—
82	C₄	—	388	2	no	—
83	C₄	—	194	1	no	—
84	C₄	#15690867	1	1	yes	characteristic
85	C₄	#15690875	1	1	yes	characteristic
86	C₂	#20261583	1	1	yes	characteristic
87	C₂	#20261648	1	1	yes	characteristic
88	C₂	#20261649	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15690872
commutator	C₈×C₉₇	C₂×C₈
Frattini	C₂×C₈	C₂×#87604
Fitting	#107×C₉₇	C₄
socle, prime radical	C₂²×C₉₇	#102
cosocle	C₈×Dic₉₇	C₂²
hypercenter	#107	#87604
hypocenter	C₉₇	#1332
LCS	G ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #107

Complements

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

N	H ≅ G/N	type
15. Dic₃₈₈	C₈	semidirect
45. C₉₇	#1332	semidirect

Maximal subgroups

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

subgroup	index
#20261621	2
#20261598	2
#20260829	2
#1332	97