Subgroups — #27158791

Order 12416.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#13095523	C₂	1	1	yes	maximal, characteristic
2	C₄×#13095523	C₂	1	1	yes	maximal, characteristic
3	#20261784	C₂	1	1	yes	maximal, characteristic
4	C₂×#13095523	—	2	1	no	—
5	C₂×#13095523	—	2	1	no	—
6	#15690877	—	2	1	no	—
7	C₄⋊#11715897	C₄	1	1	yes	characteristic
8	C₄×#11715897	—	2	1	no	—
9	C₄×#11715897	C₂²	1	1	yes	characteristic
10	#15690775	C₄	1	1	yes	characteristic
11	#13095523	—	2	2	no	—
12	#13095523	—	4	1	no	—
13	C₂×#11715897	—	2	1	no	—
14	C₂×#11715897	D₄	1	1	yes	characteristic
15	C₂×#11715897	D₄	1	1	yes	characteristic
16	C₂×Dic₁₉₄	C₈	1	1	yes	characteristic
17	#13095490	—	2	1	no	—
18	C₄×Dic₉₇	—	2	1	no	—
19	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
20	C₄⋊C₄×C₉₇	C₈	1	1	yes	characteristic
21	C₂×C₄×C₉₇	—	2	1	no	—
22	C₂×C₄×C₉₇	C₂×C₈	1	1	yes	characteristic
23	C₂×Dic₉₇	—	2	1	no	—
24	C₂×Dic₉₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×Dic₉₇	M₁₆	1	1	yes	characteristic
26	Dic₁₉₄	—	2	1	no	—
27	Dic₁₉₄	C₁₆	1	2	yes	complemented
28	#11715897	—	2	1	no	—
29	#11715897	—	4	1	no	—
30	#11715897	SD₁₆	1	1	yes	characteristic
31	#11715897	Q₁₆	1	1	yes	characteristic
32	C₂²×C₉₇	C₂²⋊C₈	1	1	yes	characteristic
33	C₄×C₉₇	—	4	1	no	—
34	C₄×C₉₇	C₂×C₁₆	1	1	yes	characteristic
35	C₄×C₉₇	M₃₂	1	1	yes	characteristic
36	Dic₉₇	—	2	2	no	—
37	Dic₉₇	—	2	1	no	—
38	Dic₉₇	#103	1	1	yes	characteristic
39	Dic₉₇	#104	1	1	yes	characteristic
40	C₂×C₉₇	Q₈⋊C₈	1	1	yes	characteristic
41	C₂×C₉₇	C₂²⋊C₁₆	1	1	yes	characteristic
42	C₂×C₉₇	#350	1	1	yes	characteristic
43	Q₈⋊C₁₆	—	97	1	no	maximal, Sylow
44	C₉₇	Q₈⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
45	C₈×Q₈	—	97	1	no	—
46	C₄⋊C₁₆	—	97	1	no	—
47	C₄×C₁₆	—	97	1	no	—
48	C₄×Q₈	—	97	1	no	—
49	C₂×C₁₆	—	194	1	no	—
50	C₂×C₁₆	—	194	1	no	—
51	C₄⋊C₈	—	97	1	no	—
52	C₄⋊C₈	—	194	1	no	—
53	C₄×C₈	—	97	1	no	—
54	C₄×C₈	—	194	1	no	—
55	C₂×Q₈	—	97	1	no	—
56	C₂×C₈	—	97	1	no	—
57	C₂×C₈	—	97	1	no	—
58	C₂×C₈	—	194	1	no	—
59	C₄⋊C₄	—	194	1	no	—
60	C₄⋊C₄	#11715906	1	1	yes	characteristic
61	C₄²	—	97	1	no	—
62	C₄²	—	194	1	no	—
63	C₁₆	—	388	1	no	—
64	C₁₆	—	194	2	no	—
65	Q₈	—	97	2	no	—
66	Q₈	—	194	1	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	97	1	no	—
69	C₂×C₄	—	97	1	no	—
70	C₂×C₄	—	194	1	no	—
71	C₂×C₄	C₂×#11715906	1	1	yes	characteristic
72	C₈	—	97	1	no	—
73	C₈	—	97	1	no	—
74	C₈	—	388	1	no	—
75	C₈	—	194	1	no	—
76	C₂²	C₂²⋊#11715906	1	1	yes	characteristic
77	C₄	—	4	1	no	—
78	C₄	—	97	1	no	—
79	C₄	—	97	1	no	—
80	C₄	—	194	2	no	—
81	C₄	—	194	1	no	—
82	C₄	#15690905	1	1	yes	characteristic
83	C₄	#15690907	1	1	yes	characteristic
84	C₂	Q₈⋊#11715906	1	1	yes	characteristic
85	C₂	#20261832	1	1	yes	characteristic
86	C₂	#20261835	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊#11715906
commutator	C₄×C₉₇	C₂×C₁₆
Frattini	C₂×C₄	C₂×#11715906
Fitting	C₄⋊C₄×C₉₇	C₈
socle, prime radical	C₂²×C₉₇	C₂²⋊C₈
cosocle	C₄×#11715897	C₂²
hypercenter	C₄⋊C₄	#11715906
hypocenter	C₉₇	Q₈⋊C₁₆
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
27. Dic₁₉₄	C₁₆	semidirect
44. C₉₇	Q₈⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#13095523	2
C₄×#13095523	2
#20261784	2
Q₈⋊C₁₆	97