Subgroups — #27158793

Order 12416.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261838	C₂	1	1	yes	maximal, characteristic
2	#20261829	C₂	1	1	yes	maximal, characteristic
3	#20261786	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	#15690865	—	2	1	no	—
9	C₄×#11715897	C₂²	1	1	yes	characteristic
10	Q₈×Dic₉₇	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	C₄⋊Dic₉₇	—	2	1	no	—
17	C₄⋊Dic₉₇	C₈	1	1	yes	characteristic
18	C₄×Dic₉₇	—	2	1	no	—
19	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
20	C₂×Q₈×C₉₇	C₈	1	1	yes	characteristic
21	Q₈×C₉₇	—	4	1	no	—
22	C₂×C₄×C₉₇	—	2	1	no	—
23	C₂×C₄×C₉₇	C₂×C₈	1	1	yes	characteristic
24	C₂×Dic₉₇	—	2	1	no	—
25	C₂×Dic₉₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×Dic₉₇	M₁₆	1	1	yes	characteristic
27	#11715897	—	2	1	no	—
28	#11715897	—	4	1	no	—
29	#11715897	SD₁₆	1	1	yes	characteristic
30	#11715897	Q₁₆	1	1	yes	characteristic
31	C₂²×C₉₇	C₂²⋊C₈	1	1	yes	characteristic
32	C₄×C₉₇	—	2	1	no	—
33	C₄×C₉₇	—	4	1	no	—
34	Dic₉₇	—	2	1	no	—
35	Dic₉₇	—	4	1	no	—
36	Dic₉₇	#103	1	1	yes	characteristic
37	Dic₉₇	#104	1	1	yes	characteristic
38	C₂×C₉₇	Q₈⋊C₈	1	1	yes	characteristic
39	C₂×C₉₇	#349	1	1	yes	characteristic
40	C₂×C₉₇	#350	1	1	yes	characteristic
41	#1338	—	97	1	no	maximal, Sylow
42	C₉₇	#1338	1	1	yes	characteristic, complemented, Sylow
43	#446	—	97	1	no	—
44	C₄⋊C₁₆	—	97	1	no	—
45	#346	—	97	1	no	—
46	C₄×Q₈	—	97	1	no	—
47	C₂×C₁₆	—	194	1	no	—
48	C₂×C₁₆	—	194	1	no	—
49	C₄⋊C₈	—	97	1	no	—
50	C₄⋊C₈	—	194	1	no	—
51	#97	—	194	1	no	—
52	C₄×C₈	—	97	1	no	—
53	C₂×Q₈	#11715906	1	1	yes	characteristic
54	C₂×C₈	—	97	1	no	—
55	C₂×C₈	—	97	1	no	—
56	C₂×C₈	—	194	1	no	—
57	C₄⋊C₄	—	97	1	no	—
58	C₄⋊C₄	—	194	1	no	—
59	C₄²	—	97	1	no	—
60	C₄²	—	194	1	no	—
61	C₁₆	—	388	1	no	—
62	C₁₆	—	194	2	no	—
63	Q₈	—	4	1	no	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	97	1	no	—
66	C₂×C₄	—	97	1	no	—
67	C₂×C₄	—	194	1	no	—
68	C₂×C₄	C₂×#11715906	1	1	yes	characteristic
69	C₈	—	97	1	no	—
70	C₈	—	97	1	no	—
71	C₈	—	388	1	no	—
72	C₈	—	194	1	no	—
73	C₂²	C₂²⋊#11715906	1	1	yes	characteristic
74	C₄	—	2	1	no	—
75	C₄	—	4	1	no	—
76	C₄	—	97	1	no	—
77	C₄	—	97	1	no	—
78	C₄	—	388	1	no	—
79	C₄	—	194	1	no	—
80	C₂	Q₈⋊#11715906	1	1	yes	characteristic
81	C₂	#20261833	1	1	yes	characteristic
82	C₂	#20261836	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊#11715906
commutator	C₂×C₄×C₉₇	C₂×C₈
Frattini	C₂×C₄	C₂×#11715906
Fitting	C₂×Q₈×C₉₇	C₈
socle, prime radical	C₂²×C₉₇	C₂²⋊C₈
cosocle	C₄×#11715897	C₂²
hypercenter	C₂×Q₈	#11715906
hypocenter	C₉₇	#1338
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
42. C₉₇	#1338	semidirect

Maximal subgroups

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

subgroup	index
#20261838	2
#20261829	2
#20261786	2
#1338	97