Subgroups — #27156642

Order 12416.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261756	C₂	1	2	yes	maximal
2	#20260758	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#11715897	—	2	2	no	—
4	C₂²⋊#11715897	C₄	1	4	yes	—
5	#15690865	—	2	2	no	—
6	C₂×#13095495	C₄	1	1	yes	characteristic
7	C₂×C₄×Dic₉₇	C₂²	1	1	yes	characteristic
8	#15690739	—	2	2	no	—
9	C₂×M₁₆×C₉₇	C₄	1	1	yes	characteristic
10	C₂×#11715897	—	2	4	no	—
11	C₂²×Dic₉₇	C₂×C₄	1	2	yes	—
12	#13095495	—	4	1	no	—
13	C₂×#11715895	—	2	1	no	—
14	C₄×Dic₉₇	D₄	1	1	yes	characteristic
15	C₄×Dic₉₇	D₄	1	2	yes	—
16	C₄×Dic₉₇	Q₈	1	1	yes	characteristic
17	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
18	M₁₆×C₉₇	—	4	1	no	—
19	C₂×C₈×C₉₇	—	2	1	no	—
20	C₂³×C₉₇	C₄²	1	1	yes	characteristic
21	C₂×C₄×C₉₇	—	2	1	no	—
22	C₂×C₄×C₉₇	—	2	1	no	—
23	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
24	C₂×C₄×C₉₇	C₄⋊C₄	1	1	yes	characteristic
25	C₂×Dic₉₇	—	4	2	no	—
26	C₂×Dic₉₇	C₂²⋊C₄	1	2	yes	—
27	C₂×Dic₉₇	C₄⋊C₄	1	2	yes	—
28	#11715897	—	4	4	no	—
29	C₈×C₉₇	—	4	1	no	—
30	#11715895	—	4	1	no	—
31	C₂²×C₉₇	—	2	1	no	—
32	C₂²×C₉₇	—	2	2	no	—
33	C₂²×C₉₇	#95	1	1	yes	characteristic
34	C₄×C₉₇	—	2	1	no	—
35	C₄×C₉₇	#100	1	1	yes	characteristic
36	C₄×C₉₇	#101	1	1	yes	characteristic
37	Dic₉₇	—	2	2	no	—
38	Dic₉₇	—	2	2	no	—
39	C₂×C₉₇	—	4	1	no	—
40	C₂×C₉₇	#343	1	1	yes	characteristic
41	C₂×C₉₇	#344	1	2	yes	—
42	#1307	—	97	1	no	maximal, Sylow
43	C₉₇	#1307	1	1	yes	characteristic, complemented, Sylow
44	#432	—	97	2	no	—
45	#423	—	97	1	no	—
46	C₂×M₁₆	—	97	1	no	—
47	C₂×M₁₆	#87604	1	1	yes	characteristic
48	C₂×C₄²	—	97	1	no	—
49	C₄⋊C₈	—	194	2	no	—
50	C₄⋊C₈	—	194	2	no	—
51	C₂²⋊C₈	—	97	4	no	—
52	#97	—	194	2	no	—
53	C₂²×C₄	—	97	2	no	—
54	C₂²×C₄	C₂×#87604	1	1	yes	characteristic
55	M₁₆	—	4	1	no	—
56	M₁₆	—	388	1	no	—
57	C₂×C₈	—	2	1	no	—
58	C₂×C₈	—	194	4	no	—
59	C₂×C₈	—	194	1	no	—
60	C₄²	—	97	2	no	—
61	C₄²	—	97	1	no	—
62	C₄²	—	97	1	no	—
63	C₂³	C₄×#87604	1	1	yes	characteristic
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	97	2	no	—
67	C₂×C₄	—	97	2	no	—
68	C₂×C₄	—	388	2	no	—
69	C₂×C₄	C₂²⋊#87604	1	1	yes	characteristic
70	C₂×C₄	C₄⋊#87604	1	1	yes	characteristic
71	C₈	—	4	1	no	—
72	C₈	—	388	4	no	—
73	C₈	—	388	1	no	—
74	C₂²	—	2	1	no	—
75	C₂²	—	2	2	no	—
76	C₂²	(C₂×#87604)⋊C₄	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	—	194	2	no	—
79	C₄	—	194	2	no	—
80	C₄	#15690870	1	1	yes	characteristic
81	C₄	#15690871	1	1	yes	characteristic
82	C₂	—	4	1	no	—
83	C₂	#20261615	1	1	yes	characteristic
84	C₂	#20261618	1	2	yes	—

	subgroup	G/H
center	C₂²	(C₂×#87604)⋊C₄
commutator, prime radical	C₂³×C₉₇	C₄²
Frattini	C₂²×C₄	C₂×#87604
Fitting	C₂×M₁₆×C₉₇	C₄
socle	C₂²×C₉₇	#95
cosocle	C₂×C₄×Dic₉₇	C₂²
hypercenter	C₂×M₁₆	#87604
hypocenter	C₉₇	#1307
LCS	G ▷ C₂³×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
43. C₉₇	#1307	semidirect

Maximal subgroups

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

subgroup	index
#20261756	2
#20260758	2
#1307	97