Subgroups — #27156644

Order 12416.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261738	C₂	1	2	yes	maximal
2	#20260758	C₂	1	1	yes	maximal, characteristic
3	C₂×#13095520	C₄	1	4	yes	—
4	#15690877	—	2	2	no	—
5	C₄⋊#11715897	—	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	#13095520	—	4	4	no	—
11	C₂×#11715897	—	2	4	no	—
12	C₂²×Dic₉₇	C₂×C₄	1	2	yes	—
13	#13095495	—	4	1	no	—
14	C₂×#11715895	—	2	1	no	—
15	C₄×Dic₉₇	D₄	1	1	yes	characteristic
16	C₄×Dic₉₇	D₄	1	2	yes	—
17	C₄×Dic₉₇	Q₈	1	1	yes	characteristic
18	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
19	M₁₆×C₉₇	—	4	1	no	—
20	C₂×C₈×C₉₇	—	2	1	no	—
21	C₂³×C₉₇	C₄²	1	1	yes	characteristic
22	C₂×C₄×C₉₇	—	2	1	no	—
23	C₂×C₄×C₉₇	—	2	1	no	—
24	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₉₇	C₄⋊C₄	1	1	yes	characteristic
26	C₂×Dic₉₇	—	2	2	no	—
27	C₂×Dic₉₇	—	2	2	no	—
28	C₂×Dic₉₇	C₂²⋊C₄	1	2	yes	—
29	C₂×Dic₉₇	C₄⋊C₄	1	2	yes	—
30	#11715897	—	4	4	no	—
31	C₈×C₉₇	—	4	1	no	—
32	#11715895	—	4	1	no	—
33	C₂²×C₉₇	—	2	2	no	—
34	C₂²×C₉₇	—	2	1	no	—
35	C₂²×C₉₇	#95	1	1	yes	characteristic
36	C₄×C₉₇	—	2	1	no	—
37	C₄×C₉₇	#100	1	1	yes	characteristic
38	C₄×C₉₇	#101	1	1	yes	characteristic
39	Dic₉₇	—	2	2	no	—
40	Dic₉₇	#100	1	2	yes	—
41	Dic₉₇	#101	1	2	yes	—
42	C₂×C₉₇	—	4	1	no	—
43	C₂×C₉₇	#343	1	2	yes	—
44	C₂×C₉₇	#343	1	1	yes	characteristic
45	#1309	—	97	1	no	maximal, Sylow
46	C₉₇	#1309	1	1	yes	characteristic, complemented, Sylow
47	#423	—	97	2	no	—
48	#423	—	97	1	no	—
49	C₂×M₁₆	—	97	4	no	—
50	C₂×M₁₆	—	97	1	no	—
51	C₂×M₁₆	#87604	1	1	yes	characteristic
52	C₂×C₄²	—	97	1	no	—
53	C₄⋊C₈	—	194	2	no	—
54	C₄⋊C₈	—	194	2	no	—
55	C₄⋊C₈	—	194	2	no	—
56	C₂²×C₄	—	97	2	no	—
57	C₂²×C₄	C₂×#87604	1	1	yes	characteristic
58	M₁₆	—	4	1	no	—
59	M₁₆	—	388	1	no	—
60	M₁₆	—	388	4	no	—
61	C₂×C₈	—	2	1	no	—
62	C₂×C₈	—	194	4	no	—
63	C₂×C₈	—	194	1	no	—
64	C₄²	—	97	2	no	—
65	C₄²	—	97	1	no	—
66	C₄²	—	97	1	no	—
67	C₂³	C₄×#87604	1	1	yes	characteristic
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	97	2	no	—
71	C₂×C₄	—	97	2	no	—
72	C₂×C₄	—	194	2	no	—
73	C₂×C₄	—	194	2	no	—
74	C₂×C₄	C₂²⋊#87604	1	1	yes	characteristic
75	C₂×C₄	C₄⋊#87604	1	1	yes	characteristic
76	C₈	—	4	1	no	—
77	C₈	—	388	4	no	—
78	C₈	—	388	1	no	—
79	C₂²	—	2	2	no	—
80	C₂²	—	2	1	no	—
81	C₂²	(C₂×#87604)⋊C₄	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	97	2	no	—
84	C₄	—	97	2	no	—
85	C₄	—	194	2	no	—
86	C₄	#15690870	1	1	yes	characteristic
87	C₄	#15690871	1	1	yes	characteristic
88	C₂	—	4	1	no	—
89	C₂	#20261614	1	2	yes	—
90	C₂	#20261615	1	1	yes	characteristic

	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	#1309
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
46. C₉₇	#1309	semidirect

Maximal subgroups

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

subgroup	index
#20261738	2
#20260758	2
#1309	97