Subgroups — #27159030

Order 12416.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261866	C₂	1	2	yes	maximal, complemented
2	#20261865	C₂	1	2	yes	maximal, complemented
3	#20261864	C₂	1	2	yes	maximal, complemented
4	C₂×#15690905	C₂	1	1	yes	maximal, characteristic
5	#15690909	C₂²	1	4	yes	complemented
6	#15690905	C₄	1	1	yes	characteristic
7	#15690905	C₄	1	1	yes	characteristic
8	#15690905	C₂²	1	1	yes	characteristic
9	#15690905	C₂²	1	1	yes	characteristic
10	C₂×#13095523	C₄	1	1	yes	characteristic
11	C₂×#13095523	C₂²	1	1	yes	characteristic
12	C₂×#13095519	C₄	1	1	yes	characteristic
13	#13095523	C₂×C₄	1	1	yes	characteristic
14	#13095523	C₂×C₄	1	1	yes	characteristic
15	#13095523	C₂×C₄	1	1	yes	characteristic
16	#13095523	C₂³	1	1	yes	characteristic
17	#13095519	C₈	1	1	yes	characteristic
18	#13095519	C₈	1	1	yes	characteristic
19	#13095519	C₂×C₄	1	1	yes	characteristic
20	#13095519	C₂×C₄	1	1	yes	characteristic
21	C₂×#11715897	C₈	1	1	yes	characteristic
22	C₂×#11715897	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×D₉₇	C₈	1	1	yes	characteristic
24	C₂²×D₉₇	C₁₆	1	1	yes	characteristic
25	C₂×C₄×C₉₇	C₁₆	1	1	yes	characteristic
26	C₂×Dic₉₇	C₂×C₈	1	1	yes	characteristic
27	C₄×D₉₇	C₁₆	1	1	yes	characteristic
28	C₄×D₉₇	C₁₆	1	1	yes	characteristic
29	C₄×D₉₇	C₂×C₈	1	1	yes	characteristic
30	C₄×D₉₇	C₂×C₈	1	1	yes	characteristic
31	#11715897	C₂×C₈	1	1	yes	characteristic
32	#11715897	C₂×C₈	1	1	yes	characteristic
33	#11715897	C₂×C₈	1	1	yes	characteristic
34	#11715897	C₂²×C₄	1	1	yes	characteristic
35	C₂²×C₉₇	C₂×C₁₆	1	1	yes	characteristic
36	C₂×D₉₇	—	2	2	no	—
37	C₂×D₉₇	C₂×C₁₆	1	1	yes	characteristic
38	C₂×D₉₇	C₂×C₁₆	1	1	yes	characteristic
39	C₄×C₉₇	C₂×C₁₆	1	1	yes	characteristic
40	C₄×C₉₇	C₂×C₁₆	1	1	yes	characteristic
41	Dic₉₇	C₂×C₁₆	1	1	yes	characteristic
42	Dic₉₇	C₂²×C₈	1	1	yes	characteristic
43	C₂×C₉₇	—	2	1	no	—
44	C₂×C₉₇	C₂²×C₁₆	1	1	yes	characteristic
45	D₉₇	—	2	1	no	—
46	D₉₇	M₆₄	1	2	yes	complemented
47	C₂×M₆₄	—	97	1	no	maximal, Sylow
48	C₉₇	C₂×M₆₄	1	1	yes	characteristic, complemented, Sylow
49	C₂²×C₁₆	—	97	1	no	—
50	M₆₄	—	97	2	no	—
51	M₆₄	—	97	2	no	—
52	C₂×C₃₂	—	97	2	no	—
53	C₂²×C₈	—	97	1	no	—
54	C₂×C₁₆	—	97	1	no	—
55	C₂×C₁₆	—	97	1	no	—
56	C₂×C₁₆	—	97	1	no	—
57	C₂×C₁₆	—	97	1	no	—
58	C₂×C₁₆	—	97	1	no	—
59	C₂×C₁₆	—	97	1	no	—
60	C₃₂	—	97	4	no	—
61	C₂²×C₄	—	97	1	no	—
62	C₂×C₈	—	97	1	no	—
63	C₂×C₈	—	97	1	no	—
64	C₂×C₈	—	97	1	no	—
65	C₂×C₈	—	97	1	no	—
66	C₂×C₈	—	97	1	no	—
67	C₂×C₈	—	97	1	no	—
68	C₁₆	—	97	1	no	—
69	C₁₆	—	97	1	no	—
70	C₁₆	—	97	1	no	—
71	C₁₆	—	97	1	no	—
72	C₂³	—	97	1	no	—
73	C₂×C₄	—	97	1	no	—
74	C₂×C₄	—	97	1	no	—
75	C₂×C₄	—	97	1	no	—
76	C₂×C₄	—	97	1	no	—
77	C₂×C₄	—	97	1	no	—
78	C₂×C₄	#13095525	1	1	yes	characteristic
79	C₈	—	97	1	no	—
80	C₈	—	97	1	no	—
81	C₈	—	97	1	no	—
82	C₈	—	97	1	no	—
83	C₂²	—	97	1	no	—
84	C₂²	—	97	1	no	—
85	C₂²	—	194	2	no	—
86	C₂²	C₂×#13095525	1	1	yes	characteristic
87	C₄	—	97	1	no	—
88	C₄	—	97	1	no	—
89	C₄	C₂×#13095525	1	1	yes	characteristic
90	C₄	C₂×#13095525	1	1	yes	characteristic
91	C₂	—	2	1	no	—
92	C₂	—	97	2	no	—
93	C₂	—	194	1	no	—
94	C₂	C₂²×#13095525	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#13095525
commutator, socle	C₂×C₉₇	C₂²×C₁₆
Frattini	C₂	C₂²×#13095525
Fitting	C₂×C₄×C₉₇	C₁₆
prime radical	C₂²×D₉₇	C₁₆
cosocle	#13095523	C₂³
hypercenter	C₂×C₄	#13095525
hypocenter	C₉₇	C₂×M₆₄
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
1. #20261866	C₂	semidirect
2. #20261865	C₂	semidirect
3. #20261864	C₂	semidirect
5. #15690909	C₂²	semidirect
46. D₉₇	M₆₄	semidirect
48. C₉₇	C₂×M₆₄	semidirect

Maximal subgroups

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

subgroup	index
#20261866	2
#20261865	2
#20261864	2
C₂×#15690905	2
C₂×M₆₄	97