Group #27140815 — subgroups

Subgroups — #27140815

Order 12416.

228 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261465	C₂	1	1	yes	maximal, characteristic, complemented
2	#20261109	C₂	1	1	yes	maximal, characteristic
3	#20260936	C₂	1	2	yes	maximal
4	#20260465	C₂	1	2	yes	maximal, complemented
5	#333×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13095500	—	2	2	no	—
7	C₂×#13095498	—	2	2	no	—
8	#15690809	—	2	2	no	—
9	#15690808	—	2	2	no	—
10	#15690807	C₂²	1	1	yes	characteristic
11	C₄×Dic₁₉₄	C₂²	1	2	yes	—
12	#15690775	—	2	2	no	—
13	#15690732	—	2	2	no	—
14	C₄×#11715895	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₉₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₉₇	C₂²	1	2	yes	—
17	C₂×Dic₁₉₄	—	2	2	no	—
18	C₂×Dic₁₉₄	D₄	1	2	yes	—
19	C₂×D₃₈₈	—	2	1	no	—
20	C₂×D₃₈₈	—	2	2	no	—
21	#13095500	—	4	4	no	—
22	#13095498	—	4	4	no	—
23	C₂×#11715895	—	2	1	no	—
24	C₄⋊Dic₉₇	D₄	1	2	yes	—
25	#13095490	—	2	4	no	—
26	#13095490	—	2	2	no	—
27	C₄×Dic₉₇	—	2	2	no	—
28	C₄×Dic₉₇	—	2	2	no	—
29	C₂×Q₈×C₉₇	D₄	1	2	yes	—
30	C₂×C₈×C₉₇	—	2	2	no	—
31	C₄⋊C₄×C₉₇	—	2	2	no	—
32	C₄²×C₉₇	C₂³	1	1	yes	characteristic
33	C₂²×D₉₇	—	4	1	no	—
34	Q₈×C₉₇	—	4	2	no	—
35	C₂×C₄×C₉₇	—	2	2	no	—
36	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
38	C₂×Dic₉₇	—	2	2	no	—
39	C₂×Dic₉₇	—	2	2	no	—
40	D₃₈₈	—	4	2	no	—
41	D₃₈₈	—	4	2	no	—
42	D₃₈₈	—	4	2	no	—
43	Dic₁₉₄	—	2	2	no	—
44	Dic₁₉₄	—	8	2	no	—
45	Dic₁₉₄	SD₁₆	1	4	yes	complemented
46	C₈×C₉₇	—	4	2	no	—
47	#11715895	—	2	2	no	—
48	C₂²×C₉₇	#120	1	1	yes	characteristic
49	C₂×D₉₇	—	8	2	no	—
50	C₂×D₉₇	—	8	1	no	—
51	C₄×C₉₇	—	2	1	no	—
52	C₄×C₉₇	—	4	2	no	—
53	C₄×C₉₇	C₂×SD₁₆	1	2	yes	—
54	C₄×C₉₇	#137	1	2	yes	—
55	Dic₉₇	—	2	4	no	—
56	Dic₉₇	—	4	2	no	—
57	C₂×C₉₇	#448	1	2	yes	—
58	C₂×C₉₇	#453	1	1	yes	characteristic
59	D₉₇	—	16	1	no	—
60	#1623	—	97	1	no	maximal, Sylow
61	C₉₇	#1623	1	1	yes	characteristic, complemented, Sylow
62	Q₈²	—	97	1	no	—
63	#492	—	97	1	no	—
64	#460	—	97	2	no	—
65	#333	D₉₇	1	1	yes	characteristic, complemented
66	Q₈⋊C₈	—	97	2	no	—
67	C₂×SD₁₆	—	194	2	no	—
68	C₂×SD₁₆	—	194	2	no	—
69	C₄⋊Q₈	—	194	2	no	—
70	C₄⋊Q₈	—	194	2	no	—
71	C₄⋊Q₈	C₂×D₉₇	1	1	yes	characteristic
72	#127	—	97	1	no	—
73	C₄×Q₈	—	97	2	no	—
74	C₄×Q₈	—	194	2	no	—
75	C₄⋊C₈	C₂×D₉₇	1	2	yes	—
76	#102	—	194	2	no	—
77	C₄×C₈	—	97	1	no	—
78	C₂×Q₈	—	97	2	no	—
79	C₂×Q₈	—	194	2	no	—
80	C₂×Q₈	D₃₈₈	1	2	yes	—
81	C₂×D₄	—	194	2	no	—
82	C₂×D₄	—	194	1	no	—
83	SD₁₆	—	388	4	no	—
84	SD₁₆	—	388	4	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	194	1	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	97	2	no	—
89	C₄⋊C₄	—	194	4	no	—
90	C₄⋊C₄	—	194	2	no	—
91	C₄²	—	194	2	no	—
92	C₄²	—	194	2	no	—
93	C₄²	C₂²×D₉₇	1	1	yes	characteristic
94	C₂³	—	388	1	no	—
95	Q₈	—	4	2	no	—
96	Q₈	—	776	2	no	—
97	Q₈	—	97	4	no	—
98	Q₈	—	194	2	no	—
99	D₄	—	388	2	no	—
100	D₄	—	388	2	no	—
101	D₄	—	388	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	194	2	no	—
104	C₂×C₄	—	194	2	no	—
105	C₂×C₄	D₄×D₉₇	1	2	yes	—
106	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
107	C₈	—	4	2	no	—
108	C₈	—	194	2	no	—
109	C₂²	—	776	2	no	—
110	C₂²	—	776	1	no	—
111	C₂²	#15690780	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	4	2	no	—
114	C₄	—	388	2	no	—
115	C₄	—	194	4	no	—
116	C₄	SD₁₆×D₉₇	1	2	yes	—
117	C₄	#15690842	1	2	yes	—
118	C₂	—	1552	1	no	—
119	C₂	#20260875	1	2	yes	—
120	C₂	#20260905	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20261465

C₂

yes

maximal, characteristic, complemented

#20261109

C₂

yes

maximal, characteristic

#20260936

C₂

yes

maximal

#20260465

C₂

yes

maximal, complemented

#333×C₉₇

C₂

yes

maximal, characteristic, complemented

—

—

—

—

C₂²

yes

characteristic

C₄×Dic₁₉₄

C₂²

yes

—

#15690775

—

#15690732

—

C₄×#11715895

C₂²

yes

characteristic

C₄⋊Q₈×C₉₇

C₂²

yes

characteristic

C₄⋊C₈×C₉₇

C₂²

yes

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

D₄

yes

—

C₂×D₃₈₈

—

C₂×D₃₈₈

—

#13095500

—

#13095498

—

C₂×#11715895

—

C₄⋊Dic₉₇

D₄

yes

—

#13095490

—

#13095490

—

C₄×Dic₉₇

—

C₄×Dic₉₇

—

C₂×Q₈×C₉₇

D₄

yes

—

C₂×C₈×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄²×C₉₇

C₂³

yes

characteristic

C₂²×D₉₇

—

Q₈×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×Dic₉₇

—

C₂×Dic₉₇

—

D₃₈₈

—

D₃₈₈

—

D₃₈₈

—

Dic₁₉₄

—

Dic₁₉₄

—

Dic₁₉₄

SD₁₆

yes

complemented

C₈×C₉₇

—

#11715895

—

C₂²×C₉₇

#120

yes

characteristic

C₂×D₉₇

—

C₂×D₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₂×SD₁₆

yes

—

C₄×C₉₇

#137

yes

—

Dic₉₇

—

Dic₉₇

—

C₂×C₉₇

#448

yes

—

C₂×C₉₇

#453

yes

characteristic

D₉₇

—

#1623

—

maximal, Sylow

C₉₇

#1623

yes

characteristic, complemented, Sylow

Q₈²

—

#492

—

#460

—

#333

D₉₇

yes

characteristic, complemented

Q₈⋊C₈

—

C₂×SD₁₆

—

194

—

C₂×SD₁₆

—

194

—

C₄⋊Q₈

—

194

—

C₄⋊Q₈

—

194

—

C₄⋊Q₈

C₂×D₉₇

yes

characteristic

#127

—

C₄×Q₈

—

C₄×Q₈

—

194

—

C₄⋊C₈

C₂×D₉₇

yes

—

#102

—

194

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

194

—

C₂×Q₈

D₃₈₈

yes

—

C₂×D₄

—

194

—

C₂×D₄

—

194

—

SD₁₆

—

388

—

SD₁₆

—

388

—

C₂×C₈

—

C₂×C₈

—

194

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

194

—

C₄⋊C₄

—

194

—

C₄²

—

194

—

C₄²

—

194

—

C₄²

C₂²×D₉₇

yes

characteristic

C₂³

—

388

—

Q₈

—

Q₈

—

776

—

Q₈

—

Q₈

—

194

—

D₄

—

388

—

100

D₄

—

388

—

101

D₄

—

388

—

102

C₂×C₄

—

103

C₂×C₄

—

194

—

104

C₂×C₄

—

194

—

105

C₂×C₄

D₄×D₉₇

yes

—

106

C₂×C₄

C₂×D₃₈₈

yes

characteristic

107

C₈

—

108

C₈

—

194

—

109

C₂²

—

776

—

110

C₂²

—

776

—

111

C₂²

#15690780

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

388

—

115

C₄

—

194

—

116

C₄

SD₁₆×D₉₇

yes

—

117

C₄

#15690842

yes

—

118

C₂

—

1552

—

119

C₂

#20260875

yes

—

120

C₂

#20260905

yes

characteristic

	subgroup	G/H
center	C₂²	#15690780
commutator, cosocle	C₄²×C₉₇	C₂³
Frattini	C₄²	C₂²×D₉₇
Fitting	#333×C₉₇	C₂
socle	C₂²×C₉₇	#120
prime radical	#15690807	C₂²
hypercenter	#333	D₉₇
hypocenter	C₉₇	#1623
LCS	G ▷ C₄²×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

#15690780

commutator, cosocle

C₄²×C₉₇

C₂³

Frattini

C₄²

C₂²×D₉₇

Fitting

#333×C₉₇

C₂

C₂²×C₉₇

C₂²

D₉₇

C₉₇

G ▷ C₄²×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
1. #20261465	C₂	semidirect
4. #20260465	C₂	semidirect
5. #333×C₉₇	C₂	semidirect
45. Dic₁₉₄	SD₁₆	semidirect
61. C₉₇	#1623	semidirect
65. #333	D₉₇	semidirect

H ≅ G/N

type

1. #20261465

C₂

semidirect

4. #20260465

C₂

semidirect

5. #333×C₉₇

C₂

semidirect

45. Dic₁₉₄

SD₁₆

semidirect

61. C₉₇

#1623

semidirect

65. #333

D₉₇

semidirect

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

subgroup	index
#20261465	2
#20261109	2
#20260936	2
#20260465	2
#333×C₉₇	2
#1623	97

subgroup

index

#333×C₉₇

Subgroups — #27140815

Conjugacy classes of subgroups

Complements

Maximal subgroups