Group #20260592 — subgroups

Subgroups — #20260592

Order 6208.

234 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 87 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	C₂²×C₄×D₉₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₉₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#13095488	C₂	1	2	yes	maximal
4	#15690717	C₂	1	2	yes	maximal, complemented
5	C₂×C₄⋊C₄×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₉₇	C₂²	1	1	yes	characteristic
7	C₂²×Dic₉₇	C₂²	1	2	yes	—
8	C₂²×Dic₉₇	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₉₇	—	2	2	no	—
10	C₂×C₄×D₉₇	—	2	2	no	—
11	C₂×C₄×D₉₇	C₄	1	4	yes	complemented
12	C₄⋊Dic₉₇	—	2	2	no	—
13	#13095488	—	2	4	no	—
14	C₂²×C₄×C₉₇	C₂²	1	2	yes	—
15	C₂²×C₄×C₉₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₄×C₉₇	—	2	2	no	—
17	C₂³×C₉₇	C₂³	1	1	yes	characteristic
18	C₂²×D₉₇	—	2	1	no	—
19	C₂²×D₉₇	—	2	1	no	—
20	C₂²×D₉₇	—	2	1	no	—
21	C₂²×D₉₇	—	2	1	no	—
22	C₂²×D₉₇	C₂×C₄	1	2	yes	—
23	C₂²×D₉₇	D₄	1	2	yes	—
24	C₂²×D₉₇	Q₈	1	2	yes	—
25	C₂×C₄×C₉₇	—	2	2	no	—
26	C₂×C₄×C₉₇	—	2	2	no	—
27	C₂×C₄×C₉₇	—	2	2	no	—
28	C₂×C₄×C₉₇	C₂×C₄	1	2	yes	—
29	C₂×C₄×C₉₇	D₄	1	2	yes	—
30	C₂×C₄×C₉₇	D₄	1	2	yes	—
31	C₂×Dic₉₇	—	2	1	no	—
32	C₂×Dic₉₇	—	2	1	no	—
33	C₂×Dic₉₇	—	2	2	no	—
34	C₂×Dic₉₇	—	2	2	no	—
35	C₂×Dic₉₇	—	2	2	no	—
36	C₂×Dic₉₇	C₂×C₄	1	2	yes	—
37	C₄×D₉₇	—	2	8	no	—
38	C₂²×C₉₇	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₉₇	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₉₇	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₉₇	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₉₇	C₂×Q₈	1	1	yes	characteristic
43	C₂²×C₉₇	#41	1	1	yes	characteristic
44	C₂²×C₉₇	#41	1	1	yes	characteristic
45	C₂×D₉₇	—	2	2	no	—
46	C₂×D₉₇	—	2	2	no	—
47	C₂×D₉₇	—	2	2	no	—
48	C₂×D₉₇	—	2	2	no	—
49	C₂×D₉₇	—	2	2	no	—
50	C₂×D₉₇	—	2	2	no	—
51	C₂×D₉₇	C₄⋊C₄	1	4	yes	complemented
52	C₄×C₉₇	—	4	2	no	—
53	C₄×C₉₇	C₂²⋊C₄	1	4	yes	complemented
54	Dic₉₇	—	2	2	no	—
55	Dic₉₇	—	4	2	no	—
56	C₂×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₉₇	#117	1	1	yes	characteristic
59	C₂×C₉₇	#121	1	1	yes	characteristic
60	C₂×C₉₇	#121	1	1	yes	characteristic
61	C₂×C₉₇	C₂²⋊Q₈	1	1	yes	characteristic
62	C₂×C₉₇	C₂²⋊Q₈	1	1	yes	characteristic
63	D₉₇	—	2	4	no	—
64	C₉₇	#380	1	1	yes	characteristic, complemented, Sylow
65	#380	—	97	1	no	maximal, Sylow
66	C₂³×C₄	—	97	1	no	—
67	C₂×C₄⋊C₄	—	97	1	no	—
68	C₂×C₄⋊C₄	D₉₇	1	1	yes	characteristic, complemented
69	C₂×C₂²⋊C₄	—	97	2	no	—
70	#95	—	97	2	no	—
71	C₂⁴	—	97	1	no	—
72	C₂²×C₄	—	97	1	no	—
73	C₂²×C₄	—	97	4	no	—
74	C₂²×C₄	—	97	2	no	—
75	C₂²×C₄	—	194	2	no	—
76	C₂²×C₄	—	194	2	no	—
77	C₂²×C₄	C₂×D₉₇	1	2	yes	—
78	C₂²×C₄	C₂×D₉₇	1	1	yes	characteristic
79	C₄⋊C₄	—	2	2	no	—
80	C₄⋊C₄	—	194	2	no	—
81	C₂²⋊C₄	—	194	4	no	—
82	C₂³	—	97	2	no	—
83	C₂³	—	97	2	no	—
84	C₂³	—	97	2	no	—
85	C₂³	—	194	1	no	—
86	C₂³	—	194	1	no	—
87	C₂³	—	194	1	no	—
88	C₂³	—	194	1	no	—
89	C₂³	C₂²×D₉₇	1	1	yes	characteristic
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	97	2	no	—
94	C₂×C₄	—	194	2	no	—
95	C₂×C₄	—	194	2	no	—
96	C₂×C₄	—	194	2	no	—
97	C₂×C₄	—	194	1	no	—
98	C₂×C₄	—	194	1	no	—
99	C₂×C₄	—	194	8	no	—
100	C₂×C₄	C₄×D₉₇	1	2	yes	—
101	C₂×C₄	D₃₈₈	1	2	yes	—
102	C₂×C₄	C₂²⋊D₉₇	1	2	yes	—
103	C₂²	—	97	4	no	—
104	C₂²	—	194	2	no	—
105	C₂²	—	194	2	no	—
106	C₂²	—	194	2	no	—
107	C₂²	—	194	2	no	—
108	C₂²	—	194	2	no	—
109	C₂²	—	194	2	no	—
110	C₂²	C₂×C₄×D₉₇	1	1	yes	characteristic
111	C₂²	D₄×D₉₇	1	1	yes	characteristic
112	C₂²	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
113	C₂²	C₂×D₃₈₈	1	1	yes	characteristic
114	C₂²	Q₈×D₉₇	1	1	yes	characteristic
115	C₂²	#13095510	1	1	yes	characteristic
116	C₂²	#13095512	1	1	yes	characteristic
117	C₄	—	4	2	no	—
118	C₄	—	388	2	no	—
119	C₄	—	194	2	no	—
120	C₄	#13095488	1	4	yes	complemented
121	C₂	—	194	4	no	—
122	C₂	C₂×#13095488	1	1	yes	characteristic
123	C₂	C₄⋊C₄×D₉₇	1	1	yes	characteristic
124	C₂	#15690768	1	1	yes	characteristic
125	C₂	#15690781	1	1	yes	characteristic
126	C₂	#15690785	1	1	yes	characteristic
127	C₂	#15690786	1	1	yes	characteristic
128	C₂	#15690790	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄×D₉₇

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₉₇

C₂

yes

maximal, characteristic, complemented

C₂×#13095488

C₂

yes

maximal

#15690717

C₂

yes

maximal, complemented

C₂×C₄⋊C₄×C₉₇

C₂

yes

maximal, characteristic, complemented

C₂³×D₉₇

C₂²

yes

characteristic

C₂²×Dic₉₇

C₂²

yes

—

C₂²×Dic₉₇

C₂²

yes

characteristic

C₂×C₄×D₉₇

—

C₂×C₄×D₉₇

—

C₂×C₄×D₉₇

C₄

yes

complemented

C₄⋊Dic₉₇

—

#13095488

—

C₂²×C₄×C₉₇

C₂²

yes

—

C₂²×C₄×C₉₇

C₂²

yes

characteristic

C₄⋊C₄×C₉₇

—

C₂³×C₉₇

C₂³

yes

characteristic

C₂²×D₉₇

—

C₂²×D₉₇

—

C₂²×D₉₇

—

C₂²×D₉₇

—

C₂²×D₉₇

C₂×C₄

yes

—

C₂²×D₉₇

D₄

yes

—

C₂²×D₉₇

Q₈

yes

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂×C₄

yes

—

C₂×C₄×C₉₇

D₄

yes

—

C₂×C₄×C₉₇

D₄

yes

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

C₂×C₄

yes

—

C₄×D₉₇

—

C₂²×C₉₇

C₂²×C₄

yes

characteristic

C₂²×C₉₇

C₂×D₄

yes

characteristic

C₂²×C₉₇

C₂×D₄

yes

characteristic

C₂²×C₉₇

C₂×D₄

yes

characteristic

C₂²×C₉₇

C₂×Q₈

yes

C₂²×C₉₇

yes

C₂²×C₉₇

yes

C₂×D₉₇

—

C₂×D₉₇

—

C₂×D₉₇

—

C₂×D₉₇

—

C₂×D₉₇

—

C₂×D₉₇

—

C₂×D₉₇

C₄⋊C₄

yes

complemented

C₄×C₉₇

—

C₄×C₉₇

C₂²⋊C₄

yes

complemented

Dic₉₇

—

Dic₉₇

—

C₂×C₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₉₇

C₂×C₄⋊C₄

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

C₂²⋊Q₈

yes

characteristic

C₂×C₉₇

C₂²⋊Q₈

yes

characteristic

D₉₇

—

C₉₇

#380

yes

characteristic, complemented, Sylow

#380

—

maximal, Sylow

C₂³×C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

D₉₇

yes

characteristic, complemented

C₂×C₂²⋊C₄

—

#95

—

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

194

—

C₂²×C₄

—

194

—

C₂²×C₄

C₂×D₉₇

yes

—

C₂²×C₄

C₂×D₉₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

194

—

C₂²⋊C₄

—

194

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

194

—

C₂³

—

194

—

C₂³

—

194

—

C₂³

—

194

—

C₂³

C₂²×D₉₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

194

—

C₂×C₄

—

194

—

C₂×C₄

—

194

—

C₂×C₄

—

194

—

C₂×C₄

—

194

—

C₂×C₄

—

194

—

100

C₂×C₄

C₄×D₉₇

yes

—

101

C₂×C₄

D₃₈₈

yes

—

102

C₂×C₄

C₂²⋊D₉₇

yes

—

103

C₂²

—

104

C₂²

—

194

—

105

C₂²

—

194

—

106

C₂²

—

194

—

107

C₂²

—

194

—

108

C₂²

—

194

—

109

C₂²

—

194

—

110

C₂²

C₂×C₄×D₉₇

yes

characteristic

111

C₂²

D₄×D₉₇

yes

characteristic

112

C₂²

C₂×C₂²⋊D₉₇

yes

characteristic

113

C₂²

C₂×D₃₈₈

yes

characteristic

114

C₂²

Q₈×D₉₇

yes

characteristic

115

C₂²

#13095510

yes

characteristic

116

C₂²

#13095512

yes

characteristic

117

C₄

—

118

C₄

—

388

—

119

C₄

—

194

—

120

C₄

#13095488

yes

complemented

121

C₂

—

194

—

122

C₂

C₂×#13095488

yes

characteristic

123

C₂

C₄⋊C₄×D₉₇

yes

characteristic

124

C₂

#15690768

yes

characteristic

125

C₂

#15690781

yes

characteristic

126

C₂

#15690785

yes

characteristic

127

C₂

#15690786

yes

characteristic

128

C₂

#15690790

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₉₇
commutator	C₂²×C₉₇	C₂²×C₄
Fitting	C₂×C₄⋊C₄×C₉₇	C₂
socle	C₂³×C₉₇	C₂³
hypercenter	C₂×C₄⋊C₄	D₉₇
hypocenter	C₉₇	#380
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂²×D₉₇

commutator

C₂²×C₉₇

C₂²×C₄

Fitting

C₂×C₄⋊C₄×C₉₇

C₂

socle

C₂³×C₉₇

C₂³

hypercenter

C₂×C₄⋊C₄

D₉₇

C₉₇

G ▷ C₂²×C₉₇ ▷ C₉₇

C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
2. C₂×C₄⋊Dic₉₇	C₂	semidirect
4. #15690717	C₂	semidirect
5. C₂×C₄⋊C₄×C₉₇	C₂	semidirect
11. C₂×C₄×D₉₇	C₄	semidirect
51. C₂×D₉₇	C₄⋊C₄	semidirect
53. C₄×C₉₇	C₂²⋊C₄	semidirect
64. C₉₇	#380	semidirect
68. C₂×C₄⋊C₄	D₉₇	semidirect
120. C₄	#13095488	semidirect

H ≅ G/N

type

2. C₂×C₄⋊Dic₉₇

C₂

semidirect

4. #15690717

C₂

semidirect

5. C₂×C₄⋊C₄×C₉₇

C₂

semidirect

11. C₂×C₄×D₉₇

C₄

semidirect

51. C₂×D₉₇

C₄⋊C₄

semidirect

53. C₄×C₉₇

C₂²⋊C₄

semidirect

64. C₉₇

#380

semidirect

68. C₂×C₄⋊C₄

D₉₇

semidirect

120. C₄

#13095488

semidirect

Subgroups — #20260592

Conjugacy classes of subgroups

Complements