Group #20261413 — subgroups

Subgroups — #20261413

Order 6208.

248 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 95 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	#15690803	C₂	1	2	yes	maximal, complemented
2	#15690797	C₂	1	1	yes	maximal, characteristic, complemented
3	#15690792	C₂	1	2	yes	maximal, complemented
4	#15690788	C₂	1	2	yes	maximal, complemented
5	#15690778	C₂	1	1	yes	maximal, characteristic
6	C₄×C₂²⋊D₉₇	C₂	1	2	yes	maximal, complemented
7	#15690768	C₂	1	2	yes	maximal, complemented
8	C₂²⋊C₄×D₉₇	C₂	1	2	yes	maximal, complemented
9	#123×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×D₉₇	C₂²	1	1	yes	characteristic
11	C₂×Dic₁₉₄	C₂²	1	1	yes	characteristic, complemented
12	C₂×C₂²⋊D₉₇	C₂²	1	2	yes	—
13	C₂×C₄×D₉₇	C₂²	1	2	yes	—
14	C₂×C₄×D₉₇	C₂²	1	2	yes	—
15	C₄⋊Dic₉₇	C₂²	1	2	yes	complemented
16	#13095490	C₂²	1	2	yes	complemented
17	#13095490	C₂²	1	2	yes	complemented
18	C₂²⋊Dic₉₇	C₂²	1	2	yes	—
19	C₂²⋊Dic₉₇	C₂²	1	1	yes	characteristic
20	C₂²⋊Dic₉₇	C₂²	1	2	yes	complemented
21	#13095488	C₂²	1	2	yes	—
22	#13095488	C₂²	1	2	yes	—
23	#13095488	C₂²	1	2	yes	—
24	C₄×Dic₉₇	C₂²	1	1	yes	characteristic, complemented
25	C₄×Dic₉₇	C₂²	1	2	yes	complemented
26	C₂×D₄×C₉₇	C₂²	1	1	yes	characteristic
27	C₂²×C₄×C₉₇	C₂²	1	1	yes	characteristic, complemented
28	C₄⋊C₄×C₉₇	C₂²	1	2	yes	complemented
29	C₂²⋊C₄×C₉₇	C₂²	1	2	yes	—
30	C₂²⋊C₄×C₉₇	C₂²	1	1	yes	characteristic, complemented
31	C₂³×C₉₇	C₂³	1	1	yes	characteristic
32	C₂³×C₉₇	C₂³	1	1	yes	characteristic
33	C₂²×D₉₇	—	2	4	no	—
34	C₂²×D₉₇	—	4	1	no	—
35	C₂²×D₉₇	C₂³	1	2	yes	—
36	D₄×C₉₇	—	4	1	no	—
37	C₂×C₄×C₉₇	—	4	1	no	—
38	C₂×C₄×C₉₇	C₂³	1	2	yes	—
39	C₂×C₄×C₉₇	C₂³	1	2	yes	—
40	C₂×C₄×C₉₇	C₂³	1	1	yes	characteristic
41	C₂²⋊D₉₇	—	2	4	no	—
42	C₂×Dic₉₇	C₂³	1	2	yes	—
43	C₂×Dic₉₇	C₂³	1	2	yes	—
44	C₂×Dic₉₇	C₂³	1	2	yes	—
45	C₄×D₉₇	—	2	4	no	—
46	C₄×D₉₇	—	4	2	no	—
47	Dic₁₉₄	—	4	1	no	—
48	C₂²×C₉₇	—	2	2	no	—
49	C₂²×C₉₇	—	2	2	no	—
50	C₂²×C₉₇	—	2	1	no	—
51	C₂²×C₉₇	—	2	1	no	—
52	C₂²×C₉₇	C₂⁴	1	1	yes	characteristic
53	C₂×D₉₇	—	2	2	no	—
54	C₂×D₉₇	—	2	2	no	—
55	C₂×D₉₇	—	4	4	no	—
56	C₂×D₉₇	#41	1	4	yes	—
57	C₄×C₉₇	—	2	2	no	—
58	C₄×C₉₇	—	2	1	no	—
59	C₄×C₉₇	—	2	2	no	—
60	Dic₉₇	—	2	2	no	—
61	Dic₉₇	—	2	2	no	—
62	Dic₉₇	—	2	2	no	—
63	C₂×C₉₇	—	4	1	no	—
64	C₂×C₉₇	—	4	1	no	—
65	C₂×C₉₇	C₂×#41	1	2	yes	—
66	C₂×C₉₇	H(5,𝔽₂)	1	1	yes	characteristic
67	D₉₇	—	2	4	no	—
68	C₉₇	#551	1	1	yes	characteristic, complemented, Sylow
69	#551	—	97	1	no	maximal, Sylow
70	#126	—	97	2	no	—
71	#124	—	97	1	no	—
72	#123	—	97	2	no	—
73	#123	D₉₇	1	1	yes	characteristic, complemented
74	C₂²⋊Q₈	—	97	2	no	—
75	#120	—	97	1	no	—
76	C₄×D₄	—	97	2	no	—
77	#117	—	97	2	no	—
78	C₂×C₂²⋊C₄	—	97	2	no	—
79	C₂⁴	—	97	1	no	—
80	C₂×Q₈	—	97	1	no	—
81	C₂×D₄	—	97	2	no	—
82	C₂×D₄	C₂×D₉₇	1	1	yes	characteristic
83	C₂²×C₄	—	97	2	no	—
84	C₂²×C₄	—	97	2	no	—
85	C₂²×C₄	C₂×D₉₇	1	1	yes	characteristic, complemented
86	C₄⋊C₄	—	97	2	no	—
87	C₄⋊C₄	—	97	2	no	—
88	C₄⋊C₄	—	97	2	no	—
89	C₄⋊C₄	C₂×D₉₇	1	2	yes	complemented
90	C₂²⋊C₄	—	97	2	no	—
91	C₂²⋊C₄	—	97	1	no	—
92	C₂²⋊C₄	—	97	2	no	—
93	C₂²⋊C₄	—	97	2	no	—
94	C₂²⋊C₄	—	97	2	no	—
95	C₂²⋊C₄	—	97	2	no	—
96	C₂²⋊C₄	C₂×D₉₇	1	2	yes	—
97	C₂²⋊C₄	C₂×D₉₇	1	1	yes	characteristic, complemented
98	C₄²	—	97	1	no	—
99	C₄²	—	97	2	no	—
100	C₂³	—	97	2	no	—
101	C₂³	—	388	1	no	—
102	C₂³	—	194	4	no	—
103	C₂³	C₂²×D₉₇	1	1	yes	characteristic
104	C₂³	C₂²×D₉₇	1	1	yes	characteristic
105	Q₈	—	388	1	no	—
106	D₄	—	4	1	no	—
107	D₄	—	194	4	no	—
108	C₂×C₄	—	4	1	no	—
109	C₂×C₄	—	97	2	no	—
110	C₂×C₄	—	97	2	no	—
111	C₂×C₄	—	97	2	no	—
112	C₂×C₄	—	388	2	no	—
113	C₂×C₄	—	194	4	no	—
114	C₂×C₄	C₂²×D₉₇	1	1	yes	characteristic
115	C₂×C₄	C₂²×D₉₇	1	2	yes	—
116	C₂×C₄	C₂²×D₉₇	1	2	yes	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	2	1	no	—
120	C₂²	—	2	1	no	—
121	C₂²	—	97	4	no	—
122	C₂²	—	388	4	no	—
123	C₂²	—	194	2	no	—
124	C₂²	—	194	2	no	—
125	C₂²	C₂³×D₉₇	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	2	2	no	—
128	C₄	—	2	1	no	—
129	C₄	—	194	2	no	—
130	C₄	—	194	2	no	—
131	C₄	—	194	2	no	—
132	C₂	—	4	1	no	—
133	C₂	—	4	1	no	—
134	C₂	—	194	4	no	—
135	C₂	#41×D₉₇	1	2	yes	—
136	C₂	#15690855	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15690803

C₂

yes

maximal, complemented

#15690797

C₂

yes

maximal, characteristic, complemented

#15690792

C₂

yes

maximal, complemented

#15690788

C₂

yes

maximal, complemented

#15690778

C₂

yes

maximal, characteristic

C₄×C₂²⋊D₉₇

C₂

yes

maximal, complemented

#15690768

C₂

yes

maximal, complemented

C₂²⋊C₄×D₉₇

C₂

yes

maximal, complemented

#123×C₉₇

C₂

yes

maximal, characteristic, complemented

C₂³×D₉₇

C₂²

yes

characteristic

C₂×Dic₁₉₄

C₂²

yes

characteristic, complemented

C₂×C₂²⋊D₉₇

C₂²

yes

—

C₂×C₄×D₉₇

C₂²

yes

—

C₂×C₄×D₉₇

C₂²

yes

—

C₄⋊Dic₉₇

C₂²

yes

complemented

#13095490

C₂²

yes

complemented

#13095490

C₂²

yes

complemented

C₂²⋊Dic₉₇

C₂²

yes

—

C₂²⋊Dic₉₇

C₂²

yes

characteristic

C₂²⋊Dic₉₇

C₂²

yes

complemented

#13095488

C₂²

yes

—

#13095488

C₂²

yes

—

#13095488

C₂²

yes

—

C₄×Dic₉₇

C₂²

yes

characteristic, complemented

C₄×Dic₉₇

C₂²

yes

complemented

C₂×D₄×C₉₇

C₂²

yes

characteristic

C₂²×C₄×C₉₇

C₂²

yes

characteristic, complemented

C₄⋊C₄×C₉₇

C₂²

yes

complemented

C₂²⋊C₄×C₉₇

C₂²

yes

—

C₂²⋊C₄×C₉₇

C₂²

yes

characteristic, complemented

C₂³×C₉₇

C₂³

yes

characteristic

C₂³×C₉₇

C₂³

yes

characteristic

C₂²×D₉₇

—

C₂²×D₉₇

—

C₂²×D₉₇

C₂³

yes

—

D₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂³

yes

—

C₂×C₄×C₉₇

C₂³

yes

—

C₂×C₄×C₉₇

C₂³

yes

characteristic

C₂²⋊D₉₇

—

C₂×Dic₉₇

C₂³

yes

—

C₂×Dic₉₇

C₂³

yes

—

C₂×Dic₉₇

C₂³

yes

—

C₄×D₉₇

—

C₄×D₉₇

—

Dic₁₉₄

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂⁴

yes

characteristic

C₂×D₉₇

—

C₂×D₉₇

—

C₂×D₉₇

—

C₂×D₉₇

#41

yes

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

Dic₉₇

—

Dic₉₇

—

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

C₂×#41

yes

—

C₂×C₉₇

H(5,𝔽₂)

yes

characteristic

D₉₇

—

C₉₇

#551

yes

characteristic, complemented, Sylow

—

—

—

—

D₉₇

yes

characteristic, complemented

C₂²⋊Q₈

—

#120

—

C₄×D₄

—

#117

—

C₂×C₂²⋊C₄

—

C₂⁴

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

C₂×D₉₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₉₇

yes

characteristic, complemented

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₉₇

yes

complemented

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₉₇

yes

—

C₂²⋊C₄

C₂×D₉₇

yes

characteristic, complemented

C₄²

—

C₄²

—

100

C₂³

—

101

C₂³

—

388

—

102

C₂³

—

194

—

103

C₂³

C₂²×D₉₇

yes

characteristic

104

C₂³

C₂²×D₉₇

yes

characteristic

105

Q₈

—

388

—

106

D₄

—

107

D₄

—

194

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

388

—

113

C₂×C₄

—

194

—

114

C₂×C₄

C₂²×D₉₇

yes

characteristic

115

C₂×C₄

C₂²×D₉₇

yes

—

116

C₂×C₄

C₂²×D₉₇

yes

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

388

—

123

C₂²

—

194

—

124

C₂²

—

194

—

125

C₂²

C₂³×D₉₇

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

194

—

130

C₄

—

194

—

131

C₄

—

194

—

132

C₂

—

133

C₂

—

134

C₂

—

194

—

135

C₂

#41×D₉₇

yes

—

136

C₂

#15690855

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₉₇
commutator, socle	C₂²×C₉₇	C₂⁴
Fitting	#123×C₉₇	C₂
hypercenter	#123	D₉₇
hypocenter	C₉₇	#551
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ #123

subgroup

G/H

center, Frattini

C₂²

C₂³×D₉₇

commutator, socle

C₂²×C₉₇

C₂⁴

Fitting

#123×C₉₇

C₂

D₉₇

C₉₇

G ▷ C₂²×C₉₇ ▷ C₉₇

C₁ ◁ C₂² ◁ #123

N	H ≅ G/N	type
1. #15690803	C₂	semidirect
2. #15690797	C₂	semidirect
3. #15690792	C₂	semidirect
4. #15690788	C₂	semidirect
6. C₄×C₂²⋊D₉₇	C₂	semidirect
7. #15690768	C₂	semidirect
8. C₂²⋊C₄×D₉₇	C₂	semidirect
9. #123×C₉₇	C₂	semidirect
11. C₂×Dic₁₉₄	C₂²	semidirect
15. C₄⋊Dic₉₇	C₂²	semidirect
16. #13095490	C₂²	semidirect
17. #13095490	C₂²	semidirect
20. C₂²⋊Dic₉₇	C₂²	semidirect
24. C₄×Dic₉₇	C₂²	semidirect
25. C₄×Dic₉₇	C₂²	semidirect
27. C₂²×C₄×C₉₇	C₂²	semidirect
28. C₄⋊C₄×C₉₇	C₂²	semidirect
30. C₂²⋊C₄×C₉₇	C₂²	semidirect
68. C₉₇	#551	semidirect
73. #123	D₉₇	semidirect
85. C₂²×C₄	C₂×D₉₇	semidirect
89. C₄⋊C₄	C₂×D₉₇	semidirect
97. C₂²⋊C₄	C₂×D₉₇	semidirect

H ≅ G/N

type

1. #15690803

C₂

semidirect

2. #15690797

C₂

semidirect

3. #15690792

C₂

semidirect

4. #15690788

C₂

semidirect

6. C₄×C₂²⋊D₉₇

C₂

semidirect

7. #15690768

C₂

semidirect

8. C₂²⋊C₄×D₉₇

C₂

semidirect

9. #123×C₉₇

C₂

semidirect

11. C₂×Dic₁₉₄

C₂²

semidirect

15. C₄⋊Dic₉₇

C₂²

semidirect

16. #13095490

C₂²

semidirect

17. #13095490

C₂²

semidirect

20. C₂²⋊Dic₉₇

C₂²

semidirect

24. C₄×Dic₉₇

C₂²

semidirect

25. C₄×Dic₉₇

C₂²

semidirect

27. C₂²×C₄×C₉₇

C₂²

semidirect

28. C₄⋊C₄×C₉₇

C₂²

semidirect

30. C₂²⋊C₄×C₉₇

C₂²

semidirect

68. C₉₇

#551

semidirect

73. #123

D₉₇

semidirect

85. C₂²×C₄

C₂×D₉₇

semidirect

89. C₄⋊C₄

C₂×D₉₇

semidirect

97. C₂²⋊C₄

C₂×D₉₇

semidirect

Subgroups — #20261413

Conjugacy classes of subgroups

Complements