Group #20261249 — subgroups

Subgroups — #124×D₉₇ #20261249

Order 6208.

330 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 111 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	C₂×Q₈×D₉₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₄×D₉₇	C₂	1	1	yes	maximal, characteristic
3	#15690799	C₂	1	1	yes	maximal, characteristic, complemented
4	#15690798	C₂	1	1	yes	maximal, characteristic, complemented
5	#15690797	C₂	1	1	yes	maximal, characteristic, complemented
6	#15690796	C₂	1	4	yes	maximal, complemented
7	C₂²⋊C₄×D₉₇	C₂	1	4	yes	maximal, complemented
8	C₄²×D₉₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#124×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×D₉₇	C₂²	1	2	yes	—
11	C₂×Dic₁₉₄	C₂²	1	2	yes	complemented
12	C₂×Dic₁₉₄	C₂²	1	1	yes	characteristic, complemented
13	Q₈×D₉₇	—	2	4	no	—
14	C₂×D₃₈₈	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊D₉₇	C₂²	1	2	yes	—
16	D₄×D₉₇	—	2	4	no	—
17	C₂×C₄×D₉₇	C₂²	1	2	yes	—
18	C₂×C₄×D₉₇	C₂²	1	2	yes	—
19	C₂×C₄×D₉₇	C₂²	1	1	yes	characteristic
20	C₂²⋊Dic₉₇	C₂²	1	4	yes	complemented
21	#13095488	C₂²	1	4	yes	complemented
22	#13095488	C₂²	1	4	yes	complemented
23	C₄×Dic₉₇	C₂²	1	2	yes	complemented
24	C₄×Dic₉₇	C₂²	1	1	yes	characteristic, complemented
25	C₂×Q₈×C₉₇	C₂²	1	1	yes	characteristic, complemented
26	C₂×D₄×C₉₇	C₂²	1	1	yes	characteristic
27	C₂²⋊C₄×C₉₇	C₂²	1	4	yes	complemented
28	C₄²×C₉₇	C₂²	1	1	yes	characteristic, complemented
29	C₂³×C₉₇	C₂³	1	2	yes	—
30	C₂²×D₉₇	—	2	4	no	—
31	C₂²×D₉₇	—	2	8	no	—
32	C₂²×D₉₇	C₂³	1	2	yes	—
33	C₂²×D₉₇	C₂³	1	1	yes	characteristic
34	Q₈×C₉₇	—	2	2	no	—
35	D₄×C₉₇	—	2	2	no	—
36	C₂×C₄×C₉₇	C₂³	1	2	yes	—
37	C₂×C₄×C₉₇	C₂³	1	2	yes	—
38	C₂×C₄×C₉₇	C₂³	1	1	yes	characteristic
39	C₂²⋊D₉₇	—	2	4	no	—
40	C₂×Dic₉₇	C₂³	1	2	yes	—
41	C₂×Dic₉₇	C₂³	1	2	yes	—
42	C₂×Dic₉₇	C₂³	1	1	yes	characteristic
43	D₃₈₈	—	2	2	no	—
44	C₄×D₉₇	—	2	4	no	—
45	C₄×D₉₇	—	2	4	no	—
46	C₄×D₉₇	D₄	1	4	yes	—
47	Dic₁₉₄	—	2	2	no	—
48	Dic₁₉₄	—	2	4	no	—
49	C₂²×C₉₇	—	2	2	no	—
50	C₂²×C₉₇	—	2	4	no	—
51	C₂²×C₉₇	C₂⁴	1	1	yes	characteristic
52	C₂×D₉₇	—	2	2	no	—
53	C₂×D₉₇	—	2	4	no	—
54	C₂×D₉₇	—	4	8	no	—
55	C₂×D₉₇	C₂×D₄	1	2	yes	—
56	C₂×D₉₇	#41	1	4	yes	—
57	C₄×C₉₇	—	2	2	no	—
58	C₄×C₉₇	—	2	2	no	—
59	C₄×C₉₇	C₂×D₄	1	2	yes	—
60	Dic₉₇	—	2	2	no	—
61	Dic₉₇	—	2	2	no	—
62	Dic₉₇	C₂×D₄	1	2	yes	—
63	C₂×C₉₇	—	4	2	no	—
64	C₂×C₉₇	C₂²×D₄	1	1	yes	characteristic
65	C₂×C₉₇	C₂×#41	1	2	yes	—
66	D₉₇	—	4	2	no	—
67	D₉₇	#124	1	4	yes	direct product factor
68	C₉₇	C₂×#124	1	1	yes	characteristic, complemented, Sylow
69	C₂×#124	—	97	1	no	maximal, Sylow
70	C₂²×Q₈	—	97	1	no	—
71	C₂²×D₄	—	97	1	no	—
72	#124	—	97	1	no	—
73	#124	—	97	1	no	—
74	#124	—	97	4	no	—
75	#124	—	97	1	no	—
76	#124	D₉₇	1	1	yes	characteristic, direct product factor
77	C₂×C₂²⋊C₄	—	97	4	no	—
78	C₂×C₄²	—	97	1	no	—
79	C₂⁴	—	97	2	no	—
80	C₂×Q₈	—	97	2	no	—
81	C₂×Q₈	—	97	1	no	—
82	C₂×Q₈	—	194	4	no	—
83	C₂×Q₈	C₂×D₉₇	1	1	yes	characteristic, complemented
84	C₂×D₄	—	97	2	no	—
85	C₂×D₄	—	97	1	no	—
86	C₂×D₄	—	194	4	no	—
87	C₂×D₄	C₂×D₉₇	1	1	yes	characteristic
88	C₂²×C₄	—	97	1	no	—
89	C₂²×C₄	—	97	2	no	—
90	C₂²×C₄	—	97	2	no	—
91	C₂²⋊C₄	—	97	4	no	—
92	C₂²⋊C₄	—	97	4	no	—
93	C₂²⋊C₄	—	97	4	no	—
94	C₂²⋊C₄	C₂×D₉₇	1	4	yes	complemented
95	C₄²	—	97	2	no	—
96	C₄²	—	97	1	no	—
97	C₄²	C₂×D₉₇	1	1	yes	characteristic, complemented
98	C₂³	—	97	1	no	—
99	C₂³	—	97	2	no	—
100	C₂³	—	194	4	no	—
101	C₂³	—	194	8	no	—
102	C₂³	C₂²×D₉₇	1	2	yes	—
103	Q₈	—	2	2	no	—
104	Q₈	—	194	4	no	—
105	Q₈	—	194	2	no	—
106	D₄	—	2	2	no	—
107	D₄	—	194	4	no	—
108	D₄	—	194	2	no	—
109	C₂×C₄	—	97	1	no	—
110	C₂×C₄	—	97	4	no	—
111	C₂×C₄	—	97	2	no	—
112	C₂×C₄	—	97	2	no	—
113	C₂×C₄	—	194	4	no	—
114	C₂×C₄	—	194	4	no	—
115	C₂×C₄	C₂²×D₉₇	1	2	yes	—
116	C₂×C₄	C₂²×D₉₇	1	2	yes	—
117	C₂×C₄	C₂²×D₉₇	1	1	yes	characteristic
118	C₂²	—	2	2	no	—
119	C₂²	—	2	4	no	—
120	C₂²	—	97	2	no	—
121	C₂²	—	97	4	no	—
122	C₂²	—	388	8	no	—
123	C₂²	—	194	2	no	—
124	C₂²	—	194	4	no	—
125	C₂²	C₂³×D₉₇	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	2	2	no	—
128	C₄	—	97	2	no	—
129	C₄	—	194	2	no	—
130	C₄	—	194	2	no	—
131	C₄	D₄×D₉₇	1	2	yes	—
132	C₂	—	4	2	no	—
133	C₂	—	97	4	no	—
134	C₂	—	388	2	no	—
135	C₂	C₂×D₄×D₉₇	1	1	yes	characteristic
136	C₂	#41×D₉₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×D₉₇

C₂

yes

maximal, characteristic, complemented

C₂×D₄×D₉₇

C₂

yes

maximal, characteristic

#15690799

C₂

yes

maximal, characteristic, complemented

#15690798

C₂

yes

maximal, characteristic, complemented

#15690797

C₂

yes

maximal, characteristic, complemented

#15690796

C₂

yes

maximal, complemented

C₂²⋊C₄×D₉₇

C₂

yes

maximal, complemented

C₄²×D₉₇

C₂

yes

maximal, characteristic, complemented

#124×C₉₇

C₂

yes

maximal, characteristic, complemented

C₂³×D₉₇

C₂²

yes

—

C₂×Dic₁₉₄

C₂²

yes

complemented

C₂×Dic₁₉₄

C₂²

yes

characteristic, complemented

Q₈×D₉₇

—

C₂×D₃₈₈

C₂²

yes

characteristic

C₂×C₂²⋊D₉₇

C₂²

yes

—

D₄×D₉₇

—

C₂×C₄×D₉₇

C₂²

yes

—

C₂×C₄×D₉₇

C₂²

yes

—

C₂×C₄×D₉₇

C₂²

yes

characteristic

C₂²⋊Dic₉₇

C₂²

yes

complemented

#13095488

C₂²

yes

complemented

#13095488

C₂²

yes

complemented

C₄×Dic₉₇

C₂²

yes

complemented

C₄×Dic₉₇

C₂²

yes

characteristic, complemented

C₂×Q₈×C₉₇

C₂²

yes

characteristic, complemented

C₂×D₄×C₉₇

C₂²

yes

characteristic

C₂²⋊C₄×C₉₇

C₂²

yes

complemented

C₄²×C₉₇

C₂²

yes

characteristic, complemented

C₂³×C₉₇

C₂³

yes

—

C₂²×D₉₇

—

C₂²×D₉₇

—

C₂²×D₉₇

C₂³

yes

—

C₂²×D₉₇

C₂³

yes

characteristic

Q₈×C₉₇

—

D₄×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

characteristic

D₃₈₈

—

C₄×D₉₇

—

C₄×D₉₇

—

C₄×D₉₇

D₄

yes

—

Dic₁₉₄

—

Dic₁₉₄

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂⁴

yes

characteristic

C₂×D₉₇

—

C₂×D₉₇

—

C₂×D₉₇

—

C₂×D₉₇

C₂×D₄

yes

—

C₂×D₉₇

#41

yes

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₂×D₄

yes

—

Dic₉₇

—

Dic₉₇

—

Dic₉₇

C₂×D₄

yes

—

C₂×C₉₇

—

C₂×C₉₇

C₂²×D₄

yes

characteristic

C₂×C₉₇

C₂×#41

yes

—

D₉₇

—

D₉₇

#124

yes

direct product factor

C₉₇

C₂×#124

yes

characteristic, complemented, Sylow

C₂×#124

—

maximal, Sylow

C₂²×Q₈

—

C₂²×D₄

—

—

—

—

—

D₉₇

yes

characteristic, direct product factor

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂⁴

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

194

—

C₂×Q₈

C₂×D₉₇

yes

characteristic, complemented

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

194

—

C₂×D₄

C₂×D₉₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₉₇

yes

complemented

C₄²

—

C₄²

—

C₄²

C₂×D₉₇

yes

characteristic, complemented

C₂³

—

C₂³

—

100

C₂³

—

194

—

101

C₂³

—

194

—

102

C₂³

C₂²×D₉₇

yes

—

103

Q₈

—

104

Q₈

—

194

—

105

Q₈

—

194

—

106

D₄

—

107

D₄

—

194

—

108

D₄

—

194

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

194

—

114

C₂×C₄

—

194

—

115

C₂×C₄

C₂²×D₉₇

yes

—

116

C₂×C₄

C₂²×D₉₇

yes

—

117

C₂×C₄

C₂²×D₉₇

yes

characteristic

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

D₄×D₉₇

yes

—

132

C₂

—

133

C₂

—

134

C₂

—

388

—

135

C₂

C₂×D₄×D₉₇

yes

characteristic

136

C₂

#41×D₉₇

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₉₇
commutator, socle	C₂²×C₉₇	C₂⁴
Fitting	#124×C₉₇	C₂
hypercenter	#124	D₉₇
hypocenter	C₉₇	C₂×#124
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×D₉₇

commutator, socle

C₂²×C₉₇

C₂⁴

Fitting

#124×C₉₇

C₂

D₉₇

C₉₇

G ▷ C₂²×C₉₇ ▷ C₉₇

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
1. C₂×Q₈×D₉₇	C₂	semidirect
3. #15690799	C₂	semidirect
4. #15690798	C₂	semidirect
5. #15690797	C₂	semidirect
6. #15690796	C₂	semidirect
7. C₂²⋊C₄×D₉₇	C₂	semidirect
8. C₄²×D₉₇	C₂	semidirect
9. #124×C₉₇	C₂	semidirect
11. C₂×Dic₁₉₄	C₂²	semidirect
12. C₂×Dic₁₉₄	C₂²	semidirect
20. C₂²⋊Dic₉₇	C₂²	semidirect
21. #13095488	C₂²	semidirect
22. #13095488	C₂²	semidirect
23. C₄×Dic₉₇	C₂²	semidirect
24. C₄×Dic₉₇	C₂²	semidirect
25. C₂×Q₈×C₉₇	C₂²	semidirect
27. C₂²⋊C₄×C₉₇	C₂²	semidirect
28. C₄²×C₉₇	C₂²	semidirect
67. D₉₇	#124	direct (N × H)
68. C₉₇	C₂×#124	semidirect
76. #124	D₉₇	direct (N × H)
83. C₂×Q₈	C₂×D₉₇	semidirect
94. C₂²⋊C₄	C₂×D₉₇	semidirect
97. C₄²	C₂×D₉₇	semidirect

H ≅ G/N

type

1. C₂×Q₈×D₉₇

C₂

semidirect

3. #15690799

C₂

semidirect

4. #15690798

C₂

semidirect

5. #15690797

C₂

semidirect

6. #15690796

C₂

semidirect

7. C₂²⋊C₄×D₉₇

C₂

semidirect

8. C₄²×D₉₇

C₂

semidirect

9. #124×C₉₇

C₂

semidirect

11. C₂×Dic₁₉₄

C₂²

semidirect

12. C₂×Dic₁₉₄

C₂²

semidirect

20. C₂²⋊Dic₉₇

C₂²

semidirect

21. #13095488

C₂²

semidirect

22. #13095488

C₂²

semidirect

23. C₄×Dic₉₇

C₂²

semidirect

24. C₄×Dic₉₇

C₂²

semidirect

25. C₂×Q₈×C₉₇

C₂²

semidirect

27. C₂²⋊C₄×C₉₇

C₂²

semidirect

28. C₄²×C₉₇

C₂²

semidirect

67. D₉₇

#124

direct (N × H)

68. C₉₇

C₂×#124

semidirect

76. #124

D₉₇

direct (N × H)

83. C₂×Q₈

C₂×D₉₇

semidirect

94. C₂²⋊C₄

C₂×D₉₇

semidirect

97. C₄²

C₂×D₉₇

semidirect

Subgroups — #124×D₉₇ #20261249

Conjugacy classes of subgroups

Complements

Subgroups — #124×D97 #20261249

Conjugacy classes of subgroups

Complements

Subgroups — #124×D₉₇ #20261249