Group #34088631 — subgroups

Subgroups — #34088631

Order 19648.

220 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 91 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	#23754252	C₂	1	1	yes	maximal, characteristic, complemented
2	#23754248	C₂	1	1	yes	maximal, characteristic, complemented
3	#23754244	C₂	1	2	yes	maximal, complemented
4	#23754242	C₂	1	2	yes	maximal, complemented
5	#23754237	C₂	1	2	yes	maximal, complemented
6	#23754236	C₂	1	2	yes	maximal, complemented
7	#23754233	C₂	1	2	yes	maximal, complemented
8	#23754232	C₂	1	2	yes	maximal, complemented
9	#124×C₃₀₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₆₁₄	C₂²	1	1	yes	characteristic
11	C₂×D₁₂₂₈	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₃₀₇	C₂²	1	2	yes	—
13	C₂×C₂²⋊D₃₀₇	C₂²	1	2	yes	—
14	C₂²×Dic₃₀₇	C₂²	1	2	yes	—
15	C₂×C₄×D₃₀₇	C₂²	1	2	yes	—
16	C₄⋊Dic₃₀₇	C₂²	1	1	yes	characteristic
17	C₄⋊Dic₃₀₇	C₂²	1	1	yes	characteristic
18	C₄⋊Dic₃₀₇	C₂²	1	2	yes	—
19	Dic₃₀₇⋊C₄	C₂²	1	2	yes	—
20	Dic₃₀₇⋊C₄	C₂²	1	2	yes	—
21	Dic₃₀₇⋊C₄	C₂²	1	2	yes	—
22	C₂²⋊Dic₃₀₇	C₂²	1	2	yes	—
23	(C₂×D₃₀₇)⋊C₄	C₂²	1	2	yes	—
24	(C₂×D₃₀₇)⋊C₄	C₂²	1	2	yes	—
25	(C₂×D₃₀₇)⋊C₄	C₂²	1	2	yes	—
26	C₂×Q₈×C₃₀₇	C₂²	1	1	yes	characteristic
27	C₂×D₄×C₃₀₇	C₂²	1	1	yes	characteristic
28	C₂²⋊C₄×C₃₀₇	C₂²	1	2	yes	—
29	C₂²⋊C₄×C₃₀₇	C₂²	1	2	yes	—
30	C₄²×C₃₀₇	C₂²	1	1	yes	characteristic
31	C₂²×D₃₀₇	C₂³	1	2	yes	—
32	Dic₆₁₄	—	4	1	no	—
33	D₁₂₂₈	—	4	1	no	—
34	C₂²⋊D₃₀₇	—	4	2	no	—
35	C₂²⋊D₃₀₇	—	4	2	no	—
36	C₂×Dic₃₀₇	—	4	2	no	—
37	C₂×Dic₃₀₇	C₂³	1	2	yes	—
38	C₂×Dic₃₀₇	C₂³	1	2	yes	—
39	C₂×Dic₃₀₇	C₂³	1	2	yes	—
40	C₄×D₃₀₇	—	4	2	no	—
41	C₂³×C₃₀₇	C₂³	1	2	yes	—
42	Q₈×C₃₀₇	—	4	1	no	—
43	D₄×C₃₀₇	—	4	1	no	—
44	C₂×C₄×C₃₀₇	C₂³	1	1	yes	characteristic
45	C₂×C₄×C₃₀₇	C₂³	1	1	yes	characteristic
46	C₂×C₄×C₃₀₇	C₂³	1	1	yes	characteristic
47	C₂×C₄×C₃₀₇	C₂³	1	2	yes	—
48	C₂²×C₃₀₇	—	2	2	no	—
49	C₂²×C₃₀₇	—	2	2	no	—
50	C₂²×C₃₀₇	—	2	2	no	—
51	C₂²×C₃₀₇	C₂⁴	1	1	yes	characteristic
52	C₂×D₃₀₇	—	2	2	no	—
53	C₂×D₃₀₇	—	2	2	no	—
54	C₂×D₃₀₇	—	2	2	no	—
55	C₄×C₃₀₇	—	2	1	no	—
56	C₄×C₃₀₇	—	2	2	no	—
57	C₄×C₃₀₇	—	2	1	no	—
58	C₄×C₃₀₇	—	2	1	no	—
59	Dic₃₀₇	—	2	2	no	—
60	Dic₃₀₇	—	2	2	no	—
61	Dic₃₀₇	—	2	2	no	—
62	C₂×C₃₀₇	—	4	2	no	—
63	C₂×C₃₀₇	H(5,𝔽₂)	1	1	yes	characteristic
64	C₂×C₃₀₇	H(5,𝔽₂)	1	1	yes	characteristic
65	C₂×C₃₀₇	2₋¹⁺⁴	1	1	yes	characteristic
66	D₃₀₇	—	4	2	no	—
67	C₃₀₇	#562	1	1	yes	characteristic, complemented, Sylow
68	#562	—	307	1	no	maximal, Sylow
69	#125	—	307	1	no	—
70	#124	—	307	1	no	—
71	#124	D₃₀₇	1	1	yes	characteristic, complemented
72	#123	—	307	2	no	—
73	#123	—	307	2	no	—
74	C₂²⋊Q₈	—	307	2	no	—
75	C₂²⋊Q₈	—	307	2	no	—
76	#121	—	307	2	no	—
77	#121	—	307	2	no	—
78	C₂×Q₈	—	307	1	no	—
79	C₂×Q₈	C₂×D₃₀₇	1	1	yes	characteristic
80	C₂×D₄	—	307	2	no	—
81	C₂×D₄	—	307	2	no	—
82	C₂×D₄	—	307	1	no	—
83	C₂×D₄	C₂×D₃₀₇	1	1	yes	characteristic
84	C₂²×C₄	—	307	2	no	—
85	C₂²×C₄	—	307	2	no	—
86	C₄⋊C₄	—	307	2	no	—
87	C₄⋊C₄	—	307	2	no	—
88	C₄⋊C₄	—	307	2	no	—
89	C₄⋊C₄	—	307	2	no	—
90	C₄⋊C₄	—	307	1	no	—
91	C₄⋊C₄	—	307	1	no	—
92	C₂²⋊C₄	—	307	2	no	—
93	C₂²⋊C₄	—	307	2	no	—
94	C₂²⋊C₄	—	307	2	no	—
95	C₂²⋊C₄	—	307	2	no	—
96	C₂²⋊C₄	C₂×D₃₀₇	1	2	yes	—
97	C₂²⋊C₄	C₂×D₃₀₇	1	2	yes	—
98	C₄²	C₂×D₃₀₇	1	1	yes	characteristic
99	C₂³	—	307	2	no	—
100	C₂³	C₂²×D₃₀₇	1	2	yes	—
101	Q₈	—	4	1	no	—
102	Q₈	—	1228	1	no	—
103	D₄	—	4	1	no	—
104	D₄	—	1228	1	no	—
105	D₄	—	1228	2	no	—
106	D₄	—	1228	2	no	—
107	C₂×C₄	—	307	2	no	—
108	C₂×C₄	—	307	2	no	—
109	C₂×C₄	—	307	2	no	—
110	C₂×C₄	—	1228	2	no	—
111	C₂×C₄	—	1228	2	no	—
112	C₂×C₄	C₂²×D₃₀₇	1	1	yes	characteristic
113	C₂×C₄	C₂²×D₃₀₇	1	1	yes	characteristic
114	C₂×C₄	C₂²×D₃₀₇	1	1	yes	characteristic
115	C₂×C₄	C₂²×D₃₀₇	1	2	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	614	2	no	—
120	C₂²	—	614	2	no	—
121	C₂²	—	614	2	no	—
122	C₂²	C₂³×D₃₀₇	1	1	yes	characteristic
123	C₄	—	2	2	no	—
124	C₄	—	2	1	no	—
125	C₄	—	2	1	no	—
126	C₄	—	2	1	no	—
127	C₄	—	614	2	no	—
128	C₄	—	614	2	no	—
129	C₄	—	614	2	no	—
130	C₂	—	4	2	no	—
131	C₂	—	1228	2	no	—
132	C₂	#23754305	1	1	yes	characteristic
133	C₂	#23754306	1	1	yes	characteristic
134	C₂	#23754307	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23754252

C₂

yes

maximal, characteristic, complemented

#23754248

C₂

yes

maximal, characteristic, complemented

#23754244

C₂

yes

maximal, complemented

#23754242

C₂

yes

maximal, complemented

#23754237

C₂

yes

maximal, complemented

#23754236

C₂

yes

maximal, complemented

#23754233

C₂

yes

maximal, complemented

#23754232

C₂

yes

maximal, complemented

#124×C₃₀₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₆₁₄

C₂²

yes

characteristic

C₂×D₁₂₂₈

C₂²

yes

characteristic

C₂×C₂²⋊D₃₀₇

C₂²

yes

—

C₂×C₂²⋊D₃₀₇

C₂²

yes

—

C₂²×Dic₃₀₇

C₂²

yes

—

C₂×C₄×D₃₀₇

C₂²

yes

—

C₄⋊Dic₃₀₇

C₂²

yes

characteristic

C₄⋊Dic₃₀₇

C₂²

yes

characteristic

C₄⋊Dic₃₀₇

C₂²

yes

—

Dic₃₀₇⋊C₄

C₂²

yes

—

Dic₃₀₇⋊C₄

C₂²

yes

—

Dic₃₀₇⋊C₄

C₂²

yes

—

C₂²⋊Dic₃₀₇

C₂²

yes

—

(C₂×D₃₀₇)⋊C₄

C₂²

yes

—

(C₂×D₃₀₇)⋊C₄

C₂²

yes

—

(C₂×D₃₀₇)⋊C₄

C₂²

yes

—

C₂×Q₈×C₃₀₇

C₂²

yes

characteristic

C₂×D₄×C₃₀₇

C₂²

yes

characteristic

C₂²⋊C₄×C₃₀₇

C₂²

yes

—

C₂²⋊C₄×C₃₀₇

C₂²

yes

—

C₄²×C₃₀₇

C₂²

yes

characteristic

C₂²×D₃₀₇

C₂³

yes

—

Dic₆₁₄

—

D₁₂₂₈

—

C₂²⋊D₃₀₇

—

C₂²⋊D₃₀₇

—

C₂×Dic₃₀₇

—

C₂×Dic₃₀₇

C₂³

yes

—

C₂×Dic₃₀₇

C₂³

yes

—

C₂×Dic₃₀₇

C₂³

yes

—

C₄×D₃₀₇

—

C₂³×C₃₀₇

C₂³

yes

—

Q₈×C₃₀₇

—

D₄×C₃₀₇

—

C₂×C₄×C₃₀₇

C₂³

yes

characteristic

C₂×C₄×C₃₀₇

C₂³

yes

characteristic

C₂×C₄×C₃₀₇

C₂³

yes

characteristic

C₂×C₄×C₃₀₇

C₂³

yes

—

C₂²×C₃₀₇

—

C₂²×C₃₀₇

—

C₂²×C₃₀₇

—

C₂²×C₃₀₇

C₂⁴

yes

characteristic

C₂×D₃₀₇

—

C₂×D₃₀₇

—

C₂×D₃₀₇

—

C₄×C₃₀₇

—

C₄×C₃₀₇

—

C₄×C₃₀₇

—

C₄×C₃₀₇

—

Dic₃₀₇

—

Dic₃₀₇

—

Dic₃₀₇

—

C₂×C₃₀₇

—

C₂×C₃₀₇

yes

C₂×C₃₀₇

yes

C₂×C₃₀₇

2₋¹⁺⁴

yes

characteristic

D₃₀₇

—

C₃₀₇

#562

yes

characteristic, complemented, Sylow

#562

—

307

maximal, Sylow

#125

—

307

—

#124

—

307

—

#124

D₃₀₇

yes

characteristic, complemented

#123

—

307

—

#123

—

307

—

C₂²⋊Q₈

—

307

—

C₂²⋊Q₈

—

307

—

#121

—

307

—

#121

—

307

—

C₂×Q₈

—

307

—

C₂×Q₈

C₂×D₃₀₇

yes

characteristic

C₂×D₄

—

307

—

C₂×D₄

—

307

—

C₂×D₄

—

307

—

C₂×D₄

C₂×D₃₀₇

yes

characteristic

C₂²×C₄

—

307

—

C₂²×C₄

—

307

—

C₄⋊C₄

—

307

—

C₄⋊C₄

—

307

—

C₄⋊C₄

—

307

—

C₄⋊C₄

—

307

—

C₄⋊C₄

—

307

—

C₄⋊C₄

—

307

—

C₂²⋊C₄

—

307

—

C₂²⋊C₄

—

307

—

C₂²⋊C₄

—

307

—

C₂²⋊C₄

—

307

—

C₂²⋊C₄

C₂×D₃₀₇

yes

—

C₂²⋊C₄

C₂×D₃₀₇

yes

—

C₄²

C₂×D₃₀₇

yes

characteristic

C₂³

—

307

—

100

C₂³

C₂²×D₃₀₇

yes

—

101

Q₈

—

102

Q₈

—

1228

—

103

D₄

—

104

D₄

—

1228

—

105

D₄

—

1228

—

106

D₄

—

1228

—

107

C₂×C₄

—

307

—

108

C₂×C₄

—

307

—

109

C₂×C₄

—

307

—

110

C₂×C₄

—

1228

—

111

C₂×C₄

—

1228

—

112

C₂×C₄

C₂²×D₃₀₇

yes

characteristic

113

C₂×C₄

C₂²×D₃₀₇

yes

characteristic

114

C₂×C₄

C₂²×D₃₀₇

yes

characteristic

115

C₂×C₄

C₂²×D₃₀₇

yes

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

614

—

120

C₂²

—

614

—

121

C₂²

—

614

—

122

C₂²

C₂³×D₃₀₇

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

614

—

128

C₄

—

614

—

129

C₄

—

614

—

130

C₂

—

131

C₂

—

1228

—

132

C₂

#23754305

yes

characteristic

133

C₂

#23754306

yes

characteristic

134

C₂

#23754307

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₀₇
commutator, socle, cosocle	C₂²×C₃₀₇	C₂⁴
Fitting	#124×C₃₀₇	C₂
hypercenter	#124	D₃₀₇
hypocenter	C₃₀₇	#562
LCS	G ▷ C₂²×C₃₀₇ ▷ C₃₀₇
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃₀₇

commutator, socle, cosocle

C₂²×C₃₀₇

C₂⁴

Fitting

#124×C₃₀₇

C₂

D₃₀₇

C₃₀₇

G ▷ C₂²×C₃₀₇ ▷ C₃₀₇

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
1. #23754252	C₂	semidirect
2. #23754248	C₂	semidirect
3. #23754244	C₂	semidirect
4. #23754242	C₂	semidirect
5. #23754237	C₂	semidirect
6. #23754236	C₂	semidirect
7. #23754233	C₂	semidirect
8. #23754232	C₂	semidirect
9. #124×C₃₀₇	C₂	semidirect
67. C₃₀₇	#562	semidirect
71. #124	D₃₀₇	semidirect

H ≅ G/N

type

1. #23754252

C₂

semidirect

2. #23754248

C₂

semidirect

3. #23754244

C₂

semidirect

4. #23754242

C₂

semidirect

5. #23754237

C₂

semidirect

6. #23754236

C₂

semidirect

7. #23754233

C₂

semidirect

8. #23754232

C₂

semidirect

9. #124×C₃₀₇

C₂

semidirect

67. C₃₀₇

#562

semidirect

71. #124

D₃₀₇

semidirect

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

subgroup	index
#23754252	2
#23754248	2
#23754244	2
#23754242	2
#23754237	2
#23754236	2
#23754233	2
#23754232	2
#124×C₃₀₇	2
#562	307

subgroup

index

#124×C₃₀₇

307

Subgroups — #34088631

Conjugacy classes of subgroups

Complements

Maximal subgroups