Group #34088641 — subgroups

Subgroups — #34088641

Order 19648.

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

normal subgroups only

This group has 196 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	#23754259	C₂	1	2	yes	maximal, complemented
2	#23754255	C₂	1	2	yes	maximal, complemented
3	#23754255	C₂	1	2	yes	maximal, complemented
4	#23754252	C₂	1	1	yes	maximal, characteristic, complemented
5	#23754248	C₂	1	1	yes	maximal, characteristic
6	#23754245	C₂	1	2	yes	maximal
7	#23754240	C₂	1	2	yes	maximal, complemented
8	#23754238	C₂	1	2	yes	maximal, complemented
9	#125×C₃₀₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₆₁₄	C₂²	1	2	yes	—
11	C₂×Dic₆₁₄	C₂²	1	1	yes	characteristic
12	C₂×D₁₂₂₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₃₀₇	C₂²	1	2	yes	—
14	C₄⋊Dic₃₀₇	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₃₀₇	C₂²	1	2	yes	—
16	C₄⋊Dic₃₀₇	C₂²	1	1	yes	characteristic
17	Dic₃₀₇⋊C₄	C₂²	1	2	yes	—
18	Dic₃₀₇⋊C₄	C₂²	1	2	yes	—
19	Dic₃₀₇⋊C₄	C₂²	1	2	yes	—
20	(C₂×D₃₀₇)⋊C₄	C₂²	1	2	yes	—
21	(C₂×D₃₀₇)⋊C₄	C₂²	1	2	yes	—
22	(C₂×D₃₀₇)⋊C₄	C₂²	1	2	yes	—
23	(C₂×D₃₀₇)⋊C₄	C₂²	1	2	yes	—
24	(C₂×D₃₀₇)⋊C₄	C₂²	1	2	yes	—
25	C₄×Dic₃₀₇	C₂²	1	2	yes	—
26	C₄⋊C₄×C₃₀₇	C₂²	1	2	yes	—
27	C₄⋊C₄×C₃₀₇	C₂²	1	2	yes	—
28	C₄⋊C₄×C₃₀₇	C₂²	1	1	yes	characteristic
29	C₄⋊C₄×C₃₀₇	C₂²	1	1	yes	characteristic
30	C₄²×C₃₀₇	C₂²	1	1	yes	characteristic
31	C₂²×D₃₀₇	C₂³	1	2	yes	—
32	Dic₆₁₄	—	4	2	no	—
33	Dic₆₁₄	—	4	1	no	—
34	D₁₂₂₈	—	4	1	no	—
35	C₂×Dic₃₀₇	C₂³	1	2	yes	—
36	C₂×Dic₃₀₇	C₂³	1	2	yes	—
37	C₂×Dic₃₀₇	C₂³	1	2	yes	—
38	C₄×D₃₀₇	—	4	2	no	—
39	C₂×C₄×C₃₀₇	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₃₀₇	C₂³	1	1	yes	characteristic
41	C₂×C₄×C₃₀₇	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₃₀₇	C₂³	1	2	yes	—
43	C₂×C₄×C₃₀₇	C₂³	1	2	yes	—
44	C₂²×C₃₀₇	C₂⁴	1	1	yes	characteristic
45	C₂×D₃₀₇	—	2	2	no	—
46	C₂×D₃₀₇	—	2	2	no	—
47	C₂×D₃₀₇	—	2	2	no	—
48	C₄×C₃₀₇	—	2	2	no	—
49	C₄×C₃₀₇	—	2	1	no	—
50	C₄×C₃₀₇	—	2	1	no	—
51	C₄×C₃₀₇	—	2	2	no	—
52	C₄×C₃₀₇	—	2	1	no	—
53	Dic₃₀₇	—	2	2	no	—
54	Dic₃₀₇	—	2	2	no	—
55	Dic₃₀₇	—	2	2	no	—
56	C₂×C₃₀₇	H(5,𝔽₂)	1	1	yes	characteristic
57	C₂×C₃₀₇	2₋¹⁺⁴	1	1	yes	characteristic
58	C₂×C₃₀₇	2₋¹⁺⁴	1	1	yes	characteristic
59	D₃₀₇	—	4	2	no	—
60	C₃₀₇	#563	1	1	yes	characteristic, complemented, Sylow
61	#563	—	307	1	no	maximal, Sylow
62	C₄⋊Q₈	—	307	2	no	—
63	#126	—	307	2	no	—
64	#126	—	307	2	no	—
65	#125	—	307	1	no	—
66	#125	D₃₀₇	1	1	yes	characteristic, complemented
67	#124	—	307	1	no	—
68	#123	—	307	2	no	—
69	C₂²⋊Q₈	—	307	2	no	—
70	C₂²⋊Q₈	—	307	2	no	—
71	C₂×Q₈	—	307	2	no	—
72	C₂×Q₈	—	307	1	no	—
73	C₂×D₄	—	307	1	no	—
74	C₂²×C₄	—	307	2	no	—
75	C₄⋊C₄	—	307	1	no	—
76	C₄⋊C₄	—	307	2	no	—
77	C₄⋊C₄	—	307	2	no	—
78	C₄⋊C₄	—	307	2	no	—
79	C₄⋊C₄	—	307	2	no	—
80	C₄⋊C₄	—	307	1	no	—
81	C₄⋊C₄	C₂×D₃₀₇	1	1	yes	characteristic
82	C₄⋊C₄	C₂×D₃₀₇	1	2	yes	—
83	C₄⋊C₄	C₂×D₃₀₇	1	2	yes	—
84	C₄⋊C₄	C₂×D₃₀₇	1	1	yes	characteristic
85	C₂²⋊C₄	—	307	2	no	—
86	C₂²⋊C₄	—	307	2	no	—
87	C₂²⋊C₄	—	307	2	no	—
88	C₂²⋊C₄	—	307	2	no	—
89	C₂²⋊C₄	—	307	2	no	—
90	C₄²	—	307	2	no	—
91	C₄²	C₂×D₃₀₇	1	1	yes	characteristic
92	C₂³	—	307	2	no	—
93	Q₈	—	1228	2	no	—
94	Q₈	—	1228	1	no	—
95	D₄	—	1228	1	no	—
96	C₂×C₄	—	307	2	no	—
97	C₂×C₄	—	307	2	no	—
98	C₂×C₄	—	307	2	no	—
99	C₂×C₄	—	1228	2	no	—
100	C₂×C₄	C₂²×D₃₀₇	1	2	yes	—
101	C₂×C₄	C₂²×D₃₀₇	1	1	yes	characteristic
102	C₂×C₄	C₂²×D₃₀₇	1	1	yes	characteristic
103	C₂×C₄	C₂²×D₃₀₇	1	1	yes	characteristic
104	C₂×C₄	C₂²×D₃₀₇	1	2	yes	—
105	C₂²	—	614	2	no	—
106	C₂²	—	614	2	no	—
107	C₂²	—	614	2	no	—
108	C₂²	C₂³×D₃₀₇	1	1	yes	characteristic
109	C₄	—	2	2	no	—
110	C₄	—	2	1	no	—
111	C₄	—	2	2	no	—
112	C₄	—	2	1	no	—
113	C₄	—	2	1	no	—
114	C₄	—	614	2	no	—
115	C₄	—	614	2	no	—
116	C₄	—	614	2	no	—
117	C₂	—	1228	2	no	—
118	C₂	#23754306	1	1	yes	characteristic
119	C₂	#23754307	1	1	yes	characteristic
120	C₂	#23754308	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23754259

C₂

yes

maximal, complemented

#23754255

C₂

yes

maximal, complemented

#23754255

C₂

yes

maximal, complemented

#23754252

C₂

yes

maximal, characteristic, complemented

#23754248

C₂

yes

maximal, characteristic

#23754245

C₂

yes

maximal

#23754240

C₂

yes

maximal, complemented

#23754238

C₂

yes

maximal, complemented

#125×C₃₀₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₆₁₄

C₂²

yes

—

C₂×Dic₆₁₄

C₂²

yes

characteristic

C₂×D₁₂₂₈

C₂²

yes

characteristic

C₂×C₄×D₃₀₇

C₂²

yes

—

C₄⋊Dic₃₀₇

C₂²

yes

characteristic

C₄⋊Dic₃₀₇

C₂²

yes

—

C₄⋊Dic₃₀₇

C₂²

yes

characteristic

Dic₃₀₇⋊C₄

C₂²

yes

—

Dic₃₀₇⋊C₄

C₂²

yes

—

Dic₃₀₇⋊C₄

C₂²

yes

—

(C₂×D₃₀₇)⋊C₄

C₂²

yes

—

(C₂×D₃₀₇)⋊C₄

C₂²

yes

—

(C₂×D₃₀₇)⋊C₄

C₂²

yes

—

(C₂×D₃₀₇)⋊C₄

C₂²

yes

—

(C₂×D₃₀₇)⋊C₄

C₂²

yes

—

C₄×Dic₃₀₇

C₂²

yes

—

C₄⋊C₄×C₃₀₇

C₂²

yes

—

C₄⋊C₄×C₃₀₇

C₂²

yes

—

C₄⋊C₄×C₃₀₇

C₂²

yes

characteristic

C₄⋊C₄×C₃₀₇

C₂²

yes

characteristic

C₄²×C₃₀₇

C₂²

yes

characteristic

C₂²×D₃₀₇

C₂³

yes

—

Dic₆₁₄

—

Dic₆₁₄

—

D₁₂₂₈

—

C₂×Dic₃₀₇

C₂³

yes

—

C₂×Dic₃₀₇

C₂³

yes

—

C₂×Dic₃₀₇

C₂³

yes

—

C₄×D₃₀₇

—

C₂×C₄×C₃₀₇

C₂³

yes

characteristic

C₂×C₄×C₃₀₇

C₂³

yes

characteristic

C₂×C₄×C₃₀₇

C₂³

yes

characteristic

C₂×C₄×C₃₀₇

C₂³

yes

—

C₂×C₄×C₃₀₇

C₂³

yes

—

C₂²×C₃₀₇

C₂⁴

yes

characteristic

C₂×D₃₀₇

—

C₂×D₃₀₇

—

C₂×D₃₀₇

—

C₄×C₃₀₇

—

C₄×C₃₀₇

—

C₄×C₃₀₇

—

C₄×C₃₀₇

—

C₄×C₃₀₇

—

Dic₃₀₇

—

Dic₃₀₇

—

Dic₃₀₇

—

C₂×C₃₀₇

H(5,𝔽₂)

yes

characteristic

C₂×C₃₀₇

2₋¹⁺⁴

yes

characteristic

C₂×C₃₀₇

2₋¹⁺⁴

yes

characteristic

D₃₀₇

—

C₃₀₇

#563

yes

characteristic, complemented, Sylow

#563

—

307

maximal, Sylow

C₄⋊Q₈

—

307

—

#126

—

307

—

#126

—

307

—

#125

—

307

—

#125

D₃₀₇

yes

characteristic, complemented

#124

—

307

—

#123

—

307

—

C₂²⋊Q₈

—

307

—

C₂²⋊Q₈

—

307

—

C₂×Q₈

—

307

—

C₂×Q₈

—

307

—

C₂×D₄

—

307

—

C₂²×C₄

—

307

—

C₄⋊C₄

—

307

—

C₄⋊C₄

—

307

—

C₄⋊C₄

—

307

—

C₄⋊C₄

—

307

—

C₄⋊C₄

—

307

—

C₄⋊C₄

—

307

—

C₄⋊C₄

C₂×D₃₀₇

yes

characteristic

C₄⋊C₄

C₂×D₃₀₇

yes

—

C₄⋊C₄

C₂×D₃₀₇

yes

—

C₄⋊C₄

C₂×D₃₀₇

yes

characteristic

C₂²⋊C₄

—

307

—

C₂²⋊C₄

—

307

—

C₂²⋊C₄

—

307

—

C₂²⋊C₄

—

307

—

C₂²⋊C₄

—

307

—

C₄²

—

307

—

C₄²

C₂×D₃₀₇

yes

characteristic

C₂³

—

307

—

Q₈

—

1228

—

Q₈

—

1228

—

D₄

—

1228

—

C₂×C₄

—

307

—

C₂×C₄

—

307

—

C₂×C₄

—

307

—

C₂×C₄

—

1228

—

100

C₂×C₄

C₂²×D₃₀₇

yes

—

101

C₂×C₄

C₂²×D₃₀₇

yes

characteristic

102

C₂×C₄

C₂²×D₃₀₇

yes

characteristic

103

C₂×C₄

C₂²×D₃₀₇

yes

characteristic

104

C₂×C₄

C₂²×D₃₀₇

yes

—

105

C₂²

—

614

—

106

C₂²

—

614

—

107

C₂²

—

614

—

108

C₂²

C₂³×D₃₀₇

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

—

614

—

115

C₄

—

614

—

116

C₄

—

614

—

117

C₂

—

1228

—

118

C₂

#23754306

yes

characteristic

119

C₂

#23754307

yes

characteristic

120

C₂

#23754308

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₀₇
commutator, socle, cosocle	C₂²×C₃₀₇	C₂⁴
Fitting	#125×C₃₀₇	C₂
prime radical	C₂×D₁₂₂₈	C₂²
hypercenter	#125	D₃₀₇
hypocenter	C₃₀₇	#563
LCS	G ▷ C₂²×C₃₀₇ ▷ C₃₀₇
UCS	C₁ ◁ C₂² ◁ #125

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃₀₇

commutator, socle, cosocle

C₂²×C₃₀₇

C₂⁴

Fitting

#125×C₃₀₇

C₂

prime radical

C₂×D₁₂₂₈

C₂²

D₃₀₇

C₃₀₇

G ▷ C₂²×C₃₀₇ ▷ C₃₀₇

C₁ ◁ C₂² ◁ #125

N	H ≅ G/N	type
1. #23754259	C₂	semidirect
2. #23754255	C₂	semidirect
3. #23754255	C₂	semidirect
4. #23754252	C₂	semidirect
7. #23754240	C₂	semidirect
8. #23754238	C₂	semidirect
9. #125×C₃₀₇	C₂	semidirect
60. C₃₀₇	#563	semidirect
66. #125	D₃₀₇	semidirect

H ≅ G/N

type

1. #23754259

C₂

semidirect

2. #23754255

C₂

semidirect

3. #23754255

C₂

semidirect

4. #23754252

C₂

semidirect

7. #23754240

C₂

semidirect

8. #23754238

C₂

semidirect

9. #125×C₃₀₇

C₂

semidirect

60. C₃₀₇

#563

semidirect

66. #125

D₃₀₇

semidirect

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

subgroup	index
#23754259	2
#23754255	2
#23754252	2
#23754248	2
#23754245	2
#23754240	2
#23754238	2
#125×C₃₀₇	2
#563	307

subgroup

index

#125×C₃₀₇

307

Subgroups — #34088641

Conjugacy classes of subgroups

Complements

Maximal subgroups