Group #30647294 — subgroups

Subgroups — #30647294

Order 16256.

484 conjugacy classes of subgroups; only the 293 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 293 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×#14523232	C₂	1	2	yes	maximal
2	#22549903	C₂	1	8	yes	maximal, complemented
3	#22549900	C₂	1	4	yes	maximal, complemented
4	C₂×#117×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523232	—	2	8	no	—
6	C₂×#14523232	C₂²	1	4	yes	—
7	C₂×#14523232	C₂²	1	4	yes	—
8	C₂²×C₁₂₇⋊C₈	C₂²	1	4	yes	—
9	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	16	yes	complemented
10	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
11	#117×C₁₂₇	C₄	1	4	yes	—
12	#117×C₁₂₇	C₂²	1	4	yes	—
13	C₂×C₄⋊C₄×C₁₂₇	C₄	1	2	yes	—
14	C₂×C₂²⋊C₄×C₁₂₇	C₄	1	2	yes	—
15	C₂×C₄²×C₁₂₇	C₂²	1	2	yes	—
16	#14523232	—	2	8	no	—
17	#14523232	—	2	8	no	—
18	C₂×C₁₂₇⋊C₈	—	2	8	no	—
19	C₂×C₁₂₇⋊C₈	C₂³	1	8	yes	—
20	C₂⁴×C₁₂₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₁₂₇	D₄	1	4	yes	—
25	C₂²×C₄×C₁₂₇	Q₈	1	4	yes	—
26	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
27	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₁₂₇	C₂×C₄	1	8	yes	—
29	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	8	yes	—
30	C₄²×C₁₂₇	C₂×C₄	1	4	yes	—
31	C₄²×C₁₂₇	C₂³	1	4	yes	—
32	C₁₂₇⋊C₈	—	2	8	no	—
33	C₂³×C₁₂₇	—	4	2	no	—
34	C₂³×C₁₂₇	C₄⋊C₄	1	4	yes	—
35	C₂³×C₁₂₇	C₂²×C₄	1	2	yes	—
36	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₂₇	—	4	4	no	—
38	C₂×C₄×C₁₂₇	C₄⋊C₄	1	4	yes	—
39	C₂×C₄×C₁₂₇	C₄⋊C₄	1	8	yes	—
40	C₂×C₄×C₁₂₇	C₂²×C₄	1	8	yes	—
41	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
43	C₂×C₄×C₁₂₇	C₂×D₄	1	4	yes	—
44	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₁₂₇	C₂×Q₈	1	4	yes	—
46	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
48	C₂²×C₁₂₇	—	2	2	no	—
49	C₂²×C₁₂₇	—	2	4	no	—
50	C₂²×C₁₂₇	—	4	4	no	—
51	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	4	yes	—
52	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
53	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
54	C₄×C₁₂₇	—	2	8	no	—
55	C₄×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
56	C₄×C₁₂₇	C₂×C₄⋊C₄	1	4	yes	—
57	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
59	C₂×C₁₂₇	—	2	2	no	—
60	C₂×C₁₂₇	—	2	4	no	—
61	C₂×C₁₂₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₁₂₇	#568	1	2	yes	—
63	#2905	—	127	1	no	maximal, Sylow
64	C₁₂₇	#2905	1	1	yes	characteristic, complemented, Sylow
65	C₂²×M₁₆	—	127	2	no	—
66	C₂×#117	D₁₂₇	1	1	yes	characteristic
67	#424	—	127	8	no	—
68	#423	—	127	4	no	—
69	C₂³×C₄	C₂×D₁₂₇	1	1	yes	characteristic
70	C₂×M₁₆	—	127	4	no	—
71	C₂×M₁₆	—	127	4	no	—
72	C₂×M₁₆	—	254	8	no	—
73	C₂²×C₈	—	127	4	no	—
74	#117	Dic₁₂₇	1	4	yes	—
75	#117	C₂×D₁₂₇	1	4	yes	—
76	C₂×C₄⋊C₄	Dic₁₂₇	1	2	yes	—
77	C₂×C₂²⋊C₄	Dic₁₂₇	1	2	yes	—
78	C₂×C₄²	C₂×D₁₂₇	1	2	yes	—
79	C₄⋊C₈	—	127	16	no	—
80	C₂⁴	C₂×Dic₁₂₇	1	1	yes	characteristic
81	C₂²×C₄	C₂×Dic₁₂₇	1	4	yes	—
82	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
83	C₂²×C₄	C₂×Dic₁₂₇	1	2	yes	—
84	C₂²×C₄	D₅₀₈	1	4	yes	—
85	C₂²×C₄	Dic₂₅₄	1	4	yes	—
86	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₁₂₇	1	2	yes	—
88	M₁₆	—	254	8	no	—
89	M₁₆	—	254	8	no	—
90	C₂×C₈	—	127	8	no	—
91	C₂×C₈	—	254	8	no	—
92	C₄⋊C₄	C₂×Dic₁₂₇	1	8	yes	—
93	C₂²⋊C₄	C₂×Dic₁₂₇	1	8	yes	—
94	C₄²	C₂×Dic₁₂₇	1	4	yes	—
95	C₄²	C₂²×D₁₂₇	1	4	yes	—
96	C₂³	—	4	2	no	—
97	C₂³	C₄⋊Dic₁₂₇	1	4	yes	—
98	C₂³	C₂²×Dic₁₂₇	1	2	yes	—
99	C₂³	C₂²×Dic₁₂₇	1	1	yes	characteristic
100	C₂×C₄	—	4	4	no	—
101	C₂×C₄	C₄⋊Dic₁₂₇	1	4	yes	—
102	C₂×C₄	C₄⋊Dic₁₂₇	1	8	yes	—
103	C₂×C₄	C₂²×Dic₁₂₇	1	8	yes	—
104	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
105	C₂×C₄	C₂²×Dic₁₂₇	1	2	yes	—
106	C₂×C₄	C₂×D₅₀₈	1	4	yes	—
107	C₂×C₄	C₂×D₅₀₈	1	2	yes	—
108	C₂×C₄	C₂×Dic₂₅₄	1	2	yes	—
109	C₂×C₄	C₂×Dic₂₅₄	1	4	yes	—
110	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
111	C₈	—	254	8	no	—
112	C₂²	—	2	2	no	—
113	C₂²	—	2	4	no	—
114	C₂²	—	4	4	no	—
115	C₂²	C₂×C₄⋊Dic₁₂₇	1	4	yes	—
116	C₂²	C₂×C₄⋊Dic₁₂₇	1	2	yes	—
117	C₂²	C₂³×Dic₁₂₇	1	1	yes	characteristic
118	C₄	—	2	8	no	—
119	C₄	C₂×C₄⋊Dic₁₂₇	1	2	yes	—
120	C₄	C₂×C₄⋊Dic₁₂₇	1	4	yes	—
121	C₄	C₂²×D₅₀₈	1	1	yes	characteristic
122	C₄	C₂²×Dic₂₅₄	1	1	yes	characteristic
123	C₂	—	2	2	no	—
124	C₂	—	2	4	no	—
125	C₂	C₂²×C₄⋊Dic₁₂₇	1	1	yes	characteristic
126	C₂	#22550643	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#14523232

C₂

yes

maximal

#22549903

C₂

yes

maximal, complemented

#22549900

C₂

yes

maximal, complemented

C₂×#117×C₁₂₇

C₂

yes

maximal, characteristic

C₂×#14523232

—

C₂×#14523232

C₂²

yes

—

C₂×#14523232

C₂²

yes

—

C₂²×C₁₂₇⋊C₈

C₂²

yes

—

C₄⋊(C₁₂₇⋊C₈)

C₂²

yes

complemented

C₂³×C₄×C₁₂₇

C₂²

yes

characteristic

#117×C₁₂₇

C₄

yes

—

#117×C₁₂₇

C₂²

yes

—

C₂×C₄⋊C₄×C₁₂₇

C₄

yes

—

C₂×C₂²⋊C₄×C₁₂₇

C₄

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

—

#14523232

—

#14523232

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

C₂³

yes

—

C₂⁴×C₁₂₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₂₇

C₂×C₄

yes

—

C₂²×C₄×C₁₂₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₂₇

C₂×C₄

yes

—

C₂²×C₄×C₁₂₇

D₄

yes

—

C₂²×C₄×C₁₂₇

Q₈

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₂₇

C₂×C₄

yes

—

C₂²⋊C₄×C₁₂₇

C₂×C₄

yes

—

C₄²×C₁₂₇

C₂×C₄

yes

—

C₄²×C₁₂₇

C₂³

yes

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

C₄⋊C₄

yes

—

C₂³×C₁₂₇

C₂²×C₄

yes

—

C₂³×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₂₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂²×C₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×Q₈

yes

—

C₂×C₄×C₁₂₇

C₂×Q₈

yes

—

C₂×C₄×C₁₂₇

C₂⁴

yes

characteristic

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×C₄⋊C₄

yes

—

C₂²×C₁₂₇

C₂×C₄⋊C₄

yes

—

C₂²×C₁₂₇

C₂³×C₄

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×C₄⋊C₄

yes

—

C₄×C₁₂₇

C₂×C₄⋊C₄

yes

—

C₄×C₁₂₇

C₂²×D₄

yes

characteristic

C₄×C₁₂₇

C₂²×Q₈

yes

characteristic

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂²×C₄⋊C₄

yes

characteristic

C₂×C₁₂₇

#568

yes

—

#2905

—

127

maximal, Sylow

C₁₂₇

#2905

yes

characteristic, complemented, Sylow

C₂²×M₁₆

—

127

—

C₂×#117

D₁₂₇

yes

characteristic

#424

—

127

—

#423

—

127

—

C₂³×C₄

C₂×D₁₂₇

yes

characteristic

C₂×M₁₆

—

127

—

C₂×M₁₆

—

127

—

C₂×M₁₆

—

254

—

C₂²×C₈

—

127

—

#117

Dic₁₂₇

yes

—

#117

C₂×D₁₂₇

yes

—

C₂×C₄⋊C₄

Dic₁₂₇

yes

—

C₂×C₂²⋊C₄

Dic₁₂₇

yes

—

C₂×C₄²

C₂×D₁₂₇

yes

—

C₄⋊C₈

—

127

—

C₂⁴

C₂×Dic₁₂₇

yes

characteristic

C₂²×C₄

C₂×Dic₁₂₇

yes

—

C₂²×C₄

C₂×Dic₁₂₇

yes

characteristic

C₂²×C₄

C₂×Dic₁₂₇

yes

—

C₂²×C₄

D₅₀₈

yes

—

C₂²×C₄

Dic₂₅₄

yes

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

C₂²×C₄

C₂²×D₁₂₇

yes

—

M₁₆

—

254

—

M₁₆

—

254

—

C₂×C₈

—

127

—

C₂×C₈

—

254

—

C₄⋊C₄

C₂×Dic₁₂₇

yes

—

C₂²⋊C₄

C₂×Dic₁₂₇

yes

—

C₄²

C₂×Dic₁₂₇

yes

—

C₄²

C₂²×D₁₂₇

yes

—

C₂³

—

C₂³

C₄⋊Dic₁₂₇

yes

—

C₂³

C₂²×Dic₁₂₇

yes

—

C₂³

C₂²×Dic₁₂₇

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

C₄⋊Dic₁₂₇

yes

—

102

C₂×C₄

C₄⋊Dic₁₂₇

yes

—

103

C₂×C₄

C₂²×Dic₁₂₇

yes

—

104

C₂×C₄

C₂²×Dic₁₂₇

yes

characteristic

105

C₂×C₄

C₂²×Dic₁₂₇

yes

—

106

C₂×C₄

C₂×D₅₀₈

yes

—

107

C₂×C₄

C₂×D₅₀₈

yes

—

108

C₂×C₄

C₂×Dic₂₅₄

yes

—

109

C₂×C₄

C₂×Dic₂₅₄

yes

—

110

C₂×C₄

C₂³×D₁₂₇

yes

characteristic

111

C₈

—

254

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

C₂×C₄⋊Dic₁₂₇

yes

—

116

C₂²

C₂×C₄⋊Dic₁₂₇

yes

—

117

C₂²

C₂³×Dic₁₂₇

yes

characteristic

118

C₄

—

119

C₄

C₂×C₄⋊Dic₁₂₇

yes

—

120

C₄

C₂×C₄⋊Dic₁₂₇

yes

—

121

C₄

C₂²×D₅₀₈

yes

characteristic

122

C₄

C₂²×Dic₂₅₄

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

C₂²×C₄⋊Dic₁₂₇

yes

characteristic

126

C₂

#22550643

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂³×C₄
Fitting	C₂×#117×C₁₂₇	C₂
prime radical	C₂⁴×C₁₂₇	C₂×C₄
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	C₂×#117	D₁₂₇
hypocenter	C₁₂₇	#2905
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₂₇

commutator, socle

C₂²×C₁₂₇

C₂³×C₄

Fitting

C₂×#117×C₁₂₇

C₂

prime radical

C₂⁴×C₁₂₇

C₂×C₄

cosocle

C₂×C₄×C₁₂₇

C₂⁴

D₁₂₇

C₁₂₇

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂×C₄ ◁ C₂×#117

N	H ≅ G/N	type
2. #22549903	C₂	semidirect
3. #22549900	C₂	semidirect
9. C₄⋊(C₁₂₇⋊C₈)	C₂²	semidirect
64. C₁₂₇	#2905	semidirect

H ≅ G/N

type

2. #22549903

C₂

semidirect

3. #22549900

C₂

semidirect

9. C₄⋊(C₁₂₇⋊C₈)

C₂²

semidirect

64. C₁₂₇

#2905

semidirect

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

subgroup	index
C₂²×#14523232	2
#22549903	2
#22549900	2
C₂×#117×C₁₂₇	2
#2905	127

subgroup

index

C₂×#117×C₁₂₇

127

Subgroups — #30647294

Conjugacy classes of subgroups

Complements

Maximal subgroups