Group #30649112 — subgroups

Subgroups — #30649112

Order 16256.

380 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 141 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×#16861209	C₂	1	1	yes	maximal, characteristic
2	#22550246	C₂	1	4	yes	maximal, complemented
3	#22550228	C₂	1	4	yes	maximal, complemented
4	#22549884	C₂	1	4	yes	maximal
5	C₄×#14523232	C₂	1	1	yes	maximal, characteristic
6	#533×C₁₂₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₅₄	C₂²	1	2	yes	—
8	C₂×#14523232	C₂²	1	2	yes	—
9	#16861209	—	2	2	no	—
10	#16861209	C₂²	1	2	yes	—
11	#16861209	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₁₂₇	—	2	2	no	—
13	#16861132	C₂²	1	16	yes	—
14	#16861120	C₂²	1	2	yes	—
15	C₄×C₁₂₇⋊C₈	C₂²	1	2	yes	—
16	C₄⋊Q₈×C₁₂₇	C₂²	1	2	yes	—
17	#125×C₁₂₇	C₂²	1	2	yes	—
18	C₂²⋊Q₈×C₁₂₇	—	2	2	no	—
19	C₄×Q₈×C₁₂₇	—	2	2	no	—
20	#117×C₁₂₇	C₂²	1	2	yes	—
21	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
22	C₂×Dic₂₅₄	—	2	2	no	—
23	C₂×Dic₂₅₄	—	2	8	no	—
24	C₂×Dic₂₅₄	C₂³	1	4	yes	—
25	C₂²×Dic₁₂₇	—	2	2	no	—
26	#14523232	—	2	4	no	—
27	C₂×C₁₂₇⋊C₈	C₂³	1	4	yes	—
28	C₄⋊Dic₁₂₇	—	2	4	no	—
29	C₄⋊Dic₁₂₇	—	2	4	no	—
30	C₂×Q₈×C₁₂₇	—	2	2	no	—
31	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
32	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₁₂₇	—	2	4	no	—
34	C₄⋊C₄×C₁₂₇	—	2	2	no	—
35	C₄⋊C₄×C₁₂₇	C₂³	1	4	yes	—
36	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
37	C₄²×C₁₂₇	—	2	2	no	—
38	C₄²×C₁₂₇	D₄	1	2	yes	—
39	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
40	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
41	Dic₂₅₄	—	2	8	no	—
42	Dic₂₅₄	—	4	4	no	—
43	C₂×Dic₁₂₇	—	2	4	no	—
44	C₂×Dic₁₂₇	—	4	4	no	—
45	C₁₂₇⋊C₈	—	2	4	no	—
46	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
47	Q₈×C₁₂₇	—	4	4	no	—
48	C₂×C₄×C₁₂₇	—	2	4	no	—
49	C₂×C₄×C₁₂₇	—	2	4	no	—
50	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₁₂₇	#41	1	4	yes	—
54	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
55	C₂²×C₁₂₇	—	2	2	no	—
56	C₂²×C₁₂₇	#124	1	2	yes	—
57	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₁₂₇	—	2	2	no	—
59	C₄×C₁₂₇	—	4	4	no	—
60	C₄×C₁₂₇	#124	1	2	yes	—
61	C₄×C₁₂₇	#137	1	4	yes	—
62	C₄×C₁₂₇	C₂×#41	1	2	yes	—
63	Dic₁₂₇	—	4	4	no	—
64	C₂×C₁₂₇	—	2	2	no	—
65	C₂×C₁₂₇	C₂×#124	1	1	yes	characteristic
66	C₂×C₁₂₇	C₂×#137	1	2	yes	—
67	#3139	—	127	1	no	maximal, Sylow
68	C₁₂₇	#3139	1	1	yes	characteristic, complemented, Sylow
69	#533	D₁₂₇	1	1	yes	characteristic
70	C₂×C₄⋊Q₈	—	127	1	no	—
71	#491	—	127	4	no	—
72	#487	—	127	4	no	—
73	#419	—	127	4	no	—
74	C₄×M₁₆	—	127	1	no	—
75	C₂²×Q₈	—	127	2	no	—
76	C₂×M₁₆	—	127	2	no	—
77	C₄⋊Q₈	—	127	2	no	—
78	C₄⋊Q₈	—	127	2	no	—
79	C₄⋊Q₈	—	254	2	no	—
80	C₄⋊Q₈	C₂×D₁₂₇	1	2	yes	—
81	#125	C₂×D₁₂₇	1	2	yes	—
82	C₂²⋊Q₈	—	2	2	no	—
83	C₄×Q₈	—	2	2	no	—
84	#117	C₂×D₁₂₇	1	2	yes	—
85	C₂×C₄⋊C₄	—	254	2	no	—
86	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
87	#103	—	127	16	no	—
88	#97	—	127	2	no	—
89	C₄×C₈	—	127	2	no	—
90	C₂×Q₈	—	2	2	no	—
91	C₂×Q₈	—	127	4	no	—
92	C₂×Q₈	—	254	8	no	—
93	C₂×Q₈	—	254	2	no	—
94	C₂²×C₄	—	254	2	no	—
95	C₂²×C₄	C₂²⋊D₁₂₇	1	2	yes	—
96	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
97	M₁₆	—	254	4	no	—
98	C₂×C₈	—	127	4	no	—
99	C₄⋊C₄	—	2	4	no	—
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	254	4	no	—
102	C₄⋊C₄	—	254	4	no	—
103	C₄⋊C₄	C₂²×D₁₂₇	1	4	yes	—
104	C₂²⋊C₄	—	2	2	no	—
105	C₄²	—	2	2	no	—
106	C₄²	C₂²⋊D₁₂₇	1	2	yes	—
107	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
108	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
109	C₂³	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
110	Q₈	—	4	4	no	—
111	Q₈	—	508	4	no	—
112	Q₈	—	254	8	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	508	4	no	—
116	C₂×C₄	—	254	4	no	—
117	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	2	yes	—
118	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
119	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	2	yes	—
120	C₂×C₄	#14523249	1	4	yes	—
121	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
122	C₈	—	254	4	no	—
123	C₂²	—	2	2	no	—
124	C₂²	#16861198	1	2	yes	—
125	C₂²	C₂²×C₂²⋊D₁₂₇	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	4	4	no	—
128	C₄	—	508	4	no	—
129	C₄	#16861198	1	2	yes	—
130	C₄	#16861239	1	4	yes	—
131	C₄	C₂×#14523249	1	2	yes	—
132	C₂	—	2	2	no	—
133	C₂	C₂×#16861198	1	1	yes	characteristic
134	C₂	C₂×#16861239	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861209

C₂

yes

maximal, characteristic

#22550246

C₂

yes

maximal, complemented

#22550228

C₂

yes

maximal, complemented

#22549884

C₂

yes

maximal

C₄×#14523232

C₂

yes

maximal, characteristic

#533×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×Dic₂₅₄

C₂²

yes

—

C₂×#14523232

C₂²

yes

—

#16861209

—

#16861209

C₂²

yes

—

#16861209

C₂²

yes

—

C₂×C₄⋊Dic₁₂₇

—

#16861132

C₂²

yes

—

#16861120

C₂²

yes

—

C₄×C₁₂₇⋊C₈

C₂²

yes

—

C₄⋊Q₈×C₁₂₇

C₂²

yes

—

#125×C₁₂₇

C₂²

yes

—

C₂²⋊Q₈×C₁₂₇

—

C₄×Q₈×C₁₂₇

—

#117×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

C₂³

yes

—

C₂²×Dic₁₂₇

—

#14523232

—

C₂×C₁₂₇⋊C₈

C₂³

yes

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₂×Q₈×C₁₂₇

—

C₂²×C₄×C₁₂₇

D₄

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

C₂³

yes

—

C₂²⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

—

C₄²×C₁₂₇

D₄

yes

—

C₄²×C₁₂₇

C₂³

yes

characteristic

C₄²×C₁₂₇

C₂³

yes

characteristic

Dic₂₅₄

—

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

C₂×D₄

yes

characteristic

Q₈×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

#41

yes

—

C₂×C₄×C₁₂₇

C₂⁴

yes

characteristic

C₂²×C₁₂₇

—

C₂²×C₁₂₇

#124

yes

—

C₂²×C₁₂₇

C₂²×D₄

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#124

yes

—

C₄×C₁₂₇

#137

yes

—

C₄×C₁₂₇

C₂×#41

yes

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂×#124

yes

characteristic

C₂×C₁₂₇

C₂×#137

yes

—

#3139

—

127

maximal, Sylow

C₁₂₇

#3139

yes

characteristic, complemented, Sylow

#533

D₁₂₇

yes

characteristic

C₂×C₄⋊Q₈

—

127

—

#491

—

127

—

#487

—

127

—

#419

—

127

—

C₄×M₁₆

—

127

—

C₂²×Q₈

—

127

—

C₂×M₁₆

—

127

—

C₄⋊Q₈

—

127

—

C₄⋊Q₈

—

127

—

C₄⋊Q₈

—

254

—

C₄⋊Q₈

C₂×D₁₂₇

yes

—

#125

C₂×D₁₂₇

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×D₁₂₇

yes

—

C₂×C₄⋊C₄

—

254

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

#103

—

127

—

#97

—

127

—

C₄×C₈

—

127

—

C₂×Q₈

—

C₂×Q₈

—

127

—

C₂×Q₈

—

254

—

C₂×Q₈

—

254

—

C₂²×C₄

—

254

—

C₂²×C₄

C₂²⋊D₁₂₇

yes

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

M₁₆

—

254

—

C₂×C₈

—

127

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

254

—

102

C₄⋊C₄

—

254

—

103

C₄⋊C₄

C₂²×D₁₂₇

yes

—

104

C₂²⋊C₄

—

105

C₄²

—

106

C₄²

C₂²⋊D₁₂₇

yes

—

107

C₄²

C₂²×D₁₂₇

yes

characteristic

108

C₄²

C₂²×D₁₂₇

yes

characteristic

109

C₂³

C₂×C₂²⋊D₁₂₇

yes

characteristic

110

Q₈

—

111

Q₈

—

508

—

112

Q₈

—

254

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

508

—

116

C₂×C₄

—

254

—

117

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

—

118

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

119

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

—

120

C₂×C₄

#14523249

yes

—

121

C₂×C₄

C₂³×D₁₂₇

yes

characteristic

122

C₈

—

254

—

123

C₂²

—

124

C₂²

#16861198

yes

—

125

C₂²

C₂²×C₂²⋊D₁₂₇

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

508

—

129

C₄

#16861198

yes

—

130

C₄

#16861239

yes

—

131

C₄

C₂×#14523249

yes

—

132

C₂

—

133

C₂

C₂×#16861198

yes

characteristic

134

C₂

C₂×#16861239

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₂₇
commutator, cosocle	C₂×C₄×C₁₂₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	#533×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²×D₄
prime radical	C₂³×C₁₂₇	C₂×D₄
hypercenter	#533	D₁₂₇
hypocenter	C₁₂₇	#3139
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₂₇

commutator, cosocle

C₂×C₄×C₁₂₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₂₇

Fitting

#533×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂²×D₄

prime radical

C₂³×C₁₂₇

C₂×D₄

D₁₂₇

C₁₂₇

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #22550246	C₂	semidirect
3. #22550228	C₂	semidirect
68. C₁₂₇	#3139	semidirect

H ≅ G/N

type

2. #22550246

C₂

semidirect

3. #22550228

C₂

semidirect

68. C₁₂₇

#3139

semidirect

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

subgroup	index
C₂×#16861209	2
#22550246	2
#22550228	2
#22549884	2
C₄×#14523232	2
#533×C₁₂₇	2
#3139	127

subgroup

index

#533×C₁₂₇

127

Subgroups — #30649112

Conjugacy classes of subgroups

Complements

Maximal subgroups