Group #30647381 — subgroups

Subgroups — #30647381

Order 16256.

384 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 213 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	#22549937	C₂	1	4	yes	maximal, complemented
2	#22549933	C₂	1	4	yes	maximal, complemented
3	#22549903	C₂	1	2	yes	maximal, complemented
4	#22549843	C₂	1	2	yes	maximal
5	C₄×#14523232	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16861120	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#41×C₁₂₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#14523232	C₂²	1	2	yes	—
9	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
10	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	8	yes	—
11	C₂²⋊(C₁₂₇⋊C₈)	C₂²	1	8	yes	—
12	#16861120	C₂²	1	2	yes	—
13	#16861120	C₂²	1	2	yes	—
14	#16861120	C₂²	1	2	yes	—
15	C₄×C₁₂₇⋊C₈	C₂²	1	2	yes	—
16	C₂×#41×C₁₂₇	C₂²	1	1	yes	characteristic
17	C₄×Q₈×C₁₂₇	C₄	1	2	yes	—
18	C₄×D₄×C₁₂₇	C₄	1	4	yes	—
19	C₄×D₄×C₁₂₇	C₄	1	2	yes	—
20	#117×C₁₂₇	C₂²	1	1	yes	characteristic
21	#117×C₁₂₇	C₂²	1	2	yes	—
22	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
23	C₂×C₄²×C₁₂₇	C₂²	1	2	yes	—
24	#14523232	—	2	4	no	—
25	C₂×C₁₂₇⋊C₈	—	2	4	no	—
26	C₂×C₁₂₇⋊C₈	C₂³	1	4	yes	—
27	C₂×C₁₂₇⋊C₈	C₂³	1	4	yes	—
28	#41×C₁₂₇	—	2	4	no	—
29	C₂×Q₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
30	C₂×D₄×C₁₂₇	C₂×C₄	1	2	yes	—
31	C₂×D₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
33	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
35	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
37	C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
38	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
39	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
40	C₄²×C₁₂₇	C₂×C₄	1	4	yes	—
41	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
42	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
43	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
44	C₄²×C₁₂₇	C₂³	1	2	yes	—
45	C₁₂₇⋊C₈	—	2	4	no	—
46	C₁₂₇⋊C₈	—	2	4	no	—
47	C₂³×C₁₂₇	C₂²×C₄	1	2	yes	—
48	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
49	Q₈×C₁₂₇	—	2	2	no	—
50	D₄×C₁₂₇	—	2	4	no	—
51	D₄×C₁₂₇	—	2	2	no	—
52	C₂×C₄×C₁₂₇	—	2	4	no	—
53	C₂×C₄×C₁₂₇	—	4	2	no	—
54	C₂×C₄×C₁₂₇	—	4	4	no	—
55	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
56	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
59	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₁₂₇	#41	1	4	yes	—
61	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
62	C₂²×C₁₂₇	—	2	2	no	—
63	C₂²×C₁₂₇	—	2	2	no	—
64	C₂²×C₁₂₇	—	2	4	no	—
65	C₂²×C₁₂₇	#117	1	2	yes	—
66	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
67	C₄×C₁₂₇	—	2	2	no	—
68	C₄×C₁₂₇	—	2	2	no	—
69	C₄×C₁₂₇	—	2	2	no	—
70	C₄×C₁₂₇	—	2	4	no	—
71	C₄×C₁₂₇	#117	1	2	yes	—
72	C₄×C₁₂₇	C₂×#41	1	2	yes	—
73	C₂×C₁₂₇	—	2	2	no	—
74	C₂×C₁₂₇	—	4	2	no	—
75	C₂×C₁₂₇	C₂×#117	1	1	yes	characteristic
76	C₂×C₁₂₇	#568	1	2	yes	—
77	#2923	—	127	1	no	maximal, Sylow
78	C₁₂₇	#2923	1	1	yes	characteristic, complemented, Sylow
79	C₄×#41	D₁₂₇	1	1	yes	characteristic
80	#433	—	127	4	no	—
81	#432	—	127	4	no	—
82	#424	—	127	2	no	—
83	#408	—	127	2	no	—
84	C₄×M₁₆	—	127	1	no	—
85	C₂×#97	—	127	1	no	—
86	C₂×#41	C₂×D₁₂₇	1	1	yes	characteristic
87	C₂×M₁₆	—	127	2	no	—
88	C₂²×C₈	—	127	2	no	—
89	C₄×Q₈	Dic₁₂₇	1	2	yes	—
90	C₄×D₄	Dic₁₂₇	1	4	yes	—
91	C₄×D₄	Dic₁₂₇	1	2	yes	—
92	#117	C₂×D₁₂₇	1	1	yes	characteristic
93	#117	C₂×D₁₂₇	1	2	yes	—
94	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
95	C₂×C₄²	C₂×D₁₂₇	1	2	yes	—
96	C₄⋊C₈	—	127	8	no	—
97	C₂²⋊C₈	—	127	8	no	—
98	#97	—	127	2	no	—
99	#97	—	127	2	no	—
100	#97	—	127	2	no	—
101	C₄×C₈	—	127	2	no	—
102	#41	—	2	4	no	—
103	C₂×Q₈	C₂×Dic₁₂₇	1	1	yes	characteristic
104	C₂×D₄	C₂×Dic₁₂₇	1	2	yes	—
105	C₂×D₄	C₂×Dic₁₂₇	1	1	yes	characteristic
106	C₂²×C₄	C₂×Dic₁₂₇	1	4	yes	—
107	C₂²×C₄	C₂×Dic₁₂₇	1	2	yes	—
108	C₂²×C₄	C₂²×D₁₂₇	1	2	yes	—
109	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
110	M₁₆	—	254	4	no	—
111	C₂×C₈	—	127	4	no	—
112	C₂×C₈	—	127	4	no	—
113	C₂×C₈	—	254	4	no	—
114	C₄⋊C₄	C₂×Dic₁₂₇	1	4	yes	—
115	C₄⋊C₄	C₂×Dic₁₂₇	1	2	yes	—
116	C₂²⋊C₄	C₂×Dic₁₂₇	1	4	yes	—
117	C₂²⋊C₄	C₂×Dic₁₂₇	1	2	yes	—
118	C₄²	C₂×Dic₁₂₇	1	4	yes	—
119	C₄²	C₂×Dic₁₂₇	1	2	yes	—
120	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
121	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
122	C₄²	C₂²×D₁₂₇	1	2	yes	—
123	C₂³	C₂²×Dic₁₂₇	1	2	yes	—
124	C₂³	C₂²×Dic₁₂₇	1	1	yes	characteristic
125	Q₈	—	2	2	no	—
126	D₄	—	2	4	no	—
127	D₄	—	2	2	no	—
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	4	2	no	—
130	C₂×C₄	—	4	4	no	—
131	C₂×C₄	C₂²×Dic₁₂₇	1	2	yes	—
132	C₂×C₄	C₂²×Dic₁₂₇	1	2	yes	—
133	C₂×C₄	C₂²×Dic₁₂₇	1	2	yes	—
134	C₂×C₄	C₂²×Dic₁₂₇	1	4	yes	—
135	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
136	C₂×C₄	#14523248	1	4	yes	—
137	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
138	C₈	—	254	4	no	—
139	C₈	—	254	4	no	—
140	C₂²	—	2	2	no	—
141	C₂²	—	2	2	no	—
142	C₂²	—	2	4	no	—
143	C₂²	#16861168	1	2	yes	—
144	C₂²	C₂³×Dic₁₂₇	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	2	2	no	—
148	C₄	—	2	4	no	—
149	C₄	#16861168	1	2	yes	—
150	C₄	C₂×#14523248	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	—	4	2	no	—
153	C₂	C₂×#16861168	1	1	yes	characteristic
154	C₂	#22550643	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22549937

C₂

yes

maximal, complemented

#22549933

C₂

yes

maximal, complemented

#22549903

C₂

yes

maximal, complemented

#22549843

C₂

yes

maximal

C₄×#14523232

C₂

yes

maximal, characteristic, complemented

C₂×#16861120

C₂

yes

maximal, characteristic, complemented

C₄×#41×C₁₂₇

C₂

yes

maximal, characteristic

C₂×#14523232

C₂²

yes

—

C₂²×C₁₂₇⋊C₈

C₂²

yes

—

C₄⋊(C₁₂₇⋊C₈)

C₂²

yes

—

C₂²⋊(C₁₂₇⋊C₈)

C₂²

yes

—

#16861120

C₂²

yes

—

#16861120

C₂²

yes

—

#16861120

C₂²

yes

—

C₄×C₁₂₇⋊C₈

C₂²

yes

—

C₂×#41×C₁₂₇

C₂²

yes

characteristic

C₄×Q₈×C₁₂₇

C₄

yes

—

C₄×D₄×C₁₂₇

C₄

yes

—

C₄×D₄×C₁₂₇

C₄

yes

—

#117×C₁₂₇

C₂²

yes

characteristic

#117×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

C₂×C₄²×C₁₂₇

C₂²

yes

—

#14523232

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

C₂³

yes

—

C₂×C₁₂₇⋊C₈

C₂³

yes

—

#41×C₁₂₇

—

C₂×Q₈×C₁₂₇

C₂×C₄

yes

characteristic

C₂×D₄×C₁₂₇

C₂×C₄

yes

—

C₂×D₄×C₁₂₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₂₇

C₂×C₄

yes

—

C₂²×C₄×C₁₂₇

C₂×C₄

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₂₇

C₂×C₄

yes

—

C₄⋊C₄×C₁₂₇

C₂×C₄

yes

—

C₂²⋊C₄×C₁₂₇

C₂×C₄

yes

—

C₂²⋊C₄×C₁₂₇

C₂×C₄

yes

—

C₄²×C₁₂₇

C₂×C₄

yes

—

C₄²×C₁₂₇

C₂×C₄

yes

—

C₄²×C₁₂₇

C₂³

yes

characteristic

C₄²×C₁₂₇

C₂³

yes

characteristic

C₄²×C₁₂₇

C₂³

yes

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

C₂²×C₄

yes

—

C₂³×C₁₂₇

C₂²×C₄

yes

characteristic

Q₈×C₁₂₇

—

D₄×C₁₂₇

—

D₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

#41

yes

—

C₂×C₄×C₁₂₇

C₂⁴

yes

characteristic

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

#117

yes

—

C₂²×C₁₂₇

C₂³×C₄

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#117

yes

—

C₄×C₁₂₇

C₂×#41

yes

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂×#117

yes

characteristic

C₂×C₁₂₇

#568

yes

—

#2923

—

127

maximal, Sylow

C₁₂₇

#2923

yes

characteristic, complemented, Sylow

C₄×#41

D₁₂₇

yes

characteristic

#433

—

127

—

#432

—

127

—

#424

—

127

—

#408

—

127

—

C₄×M₁₆

—

127

—

C₂×#97

—

127

—

C₂×#41

C₂×D₁₂₇

yes

characteristic

C₂×M₁₆

—

127

—

C₂²×C₈

—

127

—

C₄×Q₈

Dic₁₂₇

yes

—

C₄×D₄

Dic₁₂₇

yes

—

C₄×D₄

Dic₁₂₇

yes

—

#117

C₂×D₁₂₇

yes

characteristic

#117

C₂×D₁₂₇

yes

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

C₂×C₄²

C₂×D₁₂₇

yes

—

C₄⋊C₈

—

127

—

C₂²⋊C₈

—

127

—

#97

—

127

—

#97

—

127

—

100

#97

—

127

—

101

C₄×C₈

—

127

—

102

#41

—

103

C₂×Q₈

C₂×Dic₁₂₇

yes

characteristic

104

C₂×D₄

C₂×Dic₁₂₇

yes

—

105

C₂×D₄

C₂×Dic₁₂₇

yes

characteristic

106

C₂²×C₄

C₂×Dic₁₂₇

yes

—

107

C₂²×C₄

C₂×Dic₁₂₇

yes

—

108

C₂²×C₄

C₂²×D₁₂₇

yes

—

109

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

110

M₁₆

—

254

—

111

C₂×C₈

—

127

—

112

C₂×C₈

—

127

—

113

C₂×C₈

—

254

—

114

C₄⋊C₄

C₂×Dic₁₂₇

yes

—

115

C₄⋊C₄

C₂×Dic₁₂₇

yes

—

116

C₂²⋊C₄

C₂×Dic₁₂₇

yes

—

117

C₂²⋊C₄

C₂×Dic₁₂₇

yes

—

118

C₄²

C₂×Dic₁₂₇

yes

—

119

C₄²

C₂×Dic₁₂₇

yes

—

120

C₄²

C₂²×D₁₂₇

yes

characteristic

121

C₄²

C₂²×D₁₂₇

yes

characteristic

122

C₄²

C₂²×D₁₂₇

yes

—

123

C₂³

C₂²×Dic₁₂₇

yes

—

124

C₂³

C₂²×Dic₁₂₇

yes

characteristic

125

Q₈

—

126

D₄

—

127

D₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

C₂²×Dic₁₂₇

yes

—

132

C₂×C₄

C₂²×Dic₁₂₇

yes

—

133

C₂×C₄

C₂²×Dic₁₂₇

yes

—

134

C₂×C₄

C₂²×Dic₁₂₇

yes

—

135

C₂×C₄

C₂²×Dic₁₂₇

yes

characteristic

136

C₂×C₄

#14523248

yes

—

137

C₂×C₄

C₂³×D₁₂₇

yes

characteristic

138

C₈

—

254

—

139

C₈

—

254

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

#16861168

yes

—

144

C₂²

C₂³×Dic₁₂₇

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

#16861168

yes

—

150

C₄

C₂×#14523248

yes

—

151

C₂

—

152

C₂

—

153

C₂

C₂×#16861168

yes

characteristic

154

C₂

#22550643

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂³×C₄
Fitting	C₄×#41×C₁₂₇	C₂
prime radical	C₂×#41×C₁₂₇	C₂²
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	C₄×#41	D₁₂₇
hypocenter	C₁₂₇	#2923
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₂₇

commutator, socle

C₂²×C₁₂₇

C₂³×C₄

Fitting

C₄×#41×C₁₂₇

C₂

prime radical

C₂×#41×C₁₂₇

C₂²

cosocle

C₂×C₄×C₁₂₇

C₂⁴

hypercenter

C₄×#41

D₁₂₇

C₁₂₇

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂×C₄ ◁ C₄×#41

N	H ≅ G/N	type
1. #22549937	C₂	semidirect
2. #22549933	C₂	semidirect
3. #22549903	C₂	semidirect
5. C₄×#14523232	C₂	semidirect
6. C₂×#16861120	C₂	semidirect
78. C₁₂₇	#2923	semidirect

H ≅ G/N

type

1. #22549937

C₂

semidirect

2. #22549933

C₂

semidirect

3. #22549903

C₂

semidirect

5. C₄×#14523232

C₂

semidirect

6. C₂×#16861120

C₂

semidirect

78. C₁₂₇

#2923

semidirect

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

subgroup	index
#22549937	2
#22549933	2
#22549903	2
#22549843	2
C₄×#14523232	2
C₂×#16861120	2
C₄×#41×C₁₂₇	2
#2923	127

subgroup

index

C₄×#41×C₁₂₇

127

Subgroups — #30647381

Conjugacy classes of subgroups

Complements

Maximal subgroups