Group #30638488 — subgroups

Subgroups — C₄×C₂²⋊(C₁₂₇⋊C₈) #30638488

Order 16256.

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

normal subgroups only

This group has 436 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	C₂×C₂²⋊(C₁₂₇⋊C₈)	C₂	1	2	yes	maximal
2	C₂×C₄×C₁₂₇⋊C₈	C₂	1	2	yes	maximal, complemented
3	#22549631	C₂	1	2	yes	maximal, complemented
4	C₂²×C₄²×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₂₇⋊C₈	C₂²	1	4	yes	—
6	C₂²⋊(C₁₂₇⋊C₈)	C₄	1	8	yes	direct product factor
7	C₄×C₁₂₇⋊C₈	—	2	4	no	—
8	C₂³×C₄×C₁₂₇	C₄	1	2	yes	—
9	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₂×C₄²×C₁₂₇	—	2	4	no	—
11	C₂×C₄²×C₁₂₇	C₄	1	2	yes	—
12	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₂×C₁₂₇⋊C₈	—	2	4	no	—
15	C₂×C₁₂₇⋊C₈	—	2	4	no	—
16	C₂×C₁₂₇⋊C₈	C₂×C₄	1	8	yes	complemented
17	C₂⁴×C₁₂₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₁₂₇	—	2	4	no	—
19	C₂²×C₄×C₁₂₇	—	2	4	no	—
20	C₂²×C₄×C₁₂₇	—	2	2	no	—
21	C₂²×C₄×C₁₂₇	—	2	2	no	—
22	C₂²×C₄×C₁₂₇	C₈	1	8	yes	complemented
23	C₂²×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
28	C₄²×C₁₂₇	—	2	4	no	—
29	C₄²×C₁₂₇	—	2	2	no	—
30	C₄²×C₁₂₇	D₄	1	4	yes	—
31	C₁₂₇⋊C₈	—	2	8	no	—
32	C₂³×C₁₂₇	—	2	1	no	—
33	C₂³×C₁₂₇	—	2	1	no	—
34	C₂³×C₁₂₇	—	2	2	no	—
35	C₂³×C₁₂₇	C₄²	1	2	yes	—
36	C₂³×C₁₂₇	C₂×C₈	1	4	yes	—
37	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₂₇	—	2	1	no	—
39	C₂×C₄×C₁₂₇	—	2	2	no	—
40	C₂×C₄×C₁₂₇	—	2	1	no	—
41	C₂×C₄×C₁₂₇	—	2	2	no	—
42	C₂×C₄×C₁₂₇	—	2	16	no	—
43	C₂×C₄×C₁₂₇	—	2	8	no	—
44	C₂×C₄×C₁₂₇	C₄²	1	2	yes	—
45	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	4	yes	—
46	C₂×C₄×C₁₂₇	C₂×C₈	1	4	yes	—
47	C₂×C₄×C₁₂₇	C₂×C₈	1	4	yes	—
48	C₂×C₄×C₁₂₇	M₁₆	1	4	yes	—
49	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
50	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₁₂₇	#41	1	2	yes	—
52	C₂²×C₁₂₇	—	2	2	no	—
53	C₂²×C₁₂₇	—	2	2	no	—
54	C₂²×C₁₂₇	—	2	4	no	—
55	C₂²×C₁₂₇	—	2	4	no	—
56	C₂²×C₁₂₇	C₄×C₈	1	4	yes	complemented
57	C₂²×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
58	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₁₂₇	#117	1	1	yes	characteristic
60	C₂²×C₁₂₇	C₂²×C₈	1	2	yes	—
61	C₂²×C₁₂₇	C₂×M₁₆	1	2	yes	—
62	C₄×C₁₂₇	—	2	2	no	—
63	C₄×C₁₂₇	—	2	4	no	—
64	C₄×C₁₂₇	C₂²⋊C₈	1	8	yes	complemented
65	C₄×C₁₂₇	C₄×D₄	1	4	yes	—
66	C₂×C₁₂₇	—	2	4	no	—
67	C₂×C₁₂₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₁₂₇	C₂×C₄×C₈	1	1	yes	characteristic
69	C₂×C₁₂₇	C₄×M₁₆	1	1	yes	characteristic
70	C₂×C₁₂₇	C₂×C₂²⋊C₈	1	2	yes	—
71	C₂×C₁₂₇	#431	1	2	yes	—
72	C₄×C₂²⋊C₈	—	127	1	no	maximal, Sylow
73	C₁₂₇	C₄×C₂²⋊C₈	1	1	yes	characteristic, complemented, Sylow
74	C₂²×C₄²	D₁₂₇	1	1	yes	characteristic
75	C₂×C₂²⋊C₈	—	127	2	no	—
76	C₂×C₄×C₈	—	127	2	no	—
77	#336	—	127	2	no	—
78	C₂³×C₄	Dic₁₂₇	1	2	yes	—
79	C₂³×C₄	C₂×D₁₂₇	1	1	yes	characteristic
80	C₂²×C₈	—	127	4	no	—
81	C₂×C₄²	—	2	4	no	—
82	C₂×C₄²	Dic₁₂₇	1	2	yes	—
83	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
84	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
85	C₂²⋊C₈	—	127	8	no	—
86	C₄×C₈	—	254	4	no	—
87	C₂⁴	C₂×Dic₁₂₇	1	1	yes	characteristic
88	C₂²×C₄	—	2	4	no	—
89	C₂²×C₄	—	2	4	no	—
90	C₂²×C₄	—	2	2	no	—
91	C₂²×C₄	—	2	2	no	—
92	C₂²×C₄	C₁₂₇⋊C₈	1	8	yes	complemented
93	C₂²×C₄	C₄×D₁₂₇	1	4	yes	—
94	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
95	C₂²×C₄	C₂×Dic₁₂₇	1	2	yes	—
96	C₂²×C₄	C₂×Dic₁₂₇	1	2	yes	—
97	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
98	C₂×C₈	—	127	8	no	—
99	C₂×C₈	—	254	4	no	—
100	C₂×C₈	—	254	4	no	—
101	C₄²	—	2	4	no	—
102	C₄²	—	2	2	no	—
103	C₄²	C₂²⋊D₁₂₇	1	4	yes	—
104	C₂³	—	2	2	no	—
105	C₂³	—	2	1	no	—
106	C₂³	—	2	1	no	—
107	C₂³	C₄×Dic₁₂₇	1	2	yes	—
108	C₂³	C₂×C₁₂₇⋊C₈	1	4	yes	—
109	C₂³	C₂²×Dic₁₂₇	1	1	yes	characteristic
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	16	no	—
115	C₂×C₄	—	2	8	no	—
116	C₂×C₄	C₄×Dic₁₂₇	1	2	yes	—
117	C₂×C₄	C₂²⋊Dic₁₂₇	1	4	yes	—
118	C₂×C₄	C₂×C₁₂₇⋊C₈	1	4	yes	—
119	C₂×C₄	C₂×C₁₂₇⋊C₈	1	4	yes	—
120	C₂×C₄	#14523232	1	4	yes	—
121	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
122	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	2	yes	—
123	C₂×C₄	#14523248	1	2	yes	—
124	C₈	—	254	8	no	—
125	C₂²	—	2	2	no	—
126	C₂²	—	2	4	no	—
127	C₂²	—	2	4	no	—
128	C₂²	—	2	2	no	—
129	C₂²	C₄×C₁₂₇⋊C₈	1	4	yes	complemented
130	C₂²	C₂×C₄×Dic₁₂₇	1	1	yes	characteristic
131	C₂²	C₂×C₂²⋊Dic₁₂₇	1	1	yes	characteristic
132	C₂²	#16861168	1	1	yes	characteristic
133	C₂²	C₂²×C₁₂₇⋊C₈	1	2	yes	—
134	C₂²	C₂×#14523232	1	2	yes	—
135	C₄	—	2	4	no	—
136	C₄	—	2	2	no	—
137	C₄	C₂²⋊(C₁₂₇⋊C₈)	1	8	yes	direct product factor
138	C₄	C₄×C₂²⋊D₁₂₇	1	4	yes	—
139	C₂	—	2	4	no	—
140	C₂	C₄×C₂²⋊Dic₁₂₇	1	1	yes	characteristic
141	C₂	C₂×C₄×C₁₂₇⋊C₈	1	1	yes	characteristic
142	C₂	C₄×#14523232	1	1	yes	characteristic
143	C₂	C₂×C₂²⋊(C₁₂₇⋊C₈)	1	2	yes	—
144	C₂	#22549929	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂²⋊(C₁₂₇⋊C₈)

C₂

yes

maximal

C₂×C₄×C₁₂₇⋊C₈

C₂

yes

maximal, complemented

#22549631

C₂

yes

maximal, complemented

C₂²×C₄²×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×C₁₂₇⋊C₈

C₂²

yes

—

C₂²⋊(C₁₂₇⋊C₈)

C₄

yes

direct product factor

C₄×C₁₂₇⋊C₈

—

C₂³×C₄×C₁₂₇

C₄

yes

—

C₂³×C₄×C₁₂₇

C₂²

yes

characteristic

C₂×C₄²×C₁₂₇

—

C₂×C₄²×C₁₂₇

C₄

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

complemented

C₂⁴×C₁₂₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

C₈

yes

complemented

C₂²×C₄×C₁₂₇

C₂×C₄

yes

—

C₂²×C₄×C₁₂₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₂₇

C₂×C₄

yes

—

C₂²×C₄×C₁₂₇

C₂×C₄

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₄²×C₁₂₇

—

C₄²×C₁₂₇

—

C₄²×C₁₂₇

D₄

yes

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

C₄²

yes

—

C₂³×C₁₂₇

C₂×C₈

yes

—

C₂³×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₄²

yes

—

C₂×C₄×C₁₂₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₂₇

C₂×C₈

yes

—

C₂×C₄×C₁₂₇

C₂×C₈

yes

—

C₂×C₄×C₁₂₇

M₁₆

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

#41

yes

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₄×C₈

yes

complemented

C₂²×C₁₂₇

C₂×C₄²

yes

characteristic

C₂²×C₁₂₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₂₇

#117

yes

characteristic

C₂²×C₁₂₇

C₂²×C₈

yes

—

C₂²×C₁₂₇

C₂×M₁₆

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂²⋊C₈

yes

complemented

C₄×C₁₂₇

C₄×D₄

yes

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₄×C₂²⋊C₄

yes

characteristic

C₂×C₁₂₇

C₂×C₄×C₈

yes

characteristic

C₂×C₁₂₇

C₄×M₁₆

yes

characteristic

C₂×C₁₂₇

C₂×C₂²⋊C₈

yes

—

C₂×C₁₂₇

#431

yes

—

C₄×C₂²⋊C₈

—

127

maximal, Sylow

C₁₂₇

C₄×C₂²⋊C₈

yes

characteristic, complemented, Sylow

C₂²×C₄²

D₁₂₇

yes

characteristic

C₂×C₂²⋊C₈

—

127

—

C₂×C₄×C₈

—

127

—

#336

—

127

—

C₂³×C₄

Dic₁₂₇

yes

—

C₂³×C₄

C₂×D₁₂₇

yes

characteristic

C₂²×C₈

—

127

—

C₂×C₄²

—

C₂×C₄²

Dic₁₂₇

yes

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

C₂²⋊C₈

—

127

—

C₄×C₈

—

254

—

C₂⁴

C₂×Dic₁₂₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₁₂₇⋊C₈

yes

complemented

C₂²×C₄

C₄×D₁₂₇

yes

—

C₂²×C₄

C₂×Dic₁₂₇

yes

characteristic

C₂²×C₄

C₂×Dic₁₂₇

yes

—

C₂²×C₄

C₂×Dic₁₂₇

yes

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

C₂×C₈

—

127

—

C₂×C₈

—

254

—

100

C₂×C₈

—

254

—

101

C₄²

—

102

C₄²

—

103

C₄²

C₂²⋊D₁₂₇

yes

—

104

C₂³

—

105

C₂³

—

106

C₂³

—

107

C₂³

C₄×Dic₁₂₇

yes

—

108

C₂³

C₂×C₁₂₇⋊C₈

yes

—

109

C₂³

C₂²×Dic₁₂₇

yes

characteristic

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

C₄×Dic₁₂₇

yes

—

117

C₂×C₄

C₂²⋊Dic₁₂₇

yes

—

118

C₂×C₄

C₂×C₁₂₇⋊C₈

yes

—

119

C₂×C₄

C₂×C₁₂₇⋊C₈

yes

—

120

C₂×C₄

#14523232

yes

—

121

C₂×C₄

C₂×C₄×D₁₂₇

yes

—

122

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

—

123

C₂×C₄

#14523248

yes

—

124

C₈

—

254

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

C₄×C₁₂₇⋊C₈

yes

complemented

130

C₂²

C₂×C₄×Dic₁₂₇

yes

characteristic

131

C₂²

C₂×C₂²⋊Dic₁₂₇

yes

characteristic

132

C₂²

#16861168

yes

characteristic

133

C₂²

C₂²×C₁₂₇⋊C₈

yes

—

134

C₂²

C₂×#14523232

yes

—

135

C₄

—

136

C₄

—

137

C₄

C₂²⋊(C₁₂₇⋊C₈)

yes

direct product factor

138

C₄

C₄×C₂²⋊D₁₂₇

yes

—

139

C₂

—

140

C₂

C₄×C₂²⋊Dic₁₂₇

yes

characteristic

141

C₂

C₂×C₄×C₁₂₇⋊C₈

yes

characteristic

142

C₂

C₄×#14523232

yes

characteristic

143

C₂

C₂×C₂²⋊(C₁₂₇⋊C₈)

yes

—

144

C₂

#22549929

yes

—

	subgroup	G/H
center	C₂×C₄²	C₂×D₁₂₇
commutator	C₂×C₁₂₇	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	C₂²×C₄²×C₁₂₇	C₂
socle	C₂³×C₁₂₇	C₂²×C₄
prime radical	C₂⁴×C₁₂₇	C₂×C₄
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	C₂²×C₄²	D₁₂₇
hypocenter	C₁₂₇	C₄×C₂²⋊C₈
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄²

subgroup

G/H

center

C₂×C₄²

C₂×D₁₂₇

commutator

C₂×C₁₂₇

C₂×C₄×C₈

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

C₂²×C₄²×C₁₂₇

C₂

socle

C₂³×C₁₂₇

C₂²×C₄

prime radical

C₂⁴×C₁₂₇

C₂×C₄

cosocle

C₂²×C₄×C₁₂₇

C₂³

hypercenter

C₂²×C₄²

D₁₂₇

hypocenter

C₁₂₇

C₄×C₂²⋊C₈

LCS

G ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂×C₄² ◁ C₂²×C₄²

N	H ≅ G/N	type
2. C₂×C₄×C₁₂₇⋊C₈	C₂	semidirect
3. #22549631	C₂	semidirect
6. C₂²⋊(C₁₂₇⋊C₈)	C₄	direct (N × H)
16. C₂×C₁₂₇⋊C₈	C₂×C₄	semidirect
22. C₂²×C₄×C₁₂₇	C₈	semidirect
56. C₂²×C₁₂₇	C₄×C₈	semidirect
64. C₄×C₁₂₇	C₂²⋊C₈	semidirect
73. C₁₂₇	C₄×C₂²⋊C₈	semidirect
92. C₂²×C₄	C₁₂₇⋊C₈	semidirect
129. C₂²	C₄×C₁₂₇⋊C₈	semidirect
137. C₄	C₂²⋊(C₁₂₇⋊C₈)	direct (N × H)

H ≅ G/N

type

2. C₂×C₄×C₁₂₇⋊C₈

C₂

semidirect

3. #22549631

C₂

semidirect

6. C₂²⋊(C₁₂₇⋊C₈)

C₄

direct (N × H)

16. C₂×C₁₂₇⋊C₈

C₂×C₄

semidirect

22. C₂²×C₄×C₁₂₇

C₈

semidirect

56. C₂²×C₁₂₇

C₄×C₈

semidirect

64. C₄×C₁₂₇

C₂²⋊C₈

semidirect

73. C₁₂₇

C₄×C₂²⋊C₈

semidirect

92. C₂²×C₄

C₁₂₇⋊C₈

semidirect

129. C₂²

C₄×C₁₂₇⋊C₈

semidirect

137. C₄

C₂²⋊(C₁₂₇⋊C₈)

direct (N × H)

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

subgroup	index
C₂×C₂²⋊(C₁₂₇⋊C₈)	2
C₂×C₄×C₁₂₇⋊C₈	2
#22549631	2
C₂²×C₄²×C₁₂₇	2
C₄×C₂²⋊C₈	127

subgroup

index

C₂×C₂²⋊(C₁₂₇⋊C₈)

C₂×C₄×C₁₂₇⋊C₈

#22549631

C₂²×C₄²×C₁₂₇

C₄×C₂²⋊C₈

127

Subgroups — C₄×C₂²⋊(C₁₂₇⋊C₈) #30638488

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C4×C22⋊(C127⋊C8) #30638488

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₂²⋊(C₁₂₇⋊C₈) #30638488