Group #30638594 — subgroups

Subgroups — C₄⋊C₄×C₁₂₇⋊C₈ #30638594

Order 16256.

296 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 181 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×C₄⋊(C₁₂₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₁₂₇⋊C₈	C₂	1	2	yes	maximal
3	C₂×C₄×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#22549631	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
7	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
8	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₄⋊(C₁₂₇⋊C₈)	C₄	1	4	yes	complemented
10	C₄×C₁₂₇⋊C₈	—	2	4	no	—
11	C₄×C₁₂₇⋊C₈	C₄	1	4	yes	complemented
12	C₂×C₄⋊C₄×C₁₂₇	C₄	1	2	yes	—
13	C₂×C₄²×C₁₂₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
15	#95×C₁₂₇	C₄	1	2	yes	—
16	C₂×C₁₂₇⋊C₈	—	2	2	no	—
17	C₂×C₁₂₇⋊C₈	—	2	1	no	—
18	C₂×C₁₂₇⋊C₈	—	2	1	no	—
19	C₂×C₁₂₇⋊C₈	—	2	2	no	—
20	C₂×C₁₂₇⋊C₈	C₂×C₄	1	2	yes	—
21	C₂×C₁₂₇⋊C₈	C₂×C₄	1	4	yes	—
22	C₂×C₁₂₇⋊C₈	C₂×C₄	1	2	yes	—
23	C₂×C₁₂₇⋊C₈	D₄	1	2	yes	—
24	C₂×C₁₂₇⋊C₈	Q₈	1	2	yes	—
25	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
27	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₁₂₇	C₈	1	8	yes	complemented
30	C₄²×C₁₂₇	—	2	2	no	—
31	C₄²×C₁₂₇	—	2	2	no	—
32	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
33	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
34	C₁₂₇⋊C₈	—	2	2	no	—
35	C₁₂₇⋊C₈	—	2	4	no	—
36	C₁₂₇⋊C₈	C₄⋊C₄	1	4	yes	direct product factor
37	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₂₇	—	2	4	no	—
39	C₂×C₄×C₁₂₇	—	2	2	no	—
40	C₂×C₄×C₁₂₇	—	2	4	no	—
41	C₂×C₄×C₁₂₇	C₄²	1	2	yes	—
42	C₂×C₄×C₁₂₇	C₄²	1	2	yes	—
43	C₂×C₄×C₁₂₇	C₂×C₈	1	8	yes	—
44	C₂×C₄×C₁₂₇	C₂×C₈	1	4	yes	—
45	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
50	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
51	C₂²×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
52	C₂²×C₁₂₇	C₄×D₄	1	1	yes	characteristic
53	C₂²×C₁₂₇	C₄×Q₈	1	1	yes	characteristic
54	C₂²×C₁₂₇	C₂²×C₈	1	2	yes	—
55	C₂²×C₁₂₇	#131	1	2	yes	—
56	C₄×C₁₂₇	—	2	2	no	—
57	C₄×C₁₂₇	—	2	8	no	—
58	C₄×C₁₂₇	C₄×C₈	1	4	yes	complemented
59	C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₄×C₁₂₇	#117	1	1	yes	characteristic
61	C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
62	C₄×C₁₂₇	C₄×Q₈	1	1	yes	characteristic
63	C₂×C₁₂₇	C₄×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₁₂₇	C₂×C₄×C₈	1	1	yes	characteristic
65	C₂×C₁₂₇	#405	1	1	yes	characteristic
66	C₂×C₁₂₇	C₈×D₄	1	2	yes	—
67	C₂×C₁₂₇	C₈×Q₈	1	2	yes	—
68	C₈×C₄⋊C₄	—	127	1	no	maximal, Sylow
69	C₁₂₇	C₈×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
70	C₂×C₄⋊C₈	—	127	1	no	—
71	C₂×C₄×C₈	—	127	1	no	—
72	C₂×C₄×C₈	—	127	2	no	—
73	C₄×C₄⋊C₄	D₁₂₇	1	1	yes	characteristic
74	#336	—	127	2	no	—
75	C₂²×C₈	—	127	1	no	—
76	C₂²×C₈	—	127	1	no	—
77	C₂²×C₈	—	127	2	no	—
78	C₂×C₄⋊C₄	Dic₁₂₇	1	2	yes	—
79	C₂×C₄²	C₂×D₁₂₇	1	2	yes	—
80	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
81	C₄⋊C₈	—	127	4	no	—
82	C₄×C₈	—	127	4	no	—
83	C₄×C₈	—	254	4	no	—
84	#95	Dic₁₂₇	1	2	yes	—
85	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
86	C₂²×C₄	C₂×Dic₁₂₇	1	4	yes	—
87	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
88	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
89	C₂×C₈	—	127	2	no	—
90	C₂×C₈	—	127	2	no	—
91	C₂×C₈	—	127	2	no	—
92	C₂×C₈	—	127	2	no	—
93	C₂×C₈	—	127	4	no	—
94	C₂×C₈	—	254	2	no	—
95	C₂×C₈	—	254	1	no	—
96	C₂×C₈	—	254	1	no	—
97	C₂×C₈	—	254	2	no	—
98	C₄⋊C₄	C₁₂₇⋊C₈	1	8	yes	direct product factor
99	C₄²	—	2	2	no	—
100	C₄²	—	2	2	no	—
101	C₄²	C₄×D₁₂₇	1	2	yes	—
102	C₄²	C₄×D₁₂₇	1	2	yes	—
103	C₂³	C₂²×Dic₁₂₇	1	1	yes	characteristic
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	C₄×Dic₁₂₇	1	2	yes	—
108	C₂×C₄	C₄×Dic₁₂₇	1	2	yes	—
109	C₂×C₄	C₂×C₁₂₇⋊C₈	1	8	yes	—
110	C₂×C₄	C₂×C₁₂₇⋊C₈	1	4	yes	—
111	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
112	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
113	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
114	C₂×C₄	Q₈×D₁₂₇	1	1	yes	characteristic
115	C₂×C₄	#14523247	1	1	yes	characteristic
116	C₂×C₄	#14523249	1	1	yes	characteristic
117	C₈	—	127	4	no	—
118	C₈	—	254	2	no	—
119	C₈	—	254	4	no	—
120	C₂²	C₂×C₄×Dic₁₂₇	1	1	yes	characteristic
121	C₂²	D₄×Dic₁₂₇	1	1	yes	characteristic
122	C₂²	Q₈×Dic₁₂₇	1	1	yes	characteristic
123	C₂²	C₂²×C₁₂₇⋊C₈	1	2	yes	—
124	C₂²	#16861216	1	2	yes	—
125	C₄	—	2	2	no	—
126	C₄	—	2	8	no	—
127	C₄	C₄×C₁₂₇⋊C₈	1	4	yes	complemented
128	C₄	C₄⋊C₄×D₁₂₇	1	1	yes	characteristic
129	C₄	#16861167	1	1	yes	characteristic
130	C₄	#16861171	1	1	yes	characteristic
131	C₄	#16861174	1	1	yes	characteristic
132	C₂	C₄⋊C₄×Dic₁₂₇	1	1	yes	characteristic
133	C₂	C₂×C₄×C₁₂₇⋊C₈	1	1	yes	characteristic
134	C₂	#22549832	1	1	yes	characteristic
135	C₂	D₄×C₁₂₇⋊C₈	1	2	yes	—
136	C₂	Q₈×C₁₂₇⋊C₈	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₁₂₇⋊C₈)

C₂

yes

maximal, characteristic

C₂×C₄×C₁₂₇⋊C₈

C₂

yes

maximal

C₂×C₄×C₁₂₇⋊C₈

C₂

yes

maximal, characteristic

#22549631

C₂

yes

maximal

C₄×C₄⋊C₄×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₂²×C₁₂₇⋊C₈

C₂²

yes

—

C₂²×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₄⋊(C₁₂₇⋊C₈)

C₄

yes

complemented

C₄×C₁₂₇⋊C₈

—

C₄×C₁₂₇⋊C₈

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₂₇

C₄

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

#95×C₁₂₇

C₄

yes

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

—

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

—

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

—

C₂×C₁₂₇⋊C₈

D₄

yes

—

C₂×C₁₂₇⋊C₈

Q₈

yes

—

C₂²×C₄×C₁₂₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₂₇

C₂×C₄

yes

—

C₂²×C₄×C₁₂₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₂₇

C₈

yes

complemented

C₄²×C₁₂₇

—

C₄²×C₁₂₇

—

C₄²×C₁₂₇

C₂×C₄

yes

—

C₄²×C₁₂₇

C₂×C₄

yes

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

C₄⋊C₄

yes

direct product factor

C₂³×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₄²

yes

—

C₂×C₄×C₁₂₇

C₄²

yes

—

C₂×C₄×C₁₂₇

C₂×C₈

yes

—

C₂×C₄×C₁₂₇

C₂×C₈

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×Q₈

yes

C₂×C₄×C₁₂₇

yes

C₂×C₄×C₁₂₇

yes

C₂²×C₁₂₇

C₂×C₄²

yes

characteristic

C₂²×C₁₂₇

C₄×D₄

yes

characteristic

C₂²×C₁₂₇

C₄×Q₈

yes

characteristic

C₂²×C₁₂₇

C₂²×C₈

yes

—

C₂²×C₁₂₇

#131

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₄×C₈

yes

complemented

C₄×C₁₂₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₂₇

#117

yes

characteristic

C₄×C₁₂₇

C₄×D₄

yes

characteristic

C₄×C₁₂₇

C₄×Q₈

yes

characteristic

C₂×C₁₂₇

C₄×C₄⋊C₄

yes

characteristic

C₂×C₁₂₇

C₂×C₄×C₈

yes

characteristic

C₂×C₁₂₇

#405

yes

characteristic

C₂×C₁₂₇

C₈×D₄

yes

—

C₂×C₁₂₇

C₈×Q₈

yes

—

C₈×C₄⋊C₄

—

127

maximal, Sylow

C₁₂₇

C₈×C₄⋊C₄

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

127

—

C₂×C₄×C₈

—

127

—

C₂×C₄×C₈

—

127

—

C₄×C₄⋊C₄

D₁₂₇

yes

characteristic

#336

—

127

—

C₂²×C₈

—

127

—

C₂²×C₈

—

127

—

C₂²×C₈

—

127

—

C₂×C₄⋊C₄

Dic₁₂₇

yes

—

C₂×C₄²

C₂×D₁₂₇

yes

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

C₄⋊C₈

—

127

—

C₄×C₈

—

127

—

C₄×C₈

—

254

—

#95

Dic₁₂₇

yes

—

C₂²×C₄

C₂×Dic₁₂₇

yes

characteristic

C₂²×C₄

C₂×Dic₁₂₇

yes

—

C₂²×C₄

C₂×Dic₁₂₇

yes

characteristic

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₂×C₈

—

254

—

C₂×C₈

—

254

—

C₂×C₈

—

254

—

C₂×C₈

—

254

—

C₄⋊C₄

C₁₂₇⋊C₈

yes

direct product factor

C₄²

—

100

C₄²

—

101

C₄²

C₄×D₁₂₇

yes

—

102

C₄²

C₄×D₁₂₇

yes

—

103

C₂³

C₂²×Dic₁₂₇

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₄×Dic₁₂₇

yes

—

108

C₂×C₄

C₄×Dic₁₂₇

yes

—

109

C₂×C₄

C₂×C₁₂₇⋊C₈

yes

—

110

C₂×C₄

C₂×C₁₂₇⋊C₈

yes

—

111

C₂×C₄

C₂×C₄×D₁₂₇

yes

characteristic

112

C₂×C₄

C₂×C₄×D₁₂₇

yes

characteristic

113

C₂×C₄

D₄×D₁₂₇

yes

characteristic

114

C₂×C₄

Q₈×D₁₂₇

yes

characteristic

115

C₂×C₄

#14523247

yes

characteristic

116

C₂×C₄

#14523249

yes

characteristic

117

C₈

—

127

—

118

C₈

—

254

—

119

C₈

—

254

—

120

C₂²

C₂×C₄×Dic₁₂₇

yes

characteristic

121

C₂²

D₄×Dic₁₂₇

yes

characteristic

122

C₂²

Q₈×Dic₁₂₇

yes

characteristic

123

C₂²

C₂²×C₁₂₇⋊C₈

yes

—

124

C₂²

#16861216

yes

—

125

C₄

—

126

C₄

—

127

C₄

C₄×C₁₂₇⋊C₈

yes

complemented

128

C₄

C₄⋊C₄×D₁₂₇

yes

characteristic

129

C₄

#16861167

yes

characteristic

130

C₄

#16861171

yes

characteristic

131

C₄

#16861174

yes

characteristic

132

C₂

C₄⋊C₄×Dic₁₂₇

yes

characteristic

133

C₂

C₂×C₄×C₁₂₇⋊C₈

yes

characteristic

134

C₂

#22549832

yes

characteristic

135

C₂

D₄×C₁₂₇⋊C₈

yes

—

136

C₂

Q₈×C₁₂₇⋊C₈

yes

—

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

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₂₇

commutator

C₂×C₁₂₇

C₂×C₄×C₈

Fitting

C₄×C₄⋊C₄×C₁₂₇

C₂

socle, prime radical

C₂³×C₁₂₇

C₂²×C₄

cosocle

C₂²×C₄×C₁₂₇

C₂³

hypercenter

C₄×C₄⋊C₄

D₁₂₇

hypocenter

C₁₂₇

C₈×C₄⋊C₄

LCS

G ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
9. C₄⋊(C₁₂₇⋊C₈)	C₄	semidirect
11. C₄×C₁₂₇⋊C₈	C₄	semidirect
29. C₄⋊C₄×C₁₂₇	C₈	semidirect
36. C₁₂₇⋊C₈	C₄⋊C₄	direct (N × H)
58. C₄×C₁₂₇	C₄×C₈	semidirect
69. C₁₂₇	C₈×C₄⋊C₄	semidirect
98. C₄⋊C₄	C₁₂₇⋊C₈	direct (N × H)
127. C₄	C₄×C₁₂₇⋊C₈	semidirect

H ≅ G/N

type

9. C₄⋊(C₁₂₇⋊C₈)

C₄

semidirect

11. C₄×C₁₂₇⋊C₈

C₄

semidirect

29. C₄⋊C₄×C₁₂₇

C₈

semidirect

36. C₁₂₇⋊C₈

C₄⋊C₄

direct (N × H)

58. C₄×C₁₂₇

C₄×C₈

semidirect

69. C₁₂₇

C₈×C₄⋊C₄

semidirect

98. C₄⋊C₄

C₁₂₇⋊C₈

direct (N × H)

127. C₄

C₄×C₁₂₇⋊C₈

semidirect

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

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

subgroup

index

C₂×C₄⋊(C₁₂₇⋊C₈)

C₂×C₄×C₁₂₇⋊C₈

#22549631

C₄×C₄⋊C₄×C₁₂₇

C₈×C₄⋊C₄

127

Subgroups — C₄⋊C₄×C₁₂₇⋊C₈ #30638594

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C4⋊C4×C127⋊C8 #30638594

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄⋊C₄×C₁₂₇⋊C₈ #30638594