Group #30634895 — subgroups

Subgroups — C₁₂₇×#1943 #30634895

Order 16256.

240 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 128 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×#107×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#106×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#97×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	#389×C₁₂₇	C₂	1	2	yes	maximal
5	#340×C₁₂₇	C₂	1	2	yes	maximal
6	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
9	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
10	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
12	#107×C₁₂₇	C₄	1	4	yes	complemented
13	#106×C₁₂₇	—	2	2	no	—
14	#97×C₁₂₇	—	2	2	no	—
15	#95×C₁₂₇	—	2	2	no	—
16	C₂²×C₄×C₁₂₇	—	2	2	no	—
17	C₂²×C₄×C₁₂₇	—	2	2	no	—
18	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
19	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
20	C₂×C₈×C₁₂₇	—	2	1	no	—
21	C₂×C₈×C₁₂₇	—	2	1	no	—
22	C₂×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
23	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
24	C₂×C₈×C₁₂₇	Q₈	1	2	yes	—
25	C₂×C₈×C₁₂₇	Q₈	1	2	yes	—
26	C₄⋊C₄×C₁₂₇	—	2	2	no	—
27	C₄⋊C₄×C₁₂₇	—	2	2	no	—
28	C₄⋊C₄×C₁₂₇	—	2	2	no	—
29	C₄⋊C₄×C₁₂₇	—	4	4	no	—
30	C₄⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
31	C₄²×C₁₂₇	—	2	1	no	—
32	C₄²×C₁₂₇	—	2	1	no	—
33	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₂₇	—	2	4	no	—
35	C₂×C₄×C₁₂₇	—	2	2	no	—
36	C₂×C₄×C₁₂₇	—	2	2	no	—
37	C₂×C₄×C₁₂₇	—	2	2	no	—
38	C₂×C₄×C₁₂₇	—	4	2	no	—
39	C₂×C₄×C₁₂₇	—	4	2	no	—
40	C₂×C₄×C₁₂₇	—	4	2	no	—
41	C₂×C₄×C₁₂₇	—	4	2	no	—
42	C₂×C₄×C₁₂₇	—	4	2	no	—
43	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
48	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
49	C₈×C₁₂₇	—	2	2	no	—
50	C₈×C₁₂₇	C₄⋊C₄	1	4	yes	complemented
51	C₂²×C₁₂₇	C₄×D₄	1	1	yes	characteristic
52	C₂²×C₁₂₇	#121	1	1	yes	characteristic
53	C₂²×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
54	C₂²×C₁₂₇	AGL(1,ℤ/8)	1	2	yes	—
55	C₂²×C₁₂₇	#137	1	2	yes	—
56	C₄×C₁₂₇	—	4	4	no	—
57	C₄×C₁₂₇	—	4	2	no	—
58	C₄×C₁₂₇	—	8	2	no	—
59	C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₄×C₁₂₇	C₄×Q₈	1	1	yes	characteristic
61	C₄×C₁₂₇	C₂²⋊Q₈	1	1	yes	characteristic
62	C₄×C₁₂₇	#125	1	1	yes	characteristic
63	C₂×C₁₂₇	#389	1	1	yes	characteristic
64	C₂×C₁₂₇	#441	1	1	yes	characteristic
65	C₂×C₁₂₇	#442	1	1	yes	characteristic
66	C₂×C₁₂₇	#468	1	2	yes	—
67	C₂×C₁₂₇	#501	1	2	yes	—
68	#1943	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₁₂₇	#1943	1	1	yes	characteristic, direct product factor, Sylow
70	C₂×#107	C₂×C₁₂₇	1	1	yes	characteristic
71	C₂×#106	C₂×C₁₂₇	1	1	yes	characteristic
72	C₂×#97	C₂×C₁₂₇	1	1	yes	characteristic
73	#389	C₂×C₁₂₇	1	2	yes	—
74	#340	C₂×C₁₂₇	1	2	yes	—
75	C₂²×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
76	C₂²×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
77	C₂×C₄⋊C₄	—	2	2	no	—
78	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
79	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
80	C₂×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
81	#107	C₄×C₁₂₇	1	4	yes	complemented
82	#106	—	2	2	no	—
83	#97	—	2	2	no	—
84	#95	—	2	2	no	—
85	C₂²×C₄	—	2	2	no	—
86	C₂²×C₄	—	2	2	no	—
87	C₂²×C₄	D₄×C₁₂₇	1	2	yes	—
88	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
92	C₂×C₈	D₄×C₁₂₇	1	2	yes	—
93	C₂×C₈	Q₈×C₁₂₇	1	2	yes	—
94	C₂×C₈	Q₈×C₁₂₇	1	2	yes	—
95	C₄⋊C₄	—	2	2	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	4	4	no	—
99	C₄⋊C₄	C₂×C₄×C₁₂₇	1	4	yes	—
100	C₄²	—	2	1	no	—
101	C₄²	—	2	1	no	—
102	C₂³	C₂×D₄×C₁₂₇	1	1	yes	characteristic
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	—	4	2	no	—
112	C₂×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
113	C₂×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
114	C₂×C₄	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
115	C₂×C₄	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
116	C₂×C₄	#41×C₁₂₇	1	1	yes	characteristic
117	C₂×C₄	#41×C₁₂₇	1	1	yes	characteristic
118	C₈	—	2	2	no	—
119	C₈	C₄⋊C₄×C₁₂₇	1	4	yes	complemented
120	C₂²	C₄×D₄×C₁₂₇	1	1	yes	characteristic
121	C₂²	#121×C₁₂₇	1	1	yes	characteristic
122	C₂²	C₄⋊Q₈×C₁₂₇	1	1	yes	characteristic
123	C₂²	AGL(1,ℤ/8)×C₁₂₇	1	2	yes	—
124	C₂²	#137×C₁₂₇	1	2	yes	—
125	C₄	—	4	4	no	—
126	C₄	—	4	2	no	—
127	C₄	—	8	2	no	—
128	C₄	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
129	C₄	C₄×Q₈×C₁₂₇	1	1	yes	characteristic
130	C₄	C₂²⋊Q₈×C₁₂₇	1	1	yes	characteristic
131	C₄	#125×C₁₂₇	1	1	yes	characteristic
132	C₂	#389×C₁₂₇	1	1	yes	characteristic
133	C₂	#441×C₁₂₇	1	1	yes	characteristic
134	C₂	#442×C₁₂₇	1	1	yes	characteristic
135	C₂	#468×C₁₂₇	1	2	yes	—
136	C₂	#501×C₁₂₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#107×C₁₂₇

C₂

yes

maximal, characteristic

C₂×#106×C₁₂₇

C₂

yes

maximal, characteristic

C₂×#97×C₁₂₇

C₂

yes

maximal, characteristic

#389×C₁₂₇

C₂

yes

maximal

#340×C₁₂₇

C₂

yes

maximal

C₂²×C₈×C₁₂₇

C₂²

yes

characteristic

C₂²×C₈×C₁₂₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₂₇

—

C₂×C₄⋊C₄×C₁₂₇

C₂²

yes

—

C₂×C₄⋊C₄×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

#107×C₁₂₇

C₄

yes

complemented

#106×C₁₂₇

—

#97×C₁₂₇

—

#95×C₁₂₇

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

D₄

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

C₂×C₄

yes

—

C₂×C₈×C₁₂₇

D₄

yes

—

C₂×C₈×C₁₂₇

Q₈

yes

—

C₂×C₈×C₁₂₇

Q₈

yes

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

C₂×C₄

yes

—

C₄²×C₁₂₇

—

C₄²×C₁₂₇

—

C₂³×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₂₇

C₂×Q₈

yes

C₂×C₄×C₁₂₇

yes

C₂×C₄×C₁₂₇

yes

C₈×C₁₂₇

—

C₈×C₁₂₇

C₄⋊C₄

yes

complemented

C₂²×C₁₂₇

C₄×D₄

yes

characteristic

C₂²×C₁₂₇

#121

yes

characteristic

C₂²×C₁₂₇

C₄⋊Q₈

yes

characteristic

C₂²×C₁₂₇

AGL(1,ℤ/8)

yes

—

C₂²×C₁₂₇

#137

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₂₇

C₄×Q₈

yes

characteristic

C₄×C₁₂₇

C₂²⋊Q₈

yes

C₄×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

C₂×C₁₂₇

#501

yes

—

#1943

C₁₂₇

yes

maximal, characteristic, direct product factor, Sylow

C₁₂₇

#1943

yes

characteristic, direct product factor, Sylow

C₂×#107

C₂×C₁₂₇

yes

characteristic

C₂×#106

C₂×C₁₂₇

yes

characteristic

C₂×#97

C₂×C₁₂₇

yes

characteristic

#389

C₂×C₁₂₇

yes

—

#340

C₂×C₁₂₇

yes

—

C₂²×C₈

C₂²×C₁₂₇

yes

characteristic

C₂²×C₈

C₂²×C₁₂₇

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²×C₁₂₇

yes

—

C₂×C₄⋊C₄

C₂²×C₁₂₇

yes

—

C₂×C₄²

C₂²×C₁₂₇

yes

characteristic

#107

C₄×C₁₂₇

yes

—

—

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₄×C₁₂₇

yes

—

C₂²×C₄

C₂³×C₁₂₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₄×C₁₂₇

yes

—

C₂×C₈

D₄×C₁₂₇

yes

—

C₂×C₈

Q₈×C₁₂₇

yes

—

C₂×C₈

Q₈×C₁₂₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×C₄×C₁₂₇

yes

—

100

C₄²

—

101

C₄²

—

102

C₂³

C₂×D₄×C₁₂₇

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

C₂²×C₄×C₁₂₇

yes

characteristic

113

C₂×C₄

C₂×D₄×C₁₂₇

yes

characteristic

114

C₂×C₄

C₂×Q₈×C₁₂₇

yes

characteristic

115

C₂×C₄

C₂×Q₈×C₁₂₇

yes

characteristic

116

C₂×C₄

#41×C₁₂₇

yes

characteristic

117

C₂×C₄

#41×C₁₂₇

yes

characteristic

118

C₈

—

119

C₈

C₄⋊C₄×C₁₂₇

yes

complemented

120

C₂²

C₄×D₄×C₁₂₇

yes

characteristic

121

C₂²

#121×C₁₂₇

yes

characteristic

122

C₂²

C₄⋊Q₈×C₁₂₇

yes

characteristic

123

C₂²

AGL(1,ℤ/8)×C₁₂₇

yes

—

124

C₂²

#137×C₁₂₇

yes

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

C₂×C₄⋊C₄×C₁₂₇

yes

characteristic

129

C₄

C₄×Q₈×C₁₂₇

yes

characteristic

130

C₄

C₂²⋊Q₈×C₁₂₇

yes

characteristic

131

C₄

#125×C₁₂₇

yes

characteristic

132

C₂

#389×C₁₂₇

yes

characteristic

133

C₂

#441×C₁₂₇

yes

characteristic

134

C₂

#442×C₁₂₇

yes

characteristic

135

C₂

#468×C₁₂₇

yes

—

136

C₂

#501×C₁₂₇

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₂₇	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₂²×C₄	C₂³×C₁₂₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₁₂₇

C₂×D₄

commutator

C₂×C₄

C₂²×C₄×C₁₂₇

Frattini, cosocle

C₂²×C₄

C₂³×C₁₂₇

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ G

N	H ≅ G/N	type
12. #107×C₁₂₇	C₄	semidirect
50. C₈×C₁₂₇	C₄⋊C₄	semidirect
68. #1943	C₁₂₇	direct (N × H)
69. C₁₂₇	#1943	direct (N × H)
81. #107	C₄×C₁₂₇	semidirect
119. C₈	C₄⋊C₄×C₁₂₇	semidirect

H ≅ G/N

type

12. #107×C₁₂₇

C₄

semidirect

50. C₈×C₁₂₇

C₄⋊C₄

semidirect

68. #1943

C₁₂₇

direct (N × H)

69. C₁₂₇

#1943

direct (N × H)

81. #107

C₄×C₁₂₇

semidirect

119. C₈

C₄⋊C₄×C₁₂₇

semidirect

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

subgroup	index
C₂×#107×C₁₂₇	2
C₂×#106×C₁₂₇	2
C₂×#97×C₁₂₇	2
#389×C₁₂₇	2
#340×C₁₂₇	2
#1943	127

subgroup

index

C₂×#107×C₁₂₇

C₂×#106×C₁₂₇

C₂×#97×C₁₂₇

#389×C₁₂₇

#340×C₁₂₇

#1943

127

Subgroups — C₁₂₇×#1943 #30634895

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C127×#1943 #30634895

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#1943 #30634895