Group #30634694 — subgroups

Subgroups — C₁₂₇×#1742 #30634694

Order 16256.

388 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 216 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	C₂²×M₁₆×C₁₂₇	C₂	1	2	yes	maximal
2	C₂×#117×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	#343×C₁₂₇	C₂	1	4	yes	maximal, complemented
4	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
5	C₂×M₁₆×C₁₂₇	—	2	4	no	—
6	C₂×M₁₆×C₁₂₇	—	2	2	no	—
7	C₂×M₁₆×C₁₂₇	C₄	1	8	yes	complemented
8	C₂×M₁₆×C₁₂₇	C₂²	1	4	yes	—
9	C₂²×C₈×C₁₂₇	—	2	2	no	—
10	#117×C₁₂₇	—	2	2	no	—
11	#117×C₁₂₇	—	2	2	no	—
12	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
13	C₂×C₂²⋊C₄×C₁₂₇	C₄	1	2	yes	—
14	C₂×C₄²×C₁₂₇	C₄	1	2	yes	—
15	C₂⁴×C₁₂₇	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
17	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
19	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
20	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
21	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
22	C₂²×C₄×C₁₂₇	Q₈	1	2	yes	—
23	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
24	M₁₆×C₁₂₇	—	2	8	no	—
25	M₁₆×C₁₂₇	—	2	4	no	—
26	M₁₆×C₁₂₇	C₂×C₄	1	8	yes	—
27	C₂×C₈×C₁₂₇	—	2	8	no	—
28	C₂×C₈×C₁₂₇	—	2	4	no	—
29	C₄⋊C₄×C₁₂₇	—	2	4	no	—
30	C₂²⋊C₄×C₁₂₇	—	2	4	no	—
31	C₄²×C₁₂₇	—	2	2	no	—
32	C₄²×C₁₂₇	—	2	2	no	—
33	C₂³×C₁₂₇	—	4	2	no	—
34	C₂³×C₁₂₇	C₄²	1	2	yes	—
35	C₂³×C₁₂₇	C₂²⋊C₄	1	2	yes	—
36	C₂³×C₁₂₇	C₄⋊C₄	1	2	yes	—
37	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₂₇	—	2	4	no	—
39	C₂×C₄×C₁₂₇	—	4	4	no	—
40	C₂×C₄×C₁₂₇	C₄²	1	2	yes	—
41	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	4	yes	—
43	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	4	yes	—
44	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₂₇	C₄⋊C₄	1	2	yes	—
46	C₂×C₄×C₁₂₇	C₄⋊C₄	1	4	yes	—
47	C₂×C₄×C₁₂₇	C₄⋊C₄	1	4	yes	—
48	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
52	C₈×C₁₂₇	—	2	8	no	—
53	C₂²×C₁₂₇	—	2	2	no	—
54	C₂²×C₁₂₇	—	2	4	no	—
55	C₂²×C₁₂₇	—	4	2	no	—
56	C₂²×C₁₂₇	—	4	2	no	—
57	C₂²×C₁₂₇	#95	1	4	yes	—
58	C₂²×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
59	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₄×C₁₂₇	—	4	4	no	—
62	C₄×C₁₂₇	#95	1	2	yes	—
63	C₄×C₁₂₇	#95	1	2	yes	—
64	C₄×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
65	C₄×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
66	C₂×C₁₂₇	—	2	4	no	—
67	C₂×C₁₂₇	—	2	1	no	—
68	C₂×C₁₂₇	—	2	1	no	—
69	C₂×C₁₂₇	C₂×#95	1	1	yes	characteristic
70	C₂×C₁₂₇	#413	1	2	yes	—
71	#1742	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₁₂₇	#1742	1	1	yes	characteristic, direct product factor, Sylow
73	C₂²×M₁₆	C₂×C₁₂₇	1	2	yes	—
74	C₂×#117	C₂×C₁₂₇	1	1	yes	characteristic
75	#343	C₂×C₁₂₇	1	4	yes	complemented
76	C₂³×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
77	C₂×M₁₆	—	2	4	no	—
78	C₂×M₁₆	—	2	2	no	—
79	C₂×M₁₆	C₄×C₁₂₇	1	8	yes	complemented
80	C₂×M₁₆	C₂²×C₁₂₇	1	4	yes	—
81	C₂²×C₈	—	2	2	no	—
82	#117	—	2	2	no	—
83	#117	—	2	2	no	—
84	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
85	C₂×C₂²⋊C₄	C₄×C₁₂₇	1	2	yes	—
86	C₂×C₄²	C₄×C₁₂₇	1	2	yes	—
87	C₂⁴	C₂×C₄×C₁₂₇	1	1	yes	characteristic
88	C₂²×C₄	C₂×C₄×C₁₂₇	1	4	yes	—
89	C₂²×C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
90	C₂²×C₄	C₂×C₄×C₁₂₇	1	4	yes	—
91	C₂²×C₄	D₄×C₁₂₇	1	2	yes	—
92	C₂²×C₄	D₄×C₁₂₇	1	2	yes	—
93	C₂²×C₄	D₄×C₁₂₇	1	2	yes	—
94	C₂²×C₄	Q₈×C₁₂₇	1	2	yes	—
95	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
96	M₁₆	—	2	8	no	—
97	M₁₆	—	2	4	no	—
98	M₁₆	C₂×C₄×C₁₂₇	1	8	yes	—
99	C₂×C₈	—	2	8	no	—
100	C₂×C₈	—	2	4	no	—
101	C₄⋊C₄	—	2	4	no	—
102	C₂²⋊C₄	—	2	4	no	—
103	C₄²	—	2	2	no	—
104	C₄²	—	2	2	no	—
105	C₂³	—	4	2	no	—
106	C₂³	C₄²×C₁₂₇	1	2	yes	—
107	C₂³	C₂²⋊C₄×C₁₂₇	1	2	yes	—
108	C₂³	C₄⋊C₄×C₁₂₇	1	2	yes	—
109	C₂³	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
110	C₂×C₄	—	2	4	no	—
111	C₂×C₄	—	4	4	no	—
112	C₂×C₄	C₄²×C₁₂₇	1	2	yes	—
113	C₂×C₄	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
114	C₂×C₄	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
115	C₂×C₄	C₂²⋊C₄×C₁₂₇	1	4	yes	—
116	C₂×C₄	C₂²⋊C₄×C₁₂₇	1	4	yes	—
117	C₂×C₄	C₄⋊C₄×C₁₂₇	1	2	yes	—
118	C₂×C₄	C₄⋊C₄×C₁₂₇	1	4	yes	—
119	C₂×C₄	C₄⋊C₄×C₁₂₇	1	4	yes	—
120	C₂×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
121	C₂×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
122	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
123	C₂×C₄	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
124	C₈	—	2	8	no	—
125	C₂²	—	2	2	no	—
126	C₂²	—	2	4	no	—
127	C₂²	—	4	2	no	—
128	C₂²	—	4	2	no	—
129	C₂²	#95×C₁₂₇	1	4	yes	—
130	C₂²	C₂×C₄²×C₁₂₇	1	1	yes	characteristic
131	C₂²	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
132	C₂²	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
133	C₄	—	4	4	no	—
134	C₄	#95×C₁₂₇	1	2	yes	—
135	C₄	#95×C₁₂₇	1	2	yes	—
136	C₄	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
137	C₄	C₂×C₄⋊C₄×C₁₂₇	1	2	yes	—
138	C₂	—	2	4	no	—
139	C₂	—	2	1	no	—
140	C₂	—	2	1	no	—
141	C₂	C₂×#95×C₁₂₇	1	1	yes	characteristic
142	C₂	#413×C₁₂₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×M₁₆×C₁₂₇

C₂

yes

maximal

C₂×#117×C₁₂₇

C₂

yes

maximal, characteristic

#343×C₁₂₇

C₂

yes

maximal, complemented

C₂³×C₄×C₁₂₇

C₂²

yes

characteristic

C₂×M₁₆×C₁₂₇

—

C₂×M₁₆×C₁₂₇

—

C₂×M₁₆×C₁₂₇

C₄

yes

complemented

C₂×M₁₆×C₁₂₇

C₂²

yes

—

C₂²×C₈×C₁₂₇

—

#117×C₁₂₇

—

#117×C₁₂₇

—

C₂×C₄⋊C₄×C₁₂₇

C₂²

yes

—

C₂×C₂²⋊C₄×C₁₂₇

C₄

yes

—

C₂×C₄²×C₁₂₇

C₄

yes

—

C₂⁴×C₁₂₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₂₇

C₂×C₄

yes

—

C₂²×C₄×C₁₂₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₂₇

C₂×C₄

yes

—

C₂²×C₄×C₁₂₇

D₄

yes

—

C₂²×C₄×C₁₂₇

D₄

yes

—

C₂²×C₄×C₁₂₇

D₄

yes

—

C₂²×C₄×C₁₂₇

Q₈

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

M₁₆×C₁₂₇

—

M₁₆×C₁₂₇

—

M₁₆×C₁₂₇

C₂×C₄

yes

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₂²⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

—

C₄²×C₁₂₇

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

C₄²

yes

—

C₂³×C₁₂₇

C₂²⋊C₄

yes

—

C₂³×C₁₂₇

C₄⋊C₄

yes

—

C₂³×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₄²

yes

—

C₂×C₄×C₁₂₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₂₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₂₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₂₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₂₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×Q₈

yes

characteristic

C₈×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

#95

yes

—

C₂²×C₁₂₇

C₂×C₄²

yes

characteristic

C₂²×C₁₂₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₂₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

#95

yes

—

C₄×C₁₂₇

#95

yes

—

C₄×C₁₂₇

C₂×C₂²⋊C₄

yes

—

C₄×C₁₂₇

C₂×C₄⋊C₄

yes

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂×#95

yes

characteristic

C₂×C₁₂₇

#413

yes

—

#1742

C₁₂₇

yes

maximal, characteristic, direct product factor, Sylow

C₁₂₇

#1742

yes

characteristic, direct product factor, Sylow

C₂²×M₁₆

C₂×C₁₂₇

yes

—

C₂×#117

C₂×C₁₂₇

yes

characteristic

#343

C₂×C₁₂₇

yes

complemented

C₂³×C₄

C₂²×C₁₂₇

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₄×C₁₂₇

yes

complemented

C₂×M₁₆

C₂²×C₁₂₇

yes

—

C₂²×C₈

—

#117

—

#117

—

C₂×C₄⋊C₄

C₂²×C₁₂₇

yes

—

C₂×C₂²⋊C₄

C₄×C₁₂₇

yes

—

C₂×C₄²

C₄×C₁₂₇

yes

—

C₂⁴

C₂×C₄×C₁₂₇

yes

characteristic

C₂²×C₄

C₂×C₄×C₁₂₇

yes

—

C₂²×C₄

C₂×C₄×C₁₂₇

yes

characteristic

C₂²×C₄

C₂×C₄×C₁₂₇

yes

—

C₂²×C₄

D₄×C₁₂₇

yes

—

C₂²×C₄

D₄×C₁₂₇

yes

—

C₂²×C₄

D₄×C₁₂₇

yes

—

C₂²×C₄

Q₈×C₁₂₇

yes

—

C₂²×C₄

C₂³×C₁₂₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

C₂×C₄×C₁₂₇

yes

—

C₂×C₈

—

100

C₂×C₈

—

101

C₄⋊C₄

—

102

C₂²⋊C₄

—

103

C₄²

—

104

C₄²

—

105

C₂³

—

106

C₂³

C₄²×C₁₂₇

yes

—

107

C₂³

C₂²⋊C₄×C₁₂₇

yes

—

108

C₂³

C₄⋊C₄×C₁₂₇

yes

—

109

C₂³

C₂²×C₄×C₁₂₇

yes

characteristic

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

C₄²×C₁₂₇

yes

—

113

C₂×C₄

C₂²⋊C₄×C₁₂₇

yes

characteristic

114

C₂×C₄

C₂²⋊C₄×C₁₂₇

yes

characteristic

115

C₂×C₄

C₂²⋊C₄×C₁₂₇

yes

—

116

C₂×C₄

C₂²⋊C₄×C₁₂₇

yes

—

117

C₂×C₄

C₄⋊C₄×C₁₂₇

yes

—

118

C₂×C₄

C₄⋊C₄×C₁₂₇

yes

—

119

C₂×C₄

C₄⋊C₄×C₁₂₇

yes

—

120

C₂×C₄

C₂²×C₄×C₁₂₇

yes

—

121

C₂×C₄

C₂×D₄×C₁₂₇

yes

characteristic

122

C₂×C₄

C₂×D₄×C₁₂₇

yes

—

123

C₂×C₄

C₂×Q₈×C₁₂₇

yes

characteristic

124

C₈

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

#95×C₁₂₇

yes

—

130

C₂²

C₂×C₄²×C₁₂₇

yes

characteristic

131

C₂²

C₂×C₂²⋊C₄×C₁₂₇

yes

characteristic

132

C₂²

C₂×C₄⋊C₄×C₁₂₇

yes

characteristic

133

C₄

—

134

C₄

#95×C₁₂₇

yes

—

135

C₄

#95×C₁₂₇

yes

—

136

C₄

C₂×C₂²⋊C₄×C₁₂₇

yes

—

137

C₄

C₂×C₄⋊C₄×C₁₂₇

yes

—

138

C₂

—

139

C₂

—

140

C₂

—

141

C₂

C₂×#95×C₁₂₇

yes

characteristic

142

C₂

#413×C₁₂₇

yes

—

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

subgroup

G/H

center

C₂×C₄×C₁₂₇

C₂²⋊C₄

commutator

C₂²

C₂×C₄²×C₁₂₇

Frattini, cosocle

C₂²×C₄

C₂³×C₁₂₇

socle

C₂²×C₁₂₇

C₂×C₂²⋊C₄

prime radical

C₂⁴×C₁₂₇

C₂×C₄

LCS

G ▷ C₂² ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₁₂₇ ◁ C₂³×C₄×C₁₂₇ ◁ G

N	H ≅ G/N	type
3. #343×C₁₂₇	C₂	semidirect
7. C₂×M₁₆×C₁₂₇	C₄	semidirect
71. #1742	C₁₂₇	direct (N × H)
72. C₁₂₇	#1742	direct (N × H)
75. #343	C₂×C₁₂₇	semidirect
79. C₂×M₁₆	C₄×C₁₂₇	semidirect

H ≅ G/N

type

3. #343×C₁₂₇

C₂

semidirect

7. C₂×M₁₆×C₁₂₇

C₄

semidirect

71. #1742

C₁₂₇

direct (N × H)

72. C₁₂₇

#1742

direct (N × H)

75. #343

C₂×C₁₂₇

semidirect

79. C₂×M₁₆

C₄×C₁₂₇

semidirect

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

subgroup	index
C₂²×M₁₆×C₁₂₇	2
C₂×#117×C₁₂₇	2
#343×C₁₂₇	2
#1742	127

subgroup

index

C₂²×M₁₆×C₁₂₇

C₂×#117×C₁₂₇

#343×C₁₂₇

#1742

127

Subgroups — C₁₂₇×#1742 #30634694

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C127×#1742 #30634694

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#1742 #30634694