Group #30634949 — subgroups

Subgroups — C₁₂₇×#1997 #30634949

Order 16256.

260 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 120 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⋊Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	#423×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₄×M₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	#343×C₁₂₇	C₂	1	4	yes	maximal
5	C₂²×Q₈×C₁₂₇	C₄	1	2	yes	—
6	C₂×M₁₆×C₁₂₇	C₂²	1	2	yes	—
7	C₂×M₁₆×C₁₂₇	C₂²	1	2	yes	—
8	C₄⋊Q₈×C₁₂₇	—	2	1	no	—
9	C₄⋊Q₈×C₁₂₇	—	2	1	no	—
10	C₄⋊Q₈×C₁₂₇	—	2	1	no	—
11	C₄⋊Q₈×C₁₂₇	—	2	1	no	—
12	C₂×C₄⋊C₄×C₁₂₇	C₄	1	2	yes	—
13	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₁₂₇	—	2	1	no	—
16	C₄⋊C₈×C₁₂₇	—	2	1	no	—
17	#97×C₁₂₇	—	2	1	no	—
18	C₄×C₈×C₁₂₇	—	2	1	no	—
19	C₂×Q₈×C₁₂₇	—	2	2	no	—
20	C₂×Q₈×C₁₂₇	—	2	2	no	—
21	C₂×Q₈×C₁₂₇	—	4	4	no	—
22	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
25	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
26	M₁₆×C₁₂₇	—	2	2	no	—
27	M₁₆×C₁₂₇	—	2	4	no	—
28	M₁₆×C₁₂₇	Q₈	1	4	yes	complemented
29	C₂×C₈×C₁₂₇	—	2	2	no	—
30	C₂×C₈×C₁₂₇	—	2	2	no	—
31	C₄⋊C₄×C₁₂₇	—	2	2	no	—
32	C₄⋊C₄×C₁₂₇	—	2	2	no	—
33	C₄⋊C₄×C₁₂₇	—	2	2	no	—
34	C₄⋊C₄×C₁₂₇	—	2	2	no	—
35	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
36	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
37	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
38	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
39	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
40	Q₈×C₁₂₇	—	4	4	no	—
41	Q₈×C₁₂₇	—	4	4	no	—
42	C₂×C₄×C₁₂₇	—	2	2	no	—
43	C₂×C₄×C₁₂₇	—	2	2	no	—
44	C₂×C₄×C₁₂₇	—	2	4	no	—
45	C₂×C₄×C₁₂₇	—	4	4	no	—
46	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₁₂₇	#41	1	2	yes	—
54	C₈×C₁₂₇	—	2	4	no	—
55	C₈×C₁₂₇	—	4	2	no	—
56	C₂²×C₁₂₇	—	2	1	no	—
57	C₂²×C₁₂₇	—	2	1	no	—
58	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₁₂₇	C₄×Q₈	1	2	yes	—
60	C₄×C₁₂₇	—	2	1	no	—
61	C₄×C₁₂₇	—	2	1	no	—
62	C₄×C₁₂₇	—	4	4	no	—
63	C₄×C₁₂₇	#100	1	2	yes	—
64	C₄×C₁₂₇	#101	1	2	yes	—
65	C₄×C₁₂₇	C₂²⋊Q₈	1	2	yes	—
66	C₄×C₁₂₇	#124	1	1	yes	characteristic
67	C₄×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₁₂₇	—	2	2	no	—
69	C₂×C₁₂₇	#391	1	1	yes	characteristic
70	C₂×C₁₂₇	C₂×#100	1	1	yes	characteristic
71	C₂×C₁₂₇	C₂×#101	1	1	yes	characteristic
72	#1997	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₁₂₇	#1997	1	1	yes	characteristic, direct product factor, Sylow
74	C₂×C₄⋊Q₈	C₂×C₁₂₇	1	1	yes	characteristic
75	#423	C₂×C₁₂₇	1	1	yes	characteristic
76	C₄×M₁₆	C₂×C₁₂₇	1	1	yes	characteristic
77	#343	C₂×C₁₂₇	1	4	yes	—
78	C₂²×Q₈	C₄×C₁₂₇	1	2	yes	—
79	C₂×M₁₆	C₂²×C₁₂₇	1	2	yes	—
80	C₂×M₁₆	C₂²×C₁₂₇	1	2	yes	—
81	C₄⋊Q₈	—	2	1	no	—
82	C₄⋊Q₈	—	2	1	no	—
83	C₄⋊Q₈	—	2	1	no	—
84	C₄⋊Q₈	—	2	1	no	—
85	C₂×C₄⋊C₄	C₄×C₁₂₇	1	2	yes	—
86	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
87	C₂×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
88	C₄⋊C₈	—	2	1	no	—
89	C₄⋊C₈	—	2	1	no	—
90	#97	—	2	1	no	—
91	C₄×C₈	—	2	1	no	—
92	C₂×Q₈	—	2	2	no	—
93	C₂×Q₈	—	2	2	no	—
94	C₂×Q₈	—	4	4	no	—
95	C₂²×C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
96	C₂²×C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
97	C₂²×C₄	C₂×C₄×C₁₂₇	1	4	yes	—
98	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
99	M₁₆	—	2	2	no	—
100	M₁₆	—	2	4	no	—
101	M₁₆	Q₈×C₁₂₇	1	4	yes	complemented
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	2	2	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	—	2	2	no	—
107	C₄⋊C₄	—	2	2	no	—
108	C₄²	D₄×C₁₂₇	1	1	yes	characteristic
109	C₄²	D₄×C₁₂₇	1	1	yes	characteristic
110	C₄²	D₄×C₁₂₇	1	1	yes	characteristic
111	C₄²	D₄×C₁₂₇	1	1	yes	characteristic
112	C₂³	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
113	Q₈	—	4	4	no	—
114	Q₈	—	4	4	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	4	4	no	—
119	C₂×C₄	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
120	C₂×C₄	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
121	C₂×C₄	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
122	C₂×C₄	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
123	C₂×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
124	C₂×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
125	C₂×C₄	C₂×Q₈×C₁₂₇	1	2	yes	—
126	C₂×C₄	#41×C₁₂₇	1	2	yes	—
127	C₈	—	2	4	no	—
128	C₈	—	4	2	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
132	C₂²	C₄×Q₈×C₁₂₇	1	2	yes	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	4	4	no	—
136	C₄	#100×C₁₂₇	1	2	yes	—
137	C₄	#101×C₁₂₇	1	2	yes	—
138	C₄	C₂²⋊Q₈×C₁₂₇	1	2	yes	—
139	C₄	#124×C₁₂₇	1	1	yes	characteristic
140	C₄	C₄⋊Q₈×C₁₂₇	1	1	yes	characteristic
141	C₂	—	2	2	no	—
142	C₂	#391×C₁₂₇	1	1	yes	characteristic
143	C₂	C₂×#100×C₁₂₇	1	1	yes	characteristic
144	C₂	C₂×#101×C₁₂₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊Q₈×C₁₂₇

C₂

yes

maximal, characteristic

#423×C₁₂₇

C₂

yes

maximal, characteristic

C₄×M₁₆×C₁₂₇

C₂

yes

maximal, characteristic

#343×C₁₂₇

C₂

yes

maximal

C₂²×Q₈×C₁₂₇

C₄

yes

—

C₂×M₁₆×C₁₂₇

C₂²

yes

—

C₂×M₁₆×C₁₂₇

C₂²

yes

—

C₄⋊Q₈×C₁₂₇

—

C₄⋊Q₈×C₁₂₇

—

C₄⋊Q₈×C₁₂₇

—

C₄⋊Q₈×C₁₂₇

—

C₂×C₄⋊C₄×C₁₂₇

C₄

yes

—

C₂×C₄⋊C₄×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₂₇

—

C₄⋊C₈×C₁₂₇

—

#97×C₁₂₇

—

C₄×C₈×C₁₂₇

—

C₂×Q₈×C₁₂₇

—

C₂×Q₈×C₁₂₇

—

C₂×Q₈×C₁₂₇

—

C₂²×C₄×C₁₂₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₂₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₂₇

C₂×C₄

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

M₁₆×C₁₂₇

—

M₁₆×C₁₂₇

—

M₁₆×C₁₂₇

Q₈

yes

complemented

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

D₄

yes

characteristic

C₄²×C₁₂₇

D₄

yes

characteristic

C₄²×C₁₂₇

D₄

yes

characteristic

C₄²×C₁₂₇

D₄

yes

characteristic

C₂³×C₁₂₇

C₂²×C₄

yes

characteristic

Q₈×C₁₂₇

—

Q₈×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂²⋊C₄

yes

characteristic

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×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×Q₈

yes

—

C₂×C₄×C₁₂₇

#41

yes

—

C₈×C₁₂₇

—

C₈×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₂₇

C₄×Q₈

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#100

yes

—

C₄×C₁₂₇

#101

yes

—

C₄×C₁₂₇

C₂²⋊Q₈

yes

—

C₄×C₁₂₇

#124

yes

characteristic

C₄×C₁₂₇

C₄⋊Q₈

yes

characteristic

C₂×C₁₂₇

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₁₂₇

yes

maximal, characteristic, direct product factor, Sylow

C₁₂₇

#1997

yes

characteristic, direct product factor, Sylow

C₂×C₄⋊Q₈

C₂×C₁₂₇

yes

characteristic

#423

C₂×C₁₂₇

yes

characteristic

C₄×M₁₆

C₂×C₁₂₇

yes

characteristic

#343

C₂×C₁₂₇

yes

—

C₂²×Q₈

C₄×C₁₂₇

yes

—

C₂×M₁₆

C₂²×C₁₂₇

yes

—

C₂×M₁₆

C₂²×C₁₂₇

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

C₄×C₁₂₇

yes

—

C₂×C₄⋊C₄

C₂²×C₁₂₇

yes

—

C₂×C₄²

C₂²×C₁₂₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₂×C₄×C₁₂₇

yes

characteristic

C₂²×C₄

C₂×C₄×C₁₂₇

yes

characteristic

C₂²×C₄

C₂×C₄×C₁₂₇

yes

—

C₂²×C₄

C₂³×C₁₂₇

yes

characteristic

M₁₆

—

100

M₁₆

—

101

M₁₆

Q₈×C₁₂₇

yes

complemented

102

C₂×C₈

—

103

C₂×C₈

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄²

D₄×C₁₂₇

yes

characteristic

109

C₄²

D₄×C₁₂₇

yes

characteristic

110

C₄²

D₄×C₁₂₇

yes

characteristic

111

C₄²

D₄×C₁₂₇

yes

characteristic

112

C₂³

C₂²×C₄×C₁₂₇

yes

characteristic

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

C₂²⋊C₄×C₁₂₇

yes

characteristic

120

C₂×C₄

C₂²⋊C₄×C₁₂₇

yes

characteristic

121

C₂×C₄

C₂²⋊C₄×C₁₂₇

yes

characteristic

122

C₂×C₄

C₂²⋊C₄×C₁₂₇

yes

characteristic

123

C₂×C₄

C₂×D₄×C₁₂₇

yes

characteristic

124

C₂×C₄

C₂×D₄×C₁₂₇

yes

characteristic

125

C₂×C₄

C₂×Q₈×C₁₂₇

yes

—

126

C₂×C₄

#41×C₁₂₇

yes

—

127

C₈

—

128

C₈

—

129

C₂²

—

130

C₂²

—

131

C₂²

C₂×C₂²⋊C₄×C₁₂₇

yes

characteristic

132

C₂²

C₄×Q₈×C₁₂₇

yes

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

#100×C₁₂₇

yes

—

137

C₄

#101×C₁₂₇

yes

—

138

C₄

C₂²⋊Q₈×C₁₂₇

yes

—

139

C₄

#124×C₁₂₇

yes

characteristic

140

C₄

C₄⋊Q₈×C₁₂₇

yes

characteristic

141

C₂

—

142

C₂

#391×C₁₂₇

yes

characteristic

143

C₂

C₂×#100×C₁₂₇

yes

characteristic

144

C₂

C₂×#101×C₁₂₇

yes

characteristic

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

subgroup

G/H

center, socle

C₂²×C₁₂₇

C₂×C₂²⋊C₄

commutator

C₂³

C₂²×C₄×C₁₂₇

Frattini, cosocle

C₂²×C₄

C₂³×C₁₂₇

prime radical

C₂³×C₁₂₇

C₂²×C₄

LCS

G ▷ C₂³ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ G

N	H ≅ G/N	type
28. M₁₆×C₁₂₇	Q₈	semidirect
72. #1997	C₁₂₇	direct (N × H)
73. C₁₂₇	#1997	direct (N × H)
101. M₁₆	Q₈×C₁₂₇	semidirect

H ≅ G/N

type

28. M₁₆×C₁₂₇

Q₈

semidirect

72. #1997

C₁₂₇

direct (N × H)

73. C₁₂₇

#1997

direct (N × H)

101. M₁₆

Q₈×C₁₂₇

semidirect

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

subgroup	index
C₂×C₄⋊Q₈×C₁₂₇	2
#423×C₁₂₇	2
C₄×M₁₆×C₁₂₇	2
#343×C₁₂₇	2
#1997	127

subgroup

index

C₂×C₄⋊Q₈×C₁₂₇

#423×C₁₂₇

C₄×M₁₆×C₁₂₇

#343×C₁₂₇

#1997

127

Subgroups — C₁₂₇×#1997 #30634949

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C127×#1997 #30634949

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#1997 #30634949