Group #30649233 — subgroups

Subgroups — C₂×#22550277 #30649233

Order 16256.

360 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 165 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×#16861200	C₂	1	2	yes	maximal
2	#22550277	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16861143	C₂	1	1	yes	maximal, characteristic
4	C₂×#16861140	C₂	1	2	yes	maximal
5	C₂×C₈×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#107×C₁₂₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
8	#16861200	—	2	4	no	—
9	#16861200	C₂²	1	8	yes	—
10	C₂×C₄⋊Dic₁₂₇	—	2	2	no	—
11	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
12	C₂×Dic₁₂₇⋊C₄	—	2	2	no	—
13	C₂×C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
14	#16861143	C₂²	1	4	yes	—
15	#16861140	C₂²	1	8	yes	—
16	C₈×Dic₁₂₇	C₂²	1	4	yes	—
17	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
19	#107×C₁₂₇	C₂²	1	4	yes	—
20	C₂²×Dic₁₂₇	—	2	2	no	—
21	C₂²×Dic₁₂₇	D₄	1	1	yes	characteristic
22	C₂²×Dic₁₂₇	D₄	1	1	yes	characteristic
23	C₂×C₁₂₇⋊C₈	Q₈	1	4	yes	—
24	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
25	C₄⋊Dic₁₂₇	—	2	8	no	—
26	C₄⋊Dic₁₂₇	—	2	2	no	—
27	C₄⋊Dic₁₂₇	C₂³	1	4	yes	—
28	Dic₁₂₇⋊C₄	—	2	8	no	—
29	C₄×Dic₁₂₇	D₄	1	2	yes	—
30	C₄×Dic₁₂₇	C₂³	1	2	yes	—
31	C₂²×C₄×C₁₂₇	—	2	2	no	—
32	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₁₂₇	Q₈	1	4	yes	—
34	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
35	C₄⋊C₄×C₁₂₇	—	2	2	no	—
36	C₄⋊C₄×C₁₂₇	C₂³	1	4	yes	—
37	C₂×Dic₁₂₇	—	2	2	no	—
38	C₂×Dic₁₂₇	—	2	2	no	—
39	C₂×Dic₁₂₇	—	2	4	no	—
40	C₂×Dic₁₂₇	—	4	4	no	—
41	C₂×Dic₁₂₇	C₂×D₄	1	2	yes	—
42	C₂×Dic₁₂₇	C₂×D₄	1	2	yes	—
43	C₁₂₇⋊C₈	C₂×Q₈	1	4	yes	—
44	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₂₇	—	2	4	no	—
46	C₂×C₄×C₁₂₇	—	4	4	no	—
47	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
50	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
51	C₈×C₁₂₇	C₂×Q₈	1	4	yes	—
52	C₂²×C₁₂₇	C₄⋊Q₈	1	2	yes	—
53	C₂²×C₁₂₇	#135	1	2	yes	—
54	C₂²×C₁₂₇	#135	1	2	yes	—
55	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
56	C₄×C₁₂₇	—	4	4	no	—
57	C₄×C₁₂₇	C₄⋊Q₈	1	2	yes	—
58	C₄×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
59	C₄×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
60	Dic₁₂₇	—	2	2	no	—
61	Dic₁₂₇	—	2	2	no	—
62	Dic₁₂₇	—	4	4	no	—
63	C₂×C₁₂₇	#499	1	4	yes	complemented
64	C₂×C₁₂₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₁₂₇	C₂×#135	1	1	yes	characteristic
66	C₂×C₁₂₇	C₂×#135	1	1	yes	characteristic
67	C₂×#499	—	127	1	no	maximal, Sylow
68	C₁₂₇	C₂×#499	1	1	yes	characteristic, complemented, Sylow
69	C₂×#125	—	127	2	no	—
70	#499	—	127	8	no	—
71	C₂×#107	—	127	1	no	—
72	C₂×#107	D₁₂₇	1	1	yes	characteristic
73	C₂×#106	—	127	2	no	—
74	C₂×C₄×C₈	—	127	1	no	—
75	C₂²×C₈	—	127	1	no	—
76	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
77	#125	—	127	8	no	—
78	#125	—	254	4	no	—
79	C₂×C₄⋊C₄	—	127	2	no	—
80	C₂×C₄⋊C₄	—	254	2	no	—
81	C₂×C₄⋊C₄	—	254	2	no	—
82	C₂×C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
83	C₂×C₄²	—	127	1	no	—
84	#107	—	127	4	no	—
85	#107	C₂×D₁₂₇	1	4	yes	—
86	#106	—	127	8	no	—
87	C₄×C₈	—	127	4	no	—
88	C₂²×C₄	—	2	2	no	—
89	C₂²×C₄	—	127	1	no	—
90	C₂²×C₄	—	127	1	no	—
91	C₂²×C₄	—	254	2	no	—
92	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
93	C₂×C₈	—	127	4	no	—
94	C₂×C₈	—	127	2	no	—
95	C₂×C₈	Dic₂₅₄	1	4	yes	—
96	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	127	4	no	—
99	C₄⋊C₄	—	254	8	no	—
100	C₄⋊C₄	—	254	8	no	—
101	C₄⋊C₄	—	254	2	no	—
102	C₄⋊C₄	C₂²×D₁₂₇	1	4	yes	—
103	C₄²	—	127	2	no	—
104	C₄²	—	127	2	no	—
105	C₂³	D₄×D₁₂₇	1	1	yes	characteristic
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	4	4	no	—
108	C₂×C₄	—	127	2	no	—
109	C₂×C₄	—	127	2	no	—
110	C₂×C₄	—	508	4	no	—
111	C₂×C₄	—	254	4	no	—
112	C₂×C₄	—	254	2	no	—
113	C₂×C₄	—	254	2	no	—
114	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
115	C₂×C₄	Q₈×D₁₂₇	1	2	yes	—
116	C₂×C₄	C₂×Dic₂₅₄	1	2	yes	—
117	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
118	C₈	—	127	4	no	—
119	C₈	C₂×Dic₂₅₄	1	4	yes	—
120	C₂²	#16861208	1	2	yes	—
121	C₂²	#16861228	1	2	yes	—
122	C₂²	#16861232	1	2	yes	—
123	C₂²	C₂×D₄×D₁₂₇	1	1	yes	characteristic
124	C₄	—	4	4	no	—
125	C₄	—	508	4	no	—
126	C₄	—	254	2	no	—
127	C₄	—	254	2	no	—
128	C₄	#16861208	1	2	yes	—
129	C₄	C₂×Q₈×D₁₂₇	1	1	yes	characteristic
130	C₄	C₂²×Dic₂₅₄	1	1	yes	characteristic
131	C₂	#22550277	1	4	yes	direct product factor
132	C₂	C₂×#16861208	1	1	yes	characteristic
133	C₂	C₂×#16861228	1	1	yes	characteristic
134	C₂	C₂×#16861232	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861200

C₂

yes

maximal

#22550277

C₂

yes

maximal, direct product factor

C₂×#16861143

C₂

yes

maximal, characteristic

C₂×#16861140

C₂

yes

maximal

C₂×C₈×Dic₁₂₇

C₂

yes

maximal, characteristic

C₂×#107×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×C₁₂₇⋊C₈

C₂²

yes

characteristic

#16861200

—

#16861200

C₂²

yes

—

C₂×C₄⋊Dic₁₂₇

—

C₂×C₄⋊Dic₁₂₇

C₂²

yes

—

C₂×Dic₁₂₇⋊C₄

—

C₂×C₄×Dic₁₂₇

C₂²

yes

characteristic

#16861143

C₂²

yes

—

#16861140

C₂²

yes

—

C₈×Dic₁₂₇

C₂²

yes

—

C₂²×C₈×C₁₂₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₂₇

C₂²

yes

—

#107×C₁₂₇

C₂²

yes

—

C₂²×Dic₁₂₇

—

C₂²×Dic₁₂₇

D₄

yes

characteristic

C₂²×Dic₁₂₇

D₄

yes

characteristic

C₂×C₁₂₇⋊C₈

Q₈

yes

—

C₂×C₁₂₇⋊C₈

C₂³

yes

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

C₂³

yes

—

Dic₁₂₇⋊C₄

—

C₄×Dic₁₂₇

D₄

yes

—

C₄×Dic₁₂₇

C₂³

yes

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₈×C₁₂₇

Q₈

yes

—

C₂×C₈×C₁₂₇

C₂³

yes

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

C₂³

yes

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

C₂×D₄

yes

—

C₂×Dic₁₂₇

C₂×D₄

yes

—

C₁₂₇⋊C₈

C₂×Q₈

yes

—

C₂³×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×Q₈

yes

—

C₂×C₄×C₁₂₇

C₂×Q₈

yes

—

C₂×C₄×C₁₂₇

C₂⁴

yes

characteristic

C₈×C₁₂₇

C₂×Q₈

yes

—

C₂²×C₁₂₇

C₄⋊Q₈

yes

—

C₂²×C₁₂₇

#135

yes

—

C₂²×C₁₂₇

#135

yes

—

C₂²×C₁₂₇

C₂²×D₄

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

C₄⋊Q₈

yes

—

C₄×C₁₂₇

C₂²×Q₈

yes

characteristic

C₄×C₁₂₇

C₂²×Q₈

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

#499

yes

complemented

C₂×C₁₂₇

C₂×C₄⋊Q₈

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

127

maximal, Sylow

C₁₂₇

C₂×#499

yes

characteristic, complemented, Sylow

C₂×#125

—

127

—

#499

—

127

—

C₂×#107

—

127

—

C₂×#107

D₁₂₇

yes

characteristic

C₂×#106

—

127

—

C₂×C₄×C₈

—

127

—

C₂²×C₈

—

127

—

C₂²×C₈

C₂×D₁₂₇

yes

characteristic

#125

—

127

—

#125

—

254

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

—

254

—

C₂×C₄⋊C₄

—

254

—

C₂×C₄⋊C₄

C₂×D₁₂₇

yes

—

C₂×C₄²

—

127

—

#107

—

127

—

#107

C₂×D₁₂₇

yes

—

#106

—

127

—

C₄×C₈

—

127

—

C₂²×C₄

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

254

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₂×C₈

Dic₂₅₄

yes

—

C₂×C₈

C₂²×D₁₂₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

254

—

100

C₄⋊C₄

—

254

—

101

C₄⋊C₄

—

254

—

102

C₄⋊C₄

C₂²×D₁₂₇

yes

—

103

C₄²

—

127

—

104

C₄²

—

127

—

105

C₂³

D₄×D₁₂₇

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

127

—

109

C₂×C₄

—

127

—

110

C₂×C₄

—

508

—

111

C₂×C₄

—

254

—

112

C₂×C₄

—

254

—

113

C₂×C₄

—

254

—

114

C₂×C₄

D₄×D₁₂₇

yes

characteristic

115

C₂×C₄

Q₈×D₁₂₇

yes

—

116

C₂×C₄

C₂×Dic₂₅₄

yes

—

117

C₂×C₄

C₂³×D₁₂₇

yes

characteristic

118

C₈

—

127

—

119

C₈

C₂×Dic₂₅₄

yes

—

120

C₂²

#16861208

yes

—

121

C₂²

#16861228

yes

—

122

C₂²

#16861232

yes

—

123

C₂²

C₂×D₄×D₁₂₇

yes

characteristic

124

C₄

—

125

C₄

—

508

—

126

C₄

—

254

—

127

C₄

—

254

—

128

C₄

#16861208

yes

—

129

C₄

C₂×Q₈×D₁₂₇

yes

characteristic

130

C₄

C₂²×Dic₂₅₄

yes

characteristic

131

C₂

#22550277

yes

direct product factor

132

C₂

C₂×#16861208

yes

characteristic

133

C₂

C₂×#16861228

yes

characteristic

134

C₂

C₂×#16861232

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₁₂₇
commutator, cosocle	C₂×C₄×C₁₂₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	C₂×#107×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	C₂×D₄
hypercenter	C₂×#107	D₁₂₇
hypocenter	C₁₂₇	C₂×#499
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₁₂₇

commutator, cosocle

C₂×C₄×C₁₂₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₂₇

Fitting

C₂×#107×C₁₂₇

C₂

socle, prime radical

C₂³×C₁₂₇

C₂×D₄

D₁₂₇

C₁₂₇

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
2. #22550277	C₂	direct (N × H)
63. C₂×C₁₂₇	#499	semidirect
68. C₁₂₇	C₂×#499	semidirect
131. C₂	#22550277	direct (N × H)

H ≅ G/N

type

2. #22550277

C₂

direct (N × H)

63. C₂×C₁₂₇

#499

semidirect

68. C₁₂₇

C₂×#499

semidirect

131. C₂

#22550277

direct (N × H)

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

subgroup	index
C₂×#16861200	2
#22550277	2
C₂×#16861143	2
C₂×#16861140	2
C₂×C₈×Dic₁₂₇	2
C₂×#107×C₁₂₇	2
C₂×#499	127

subgroup

index

C₂×C₈×Dic₁₂₇

C₂×#107×C₁₂₇

C₂×#499

127

Subgroups — C₂×#22550277 #30649233

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#22550277 #30649233

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22550277 #30649233