Group #30651397 — subgroups

Subgroups — #30651397

Order 16256.

392 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 139 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	Q₈×Dic₂₅₄	C₂	1	1	yes	maximal, characteristic, complemented
2	#22550222	C₂	1	3	yes	maximal, complemented
3	#22550166	C₂	1	3	yes	maximal, complemented
4	#22550073	C₂	1	3	yes	maximal, complemented
5	Q₈×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#14523236	C₂	1	3	yes	maximal, complemented
7	#550×C₁₂₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#14523236	C₂²	1	3	yes	—
9	#16861209	C₂²	1	3	yes	—
10	#16861208	—	2	3	no	—
11	Q₈×Dic₁₂₇	—	2	1	no	—
12	C₄×Dic₂₅₄	C₂²	1	3	yes	—
13	#16861140	C₂²	1	3	yes	—
14	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	3	yes	—
15	#16861132	C₂²	1	1	yes	characteristic
16	#16861130	C₂²	1	6	yes	—
17	#16861130	C₂²	1	3	yes	—
18	C₄×C₁₂₇⋊C₈	C₂²	1	3	yes	—
19	C₂×#41×C₁₂₇	—	2	1	no	—
20	#127×C₁₂₇	C₂²	1	3	yes	—
21	#121×C₁₂₇	—	2	3	no	—
22	C₄×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₁₂₇	C₂²	1	3	yes	—
24	C₂×Dic₂₅₄	—	2	3	no	—
25	C₂×Dic₂₅₄	C₂³	1	1	yes	characteristic
26	#14523236	—	2	6	no	—
27	C₂×C₁₂₇⋊C₈	C₂³	1	3	yes	—
28	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
29	C₄⋊Dic₁₂₇	—	2	3	no	—
30	C₄⋊Dic₁₂₇	C₂³	1	3	yes	—
31	Dic₁₂₇⋊C₄	—	2	3	no	—
32	C₄×Dic₁₂₇	—	2	3	no	—
33	#41×C₁₂₇	—	4	4	no	—
34	C₂×Q₈×C₁₂₇	D₄	1	1	yes	characteristic
35	C₂×D₄×C₁₂₇	—	2	3	no	—
36	C₂×D₄×C₁₂₇	—	2	3	no	—
37	C₂×D₄×C₁₂₇	C₂³	1	3	yes	—
38	C₂²×C₄×C₁₂₇	—	2	3	no	—
39	C₄⋊C₄×C₁₂₇	D₄	1	3	yes	—
40	C₄⋊C₄×C₁₂₇	C₂³	1	1	yes	characteristic
41	C₂²⋊C₄×C₁₂₇	—	2	3	no	—
42	C₄²×C₁₂₇	C₂³	1	3	yes	—
43	Dic₂₅₄	—	2	1	no	—
44	Dic₂₅₄	—	8	3	no	—
45	Dic₂₅₄	#41	1	2	yes	complemented
46	C₂×Dic₁₂₇	—	2	1	no	—
47	C₂×Dic₁₂₇	—	2	3	no	—
48	C₁₂₇⋊C₈	—	2	3	no	—
49	C₁₂₇⋊C₈	—	2	1	no	—
50	C₂³×C₁₂₇	—	2	3	no	—
51	Q₈×C₁₂₇	SD₁₆	1	4	yes	complemented
52	D₄×C₁₂₇	—	2	3	no	—
53	D₄×C₁₂₇	—	2	3	no	—
54	D₄×C₁₂₇	—	4	6	no	—
55	D₄×C₁₂₇	—	8	3	no	—
56	C₂×C₄×C₁₂₇	—	2	1	no	—
57	C₂×C₄×C₁₂₇	—	4	6	no	—
58	C₂×C₄×C₁₂₇	C₂×D₄	1	3	yes	—
59	C₂×C₄×C₁₂₇	C₂×D₄	1	3	yes	—
60	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
61	C₂²×C₁₂₇	—	4	3	no	—
62	C₂²×C₁₂₇	—	4	3	no	—
63	C₂²×C₁₂₇	—	4	3	no	—
64	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₁₂₇	—	2	3	no	—
66	C₄×C₁₂₇	—	4	1	no	—
67	C₄×C₁₂₇	C₂×SD₁₆	1	6	yes	—
68	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
69	C₄×C₁₂₇	2₋¹⁺⁴	1	1	yes	characteristic
70	Dic₁₂₇	—	2	2	no	—
71	Dic₁₂₇	—	4	3	no	—
72	C₂×C₁₂₇	—	8	3	no	—
73	C₂×C₁₂₇	#548	1	1	yes	characteristic
74	C₂×C₁₂₇	C₂²×SD₁₆	1	1	yes	characteristic
75	C₂×C₁₂₇	#576	1	1	yes	characteristic
76	#3362	—	127	1	no	maximal, Sylow
77	C₁₂₇	#3362	1	1	yes	characteristic, complemented, Sylow
78	Q₈²	—	127	1	no	—
79	#550	D₁₂₇	1	1	yes	characteristic
80	#486	—	127	3	no	—
81	#476	—	127	3	no	—
82	#460	—	127	3	no	—
83	C₈×Q₈	—	127	1	no	—
84	C₄×SD₁₆	—	127	3	no	—
85	C₂×#41	—	2	1	no	—
86	C₂×SD₁₆	—	127	3	no	—
87	C₄⋊Q₈	—	127	3	no	—
88	C₄⋊Q₈	—	254	3	no	—
89	#127	C₂×D₁₂₇	1	3	yes	—
90	#121	—	2	3	no	—
91	C₄×Q₈	—	127	3	no	—
92	C₄×Q₈	—	254	1	no	—
93	C₄×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
94	C₄×D₄	C₂×D₁₂₇	1	3	yes	—
95	#106	—	127	3	no	—
96	C₄⋊C₈	—	127	3	no	—
97	#103	—	127	1	no	—
98	#102	—	127	3	no	—
99	#102	—	127	6	no	—
100	C₄×C₈	—	127	3	no	—
101	#41	—	4	4	no	—
102	C₂×Q₈	—	127	1	no	—
103	C₂×Q₈	—	254	3	no	—
104	C₂×Q₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
105	C₂×D₄	—	2	3	no	—
106	C₂×D₄	—	2	3	no	—
107	C₂×D₄	C₂²×D₁₂₇	1	3	yes	—
108	C₂²×C₄	—	2	3	no	—
109	SD₁₆	—	254	6	no	—
110	C₂×C₈	—	127	1	no	—
111	C₂×C₈	—	127	3	no	—
112	C₄⋊C₄	—	127	3	no	—
113	C₄⋊C₄	—	254	3	no	—
114	C₄⋊C₄	—	254	3	no	—
115	C₄⋊C₄	C₂²⋊D₁₂₇	1	3	yes	—
116	C₄⋊C₄	C₂²×D₁₂₇	1	1	yes	characteristic
117	C₂²⋊C₄	—	2	3	no	—
118	C₄²	—	254	3	no	—
119	C₄²	C₂²×D₁₂₇	1	3	yes	—
120	C₂³	—	2	3	no	—
121	Q₈	—	127	2	no	—
122	Q₈	—	1016	3	no	—
123	Q₈	—	254	1	no	—
124	Q₈	#14523236	1	4	yes	complemented
125	D₄	—	2	3	no	—
126	D₄	—	2	3	no	—
127	D₄	—	4	6	no	—
128	D₄	—	8	3	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	4	6	no	—
131	C₂×C₄	—	254	3	no	—
132	C₂×C₄	—	254	1	no	—
133	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	3	yes	—
134	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	3	yes	—
135	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
136	C₈	—	254	3	no	—
137	C₈	—	254	1	no	—
138	C₂²	—	4	3	no	—
139	C₂²	—	4	3	no	—
140	C₂²	—	4	3	no	—
141	C₂²	C₂²×C₂²⋊D₁₂₇	1	1	yes	characteristic
142	C₄	—	2	3	no	—
143	C₄	—	4	1	no	—
144	C₄	—	508	3	no	—
145	C₄	—	254	2	no	—
146	C₄	C₂×#14523236	1	6	yes	—
147	C₄	#41×D₁₂₇	1	1	yes	characteristic
148	C₄	#16861256	1	1	yes	characteristic
149	C₂	—	8	3	no	—
150	C₂	#22550533	1	1	yes	characteristic
151	C₂	C₂²×#14523236	1	1	yes	characteristic
152	C₂	#22550679	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₂₅₄

C₂

yes

maximal, characteristic, complemented

#22550222

C₂

yes

maximal, complemented

#22550166

C₂

yes

maximal, complemented

#22550073

C₂

yes

maximal, complemented

Q₈×C₁₂₇⋊C₈

C₂

yes

maximal, characteristic, complemented

C₄×#14523236

C₂

yes

maximal, complemented

#550×C₁₂₇

C₂

yes

maximal, characteristic

C₂×#14523236

C₂²

yes

—

#16861209

C₂²

yes

—

#16861208

—

Q₈×Dic₁₂₇

—

C₄×Dic₂₅₄

C₂²

yes

—

#16861140

C₂²

yes

—

C₄⋊(C₁₂₇⋊C₈)

C₂²

yes

—

#16861132

C₂²

yes

characteristic

#16861130

C₂²

yes

—

#16861130

C₂²

yes

—

C₄×C₁₂₇⋊C₈

C₂²

yes

—

C₂×#41×C₁₂₇

—

#127×C₁₂₇

C₂²

yes

—

#121×C₁₂₇

—

C₄×Q₈×C₁₂₇

C₂²

yes

characteristic

C₄×D₄×C₁₂₇

C₂²

yes

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

C₂³

yes

characteristic

#14523236

—

C₂×C₁₂₇⋊C₈

C₂³

yes

—

C₂×C₁₂₇⋊C₈

C₂³

yes

characteristic

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

C₂³

yes

—

Dic₁₂₇⋊C₄

—

C₄×Dic₁₂₇

—

#41×C₁₂₇

—

C₂×Q₈×C₁₂₇

D₄

yes

characteristic

C₂×D₄×C₁₂₇

—

C₂×D₄×C₁₂₇

—

C₂×D₄×C₁₂₇

C₂³

yes

—

C₂²×C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

D₄

yes

—

C₄⋊C₄×C₁₂₇

C₂³

yes

characteristic

C₂²⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

C₂³

yes

—

Dic₂₅₄

—

Dic₂₅₄

—

Dic₂₅₄

#41

yes

complemented

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

—

Q₈×C₁₂₇

SD₁₆

yes

complemented

D₄×C₁₂₇

—

D₄×C₁₂₇

—

D₄×C₁₂₇

—

D₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂⁴

yes

characteristic

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂²×D₄

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×SD₁₆

yes

—

C₄×C₁₂₇

C₂×#41

yes

characteristic

C₄×C₁₂₇

2₋¹⁺⁴

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#548

yes

characteristic

C₂×C₁₂₇

C₂²×SD₁₆

yes

C₂×C₁₂₇

yes

—

127

maximal, Sylow

C₁₂₇

#3362

yes

characteristic, complemented, Sylow

Q₈²

—

127

—

#550

D₁₂₇

yes

characteristic

#486

—

127

—

#476

—

127

—

#460

—

127

—

C₈×Q₈

—

127

—

C₄×SD₁₆

—

127

—

C₂×#41

—

C₂×SD₁₆

—

127

—

C₄⋊Q₈

—

127

—

C₄⋊Q₈

—

254

—

#127

C₂×D₁₂₇

yes

—

#121

—

C₄×Q₈

—

127

—

C₄×Q₈

—

254

—

C₄×Q₈

C₂×D₁₂₇

yes

characteristic

C₄×D₄

C₂×D₁₂₇

yes

—

#106

—

127

—

C₄⋊C₈

—

127

—

#103

—

127

—

#102

—

127

—

#102

—

127

—

100

C₄×C₈

—

127

—

101

#41

—

102

C₂×Q₈

—

127

—

103

C₂×Q₈

—

254

—

104

C₂×Q₈

C₂²⋊D₁₂₇

yes

characteristic

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

C₂²×D₁₂₇

yes

—

108

C₂²×C₄

—

109

SD₁₆

—

254

—

110

C₂×C₈

—

127

—

111

C₂×C₈

—

127

—

112

C₄⋊C₄

—

127

—

113

C₄⋊C₄

—

254

—

114

C₄⋊C₄

—

254

—

115

C₄⋊C₄

C₂²⋊D₁₂₇

yes

—

116

C₄⋊C₄

C₂²×D₁₂₇

yes

characteristic

117

C₂²⋊C₄

—

118

C₄²

—

254

—

119

C₄²

C₂²×D₁₂₇

yes

—

120

C₂³

—

121

Q₈

—

127

—

122

Q₈

—

1016

—

123

Q₈

—

254

—

124

Q₈

#14523236

yes

complemented

125

D₄

—

126

D₄

—

127

D₄

—

128

D₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

254

—

132

C₂×C₄

—

254

—

133

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

—

134

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

—

135

C₂×C₄

C₂³×D₁₂₇

yes

characteristic

136

C₈

—

254

—

137

C₈

—

254

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

C₂²×C₂²⋊D₁₂₇

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

508

—

145

C₄

—

254

—

146

C₄

C₂×#14523236

yes

—

147

C₄

#41×D₁₂₇

yes

characteristic

148

C₄

#16861256

yes

characteristic

149

C₂

—

150

C₂

#22550533

yes

characteristic

151

C₂

C₂²×#14523236

yes

characteristic

152

C₂

#22550679

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₂₇
commutator, cosocle	C₂×C₄×C₁₂₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting, prime radical	#550×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²×D₄
hypercenter	#550	D₁₂₇
hypocenter	C₁₂₇	#3362
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #550

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₂₇

commutator, cosocle

C₂×C₄×C₁₂₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₂₇

Fitting, prime radical

#550×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂²×D₄

D₁₂₇

C₁₂₇

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #550

N	H ≅ G/N	type
1. Q₈×Dic₂₅₄	C₂	semidirect
2. #22550222	C₂	semidirect
3. #22550166	C₂	semidirect
4. #22550073	C₂	semidirect
5. Q₈×C₁₂₇⋊C₈	C₂	semidirect
6. C₄×#14523236	C₂	semidirect
45. Dic₂₅₄	#41	semidirect
51. Q₈×C₁₂₇	SD₁₆	semidirect
77. C₁₂₇	#3362	semidirect
124. Q₈	#14523236	semidirect

H ≅ G/N

type

1. Q₈×Dic₂₅₄

C₂

semidirect

2. #22550222

C₂

semidirect

3. #22550166

C₂

semidirect

4. #22550073

C₂

semidirect

5. Q₈×C₁₂₇⋊C₈

C₂

semidirect

6. C₄×#14523236

C₂

semidirect

45. Dic₂₅₄

#41

semidirect

51. Q₈×C₁₂₇

SD₁₆

semidirect

77. C₁₂₇

#3362

semidirect

124. Q₈

#14523236

semidirect

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

subgroup	index
Q₈×Dic₂₅₄	2
#22550222	2
#22550166	2
#22550073	2
Q₈×C₁₂₇⋊C₈	2
C₄×#14523236	2
#550×C₁₂₇	2
#3362	127

subgroup

index

Q₈×Dic₂₅₄

Q₈×C₁₂₇⋊C₈

#550×C₁₂₇

127

Subgroups — #30651397

Conjugacy classes of subgroups

Complements

Maximal subgroups