Group #30639878 — subgroups

Subgroups — #30639878

Order 16256.

258 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 87 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×#16861132	C₂	1	2	yes	maximal
2	#22549782	C₂	1	1	yes	maximal, characteristic
3	#22549638	C₂	1	2	yes	maximal
4	#22549631	C₂	1	1	yes	maximal, characteristic
5	C₂×#125×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₅₄	C₂²	1	1	yes	characteristic
7	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
9	#16861132	—	2	4	no	—
10	C₁₂₇⋊#95	—	2	2	no	—
11	#125×C₁₂₇	—	2	4	no	—
12	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
13	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₅₄	—	2	2	no	—
16	C₂×Dic₂₅₄	—	2	1	no	—
17	C₂×Dic₂₅₄	—	2	2	no	—
18	C₂×Dic₂₅₄	Q₈	1	2	yes	—
19	C₂²×Dic₁₂₇	—	2	1	no	—
20	C₂²×Dic₁₂₇	—	2	1	no	—
21	C₂×C₁₂₇⋊C₈	—	2	2	no	—
22	C₂×C₁₂₇⋊C₈	—	2	2	no	—
23	C₂×C₁₂₇⋊C₈	—	2	2	no	—
24	C₄⋊Dic₁₂₇	—	2	2	no	—
25	C₂²×C₄×C₁₂₇	—	2	2	no	—
26	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
27	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₁₂₇	—	2	8	no	—
29	C₄⋊C₄×C₁₂₇	—	2	2	no	—
30	C₄⋊C₄×C₁₂₇	D₄	1	4	yes	—
31	C₄²×C₁₂₇	—	2	2	no	—
32	Dic₂₅₄	—	2	4	no	—
33	Dic₂₅₄	—	4	2	no	—
34	C₂×Dic₁₂₇	—	2	2	no	—
35	C₂×Dic₁₂₇	—	4	1	no	—
36	C₂×Dic₁₂₇	—	4	1	no	—
37	C₂×Dic₁₂₇	—	4	1	no	—
38	C₂×Dic₁₂₇	—	4	2	no	—
39	C₁₂₇⋊C₈	—	4	2	no	—
40	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₂₇	—	2	2	no	—
42	C₂×C₄×C₁₂₇	—	2	4	no	—
43	C₂×C₄×C₁₂₇	—	2	2	no	—
44	C₂×C₄×C₁₂₇	—	2	2	no	—
45	C₂×C₄×C₁₂₇	—	4	2	no	—
46	C₂×C₄×C₁₂₇	—	4	2	no	—
47	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₄×C₁₂₇	#41	1	2	yes	—
50	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
51	C₂²×C₁₂₇	#120	1	1	yes	characteristic
52	C₂²×C₁₂₇	C₂²⋊Q₈	1	1	yes	characteristic
53	C₂²×C₁₂₇	#124	1	1	yes	characteristic
54	C₂²×C₁₂₇	#135	1	2	yes	—
55	C₂²×C₁₂₇	#137	1	2	yes	—
56	C₄×C₁₂₇	—	4	2	no	—
57	C₄×C₁₂₇	—	4	4	no	—
58	C₄×C₁₂₇	C₂²⋊Q₈	1	2	yes	—
59	C₄×C₁₂₇	#124	1	2	yes	—
60	Dic₁₂₇	—	4	2	no	—
61	Dic₁₂₇	—	8	1	no	—
62	C₂×C₁₂₇	#393	1	1	yes	characteristic
63	C₂×C₁₂₇	#452	1	2	yes	—
64	C₂×C₁₂₇	#479	1	2	yes	—
65	C₂×C₁₂₇	#488	1	1	yes	characteristic
66	C₂×C₁₂₇	#491	1	1	yes	characteristic
67	#2026	—	127	1	no	maximal, Sylow
68	C₁₂₇	#2026	1	1	yes	characteristic, complemented, Sylow
69	C₂×#125	D₁₂₇	1	1	yes	characteristic
70	C₂×#103	—	127	2	no	—
71	#391	—	127	1	no	—
72	#340	—	127	2	no	—
73	#336	—	127	1	no	—
74	C₂²×Q₈	—	127	1	no	—
75	C₂²×C₈	—	127	2	no	—
76	#125	—	2	4	no	—
77	C₂×C₄⋊C₄	—	2	2	no	—
78	C₂×C₄⋊C₄	—	127	1	no	—
79	C₂×C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
80	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
81	#103	—	254	4	no	—
82	#95	—	254	2	no	—
83	C₂×Q₈	—	127	2	no	—
84	C₂×Q₈	—	254	2	no	—
85	C₂×Q₈	—	254	2	no	—
86	C₂×Q₈	—	254	1	no	—
87	C₂²×C₄	—	2	2	no	—
88	C₂²×C₄	—	254	1	no	—
89	C₂²×C₄	—	254	1	no	—
90	C₂²×C₄	C₂²⋊D₁₂₇	1	2	yes	—
91	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
92	C₂×C₈	—	254	2	no	—
93	C₂×C₈	—	254	2	no	—
94	C₂×C₈	—	254	2	no	—
95	C₄⋊C₄	—	2	8	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	254	2	no	—
98	C₄⋊C₄	C₂²⋊D₁₂₇	1	4	yes	—
99	C₄²	—	2	2	no	—
100	C₂³	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
101	Q₈	—	508	2	no	—
102	Q₈	—	254	4	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	508	1	no	—
110	C₂×C₄	—	508	2	no	—
111	C₂×C₄	—	508	1	no	—
112	C₂×C₄	—	508	1	no	—
113	C₂×C₄	—	254	2	no	—
114	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	2	yes	—
115	C₂×C₄	Q₈×D₁₂₇	1	1	yes	characteristic
116	C₂×C₄	#14523249	1	2	yes	—
117	C₂×C₄	#14523249	1	1	yes	characteristic
118	C₈	—	508	2	no	—
119	C₂²	#16861178	1	1	yes	characteristic
120	C₂²	#16861185	1	1	yes	characteristic
121	C₂²	#16861198	1	1	yes	characteristic
122	C₂²	#16861229	1	2	yes	—
123	C₂²	#16861239	1	2	yes	—
124	C₄	—	4	2	no	—
125	C₄	—	4	4	no	—
126	C₄	—	1016	1	no	—
127	C₄	—	508	2	no	—
128	C₄	#16861185	1	2	yes	—
129	C₄	#16861198	1	2	yes	—
130	C₂	#22549787	1	1	yes	characteristic
131	C₂	#22550038	1	2	yes	—
132	C₂	#22550180	1	2	yes	—
133	C₂	#22550233	1	1	yes	characteristic
134	C₂	#22550246	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861132

C₂

yes

maximal

#22549782

C₂

yes

maximal, characteristic

#22549638

C₂

yes

maximal

#22549631

C₂

yes

maximal, characteristic

C₂×#125×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×Dic₂₅₄

C₂²

yes

characteristic

C₂²×C₁₂₇⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₂₇

C₂²

yes

characteristic

#16861132

—

C₁₂₇⋊#95

—

#125×C₁₂₇

—

C₂×C₄⋊C₄×C₁₂₇

—

C₂×C₄⋊C₄×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

Q₈

yes

—

C₂²×Dic₁₂₇

—

C₂²×Dic₁₂₇

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₄⋊Dic₁₂₇

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

D₄

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

D₄

yes

—

C₄²×C₁₂₇

—

Dic₂₅₄

—

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

—

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×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₂₇

#41

yes

—

C₂×C₄×C₁₂₇

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

C₂²⋊Q₈

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

—

C₂²×C₁₂₇

#137

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂²⋊Q₈

yes

—

C₄×C₁₂₇

#124

yes

—

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

#393

yes

characteristic

C₂×C₁₂₇

#452

yes

—

C₂×C₁₂₇

#479

yes

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

127

maximal, Sylow

C₁₂₇

#2026

yes

characteristic, complemented, Sylow

C₂×#125

D₁₂₇

yes

characteristic

C₂×#103

—

127

—

#391

—

127

—

#340

—

127

—

#336

—

127

—

C₂²×Q₈

—

127

—

C₂²×C₈

—

127

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

C₂×D₁₂₇

yes

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

#103

—

254

—

#95

—

254

—

C₂×Q₈

—

127

—

C₂×Q₈

—

254

—

C₂×Q₈

—

254

—

C₂×Q₈

—

254

—

C₂²×C₄

—

C₂²×C₄

—

254

—

C₂²×C₄

—

254

—

C₂²×C₄

C₂²⋊D₁₂₇

yes

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

C₂×C₈

—

254

—

C₂×C₈

—

254

—

C₂×C₈

—

254

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

254

—

C₄⋊C₄

C₂²⋊D₁₂₇

yes

—

C₄²

—

100

C₂³

C₂×C₂²⋊D₁₂₇

yes

characteristic

101

Q₈

—

508

—

102

Q₈

—

254

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

508

—

110

C₂×C₄

—

508

—

111

C₂×C₄

—

508

—

112

C₂×C₄

—

508

—

113

C₂×C₄

—

254

—

114

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

—

115

C₂×C₄

Q₈×D₁₂₇

yes

characteristic

116

C₂×C₄

#14523249

yes

—

117

C₂×C₄

#14523249

yes

characteristic

118

C₈

—

508

—

119

C₂²

#16861178

yes

characteristic

120

C₂²

#16861185

yes

characteristic

121

C₂²

#16861198

yes

characteristic

122

C₂²

#16861229

yes

—

123

C₂²

#16861239

yes

—

124

C₄

—

125

C₄

—

126

C₄

—

1016

—

127

C₄

—

508

—

128

C₄

#16861185

yes

—

129

C₄

#16861198

yes

—

130

C₂

#22549787

yes

characteristic

131

C₂

#22550038

yes

—

132

C₂

#22550180

yes

—

133

C₂

#22550233

yes

characteristic

134

C₂

#22550246

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₂₇
commutator, cosocle	C₂²×C₄×C₁₂₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	C₂×#125×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	C₂×D₄
hypercenter	C₂×#125	D₁₂₇
hypocenter	C₁₂₇	#2026
LCS	G ▷ C₂²×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₂₇

commutator, cosocle

C₂²×C₄×C₁₂₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

C₂×#125×C₁₂₇

C₂

socle, prime radical

C₂³×C₁₂₇

C₂×D₄

D₁₂₇

C₁₂₇

G ▷ C₂²×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
68. C₁₂₇	#2026	semidirect

H ≅ G/N

type

68. C₁₂₇

#2026

semidirect

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

subgroup	index
C₂×#16861132	2
#22549782	2
#22549638	2
#22549631	2
C₂×#125×C₁₂₇	2
#2026	127

subgroup

index

C₂×#125×C₁₂₇

127

Subgroups — #30639878

Conjugacy classes of subgroups

Complements

Maximal subgroups