Group #30647092 — subgroups

Subgroups — #30647092

Order 16256.

472 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 181 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	#22549773	C₂	1	8	yes	maximal
3	#22549745	C₂	1	2	yes	maximal
4	#22549742	C₂	1	1	yes	maximal, characteristic
5	C₄²×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
6	#382×C₁₂₇	C₂	1	1	yes	maximal, characteristic
7	#16861200	—	2	8	no	—
8	C₂×C₄⋊Dic₁₂₇	C₂²	1	4	yes	—
9	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
10	C₂×Dic₁₂₇⋊C₄	C₂²	1	4	yes	—
11	C₂×Dic₁₂₇⋊C₄	C₂²	1	8	yes	—
12	C₂×C₄×Dic₁₂₇	C₂²	1	4	yes	—
13	C₂×C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
15	C₁₂₇⋊#95	C₂²	1	4	yes	—
16	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
17	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
18	#95×C₁₂₇	C₂²	1	4	yes	—
19	C₂²×Dic₁₂₇	C₂³	1	4	yes	—
20	C₂²×Dic₁₂₇	C₂³	1	2	yes	—
21	C₂²×Dic₁₂₇	C₂³	1	2	yes	—
22	C₄⋊Dic₁₂₇	—	2	8	no	—
23	C₄⋊Dic₁₂₇	—	2	4	no	—
24	Dic₁₂₇⋊C₄	—	2	8	no	—
25	Dic₁₂₇⋊C₄	—	2	16	no	—
26	C₄×Dic₁₂₇	—	2	8	no	—
27	C₄×Dic₁₂₇	D₄	1	4	yes	—
28	C₄×Dic₁₂₇	Q₈	1	4	yes	—
29	C₂²×C₄×C₁₂₇	C₂³	1	4	yes	—
30	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
31	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
32	C₄⋊C₄×C₁₂₇	—	2	4	no	—
33	C₄²×C₁₂₇	Q₈	1	4	yes	—
34	C₂×Dic₁₂₇	—	2	4	no	—
35	C₂×Dic₁₂₇	—	2	4	no	—
36	C₂×Dic₁₂₇	—	2	8	no	—
37	C₂×Dic₁₂₇	C₂×D₄	1	4	yes	—
38	C₂×Dic₁₂₇	C₂×Q₈	1	4	yes	—
39	C₂×Dic₁₂₇	#41	1	8	yes	—
40	C₂³×C₁₂₇	C₂⁴	1	1	yes	characteristic
41	C₂×C₄×C₁₂₇	—	2	4	no	—
42	C₂×C₄×C₁₂₇	—	2	4	no	—
43	C₂×C₄×C₁₂₇	—	2	8	no	—
44	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
46	C₂×C₄×C₁₂₇	C₂×Q₈	1	4	yes	—
47	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
48	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
49	C₂²×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
50	C₂²×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
51	C₂²×C₁₂₇	C₂×#41	1	4	yes	—
52	C₄×C₁₂₇	—	2	4	no	—
53	C₄×C₁₂₇	—	4	4	no	—
54	C₄×C₁₂₇	C₄⋊Q₈	1	4	yes	—
55	Dic₁₂₇	—	2	4	no	—
56	Dic₁₂₇	—	4	4	no	—
57	Dic₁₂₇	#125	1	8	yes	complemented
58	C₂×C₁₂₇	C₂×#125	1	2	yes	—
59	C₂×C₁₂₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
60	C₂×C₁₂₇	#532	1	2	yes	—
61	C₂×C₁₂₇	#533	1	2	yes	—
62	#2859	—	127	1	no	maximal, Sylow
63	C₁₂₇	#2859	1	1	yes	characteristic, complemented, Sylow
64	C₂×#125	—	127	2	no	—
65	#389	—	127	8	no	—
66	#384	—	127	2	no	—
67	#382	—	127	1	no	—
68	#382	D₁₂₇	1	1	yes	characteristic
69	C₄³	—	127	1	no	—
70	#125	—	254	8	no	—
71	C₂×C₄⋊C₄	—	127	2	no	—
72	C₂×C₄⋊C₄	—	127	4	no	—
73	C₂×C₄⋊C₄	—	127	4	no	—
74	C₂×C₄⋊C₄	—	127	8	no	—
75	C₂×C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
76	C₂×C₄²	—	127	1	no	—
77	C₂×C₄²	—	127	1	no	—
78	C₂×C₄²	—	127	4	no	—
79	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
80	#95	—	127	4	no	—
81	#95	C₂×D₁₂₇	1	4	yes	—
82	C₂²×C₄	—	127	2	no	—
83	C₂²×C₄	—	127	2	no	—
84	C₂²×C₄	—	127	4	no	—
85	C₂²×C₄	C₂²×D₁₂₇	1	4	yes	—
86	C₂²×C₄	C₂²×D₁₂₇	1	2	yes	—
87	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
88	C₄⋊C₄	—	2	4	no	—
89	C₄⋊C₄	—	254	16	no	—
90	C₄⋊C₄	—	254	8	no	—
91	C₄⋊C₄	—	254	8	no	—
92	C₄⋊C₄	—	254	4	no	—
93	C₄²	—	127	4	no	—
94	C₄²	—	127	4	no	—
95	C₄²	—	254	8	no	—
96	C₄²	Dic₂₅₄	1	4	yes	—
97	C₂³	C₂³×D₁₂₇	1	1	yes	characteristic
98	C₂×C₄	—	2	4	no	—
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	2	8	no	—
101	C₂×C₄	—	127	4	no	—
102	C₂×C₄	—	127	8	no	—
103	C₂×C₄	—	127	4	no	—
104	C₂×C₄	—	254	4	no	—
105	C₂×C₄	—	254	8	no	—
106	C₂×C₄	—	254	4	no	—
107	C₂×C₄	D₄×D₁₂₇	1	2	yes	—
108	C₂×C₄	Q₈×D₁₂₇	1	2	yes	—
109	C₂×C₄	C₂×Dic₂₅₄	1	4	yes	—
110	C₂×C₄	C₂×Dic₂₅₄	1	2	yes	—
111	C₂²	C₂×D₄×D₁₂₇	1	1	yes	characteristic
112	C₂²	C₂×Q₈×D₁₂₇	1	1	yes	characteristic
113	C₂²	C₂²×Dic₂₅₄	1	1	yes	characteristic
114	C₂²	#41×D₁₂₇	1	4	yes	—
115	C₄	—	2	4	no	—
116	C₄	—	4	4	no	—
117	C₄	—	127	8	no	—
118	C₄	—	508	4	no	—
119	C₄	—	254	4	no	—
120	C₄	#16861208	1	4	yes	—
121	C₂	#125×D₁₂₇	1	2	yes	—
122	C₂	C₂×#16861208	1	1	yes	characteristic
123	C₂	#22550425	1	2	yes	—
124	C₂	#22550429	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861200

C₂

yes

maximal

#22549773

C₂

yes

maximal

#22549745

C₂

yes

maximal

#22549742

C₂

yes

maximal, characteristic

C₄²×Dic₁₂₇

C₂

yes

maximal, characteristic

#382×C₁₂₇

C₂

yes

maximal, characteristic

#16861200

—

C₂×C₄⋊Dic₁₂₇

C₂²

yes

—

C₂×C₄⋊Dic₁₂₇

C₂²

yes

—

C₂×Dic₁₂₇⋊C₄

C₂²

yes

—

C₂×Dic₁₂₇⋊C₄

C₂²

yes

—

C₂×C₄×Dic₁₂₇

C₂²

yes

—

C₂×C₄×Dic₁₂₇

C₂²

yes

characteristic

C₂×C₄×Dic₁₂₇

C₂²

yes

characteristic

C₁₂₇⋊#95

C₂²

yes

—

C₂×C₄⋊C₄×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

#95×C₁₂₇

C₂²

yes

—

C₂²×Dic₁₂₇

C₂³

yes

—

C₂²×Dic₁₂₇

C₂³

yes

—

C₂²×Dic₁₂₇

C₂³

yes

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

Dic₁₂₇⋊C₄

—

Dic₁₂₇⋊C₄

—

C₄×Dic₁₂₇

—

C₄×Dic₁₂₇

D₄

yes

—

C₄×Dic₁₂₇

Q₈

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

Q₈

yes

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

C₂×D₄

yes

—

C₂×Dic₁₂₇

C₂×Q₈

yes

—

C₂×Dic₁₂₇

#41

yes

—

C₂³×C₁₂₇

C₂⁴

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×Q₈

yes

—

C₂×C₄×C₁₂₇

C₂×Q₈

yes

—

C₂×C₄×C₁₂₇

C₂×Q₈

yes

—

C₂²×C₁₂₇

C₂²×D₄

yes

characteristic

C₂²×C₁₂₇

C₂²×Q₈

yes

characteristic

C₂²×C₁₂₇

C₂²×Q₈

yes

characteristic

C₂²×C₁₂₇

C₂×#41

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₄⋊Q₈

yes

—

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

#125

yes

complemented

C₂×C₁₂₇

C₂×#125

yes

—

C₂×C₁₂₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₂₇

#532

yes

—

C₂×C₁₂₇

#533

yes

—

#2859

—

127

maximal, Sylow

C₁₂₇

#2859

yes

characteristic, complemented, Sylow

C₂×#125

—

127

—

#389

—

127

—

#384

—

127

—

#382

—

127

—

#382

D₁₂₇

yes

characteristic

C₄³

—

127

—

#125

—

254

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

C₂×D₁₂₇

yes

—

C₂×C₄²

—

127

—

C₂×C₄²

—

127

—

C₂×C₄²

—

127

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

#95

—

127

—

#95

C₂×D₁₂₇

yes

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

C₂²×D₁₂₇

yes

—

C₂²×C₄

C₂²×D₁₂₇

yes

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₄²

—

127

—

C₄²

—

127

—

C₄²

—

254

—

C₄²

Dic₂₅₄

yes

—

C₂³

C₂³×D₁₂₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

127

—

102

C₂×C₄

—

127

—

103

C₂×C₄

—

127

—

104

C₂×C₄

—

254

—

105

C₂×C₄

—

254

—

106

C₂×C₄

—

254

—

107

C₂×C₄

D₄×D₁₂₇

yes

—

108

C₂×C₄

Q₈×D₁₂₇

yes

—

109

C₂×C₄

C₂×Dic₂₅₄

yes

—

110

C₂×C₄

C₂×Dic₂₅₄

yes

—

111

C₂²

C₂×D₄×D₁₂₇

yes

characteristic

112

C₂²

C₂×Q₈×D₁₂₇

yes

characteristic

113

C₂²

C₂²×Dic₂₅₄

yes

characteristic

114

C₂²

#41×D₁₂₇

yes

—

115

C₄

—

116

C₄

—

117

C₄

—

127

—

118

C₄

—

508

—

119

C₄

—

254

—

120

C₄

#16861208

yes

—

121

C₂

#125×D₁₂₇

yes

—

122

C₂

C₂×#16861208

yes

characteristic

123

C₂

#22550425

yes

—

124

C₂

#22550429

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₂₇
commutator, socle, prime radical, cosocle	C₂³×C₁₂₇	C₂⁴
Fitting	#382×C₁₂₇	C₂
hypercenter	#382	D₁₂₇
hypocenter	C₁₂₇	#2859
LCS	G ▷ C₂³×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ #382

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₂₇

commutator, socle, prime radical, cosocle

C₂³×C₁₂₇

C₂⁴

Fitting

#382×C₁₂₇

C₂

D₁₂₇

C₁₂₇

G ▷ C₂³×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂³ ◁ #382

N	H ≅ G/N	type
57. Dic₁₂₇	#125	semidirect
63. C₁₂₇	#2859	semidirect

H ≅ G/N

type

57. Dic₁₂₇

#125

semidirect

63. C₁₂₇

#2859

semidirect

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

subgroup	index
C₂×#16861200	2
#22549773	2
#22549745	2
#22549742	2
C₄²×Dic₁₂₇	2
#382×C₁₂₇	2
#2859	127

subgroup

index

C₄²×Dic₁₂₇

#382×C₁₂₇

#2859

127

Subgroups — #30647092

Conjugacy classes of subgroups

Complements

Maximal subgroups