Group #30647106 — subgroups

Subgroups — #30647106

Order 16256.

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

normal subgroups only

This group has 504 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×#16861208	C₂	1	2	yes	maximal
2	#22549780	C₂	1	8	yes	maximal
3	#22549742	C₂	1	1	yes	maximal, characteristic
4	#22549741	C₂	1	2	yes	maximal
5	C₄²×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
6	#382×C₁₂₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₅₄	C₂²	1	4	yes	—
8	#16861208	—	2	8	no	—
9	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
10	C₂×Dic₁₂₇⋊C₄	C₂²	1	4	yes	—
11	C₂×C₄×Dic₁₂₇	C₂²	1	4	yes	—
12	C₂×C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
14	C₁₂₇⋊#95	C₂²	1	8	yes	—
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	C₂×Dic₂₅₄	—	2	8	no	—
20	C₂×Dic₂₅₄	—	2	16	no	—
21	C₂²×Dic₁₂₇	C₂³	1	4	yes	—
22	C₂²×Dic₁₂₇	C₂³	1	2	yes	—
23	C₂²×Dic₁₂₇	C₂³	1	2	yes	—
24	C₄⋊Dic₁₂₇	—	2	4	no	—
25	Dic₁₂₇⋊C₄	—	2	8	no	—
26	C₄×Dic₁₂₇	—	2	8	no	—
27	C₄×Dic₁₂₇	D₄	1	4	yes	—
28	C₄×Dic₁₂₇	Q₈	1	4	yes	complemented
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	complemented
34	Dic₂₅₄	—	4	16	no	—
35	C₂×Dic₁₂₇	—	2	4	no	—
36	C₂×Dic₁₂₇	—	2	4	no	—
37	C₂×Dic₁₂₇	—	2	8	no	—
38	C₂×Dic₁₂₇	C₂×D₄	1	4	yes	—
39	C₂×Dic₁₂₇	C₂×Q₈	1	4	yes	—
40	C₂×Dic₁₂₇	#41	1	8	yes	—
41	C₂³×C₁₂₇	C₂⁴	1	1	yes	characteristic
42	C₂×C₄×C₁₂₇	—	2	4	no	—
43	C₂×C₄×C₁₂₇	—	2	4	no	—
44	C₂×C₄×C₁₂₇	—	2	8	no	—
45	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₁₂₇	C₂×Q₈	1	4	yes	—
48	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
49	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
50	C₂²×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
51	C₂²×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
52	C₂²×C₁₂₇	C₂×#41	1	4	yes	—
53	C₄×C₁₂₇	—	2	4	no	—
54	C₄×C₁₂₇	—	4	4	no	—
55	C₄×C₁₂₇	C₄⋊Q₈	1	4	yes	—
56	Dic₁₂₇	—	2	4	no	—
57	Dic₁₂₇	—	4	4	no	—
58	Dic₁₂₇	#124	1	8	yes	—
59	C₂×C₁₂₇	C₂×#124	1	2	yes	—
60	C₂×C₁₂₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
61	C₂×C₁₂₇	#533	1	2	yes	—
62	C₂×C₁₂₇	#533	1	2	yes	—
63	#2862	—	127	1	no	maximal, Sylow
64	C₁₂₇	#2862	1	1	yes	characteristic, complemented, Sylow
65	C₂×C₄⋊Q₈	—	127	2	no	—
66	#391	—	127	8	no	—
67	#382	—	127	2	no	—
68	#382	—	127	1	no	—
69	#382	D₁₂₇	1	1	yes	characteristic
70	C₄³	—	127	1	no	—
71	C₂²×Q₈	—	127	4	no	—
72	C₄⋊Q₈	—	254	8	no	—
73	C₂×C₄⋊C₄	—	127	2	no	—
74	C₂×C₄⋊C₄	—	127	4	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	8	no	—
81	#95	—	127	4	no	—
82	#95	C₂×D₁₂₇	1	4	yes	—
83	C₂×Q₈	—	254	8	no	—
84	C₂×Q₈	—	254	16	no	—
85	C₂²×C₄	—	127	2	no	—
86	C₂²×C₄	—	127	2	no	—
87	C₂²×C₄	—	127	4	no	—
88	C₂²×C₄	C₂²×D₁₂₇	1	4	yes	—
89	C₂²×C₄	C₂²×D₁₂₇	1	2	yes	—
90	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
91	C₄⋊C₄	—	2	4	no	—
92	C₄⋊C₄	—	254	8	no	—
93	C₄⋊C₄	—	254	4	no	—
94	C₄²	—	127	4	no	—
95	C₄²	—	127	4	no	—
96	C₄²	—	254	8	no	—
97	C₄²	Dic₂₅₄	1	4	yes	complemented
98	C₂³	C₂³×D₁₂₇	1	1	yes	characteristic
99	Q₈	—	508	16	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	2	8	no	—
103	C₂×C₄	—	127	4	no	—
104	C₂×C₄	—	127	8	no	—
105	C₂×C₄	—	127	4	no	—
106	C₂×C₄	—	254	4	no	—
107	C₂×C₄	—	254	8	no	—
108	C₂×C₄	—	254	4	no	—
109	C₂×C₄	D₄×D₁₂₇	1	2	yes	—
110	C₂×C₄	Q₈×D₁₂₇	1	2	yes	—
111	C₂×C₄	C₂×Dic₂₅₄	1	4	yes	—
112	C₂×C₄	C₂×Dic₂₅₄	1	2	yes	—
113	C₂²	C₂×D₄×D₁₂₇	1	1	yes	characteristic
114	C₂²	C₂×Q₈×D₁₂₇	1	1	yes	characteristic
115	C₂²	C₂²×Dic₂₅₄	1	1	yes	characteristic
116	C₂²	#41×D₁₂₇	1	4	yes	—
117	C₄	—	2	4	no	—
118	C₄	—	4	4	no	—
119	C₄	—	127	8	no	—
120	C₄	—	508	4	no	—
121	C₄	—	254	4	no	—
122	C₄	#16861208	1	4	yes	—
123	C₂	#124×D₁₂₇	1	2	yes	—
124	C₂	C₂×#16861208	1	1	yes	characteristic
125	C₂	#22550429	1	2	yes	—
126	C₂	#22550432	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861208

C₂

yes

maximal

#22549780

C₂

yes

maximal

#22549742

C₂

yes

maximal, characteristic

#22549741

C₂

yes

maximal

C₄²×Dic₁₂₇

C₂

yes

maximal, characteristic

#382×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×Dic₂₅₄

C₂²

yes

—

#16861208

—

C₂×C₄⋊Dic₁₂₇

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

—

C₁₂₇⋊#95

C₂²

yes

—

C₂×C₄⋊C₄×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

#95×C₁₂₇

C₂²

yes

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂²×Dic₁₂₇

C₂³

yes

—

C₂²×Dic₁₂₇

C₂³

yes

—

C₂²×Dic₁₂₇

C₂³

yes

—

C₄⋊Dic₁₂₇

—

Dic₁₂₇⋊C₄

—

C₄×Dic₁₂₇

—

C₄×Dic₁₂₇

D₄

yes

—

C₄×Dic₁₂₇

Q₈

yes

complemented

C₂²×C₄×C₁₂₇

C₂³

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

Q₈

yes

complemented

Dic₂₅₄

—

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

#124

yes

—

C₂×C₁₂₇

C₂×#124

yes

—

C₂×C₁₂₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₂₇

#533

yes

—

C₂×C₁₂₇

#533

yes

—

#2862

—

127

maximal, Sylow

C₁₂₇

#2862

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

127

—

#391

—

127

—

#382

—

127

—

#382

—

127

—

#382

D₁₂₇

yes

characteristic

C₄³

—

127

—

C₂²×Q₈

—

127

—

C₄⋊Q₈

—

254

—

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

—

127

—

#95

C₂×D₁₂₇

yes

—

C₂×Q₈

—

254

—

C₂×Q₈

—

254

—

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

—

C₄²

—

127

—

C₄²

—

127

—

C₄²

—

254

—

C₄²

Dic₂₅₄

yes

complemented

C₂³

C₂³×D₁₂₇

yes

characteristic

Q₈

—

508

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

127

—

104

C₂×C₄

—

127

—

105

C₂×C₄

—

127

—

106

C₂×C₄

—

254

—

107

C₂×C₄

—

254

—

108

C₂×C₄

—

254

—

109

C₂×C₄

D₄×D₁₂₇

yes

—

110

C₂×C₄

Q₈×D₁₂₇

yes

—

111

C₂×C₄

C₂×Dic₂₅₄

yes

—

112

C₂×C₄

C₂×Dic₂₅₄

yes

—

113

C₂²

C₂×D₄×D₁₂₇

yes

characteristic

114

C₂²

C₂×Q₈×D₁₂₇

yes

characteristic

115

C₂²

C₂²×Dic₂₅₄

yes

characteristic

116

C₂²

#41×D₁₂₇

yes

—

117

C₄

—

118

C₄

—

119

C₄

—

127

—

120

C₄

—

508

—

121

C₄

—

254

—

122

C₄

#16861208

yes

—

123

C₂

#124×D₁₂₇

yes

—

124

C₂

C₂×#16861208

yes

characteristic

125

C₂

#22550429

yes

—

126

C₂

#22550432

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	#2862
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
28. C₄×Dic₁₂₇	Q₈	semidirect
33. C₄²×C₁₂₇	Q₈	semidirect
64. C₁₂₇	#2862	semidirect
97. C₄²	Dic₂₅₄	semidirect

H ≅ G/N

type

28. C₄×Dic₁₂₇

Q₈

semidirect

33. C₄²×C₁₂₇

Q₈

semidirect

64. C₁₂₇

#2862

semidirect

97. C₄²

Dic₂₅₄

semidirect

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

subgroup	index
C₂×#16861208	2
#22549780	2
#22549742	2
#22549741	2
C₄²×Dic₁₂₇	2
#382×C₁₂₇	2
#2862	127

subgroup

index

C₄²×Dic₁₂₇

#382×C₁₂₇

#2862

127

Subgroups — #30647106

Conjugacy classes of subgroups

Complements

Maximal subgroups