Group #30635932 — subgroups

Subgroups — C₁₂₇×#2980 #30635932

Order 16256.

394 conjugacy classes of subgroups; only the 248 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 248 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	#532×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	#435×C₁₂₇	C₂	1	8	yes	maximal, complemented
3	#432×C₁₂₇	C₂	1	2	yes	maximal, complemented
4	#408×C₁₂₇	C₂	1	4	yes	maximal, complemented
5	C₂×#41×C₁₂₇	C₂²	1	2	yes	—
6	C₂×M₁₆×C₁₂₇	C₂²	1	4	yes	—
7	C₂²×C₈×C₁₂₇	C₂²	1	4	yes	—
8	C₄⋊Q₈×C₁₂₇	C₄	1	1	yes	characteristic
9	#127×C₁₂₇	C₄	1	1	yes	characteristic
10	#124×C₁₂₇	C₄	1	2	yes	—
11	#121×C₁₂₇	C₄	1	4	yes	—
12	C₄×D₄×C₁₂₇	C₂²	1	4	yes	—
13	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₁₂₇	C₂²	1	4	yes	—
15	C₂²⋊C₈×C₁₂₇	C₂²	1	4	yes	—
16	C₂²⋊C₈×C₁₂₇	C₂²	1	8	yes	—
17	#97×C₁₂₇	C₂²	1	4	yes	—
18	#41×C₁₂₇	—	4	2	no	—
19	#41×C₁₂₇	—	4	2	no	—
20	C₂×Q₈×C₁₂₇	C₂×C₄	1	2	yes	—
21	C₂×D₄×C₁₂₇	C₂×C₄	1	4	yes	—
22	C₂×D₄×C₁₂₇	C₂×C₄	1	4	yes	—
23	C₂×D₄×C₁₂₇	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₁₂₇	C₂³	1	4	yes	—
27	M₁₆×C₁₂₇	—	4	4	no	—
28	C₂×C₈×C₁₂₇	—	4	4	no	—
29	C₂×C₈×C₁₂₇	C₂³	1	8	yes	—
30	C₄⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
31	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	8	yes	—
32	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
33	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₁₂₇	C₂³	1	2	yes	—
35	C₂³×C₁₂₇	C₂²×C₄	1	4	yes	—
36	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
37	Q₈×C₁₂₇	—	4	2	no	—
38	D₄×C₁₂₇	—	4	4	no	—
39	D₄×C₁₂₇	—	4	2	no	—
40	D₄×C₁₂₇	—	4	4	no	—
41	C₂×C₄×C₁₂₇	—	2	4	no	—
42	C₂×C₄×C₁₂₇	—	2	4	no	—
43	C₂×C₄×C₁₂₇	—	2	2	no	—
44	C₂×C₄×C₁₂₇	—	4	2	no	—
45	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
48	C₂×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
49	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
50	C₈×C₁₂₇	—	2	8	no	—
51	C₂²×C₁₂₇	—	2	1	no	—
52	C₂²×C₁₂₇	—	2	4	no	—
53	C₂²×C₁₂₇	—	2	2	no	—
54	C₂²×C₁₂₇	—	2	8	no	—
55	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
56	C₄×C₁₂₇	—	2	1	no	—
57	C₄×C₁₂₇	—	2	4	no	—
58	C₄×C₁₂₇	—	2	2	no	—
59	C₄×C₁₂₇	—	2	2	no	—
60	C₄×C₁₂₇	H(5,𝔽₂)	1	2	yes	—
61	C₂×C₁₂₇	—	4	1	no	—
62	C₂×C₁₂₇	—	4	4	no	—
63	C₂×C₁₂₇	#518	1	1	yes	characteristic
64	C₂×C₁₂₇	#568	1	2	yes	—
65	#2980	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₁₂₇	#2980	1	1	yes	characteristic, direct product factor, Sylow
67	#532	C₂×C₁₂₇	1	1	yes	characteristic
68	#435	C₂×C₁₂₇	1	8	yes	complemented
69	#432	C₂×C₁₂₇	1	2	yes	complemented
70	#408	C₂×C₁₂₇	1	4	yes	complemented
71	C₂×#41	C₂²×C₁₂₇	1	2	yes	—
72	C₂×M₁₆	C₂²×C₁₂₇	1	4	yes	—
73	C₂²×C₈	C₂²×C₁₂₇	1	4	yes	—
74	C₄⋊Q₈	C₄×C₁₂₇	1	1	yes	characteristic
75	#127	C₄×C₁₂₇	1	1	yes	characteristic
76	#124	C₄×C₁₂₇	1	2	yes	—
77	#121	C₄×C₁₂₇	1	4	yes	—
78	C₄×D₄	C₂²×C₁₂₇	1	4	yes	—
79	C₂×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
80	C₄⋊C₈	C₂²×C₁₂₇	1	4	yes	—
81	C₂²⋊C₈	C₂²×C₁₂₇	1	4	yes	—
82	C₂²⋊C₈	C₂²×C₁₂₇	1	8	yes	—
83	#97	C₂²×C₁₂₇	1	4	yes	—
84	#41	—	4	2	no	—
85	#41	—	4	2	no	—
86	C₂×Q₈	C₂×C₄×C₁₂₇	1	2	yes	—
87	C₂×D₄	C₂×C₄×C₁₂₇	1	4	yes	—
88	C₂×D₄	C₂×C₄×C₁₂₇	1	4	yes	—
89	C₂×D₄	C₂×C₄×C₁₂₇	1	2	yes	—
90	C₂²×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
91	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
92	C₂²×C₄	C₂³×C₁₂₇	1	4	yes	—
93	M₁₆	—	4	4	no	—
94	C₂×C₈	—	4	4	no	—
95	C₂×C₈	C₂³×C₁₂₇	1	8	yes	—
96	C₄⋊C₄	C₂×C₄×C₁₂₇	1	4	yes	—
97	C₂²⋊C₄	C₂×C₄×C₁₂₇	1	8	yes	—
98	C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
99	C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
100	C₄²	C₂³×C₁₂₇	1	2	yes	—
101	C₂³	C₂²×C₄×C₁₂₇	1	4	yes	—
102	C₂³	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
103	Q₈	—	4	2	no	—
104	D₄	—	4	2	no	—
105	D₄	—	4	4	no	—
106	D₄	—	4	4	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
112	C₂×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
113	C₂×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
114	C₂×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
115	C₂×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
116	C₈	—	2	8	no	—
117	C₂²	—	2	1	no	—
118	C₂²	—	2	4	no	—
119	C₂²	—	2	2	no	—
120	C₂²	—	2	8	no	—
121	C₂²	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
122	C₄	—	2	1	no	—
123	C₄	—	2	4	no	—
124	C₄	—	2	2	no	—
125	C₄	—	2	2	no	—
126	C₄	H(5,𝔽₂)×C₁₂₇	1	2	yes	—
127	C₂	—	4	1	no	—
128	C₂	—	4	4	no	—
129	C₂	#518×C₁₂₇	1	1	yes	characteristic
130	C₂	#568×C₁₂₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#532×C₁₂₇

C₂

yes

maximal, characteristic

#435×C₁₂₇

C₂

yes

maximal, complemented

#432×C₁₂₇

C₂

yes

maximal, complemented

#408×C₁₂₇

C₂

yes

maximal, complemented

C₂×#41×C₁₂₇

C₂²

yes

—

C₂×M₁₆×C₁₂₇

C₂²

yes

—

C₂²×C₈×C₁₂₇

C₂²

yes

—

C₄⋊Q₈×C₁₂₇

C₄

yes

characteristic

#127×C₁₂₇

C₄

yes

characteristic

#124×C₁₂₇

C₄

yes

—

#121×C₁₂₇

C₄

yes

—

C₄×D₄×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₂₇

C₂²

yes

—

C₂²⋊C₈×C₁₂₇

C₂²

yes

—

C₂²⋊C₈×C₁₂₇

C₂²

yes

—

#97×C₁₂₇

C₂²

yes

—

#41×C₁₂₇

—

#41×C₁₂₇

—

C₂×Q₈×C₁₂₇

C₂×C₄

yes

—

C₂×D₄×C₁₂₇

C₂×C₄

yes

—

C₂×D₄×C₁₂₇

C₂×C₄

yes

—

C₂×D₄×C₁₂₇

C₂×C₄

yes

—

C₂²×C₄×C₁₂₇

C₂×C₄

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₂²×C₄×C₁₂₇

C₂³

yes

—

M₁₆×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

C₂³

yes

—

C₄⋊C₄×C₁₂₇

C₂×C₄

yes

—

C₂²⋊C₄×C₁₂₇

C₂×C₄

yes

—

C₄²×C₁₂₇

C₂×C₄

yes

characteristic

C₄²×C₁₂₇

C₂×C₄

yes

characteristic

C₄²×C₁₂₇

C₂³

yes

—

C₂³×C₁₂₇

C₂²×C₄

yes

—

C₂³×C₁₂₇

C₂²×C₄

yes

characteristic

Q₈×C₁₂₇

—

D₄×C₁₂₇

—

D₄×C₁₂₇

—

D₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂²×C₄

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

—

C₂×C₄×C₁₂₇

C₂⁴

yes

characteristic

C₈×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂³×C₄

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

H(5,𝔽₂)

yes

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#518

yes

characteristic

C₂×C₁₂₇

#568

yes

—

#2980

C₁₂₇

yes

maximal, characteristic, direct product factor, Sylow

C₁₂₇

#2980

yes

characteristic, direct product factor, Sylow

#532

C₂×C₁₂₇

yes

characteristic

#435

C₂×C₁₂₇

yes

complemented

#432

C₂×C₁₂₇

yes

complemented

#408

C₂×C₁₂₇

yes

complemented

C₂×#41

C₂²×C₁₂₇

yes

—

C₂×M₁₆

C₂²×C₁₂₇

yes

—

C₂²×C₈

C₂²×C₁₂₇

yes

—

C₄⋊Q₈

C₄×C₁₂₇

yes

characteristic

#127

C₄×C₁₂₇

yes

characteristic

#124

C₄×C₁₂₇

yes

—

#121

C₄×C₁₂₇

yes

—

C₄×D₄

C₂²×C₁₂₇

yes

—

C₂×C₄²

C₂²×C₁₂₇

yes

characteristic

C₄⋊C₈

C₂²×C₁₂₇

yes

—

C₂²⋊C₈

C₂²×C₁₂₇

yes

—

C₂²⋊C₈

C₂²×C₁₂₇

yes

—

#97

C₂²×C₁₂₇

yes

—

#41

—

#41

—

C₂×Q₈

C₂×C₄×C₁₂₇

yes

—

C₂×D₄

C₂×C₄×C₁₂₇

yes

—

C₂×D₄

C₂×C₄×C₁₂₇

yes

—

C₂×D₄

C₂×C₄×C₁₂₇

yes

—

C₂²×C₄

C₂×C₄×C₁₂₇

yes

—

C₂²×C₄

C₂³×C₁₂₇

yes

characteristic

C₂²×C₄

C₂³×C₁₂₇

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂³×C₁₂₇

yes

—

C₄⋊C₄

C₂×C₄×C₁₂₇

yes

—

C₂²⋊C₄

C₂×C₄×C₁₂₇

yes

—

C₄²

C₂×C₄×C₁₂₇

yes

characteristic

C₄²

C₂×C₄×C₁₂₇

yes

characteristic

100

C₄²

C₂³×C₁₂₇

yes

—

101

C₂³

C₂²×C₄×C₁₂₇

yes

—

102

C₂³

C₂²×C₄×C₁₂₇

yes

characteristic

103

Q₈

—

104

D₄

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

C₂²×C₄×C₁₂₇

yes

characteristic

112

C₂×C₄

C₂²×C₄×C₁₂₇

yes

—

113

C₂×C₄

C₂²×C₄×C₁₂₇

yes

—

114

C₂×C₄

C₂²×C₄×C₁₂₇

yes

—

115

C₂×C₄

C₂⁴×C₁₂₇

yes

characteristic

116

C₈

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂³×C₄×C₁₂₇

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

H(5,𝔽₂)×C₁₂₇

yes

—

127

C₂

—

128

C₂

—

129

C₂

#518×C₁₂₇

yes

characteristic

130

C₂

#568×C₁₂₇

yes

—

	subgroup	G/H
center	C₂×C₄×C₁₂₇	C₂⁴
commutator	C₂²	C₂³×C₄×C₁₂₇
Frattini, cosocle	C₂×C₄	C₂⁴×C₁₂₇
socle	C₂²×C₁₂₇	C₂³×C₄
prime radical	#532×C₁₂₇	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₂₇ ◁ G

subgroup

G/H

center

C₂×C₄×C₁₂₇

C₂⁴

commutator

C₂²

C₂³×C₄×C₁₂₇

Frattini, cosocle

C₂×C₄

C₂⁴×C₁₂₇

socle

C₂²×C₁₂₇

C₂³×C₄

prime radical

#532×C₁₂₇

C₂

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₁₂₇ ◁ G

N	H ≅ G/N	type
2. #435×C₁₂₇	C₂	semidirect
3. #432×C₁₂₇	C₂	semidirect
4. #408×C₁₂₇	C₂	semidirect
65. #2980	C₁₂₇	direct (N × H)
66. C₁₂₇	#2980	direct (N × H)
68. #435	C₂×C₁₂₇	semidirect
69. #432	C₂×C₁₂₇	semidirect
70. #408	C₂×C₁₂₇	semidirect

H ≅ G/N

type

2. #435×C₁₂₇

C₂

semidirect

3. #432×C₁₂₇

C₂

semidirect

4. #408×C₁₂₇

C₂

semidirect

65. #2980

C₁₂₇

direct (N × H)

66. C₁₂₇

#2980

direct (N × H)

68. #435

C₂×C₁₂₇

semidirect

69. #432

C₂×C₁₂₇

semidirect

70. #408

C₂×C₁₂₇

semidirect

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

subgroup	index
#532×C₁₂₇	2
#435×C₁₂₇	2
#432×C₁₂₇	2
#408×C₁₂₇	2
#2980	127

subgroup

index

#532×C₁₂₇

#435×C₁₂₇

#432×C₁₂₇

#408×C₁₂₇

#2980

127

Subgroups — C₁₂₇×#2980 #30635932

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C127×#2980 #30635932

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#2980 #30635932