Group #30637389 — subgroups

Subgroups — #30637389

Order 16256.

246 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 67 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	#22550424	C₂	1	1	yes	maximal, characteristic
2	#22549928	C₂	1	2	yes	maximal, complemented
3	#22549624	C₂	1	2	yes	maximal, complemented
4	#22549620	C₂	1	1	yes	maximal, characteristic, complemented
5	#331×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14523247	C₄	1	2	yes	—
7	#16861209	C₂²	1	1	yes	characteristic
8	#16861196	C₄	1	2	yes	—
9	#16861180	—	2	2	no	—
10	D₄×Dic₁₂₇	—	2	2	no	—
11	C₄²×D₁₂₇	C₂²	1	1	yes	characteristic
12	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	2	yes	—
13	(C₂×D₁₂₇)⋊C₈	—	2	2	no	—
14	C₈×Dic₁₂₇	—	2	2	no	—
15	#127×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
17	#14523247	—	4	4	no	—
18	C₂×Dic₂₅₄	C₂×C₄	1	2	yes	—
19	C₂×C₂²⋊D₁₂₇	—	2	2	no	—
20	C₂²×Dic₁₂₇	—	2	2	no	—
21	C₂×C₄×D₁₂₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×D₁₂₇	D₄	1	2	yes	—
23	C₂×C₁₂₇⋊C₈	—	2	2	no	—
24	C₄⋊Dic₁₂₇	—	2	2	no	—
25	C₂²⋊Dic₁₂₇	—	2	4	no	—
26	C₄×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₁₂₇	D₄	1	2	yes	—
28	C₂×D₄×C₁₂₇	—	2	2	no	—
29	C₂×D₄×C₁₂₇	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₁₂₇	—	2	2	no	—
31	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
32	C₂²×D₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
33	Dic₂₅₄	—	4	2	no	—
34	C₂²⋊D₁₂₇	—	4	4	no	—
35	C₂×Dic₁₂₇	—	2	2	no	—
36	C₂×Dic₁₂₇	—	4	4	no	—
37	C₂×Dic₁₂₇	C₂²⋊C₄	1	2	yes	—
38	C₂×Dic₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
39	C₄×D₁₂₇	—	2	2	no	—
40	C₄×D₁₂₇	—	2	2	no	—
41	C₄×D₁₂₇	—	2	2	no	—
42	C₁₂₇⋊C₈	—	4	2	no	—
43	C₂³×C₁₂₇	—	2	2	no	—
44	D₄×C₁₂₇	—	4	2	no	—
45	D₄×C₁₂₇	—	4	2	no	—
46	D₄×C₁₂₇	—	4	2	no	—
47	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
49	C₈×C₁₂₇	—	4	2	no	—
50	C₂²×C₁₂₇	—	4	2	no	—
51	C₂²×C₁₂₇	—	4	4	no	—
52	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₂×D₁₂₇	—	2	1	no	—
54	C₂×D₁₂₇	—	2	2	no	—
55	C₄×C₁₂₇	—	2	1	no	—
56	C₄×C₁₂₇	#135	1	2	yes	—
57	C₄×C₁₂₇	#135	1	2	yes	—
58	Dic₁₂₇	—	2	2	no	—
59	Dic₁₂₇	—	4	2	no	—
60	Dic₁₂₇	#100	1	2	yes	—
61	C₂×C₁₂₇	—	8	2	no	—
62	C₂×C₁₂₇	C₂×#100	1	1	yes	characteristic
63	C₂×C₁₂₇	#416	1	2	yes	—
64	D₁₂₇	—	4	1	no	—
65	#1543	—	127	1	no	maximal, Sylow
66	C₁₂₇	#1543	1	1	yes	characteristic, complemented, Sylow
67	#532	—	127	1	no	—
68	#431	—	127	2	no	—
69	#333	—	127	2	no	—
70	#331	—	127	1	no	—
71	#331	D₁₂₇	1	1	yes	characteristic, complemented
72	C₂×#41	—	127	2	no	—
73	C₄⋊Q₈	—	127	1	no	—
74	#127	C₂×D₁₂₇	1	1	yes	characteristic
75	#124	—	127	2	no	—
76	#121	—	254	2	no	—
77	C₄×D₄	—	254	2	no	—
78	C₂×C₄²	—	127	1	no	—
79	C₄⋊C₈	—	127	2	no	—
80	C₄⋊C₈	C₂×D₁₂₇	1	2	yes	—
81	C₂²⋊C₈	—	254	2	no	—
82	C₄×C₈	—	254	2	no	—
83	#41	—	508	4	no	—
84	C₂×Q₈	—	127	2	no	—
85	C₂×D₄	—	2	2	no	—
86	C₂×D₄	—	254	2	no	—
87	C₂×D₄	C₄×D₁₂₇	1	2	yes	—
88	C₂²×C₄	—	127	1	no	—
89	C₂²×C₄	—	127	2	no	—
90	C₂²×C₄	—	254	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	254	2	no	—
93	C₄⋊C₄	—	254	2	no	—
94	C₂²⋊C₄	—	254	4	no	—
95	C₄²	—	127	1	no	—
96	C₄²	—	127	2	no	—
97	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
98	C₂³	—	2	2	no	—
99	C₂³	—	127	1	no	—
100	Q₈	—	508	2	no	—
101	D₄	—	4	2	no	—
102	D₄	—	4	2	no	—
103	D₄	—	4	2	no	—
104	D₄	—	508	4	no	—
105	C₂×C₄	—	127	1	no	—
106	C₂×C₄	—	127	2	no	—
107	C₂×C₄	—	508	4	no	—
108	C₂×C₄	—	254	2	no	—
109	C₂×C₄	—	254	2	no	—
110	C₂×C₄	—	254	2	no	—
111	C₂×C₄	—	254	2	no	—
112	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
113	C₂×C₄	D₄×D₁₂₇	1	2	yes	—
114	C₈	—	4	2	no	—
115	C₈	—	508	2	no	—
116	C₂²	—	4	2	no	—
117	C₂²	—	4	4	no	—
118	C₂²	—	254	1	no	—
119	C₂²	—	254	2	no	—
120	C₂²	C₂²⋊C₄×D₁₂₇	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	127	2	no	—
123	C₄	—	508	2	no	—
124	C₄	—	254	2	no	—
125	C₄	#16861228	1	2	yes	—
126	C₄	#16861230	1	2	yes	—
127	C₂	—	8	2	no	—
128	C₂	—	508	1	no	—
129	C₂	#100×D₁₂₇	1	1	yes	characteristic
130	C₂	#22549869	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22550424

C₂

yes

maximal, characteristic

#22549928

C₂

yes

maximal, complemented

#22549624

C₂

yes

maximal, complemented

#22549620

C₂

yes

maximal, characteristic, complemented

#331×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×#14523247

C₄

yes

—

#16861209

C₂²

yes

characteristic

#16861196

C₄

yes

—

#16861180

—

D₄×Dic₁₂₇

—

C₄²×D₁₂₇

C₂²

yes

characteristic

C₄⋊(C₁₂₇⋊C₈)

C₂²

yes

—

(C₂×D₁₂₇)⋊C₈

—

C₈×Dic₁₂₇

—

#127×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₂₇

C₂²

yes

—

#14523247

—

C₂×Dic₂₅₄

C₂×C₄

yes

—

C₂×C₂²⋊D₁₂₇

—

C₂²×Dic₁₂₇

—

C₂×C₄×D₁₂₇

C₂×C₄

yes

characteristic

C₂×C₄×D₁₂₇

D₄

yes

—

C₂×C₁₂₇⋊C₈

—

C₄⋊Dic₁₂₇

—

C₂²⋊Dic₁₂₇

—

C₄×Dic₁₂₇

C₂×C₄

yes

characteristic

C₄×Dic₁₂₇

D₄

yes

—

C₂×D₄×C₁₂₇

—

C₂×D₄×C₁₂₇

C₂×C₄

yes

—

C₂×C₈×C₁₂₇

—

C₄²×C₁₂₇

C₂³

yes

characteristic

C₂²×D₁₂₇

C₂²⋊C₄

yes

characteristic

Dic₂₅₄

—

C₂²⋊D₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

C₂²⋊C₄

yes

—

C₂×Dic₁₂₇

C₂²⋊C₄

yes

characteristic

C₄×D₁₂₇

—

C₄×D₁₂₇

—

C₄×D₁₂₇

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

—

D₄×C₁₂₇

—

D₄×C₁₂₇

—

D₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₈×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#135

yes

—

C₄×C₁₂₇

#135

yes

—

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

#100

yes

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂×#100

yes

characteristic

C₂×C₁₂₇

#416

yes

—

D₁₂₇

—

#1543

—

127

maximal, Sylow

C₁₂₇

#1543

yes

characteristic, complemented, Sylow

#532

—

127

—

#431

—

127

—

#333

—

127

—

#331

—

127

—

#331

D₁₂₇

yes

characteristic, complemented

C₂×#41

—

127

—

C₄⋊Q₈

—

127

—

#127

C₂×D₁₂₇

yes

characteristic

#124

—

127

—

#121

—

254

—

C₄×D₄

—

254

—

C₂×C₄²

—

127

—

C₄⋊C₈

—

127

—

C₄⋊C₈

C₂×D₁₂₇

yes

—

C₂²⋊C₈

—

254

—

C₄×C₈

—

254

—

#41

—

508

—

C₂×Q₈

—

127

—

C₂×D₄

—

C₂×D₄

—

254

—

C₂×D₄

C₄×D₁₂₇

yes

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

254

—

C₂×C₈

—

C₂×C₈

—

254

—

C₄⋊C₄

—

254

—

C₂²⋊C₄

—

254

—

C₄²

—

127

—

C₄²

—

127

—

C₄²

C₂²×D₁₂₇

yes

characteristic

C₂³

—

C₂³

—

127

—

100

Q₈

—

508

—

101

D₄

—

102

D₄

—

103

D₄

—

104

D₄

—

508

—

105

C₂×C₄

—

127

—

106

C₂×C₄

—

127

—

107

C₂×C₄

—

508

—

108

C₂×C₄

—

254

—

109

C₂×C₄

—

254

—

110

C₂×C₄

—

254

—

111

C₂×C₄

—

254

—

112

C₂×C₄

C₂×C₄×D₁₂₇

yes

characteristic

113

C₂×C₄

D₄×D₁₂₇

yes

—

114

C₈

—

115

C₈

—

508

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

254

—

119

C₂²

—

254

—

120

C₂²

C₂²⋊C₄×D₁₂₇

yes

characteristic

121

C₄

—

122

C₄

—

127

—

123

C₄

—

508

—

124

C₄

—

254

—

125

C₄

#16861228

yes

—

126

C₄

#16861230

yes

—

127

C₂

—

128

C₂

—

508

—

129

C₂

#100×D₁₂₇

yes

characteristic

130

C₂

#22549869

yes

—

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₁₂₇
commutator	C₂×C₄×C₁₂₇	C₂²×C₄
Frattini	C₄²	C₂²×D₁₂₇
Fitting	#331×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×C₂²⋊C₄
prime radical	#22550424	C₂
cosocle	C₄²×C₁₂₇	C₂³
hypercenter	#331	D₁₂₇
hypocenter	C₁₂₇	#1543
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #331

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₁₂₇

commutator

C₂×C₄×C₁₂₇

C₂²×C₄

Frattini

C₄²

C₂²×D₁₂₇

Fitting

#331×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂×C₂²⋊C₄

prime radical

#22550424

C₂

cosocle

C₄²×C₁₂₇

C₂³

D₁₂₇

C₁₂₇

G ▷ C₂×C₄×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #331

N	H ≅ G/N	type
2. #22549928	C₂	semidirect
3. #22549624	C₂	semidirect
4. #22549620	C₂	semidirect
5. #331×C₁₂₇	C₂	semidirect
66. C₁₂₇	#1543	semidirect
71. #331	D₁₂₇	semidirect

H ≅ G/N

type

2. #22549928

C₂

semidirect

3. #22549624

C₂

semidirect

4. #22549620

C₂

semidirect

5. #331×C₁₂₇

C₂

semidirect

66. C₁₂₇

#1543

semidirect

71. #331

D₁₂₇

semidirect

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

subgroup	index
#22550424	2
#22549928	2
#22549624	2
#22549620	2
#331×C₁₂₇	2
#1543	127

subgroup

index

#331×C₁₂₇

127

Subgroups — #30637389

Conjugacy classes of subgroups

Complements

Maximal subgroups