Group #30637871 — subgroups

Subgroups — #30637871

Order 16256.

206 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 57 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	#22550606	C₂	1	1	yes	maximal, characteristic
2	#22550255	C₂	1	1	yes	maximal, characteristic
3	#22550172	C₂	1	2	yes	maximal
4	#22549606	C₂	1	2	yes	maximal
5	#333×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523238	—	2	2	no	—
7	#16861209	C₂²	1	1	yes	characteristic
8	#16861208	—	2	2	no	—
9	#16861207	—	2	2	no	—
10	C₄×Dic₂₅₄	C₂²	1	2	yes	—
11	#16861174	—	2	2	no	—
12	#16861143	—	2	2	no	—
13	#16861134	—	2	2	no	—
14	C₄×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
17	C₂×Dic₂₅₄	—	2	2	no	—
18	C₂×Dic₂₅₄	—	2	1	no	—
19	C₂×Dic₂₅₄	D₄	1	2	yes	—
20	#14523238	—	4	4	no	—
21	C₂×C₁₂₇⋊C₈	—	2	1	no	—
22	C₄⋊Dic₁₂₇	—	2	2	no	—
23	C₄⋊Dic₁₂₇	D₄	1	2	yes	—
24	Dic₁₂₇⋊C₄	—	2	4	no	—
25	Dic₁₂₇⋊C₄	—	2	2	no	—
26	C₄×Dic₁₂₇	—	2	2	no	—
27	C₄×Dic₁₂₇	—	2	2	no	—
28	C₂×Q₈×C₁₂₇	D₄	1	2	yes	—
29	C₂×C₈×C₁₂₇	—	2	2	no	—
30	C₄⋊C₄×C₁₂₇	—	2	2	no	—
31	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
32	Dic₂₅₄	—	2	2	no	—
33	Dic₂₅₄	—	4	2	no	—
34	Dic₂₅₄	—	8	2	no	—
35	Dic₂₅₄	Q₁₆	1	4	yes	complemented
36	C₂×Dic₁₂₇	—	2	2	no	—
37	C₂×Dic₁₂₇	—	2	2	no	—
38	C₂×Dic₁₂₇	—	4	1	no	—
39	C₁₂₇⋊C₈	—	2	2	no	—
40	Q₈×C₁₂₇	—	4	2	no	—
41	C₂×C₄×C₁₂₇	—	2	2	no	—
42	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
44	C₈×C₁₂₇	—	4	2	no	—
45	C₂²×C₁₂₇	#120	1	1	yes	characteristic
46	C₄×C₁₂₇	—	2	1	no	—
47	C₄×C₁₂₇	—	4	2	no	—
48	C₄×C₁₂₇	C₂×Q₁₆	1	2	yes	—
49	C₄×C₁₂₇	#137	1	2	yes	—
50	Dic₁₂₇	—	2	4	no	—
51	Dic₁₂₇	—	4	2	no	—
52	Dic₁₂₇	—	8	1	no	—
53	C₂×C₁₂₇	#451	1	2	yes	—
54	C₂×C₁₂₇	#453	1	1	yes	characteristic
55	#1633	—	127	1	no	maximal, Sylow
56	C₁₂₇	#1633	1	1	yes	characteristic, complemented, Sylow
57	Q₈²	—	127	1	no	—
58	#494	—	127	1	no	—
59	#477	—	127	2	no	—
60	#333	D₁₂₇	1	1	yes	characteristic
61	Q₈⋊C₈	—	127	2	no	—
62	C₂×Q₁₆	—	254	2	no	—
63	C₄⋊Q₈	—	127	1	no	—
64	C₄⋊Q₈	—	254	2	no	—
65	C₄⋊Q₈	—	254	2	no	—
66	C₄⋊Q₈	C₂×D₁₂₇	1	1	yes	characteristic
67	C₄×Q₈	—	127	2	no	—
68	C₄×Q₈	—	254	2	no	—
69	#107	—	254	2	no	—
70	C₄⋊C₈	C₂×D₁₂₇	1	2	yes	—
71	#103	—	254	2	no	—
72	C₄×C₈	—	127	1	no	—
73	C₂×Q₈	—	127	2	no	—
74	C₂×Q₈	—	254	2	no	—
75	C₂×Q₈	—	254	1	no	—
76	C₂×Q₈	D₅₀₈	1	2	yes	—
77	Q₁₆	—	508	4	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	254	1	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	127	2	no	—
82	C₄⋊C₄	—	254	4	no	—
83	C₄⋊C₄	—	254	2	no	—
84	C₄⋊C₄	—	254	2	no	—
85	C₄²	—	254	2	no	—
86	C₄²	—	254	2	no	—
87	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
88	Q₈	—	4	2	no	—
89	Q₈	—	127	4	no	—
90	Q₈	—	1016	2	no	—
91	Q₈	—	508	2	no	—
92	Q₈	—	254	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	508	1	no	—
95	C₂×C₄	—	254	2	no	—
96	C₂×C₄	—	254	2	no	—
97	C₂×C₄	D₄×D₁₂₇	1	2	yes	—
98	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
99	C₈	—	4	2	no	—
100	C₈	—	254	2	no	—
101	C₂²	#16861179	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	4	2	no	—
104	C₄	—	1016	1	no	—
105	C₄	—	508	2	no	—
106	C₄	—	254	4	no	—
107	C₄	Q₁₆×D₁₂₇	1	2	yes	—
108	C₄	#16861238	1	2	yes	—
109	C₂	#22550033	1	2	yes	—
110	C₂	#22550046	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22550606

C₂

yes

maximal, characteristic

#22550255

C₂

yes

maximal, characteristic

#22550172

C₂

yes

maximal

#22549606

C₂

yes

maximal

#333×C₁₂₇

C₂

yes

maximal, characteristic

C₂×#14523238

—

#16861209

C₂²

yes

characteristic

#16861208

—

#16861207

—

C₄×Dic₂₅₄

C₂²

yes

—

#16861174

—

#16861143

—

#16861134

—

C₄×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₂₇

C₂²

yes

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

D₄

yes

—

#14523238

—

C₂×C₁₂₇⋊C₈

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

D₄

yes

—

Dic₁₂₇⋊C₄

—

Dic₁₂₇⋊C₄

—

C₄×Dic₁₂₇

—

C₄×Dic₁₂₇

—

C₂×Q₈×C₁₂₇

D₄

yes

—

C₂×C₈×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

C₂³

yes

characteristic

Dic₂₅₄

—

Dic₂₅₄

—

Dic₂₅₄

—

Dic₂₅₄

Q₁₆

yes

complemented

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

—

Q₈×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₈×C₁₂₇

—

C₂²×C₁₂₇

#120

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×Q₁₆

yes

—

C₄×C₁₂₇

#137

yes

—

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

#451

yes

—

C₂×C₁₂₇

#453

yes

characteristic

#1633

—

127

maximal, Sylow

C₁₂₇

#1633

yes

characteristic, complemented, Sylow

Q₈²

—

127

—

#494

—

127

—

#477

—

127

—

#333

D₁₂₇

yes

characteristic

Q₈⋊C₈

—

127

—

C₂×Q₁₆

—

254

—

C₄⋊Q₈

—

127

—

C₄⋊Q₈

—

254

—

C₄⋊Q₈

—

254

—

C₄⋊Q₈

C₂×D₁₂₇

yes

characteristic

C₄×Q₈

—

127

—

C₄×Q₈

—

254

—

#107

—

254

—

C₄⋊C₈

C₂×D₁₂₇

yes

—

#103

—

254

—

C₄×C₈

—

127

—

C₂×Q₈

—

127

—

C₂×Q₈

—

254

—

C₂×Q₈

—

254

—

C₂×Q₈

D₅₀₈

yes

—

Q₁₆

—

508

—

C₂×C₈

—

C₂×C₈

—

254

—

C₄⋊C₄

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₄²

—

254

—

C₄²

—

254

—

C₄²

C₂²×D₁₂₇

yes

characteristic

Q₈

—

Q₈

—

127

—

Q₈

—

1016

—

Q₈

—

508

—

Q₈

—

254

—

C₂×C₄

—

C₂×C₄

—

508

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

D₄×D₁₂₇

yes

—

C₂×C₄

C₂×D₅₀₈

yes

characteristic

C₈

—

100

C₈

—

254

—

101

C₂²

#16861179

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

1016

—

105

C₄

—

508

—

106

C₄

—

254

—

107

C₄

Q₁₆×D₁₂₇

yes

—

108

C₄

#16861238

yes

—

109

C₂

#22550033

yes

—

110

C₂

#22550046

yes

characteristic

	subgroup	G/H
center	C₂²	#16861179
commutator, cosocle	C₄²×C₁₂₇	C₂³
Frattini	C₄²	C₂²×D₁₂₇
Fitting	#333×C₁₂₇	C₂
socle, prime radical	C₂²×C₁₂₇	#120
hypercenter	#333	D₁₂₇
hypocenter	C₁₂₇	#1633
LCS	G ▷ C₄²×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

#16861179

commutator, cosocle

C₄²×C₁₂₇

C₂³

Frattini

C₄²

C₂²×D₁₂₇

Fitting

#333×C₁₂₇

C₂

C₂²×C₁₂₇

D₁₂₇

C₁₂₇

G ▷ C₄²×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
35. Dic₂₅₄	Q₁₆	semidirect
56. C₁₂₇	#1633	semidirect

H ≅ G/N

type

35. Dic₂₅₄

Q₁₆

semidirect

56. C₁₂₇

#1633

semidirect

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

subgroup	index
#22550606	2
#22550255	2
#22550172	2
#22549606	2
#333×C₁₂₇	2
#1633	127

subgroup

index

#333×C₁₂₇

127

Subgroups — #30637871

Conjugacy classes of subgroups

Complements

Maximal subgroups