Group #30639706 — subgroups

Subgroups — #30639706

Order 16256.

284 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 97 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	#22550433	C₂	1	1	yes	maximal, characteristic
2	C₄²×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
3	#22549637	C₂	1	4	yes	maximal
4	#423×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	#16861209	C₄	1	2	yes	complemented
6	#16861201	C₄	1	2	yes	complemented
7	#16861188	—	2	2	no	—
8	C₄×Dic₂₅₄	—	2	2	no	—
9	#16861168	C₂²	1	2	yes	—
10	C₂×C₄×Dic₁₂₇	—	2	2	no	—
11	C₂×C₄×Dic₁₂₇	C₂²	1	2	yes	—
12	C₂×M₁₆×C₁₂₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₁₂₇	—	2	2	no	—
15	C₂×Dic₂₅₄	—	2	2	no	—
16	C₂²×Dic₁₂₇	—	2	2	no	—
17	C₄⋊Dic₁₂₇	—	2	2	no	—
18	C₄⋊Dic₁₂₇	C₂×C₄	1	4	yes	—
19	Dic₁₂₇⋊C₄	—	2	4	no	—
20	C₂²⋊Dic₁₂₇	—	2	2	no	—
21	C₄×Dic₁₂₇	—	2	8	no	—
22	C₄×Dic₁₂₇	—	2	2	no	—
23	C₄×Dic₁₂₇	—	2	2	no	—
24	C₄×Dic₁₂₇	Q₈	1	4	yes	complemented
25	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
26	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
27	M₁₆×C₁₂₇	—	2	4	no	—
28	C₂×C₈×C₁₂₇	—	2	2	no	—
29	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₁₂₇	D₄	1	2	yes	—
32	Dic₂₅₄	—	4	4	no	—
33	C₂×Dic₁₂₇	—	2	8	no	—
34	C₂×Dic₁₂₇	—	2	4	no	—
35	C₂×Dic₁₂₇	—	2	4	no	—
36	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₂₇	—	2	4	no	—
38	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
43	C₂×C₄×C₁₂₇	#41	1	2	yes	—
44	C₈×C₁₂₇	—	4	2	no	—
45	C₂²×C₁₂₇	—	2	2	no	—
46	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₁₂₇	C₂²⋊Q₈	1	2	yes	—
48	C₄×C₁₂₇	—	2	2	no	—
49	C₄×C₁₂₇	#104	1	4	yes	—
50	C₄×C₁₂₇	C₄×Q₈	1	2	yes	—
51	C₄×C₁₂₇	#124	1	1	yes	characteristic
52	C₄×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
53	Dic₁₂₇	—	2	8	no	—
54	Dic₁₂₇	—	4	4	no	—
55	C₂×C₁₂₇	—	2	2	no	—
56	C₂×C₁₂₇	#391	1	1	yes	characteristic
57	C₂×C₁₂₇	C₂×#104	1	2	yes	—
58	#1994	—	127	1	no	maximal, Sylow
59	C₁₂₇	#1994	1	1	yes	characteristic, complemented, Sylow
60	#533	—	127	1	no	—
61	#423	D₁₂₇	1	1	yes	characteristic
62	C₄³	—	127	1	no	—
63	#339	—	127	4	no	—
64	C₂×M₁₆	C₂×D₁₂₇	1	2	yes	—
65	C₄⋊Q₈	—	127	2	no	—
66	#125	—	127	2	no	—
67	C₂²⋊Q₈	—	254	2	no	—
68	C₄×Q₈	—	254	2	no	—
69	#117	—	127	2	no	—
70	C₂×C₄²	—	127	2	no	—
71	C₂×C₄²	—	254	2	no	—
72	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
73	C₄⋊C₈	—	2	2	no	—
74	C₂×Q₈	—	254	2	no	—
75	C₂²×C₄	—	254	2	no	—
76	C₂²×C₄	D₅₀₈	1	2	yes	—
77	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
78	M₁₆	—	2	4	no	—
79	C₂×C₈	—	2	2	no	—
80	C₄⋊C₄	—	127	4	no	—
81	C₄⋊C₄	—	254	4	no	—
82	C₄⋊C₄	—	254	2	no	—
83	C₂²⋊C₄	—	254	2	no	—
84	C₄²	—	127	4	no	—
85	C₄²	—	254	8	no	—
86	C₄²	—	254	2	no	—
87	C₄²	—	254	2	no	—
88	C₄²	C₄×D₁₂₇	1	1	yes	characteristic
89	C₄²	C₄×D₁₂₇	1	1	yes	characteristic
90	C₄²	C₂²⋊D₁₂₇	1	2	yes	—
91	C₂³	C₂×D₅₀₈	1	1	yes	characteristic
92	Q₈	—	508	4	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	254	4	no	—
95	C₂×C₄	—	254	8	no	—
96	C₂×C₄	—	254	4	no	—
97	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
98	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
99	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
100	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
101	C₂×C₄	Q₈×D₁₂₇	1	2	yes	—
102	C₂×C₄	#14523247	1	2	yes	—
103	C₈	—	4	2	no	—
104	C₂²	—	2	2	no	—
105	C₂²	C₂×(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
106	C₂²	#16861189	1	2	yes	—
107	C₄	—	2	2	no	—
108	C₄	—	508	4	no	—
109	C₄	—	254	8	no	—
110	C₄	#16861135	1	4	yes	—
111	C₄	#16861174	1	2	yes	—
112	C₄	#16861196	1	1	yes	characteristic
113	C₄	#16861207	1	1	yes	characteristic
114	C₂	—	2	2	no	—
115	C₂	#22549781	1	1	yes	characteristic
116	C₂	C₂×#16861135	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22550433

C₂

yes

maximal, characteristic

C₄²×Dic₁₂₇

C₂

yes

maximal, characteristic

#22549637

C₂

yes

maximal

#423×C₁₂₇

C₂

yes

maximal, characteristic

#16861209

C₄

yes

complemented

#16861201

C₄

yes

complemented

#16861188

—

C₄×Dic₂₅₄

—

#16861168

C₂²

yes

—

C₂×C₄×Dic₁₂₇

—

C₂×C₄×Dic₁₂₇

C₂²

yes

—

C₂×M₁₆×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₂₇

—

C₂×Dic₂₅₄

—

C₂²×Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

C₂×C₄

yes

—

Dic₁₂₇⋊C₄

—

C₂²⋊Dic₁₂₇

—

C₄×Dic₁₂₇

—

C₄×Dic₁₂₇

—

C₄×Dic₁₂₇

—

C₄×Dic₁₂₇

Q₈

yes

complemented

C₂²×C₄×C₁₂₇

D₄

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

M₁₆×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₄²×C₁₂₇

C₂×C₄

yes

characteristic

C₄²×C₁₂₇

C₂×C₄

yes

characteristic

C₄²×C₁₂₇

D₄

yes

—

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂³×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₂₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×Q₈

yes

—

C₂×C₄×C₁₂₇

#41

yes

—

C₈×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₂₇

C₂²⋊Q₈

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#104

yes

—

C₄×C₁₂₇

C₄×Q₈

yes

—

C₄×C₁₂₇

#124

yes

characteristic

C₄×C₁₂₇

C₄⋊Q₈

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#391

yes

characteristic

C₂×C₁₂₇

C₂×#104

yes

—

#1994

—

127

maximal, Sylow

C₁₂₇

#1994

yes

characteristic, complemented, Sylow

#533

—

127

—

#423

D₁₂₇

yes

characteristic

C₄³

—

127

—

#339

—

127

—

C₂×M₁₆

C₂×D₁₂₇

yes

—

C₄⋊Q₈

—

127

—

#125

—

127

—

C₂²⋊Q₈

—

254

—

C₄×Q₈

—

254

—

#117

—

127

—

C₂×C₄²

—

127

—

C₂×C₄²

—

254

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

C₄⋊C₈

—

C₂×Q₈

—

254

—

C₂²×C₄

—

254

—

C₂²×C₄

D₅₀₈

yes

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₂²⋊C₄

—

254

—

C₄²

—

127

—

C₄²

—

254

—

C₄²

—

254

—

C₄²

—

254

—

C₄²

C₄×D₁₂₇

yes

characteristic

C₄²

C₄×D₁₂₇

yes

characteristic

C₄²

C₂²⋊D₁₂₇

yes

—

C₂³

C₂×D₅₀₈

yes

characteristic

Q₈

—

508

—

C₂×C₄

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

(C₂×D₁₂₇)⋊C₄

yes

—

C₂×C₄

(C₂×D₁₂₇)⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₁₂₇

yes

characteristic

100

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

101

C₂×C₄

Q₈×D₁₂₇

yes

—

102

C₂×C₄

#14523247

yes

—

103

C₈

—

104

C₂²

—

105

C₂²

C₂×(C₂×D₁₂₇)⋊C₄

yes

characteristic

106

C₂²

#16861189

yes

—

107

C₄

—

108

C₄

—

508

—

109

C₄

—

254

—

110

C₄

#16861135

yes

—

111

C₄

#16861174

yes

—

112

C₄

#16861196

yes

characteristic

113

C₄

#16861207

yes

characteristic

114

C₂

—

115

C₂

#22549781

yes

characteristic

116

C₂

C₂×#16861135

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₂₇
commutator	C₂×C₄×C₁₂₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	#423×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×C₂²⋊C₄
prime radical	C₂³×C₁₂₇	C₂×D₄
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	#423	D₁₂₇
hypocenter	C₁₂₇	#1994
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₂₇

commutator

C₂×C₄×C₁₂₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

#423×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂×C₂²⋊C₄

prime radical

C₂³×C₁₂₇

C₂×D₄

cosocle

C₂²×C₄×C₁₂₇

C₂³

D₁₂₇

C₁₂₇

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
5. #16861209	C₄	semidirect
6. #16861201	C₄	semidirect
24. C₄×Dic₁₂₇	Q₈	semidirect
59. C₁₂₇	#1994	semidirect

H ≅ G/N

type

5. #16861209

C₄

semidirect

6. #16861201

C₄

semidirect

24. C₄×Dic₁₂₇

Q₈

semidirect

59. C₁₂₇

#1994

semidirect

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

subgroup	index
#22550433	2
C₄²×Dic₁₂₇	2
#22549637	2
#423×C₁₂₇	2
#1994	127

subgroup

index

#22550433

C₄²×Dic₁₂₇

#22549637

#423×C₁₂₇

#1994

127

Subgroups — #30639706

Conjugacy classes of subgroups

Complements

Maximal subgroups