Group #30649069 — subgroups

Subgroups — C₂×#22550244 #30649069

Order 16256.

384 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16861201	C₂	1	1	yes	maximal, characteristic
2	#22550244	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16861133	C₂	1	4	yes	maximal
4	C₂×#16861120	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
7	#16861201	—	2	2	no	—
8	#16861201	C₂²	1	4	yes	—
9	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
10	C₂×Dic₁₂₇⋊C₄	—	2	2	no	—
11	#16861133	C₂²	1	16	yes	—
12	#16861120	C₂²	1	4	yes	—
13	C₂²×Q₈×C₁₂₇	C₂²	1	2	yes	—
14	C₄⋊Q₈×C₁₂₇	—	2	2	no	—
15	C₄⋊Q₈×C₁₂₇	C₂²	1	4	yes	—
16	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
17	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
18	C₂²×Dic₁₂₇	—	2	2	no	—
19	C₂×C₁₂₇⋊C₈	—	2	4	no	—
20	C₂×C₁₂₇⋊C₈	C₂³	1	4	yes	—
21	C₄⋊Dic₁₂₇	—	2	2	no	—
22	C₄⋊Dic₁₂₇	C₂³	1	4	yes	—
23	Dic₁₂₇⋊C₄	—	2	8	no	—
24	C₂×Q₈×C₁₂₇	—	2	2	no	—
25	C₂×Q₈×C₁₂₇	—	2	8	no	—
26	C₂×Q₈×C₁₂₇	C₂³	1	4	yes	—
27	C₂²×C₄×C₁₂₇	—	2	2	no	—
28	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
29	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₁₂₇	—	2	8	no	—
31	C₄²×C₁₂₇	D₄	1	2	yes	—
32	C₄²×C₁₂₇	C₂³	1	2	yes	—
33	C₂×Dic₁₂₇	—	2	4	no	—
34	C₂×Dic₁₂₇	—	4	4	no	—
35	C₁₂₇⋊C₈	—	2	4	no	—
36	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
37	Q₈×C₁₂₇	—	2	8	no	—
38	Q₈×C₁₂₇	—	4	4	no	—
39	C₂×C₄×C₁₂₇	—	2	4	no	—
40	C₂×C₄×C₁₂₇	—	2	4	no	—
41	C₂×C₄×C₁₂₇	—	4	4	no	—
42	C₂×C₄×C₁₂₇	C₂×D₄	1	4	yes	—
43	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₂₇	#41	1	4	yes	—
45	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
46	C₂²×C₁₂₇	#124	1	2	yes	—
47	C₂²×C₁₂₇	#137	1	4	yes	—
48	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
49	C₄×C₁₂₇	—	2	4	no	—
50	C₄×C₁₂₇	—	4	4	no	—
51	C₄×C₁₂₇	#124	1	2	yes	—
52	C₄×C₁₂₇	C₂×#41	1	2	yes	—
53	Dic₁₂₇	—	4	4	no	—
54	C₂×C₁₂₇	#491	1	4	yes	complemented
55	C₂×C₁₂₇	C₂×#124	1	1	yes	characteristic
56	C₂×C₁₂₇	C₂×#137	1	2	yes	—
57	C₂×#491	—	127	1	no	maximal, Sylow
58	C₁₂₇	C₂×#491	1	1	yes	characteristic, complemented, Sylow
59	C₂×C₄⋊Q₈	D₁₂₇	1	1	yes	characteristic
60	C₂×#125	—	127	1	no	—
61	#491	—	127	8	no	—
62	C₂×#103	—	127	4	no	—
63	C₂×#97	—	127	1	no	—
64	C₂²×Q₈	C₂×D₁₂₇	1	2	yes	—
65	C₂²×C₈	—	127	2	no	—
66	C₄⋊Q₈	—	2	2	no	—
67	C₄⋊Q₈	C₂×D₁₂₇	1	4	yes	—
68	#125	—	127	4	no	—
69	#125	—	254	2	no	—
70	C₂×C₄⋊C₄	—	2	2	no	—
71	C₂×C₄⋊C₄	—	127	2	no	—
72	C₂×C₄⋊C₄	—	254	2	no	—
73	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
74	#103	—	127	16	no	—
75	#97	—	127	4	no	—
76	C₂×Q₈	—	2	2	no	—
77	C₂×Q₈	—	2	8	no	—
78	C₂×Q₈	C₂²×D₁₂₇	1	4	yes	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	—	254	2	no	—
81	C₂²×C₄	C₂²⋊D₁₂₇	1	2	yes	—
82	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
83	C₂×C₈	—	127	4	no	—
84	C₂×C₈	—	254	4	no	—
85	C₄⋊C₄	—	2	8	no	—
86	C₄⋊C₄	—	127	4	no	—
87	C₄⋊C₄	—	254	8	no	—
88	C₄⋊C₄	—	254	2	no	—
89	C₄²	C₂²⋊D₁₂₇	1	2	yes	—
90	C₄²	C₂²×D₁₂₇	1	2	yes	—
91	C₂³	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
92	Q₈	—	2	8	no	—
93	Q₈	—	4	4	no	—
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	4	4	no	—
97	C₂×C₄	—	508	4	no	—
98	C₂×C₄	—	254	4	no	—
99	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	4	yes	—
100	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
101	C₂×C₄	#14523247	1	4	yes	—
102	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
103	C₈	—	254	4	no	—
104	C₂²	#16861196	1	2	yes	—
105	C₂²	#16861240	1	4	yes	—
106	C₂²	C₂²×C₂²⋊D₁₂₇	1	1	yes	characteristic
107	C₄	—	2	4	no	—
108	C₄	—	4	4	no	—
109	C₄	—	508	4	no	—
110	C₄	#16861196	1	2	yes	—
111	C₄	C₂×#14523247	1	2	yes	—
112	C₂	#22550244	1	4	yes	direct product factor
113	C₂	C₂×#16861196	1	1	yes	characteristic
114	C₂	C₂×#16861240	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861201

C₂

yes

maximal, characteristic

#22550244

C₂

yes

maximal, direct product factor

C₂×#16861133

C₂

yes

maximal

C₂×#16861120

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×C₁₂₇⋊C₈

C₂²

yes

—

#16861201

—

#16861201

C₂²

yes

—

C₂×C₄⋊Dic₁₂₇

C₂²

yes

—

C₂×Dic₁₂₇⋊C₄

—

#16861133

C₂²

yes

—

#16861120

C₂²

yes

—

C₂²×Q₈×C₁₂₇

C₂²

yes

—

C₄⋊Q₈×C₁₂₇

—

C₄⋊Q₈×C₁₂₇

C₂²

yes

—

C₂×C₄⋊C₄×C₁₂₇

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

C₂²×Dic₁₂₇

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

C₂³

yes

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

C₂³

yes

—

Dic₁₂₇⋊C₄

—

C₂×Q₈×C₁₂₇

—

C₂×Q₈×C₁₂₇

—

C₂×Q₈×C₁₂₇

C₂³

yes

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

D₄

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

D₄

yes

—

C₄²×C₁₂₇

C₂³

yes

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

C₂×D₄

yes

characteristic

Q₈×C₁₂₇

—

Q₈×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

#41

yes

—

C₂×C₄×C₁₂₇

C₂⁴

yes

characteristic

C₂²×C₁₂₇

#124

yes

—

C₂²×C₁₂₇

#137

yes

—

C₂²×C₁₂₇

C₂²×D₄

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#124

yes

—

C₄×C₁₂₇

C₂×#41

yes

—

Dic₁₂₇

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

C₂×#491

—

127

maximal, Sylow

C₁₂₇

C₂×#491

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

D₁₂₇

yes

characteristic

C₂×#125

—

127

—

#491

—

127

—

C₂×#103

—

127

—

C₂×#97

—

127

—

C₂²×Q₈

C₂×D₁₂₇

yes

—

C₂²×C₈

—

127

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₁₂₇

yes

—

#125

—

127

—

#125

—

254

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

—

254

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

#103

—

127

—

#97

—

127

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²×D₁₂₇

yes

—

C₂²×C₄

—

C₂²×C₄

—

254

—

C₂²×C₄

C₂²⋊D₁₂₇

yes

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

C₂×C₈

—

127

—

C₂×C₈

—

254

—

C₄⋊C₄

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₄²

C₂²⋊D₁₂₇

yes

—

C₄²

C₂²×D₁₂₇

yes

—

C₂³

C₂×C₂²⋊D₁₂₇

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

508

—

C₂×C₄

—

254

—

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

—

100

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

101

C₂×C₄

#14523247

yes

—

102

C₂×C₄

C₂³×D₁₂₇

yes

characteristic

103

C₈

—

254

—

104

C₂²

#16861196

yes

—

105

C₂²

#16861240

yes

—

106

C₂²

C₂²×C₂²⋊D₁₂₇

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

508

—

110

C₄

#16861196

yes

—

111

C₄

C₂×#14523247

yes

—

112

C₂

#22550244

yes

direct product factor

113

C₂

C₂×#16861196

yes

characteristic

114

C₂

C₂×#16861240

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₂₇
commutator, cosocle	C₂×C₄×C₁₂₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	C₂×C₄⋊Q₈×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₁₂₇
hypocenter	C₁₂₇	C₂×#491
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₂₇

commutator, cosocle

C₂×C₄×C₁₂₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₂₇

Fitting

C₂×C₄⋊Q₈×C₁₂₇

C₂

socle, prime radical

C₂³×C₁₂₇

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₁₂₇

hypocenter

C₁₂₇

C₂×#491

LCS

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. #22550244	C₂	direct (N × H)
54. C₂×C₁₂₇	#491	semidirect
58. C₁₂₇	C₂×#491	semidirect
112. C₂	#22550244	direct (N × H)

H ≅ G/N

type

2. #22550244

C₂

direct (N × H)

54. C₂×C₁₂₇

#491

semidirect

58. C₁₂₇

C₂×#491

semidirect

112. C₂

#22550244

direct (N × H)

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

subgroup	index
C₂×#16861201	2
#22550244	2
C₂×#16861133	2
C₂×#16861120	2
C₂×C₄⋊Q₈×C₁₂₇	2
C₂×#491	127

subgroup

index

C₂×C₄⋊Q₈×C₁₂₇

127

Subgroups — C₂×#22550244 #30649069

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#22550244 #30649069

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22550244 #30649069