Group #30652305 — subgroups

Subgroups — C₂×#22550606 #30652305

Order 16256.

1112 conjugacy classes of subgroups; only the 581 normal subgroups are shown.

normal subgroups only

This group has 1112 conjugacy classes of subgroups — too many to draw. Only its 581 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	16	yes	maximal, direct product factor
2	C₂×#16861208	C₂	1	4	yes	maximal
3	C₂×#16861207	C₂	1	4	yes	maximal
4	C₂×C₄×Dic₂₅₄	C₂	1	2	yes	maximal
5	C₂×#16861174	C₂	1	4	yes	maximal
6	C₂×C₄⋊Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₅₄	C₂²	1	4	yes	—
8	C₂²×Dic₂₅₄	C₂²	1	2	yes	—
9	#16861208	C₂²	1	32	yes	—
10	#16861207	C₂²	1	32	yes	—
11	C₄×Dic₂₅₄	C₂²	1	16	yes	—
12	#16861174	C₂²	1	32	yes	—
13	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
14	C₂×Dic₁₂₇⋊C₄	C₂²	1	8	yes	—
15	C₂×Dic₁₂₇⋊C₄	C₂²	1	4	yes	—
16	C₂×C₄×Dic₁₂₇	C₂²	1	4	yes	—
17	C₂×C₄×Dic₁₂₇	C₂²	1	4	yes	—
18	C₂²×Q₈×C₁₂₇	C₂²	1	2	yes	—
19	C₄⋊Q₈×C₁₂₇	C₂²	1	8	yes	—
20	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	4	yes	—
21	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
22	C₂×Dic₂₅₄	—	4	8	no	—
23	C₂×Dic₂₅₄	Q₈	1	16	yes	complemented
24	C₂×Dic₂₅₄	C₂³	1	16	yes	—
25	C₂×Dic₂₅₄	C₂³	1	8	yes	—
26	C₂²×Dic₁₂₇	C₂³	1	4	yes	—
27	C₂²×Dic₁₂₇	C₂³	1	4	yes	—
28	C₄⋊Dic₁₂₇	C₂³	1	8	yes	—
29	Dic₁₂₇⋊C₄	C₂³	1	16	yes	—
30	Dic₁₂₇⋊C₄	C₂³	1	32	yes	—
31	C₄×Dic₁₂₇	C₂³	1	16	yes	—
32	C₄×Dic₁₂₇	C₂³	1	16	yes	—
33	C₂×Q₈×C₁₂₇	—	4	4	no	—
34	C₂×Q₈×C₁₂₇	C₂³	1	8	yes	—
35	C₂²×C₄×C₁₂₇	C₂³	1	4	yes	—
36	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
38	C₄⋊C₄×C₁₂₇	C₂³	1	16	yes	—
39	C₄²×C₁₂₇	C₂³	1	4	yes	—
40	Dic₂₅₄	—	4	16	no	—
41	Dic₂₅₄	C₂×Q₈	1	32	yes	complemented
42	C₂×Dic₁₂₇	—	2	8	no	—
43	C₂×Dic₁₂₇	C₂×Q₈	1	16	yes	—
44	C₂×Dic₁₂₇	C₂⁴	1	8	yes	—
45	C₂×Dic₁₂₇	C₂⁴	1	8	yes	—
46	C₂³×C₁₂₇	C₂⁴	1	1	yes	characteristic
47	Q₈×C₁₂₇	—	4	8	no	—
48	C₂×C₄×C₁₂₇	—	2	2	no	—
49	C₂×C₄×C₁₂₇	—	2	8	no	—
50	C₂×C₄×C₁₂₇	C₂×Q₈	1	8	yes	—
51	C₂×C₄×C₁₂₇	C₂⁴	1	2	yes	—
52	C₂×C₄×C₁₂₇	C₂⁴	1	8	yes	—
53	C₂×C₄×C₁₂₇	C₂⁴	1	4	yes	—
54	C₂²×C₁₂₇	C₂²×Q₈	1	4	yes	—
55	C₂²×C₁₂₇	H(5,𝔽₂)	1	2	yes	—
56	C₂²×C₁₂₇	C₂⁵	1	1	yes	characteristic
57	C₄×C₁₂₇	—	2	2	no	—
58	C₄×C₁₂₇	—	2	8	no	—
59	C₄×C₁₂₇	C₂²×Q₈	1	8	yes	—
60	Dic₁₂₇	—	2	8	no	—
61	Dic₁₂₇	C₂²×Q₈	1	16	yes	—
62	C₂×C₁₂₇	Q₈²	1	4	yes	complemented
63	C₂×C₁₂₇	C₂³×Q₈	1	2	yes	—
64	C₂×C₁₂₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
65	C₂×Q₈²	—	127	1	no	maximal, Sylow
66	C₁₂₇	C₂×Q₈²	1	1	yes	characteristic, complemented, Sylow
67	Q₈²	—	127	16	no	—
68	C₂×C₄⋊Q₈	—	127	4	no	—
69	C₂×C₄⋊Q₈	—	127	4	no	—
70	C₂×C₄⋊Q₈	D₁₂₇	1	1	yes	characteristic
71	C₂×C₄×Q₈	—	127	4	no	—
72	C₂×C₄×Q₈	—	127	2	no	—
73	C₂²×Q₈	—	127	2	no	—
74	C₂²×Q₈	—	127	4	no	—
75	C₂²×Q₈	C₂×D₁₂₇	1	2	yes	—
76	C₄⋊Q₈	—	127	32	no	—
77	C₄⋊Q₈	—	127	32	no	—
78	C₄⋊Q₈	C₂×D₁₂₇	1	8	yes	—
79	C₄×Q₈	—	127	32	no	—
80	C₄×Q₈	—	127	16	no	—
81	C₂×C₄⋊C₄	—	127	8	no	—
82	C₂×C₄⋊C₄	—	127	4	no	—
83	C₂×C₄⋊C₄	—	127	2	no	—
84	C₂×C₄⋊C₄	C₂×D₁₂₇	1	4	yes	—
85	C₂×C₄²	—	127	4	no	—
86	C₂×C₄²	—	127	4	no	—
87	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
88	C₂×Q₈	—	4	4	no	—
89	C₂×Q₈	—	127	16	no	—
90	C₂×Q₈	—	127	8	no	—
91	C₂×Q₈	—	127	16	no	—
92	C₂×Q₈	—	508	8	no	—
93	C₂×Q₈	C₂²×D₁₂₇	1	8	yes	—
94	C₂²×C₄	—	127	4	no	—
95	C₂²×C₄	—	127	4	no	—
96	C₂²×C₄	C₂²×D₁₂₇	1	4	yes	—
97	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₁₂₇	1	2	yes	—
99	C₄⋊C₄	—	127	32	no	—
100	C₄⋊C₄	—	127	16	no	—
101	C₄⋊C₄	—	127	8	no	—
102	C₄⋊C₄	C₂²×D₁₂₇	1	16	yes	—
103	C₄²	—	127	16	no	—
104	C₄²	—	127	16	no	—
105	C₄²	C₂²×D₁₂₇	1	4	yes	—
106	C₂³	C₂³×D₁₂₇	1	1	yes	characteristic
107	Q₈	—	4	8	no	—
108	Q₈	—	127	32	no	—
109	Q₈	—	508	16	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	8	no	—
112	C₂×C₄	—	127	16	no	—
113	C₂×C₄	—	127	8	no	—
114	C₂×C₄	—	127	8	no	—
115	C₂×C₄	—	254	8	no	—
116	C₂×C₄	Q₈×D₁₂₇	1	8	yes	—
117	C₂×C₄	C₂³×D₁₂₇	1	8	yes	—
118	C₂×C₄	C₂³×D₁₂₇	1	2	yes	—
119	C₂×C₄	C₂³×D₁₂₇	1	4	yes	—
120	C₂²	C₂×Q₈×D₁₂₇	1	4	yes	—
121	C₂²	#16861254	1	2	yes	—
122	C₂²	C₂⁴×D₁₂₇	1	1	yes	characteristic
123	C₄	—	2	2	no	—
124	C₄	—	2	8	no	—
125	C₄	—	127	16	no	—
126	C₄	—	254	8	no	—
127	C₄	C₂×Q₈×D₁₂₇	1	8	yes	—
128	C₂	#22550606	1	4	yes	direct product factor
129	C₂	C₂²×Q₈×D₁₂₇	1	2	yes	—
130	C₂	C₂×#16861254	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22550606

C₂

yes

maximal, direct product factor

C₂×#16861208

C₂

yes

maximal

C₂×#16861207

C₂

yes

maximal

C₂×C₄×Dic₂₅₄

C₂

yes

maximal

C₂×#16861174

C₂

yes

maximal

C₂×C₄⋊Q₈×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×Dic₂₅₄

C₂²

yes

—

C₂²×Dic₂₅₄

C₂²

yes

—

#16861208

C₂²

yes

—

#16861207

C₂²

yes

—

C₄×Dic₂₅₄

C₂²

yes

—

#16861174

C₂²

yes

—

C₂×C₄⋊Dic₁₂₇

C₂²

yes

—

C₂×Dic₁₂₇⋊C₄

C₂²

yes

—

C₂×Dic₁₂₇⋊C₄

C₂²

yes

—

C₂×C₄×Dic₁₂₇

C₂²

yes

—

C₂×C₄×Dic₁₂₇

C₂²

yes

—

C₂²×Q₈×C₁₂₇

C₂²

yes

—

C₄⋊Q₈×C₁₂₇

C₂²

yes

—

C₂×C₄⋊C₄×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

Q₈

yes

complemented

C₂×Dic₂₅₄

C₂³

yes

—

C₂×Dic₂₅₄

C₂³

yes

—

C₂²×Dic₁₂₇

C₂³

yes

—

C₂²×Dic₁₂₇

C₂³

yes

—

C₄⋊Dic₁₂₇

C₂³

yes

—

Dic₁₂₇⋊C₄

C₂³

yes

—

Dic₁₂₇⋊C₄

C₂³

yes

—

C₄×Dic₁₂₇

C₂³

yes

—

C₄×Dic₁₂₇

C₂³

yes

—

C₂×Q₈×C₁₂₇

—

C₂×Q₈×C₁₂₇

C₂³

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₂²×C₄×C₁₂₇

C₂³

yes

—

C₄⋊C₄×C₁₂₇

C₂³

yes

—

C₄²×C₁₂₇

C₂³

yes

—

Dic₂₅₄

—

Dic₂₅₄

C₂×Q₈

yes

complemented

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

C₂×Q₈

yes

—

C₂×Dic₁₂₇

C₂⁴

yes

—

C₂×Dic₁₂₇

C₂⁴

yes

—

C₂³×C₁₂₇

C₂⁴

yes

characteristic

Q₈×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×Q₈

yes

—

C₂×C₄×C₁₂₇

C₂⁴

yes

—

C₂×C₄×C₁₂₇

C₂⁴

yes

—

C₂×C₄×C₁₂₇

C₂⁴

yes

—

C₂²×C₁₂₇

C₂²×Q₈

yes

—

C₂²×C₁₂₇

H(5,𝔽₂)

yes

—

C₂²×C₁₂₇

C₂⁵

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂²×Q₈

yes

—

Dic₁₂₇

—

Dic₁₂₇

C₂²×Q₈

yes

—

C₂×C₁₂₇

Q₈²

yes

complemented

C₂×C₁₂₇

C₂³×Q₈

yes

—

C₂×C₁₂₇

C₂×H(5,𝔽₂)

yes

characteristic

C₂×Q₈²

—

127

maximal, Sylow

C₁₂₇

C₂×Q₈²

yes

characteristic, complemented, Sylow

Q₈²

—

127

—

C₂×C₄⋊Q₈

—

127

—

C₂×C₄⋊Q₈

—

127

—

C₂×C₄⋊Q₈

D₁₂₇

yes

characteristic

C₂×C₄×Q₈

—

127

—

C₂×C₄×Q₈

—

127

—

C₂²×Q₈

—

127

—

C₂²×Q₈

—

127

—

C₂²×Q₈

C₂×D₁₂₇

yes

—

C₄⋊Q₈

—

127

—

C₄⋊Q₈

—

127

—

C₄⋊Q₈

C₂×D₁₂₇

yes

—

C₄×Q₈

—

127

—

C₄×Q₈

—

127

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

C₂×D₁₂₇

yes

—

C₂×C₄²

—

127

—

C₂×C₄²

—

127

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

127

—

C₂×Q₈

—

127

—

C₂×Q₈

—

127

—

C₂×Q₈

—

508

—

C₂×Q₈

C₂²×D₁₂₇

yes

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

C₂²×D₁₂₇

yes

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

C₂²×C₄

C₂²×D₁₂₇

yes

—

C₄⋊C₄

—

127

—

100

C₄⋊C₄

—

127

—

101

C₄⋊C₄

—

127

—

102

C₄⋊C₄

C₂²×D₁₂₇

yes

—

103

C₄²

—

127

—

104

C₄²

—

127

—

105

C₄²

C₂²×D₁₂₇

yes

—

106

C₂³

C₂³×D₁₂₇

yes

characteristic

107

Q₈

—

108

Q₈

—

127

—

109

Q₈

—

508

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

127

—

113

C₂×C₄

—

127

—

114

C₂×C₄

—

127

—

115

C₂×C₄

—

254

—

116

C₂×C₄

Q₈×D₁₂₇

yes

—

117

C₂×C₄

C₂³×D₁₂₇

yes

—

118

C₂×C₄

C₂³×D₁₂₇

yes

—

119

C₂×C₄

C₂³×D₁₂₇

yes

—

120

C₂²

C₂×Q₈×D₁₂₇

yes

—

121

C₂²

#16861254

yes

—

122

C₂²

C₂⁴×D₁₂₇

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

127

—

126

C₄

—

254

—

127

C₄

C₂×Q₈×D₁₂₇

yes

—

128

C₂

#22550606

yes

direct product factor

129

C₂

C₂²×Q₈×D₁₂₇

yes

—

130

C₂

C₂×#16861254

yes

characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₂₇
commutator, cosocle	C₂²×C₁₂₇	C₂⁵
Frattini	C₂²	C₂⁴×D₁₂₇
Fitting	C₂×C₄⋊Q₈×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	C₂⁴
hypercenter	C₂×C₄⋊Q₈	D₁₂₇
hypocenter	C₁₂₇	C₂×Q₈²
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂³×D₁₂₇

commutator, cosocle

C₂²×C₁₂₇

C₂⁵

Frattini

C₂²

C₂⁴×D₁₂₇

Fitting

C₂×C₄⋊Q₈×C₁₂₇

C₂

socle, prime radical

C₂³×C₁₂₇

C₂⁴

hypercenter

C₂×C₄⋊Q₈

D₁₂₇

hypocenter

C₁₂₇

C₂×Q₈²

LCS

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
1. #22550606	C₂	direct (N × H)
23. C₂×Dic₂₅₄	Q₈	semidirect
41. Dic₂₅₄	C₂×Q₈	semidirect
62. C₂×C₁₂₇	Q₈²	semidirect
66. C₁₂₇	C₂×Q₈²	semidirect
128. C₂	#22550606	direct (N × H)

H ≅ G/N

type

1. #22550606

C₂

direct (N × H)

23. C₂×Dic₂₅₄

Q₈

semidirect

41. Dic₂₅₄

C₂×Q₈

semidirect

62. C₂×C₁₂₇

Q₈²

semidirect

66. C₁₂₇

C₂×Q₈²

semidirect

128. C₂

#22550606

direct (N × H)

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

subgroup	index
#22550606	2
C₂×#16861208	2
C₂×#16861207	2
C₂×C₄×Dic₂₅₄	2
C₂×#16861174	2
C₂×C₄⋊Q₈×C₁₂₇	2
C₂×Q₈²	127

subgroup

index

C₂×C₄×Dic₂₅₄

C₂×C₄⋊Q₈×C₁₂₇

C₂×Q₈²

127

Subgroups — C₂×#22550606 #30652305

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#22550606 #30652305

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22550606 #30652305