Group #22581583 — subgroups

Subgroups — #22581583

Order 8896.

206 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 103 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	#17984842	C₂	1	1	yes	maximal, characteristic, complemented
2	#17984841	C₂	1	1	yes	maximal, characteristic, complemented
3	#17984841	C₂	1	2	yes	maximal, complemented
4	#17984833	C₂	1	1	yes	maximal, characteristic, complemented
5	#17984832	C₂	1	2	yes	maximal, complemented
6	#17984822	C₂	1	2	yes	maximal
7	#17984820	C₂	1	2	yes	maximal
8	#17984800	C₂	1	1	yes	maximal, characteristic
9	#17984799	C₂	1	1	yes	maximal, characteristic
10	C₄⋊C₄×D₁₃₉	C₂	1	1	yes	maximal, characteristic
11	#125×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Dic₂₇₈	C₂²	1	2	yes	—
13	C₂×Dic₂₇₈	C₂²	1	2	yes	—
14	C₂×C₄×D₁₃₉	C₂²	1	2	yes	—
15	C₂×C₄×D₁₃₉	C₂²	1	1	yes	characteristic
16	C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
17	C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
18	C₄⋊Dic₁₃₉	C₂²	1	2	yes	—
19	C₄⋊Dic₁₃₉	C₂²	1	2	yes	—
20	Dic₁₃₉⋊C₄	C₂²	1	2	yes	—
21	Dic₁₃₉⋊C₄	C₂²	1	1	yes	characteristic
22	Dic₁₃₉⋊C₄	C₂²	1	1	yes	characteristic
23	Dic₁₃₉⋊C₄	C₂²	1	2	yes	—
24	Dic₁₃₉⋊C₄	C₂²	1	2	yes	—
25	(C₂×D₁₃₉)⋊C₄	C₂²	1	1	yes	characteristic
26	(C₂×D₁₃₉)⋊C₄	C₂²	1	1	yes	characteristic
27	(C₂×D₁₃₉)⋊C₄	C₂²	1	2	yes	—
28	C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
29	C₄×Dic₁₃₉	C₂²	1	2	yes	—
30	C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
31	C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
32	C₄⋊C₄×C₁₃₉	C₂²	1	2	yes	—
33	C₄⋊C₄×C₁₃₉	C₂²	1	2	yes	—
34	C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
35	C₂²×D₁₃₉	C₂³	1	1	yes	characteristic
36	Dic₂₇₈	—	4	2	no	—
37	Dic₂₇₈	—	4	2	no	—
38	C₂×Dic₁₃₉	C₂³	1	1	yes	characteristic
39	C₂×Dic₁₃₉	C₂³	1	1	yes	characteristic
40	C₂×Dic₁₃₉	C₂³	1	2	yes	—
41	C₂×Dic₁₃₉	C₂³	1	2	yes	—
42	C₂×Dic₁₃₉	C₂³	1	1	yes	characteristic
43	C₄×D₁₃₉	—	4	2	no	—
44	C₄×D₁₃₉	Q₈	1	4	yes	complemented
45	C₂×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
46	C₂×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
47	C₂×C₄×C₁₃₉	C₂³	1	2	yes	—
48	C₂×C₄×C₁₃₉	C₂³	1	2	yes	—
49	C₂×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
50	C₂²×C₁₃₉	C₂⁴	1	1	yes	characteristic
51	C₂×D₁₃₉	—	2	1	no	—
52	C₂×D₁₃₉	—	2	1	no	—
53	C₂×D₁₃₉	C₂×Q₈	1	2	yes	—
54	C₄×C₁₃₉	—	2	2	no	—
55	C₄×C₁₃₉	—	2	1	no	—
56	C₄×C₁₃₉	—	2	1	no	—
57	C₄×C₁₃₉	—	2	2	no	—
58	C₄×C₁₃₉	C₂×Q₈	1	2	yes	—
59	Dic₁₃₉	—	2	2	no	—
60	Dic₁₃₉	—	2	1	no	—
61	Dic₁₃₉	—	2	2	no	—
62	Dic₁₃₉	—	2	1	no	—
63	Dic₁₃₉	C₂×Q₈	1	2	yes	—
64	C₂×C₁₃₉	C₂²×Q₈	1	1	yes	characteristic
65	C₂×C₁₃₉	H(5,𝔽₂)	1	1	yes	characteristic
66	C₂×C₁₃₉	2₋¹⁺⁴	1	1	yes	characteristic
67	D₁₃₉	—	2	2	no	—
68	C₁₃₉	#544	1	1	yes	characteristic, complemented, Sylow
69	#544	—	139	1	no	maximal, Sylow
70	C₄⋊Q₈	—	139	2	no	—
71	C₄⋊Q₈	—	139	1	no	—
72	C₄⋊Q₈	—	139	1	no	—
73	#125	—	139	2	no	—
74	#125	—	139	1	no	—
75	#125	D₁₃₉	1	1	yes	characteristic, complemented
76	C₂²⋊Q₈	—	139	2	no	—
77	C₂²⋊Q₈	—	139	2	no	—
78	#117	—	139	1	no	—
79	#117	—	139	1	no	—
80	C₂×C₄⋊C₄	—	139	1	no	—
81	C₂×Q₈	—	139	2	no	—
82	C₂×Q₈	—	139	2	no	—
83	C₂²×C₄	—	139	2	no	—
84	C₂²×C₄	—	139	1	no	—
85	C₄⋊C₄	—	139	2	no	—
86	C₄⋊C₄	—	139	1	no	—
87	C₄⋊C₄	—	139	1	no	—
88	C₄⋊C₄	—	139	1	no	—
89	C₄⋊C₄	—	139	2	no	—
90	C₄⋊C₄	—	139	2	no	—
91	C₄⋊C₄	—	139	1	no	—
92	C₄⋊C₄	—	139	2	no	—
93	C₄⋊C₄	—	139	2	no	—
94	C₄⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
95	C₄⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
96	C₄⋊C₄	C₂×D₁₃₉	1	2	yes	—
97	C₄⋊C₄	C₂×D₁₃₉	1	2	yes	—
98	C₂²⋊C₄	—	139	1	no	—
99	C₂²⋊C₄	—	139	2	no	—
100	C₂²⋊C₄	—	139	1	no	—
101	C₄²	—	139	1	no	—
102	C₄²	—	139	2	no	—
103	C₄²	C₂×D₁₃₉	1	1	yes	characteristic
104	C₂³	—	139	1	no	—
105	Q₈	—	556	2	no	—
106	Q₈	—	556	2	no	—
107	C₂×C₄	—	556	2	no	—
108	C₂×C₄	—	139	1	no	—
109	C₂×C₄	—	139	1	no	—
110	C₂×C₄	—	139	2	no	—
111	C₂×C₄	—	139	2	no	—
112	C₂×C₄	—	139	1	no	—
113	C₂×C₄	—	139	4	no	—
114	C₂×C₄	C₂²×D₁₃₉	1	2	yes	—
115	C₂×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
116	C₂×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
117	C₂×C₄	C₂²×D₁₃₉	1	2	yes	—
118	C₂×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
119	C₂²	—	278	1	no	—
120	C₂²	—	278	1	no	—
121	C₂²	—	139	2	no	—
122	C₂²	C₂³×D₁₃₉	1	1	yes	characteristic
123	C₄	—	2	2	no	—
124	C₄	—	2	1	no	—
125	C₄	—	2	2	no	—
126	C₄	—	2	1	no	—
127	C₄	—	278	2	no	—
128	C₄	—	278	1	no	—
129	C₄	—	278	2	no	—
130	C₄	—	278	1	no	—
131	C₄	—	139	2	no	—
132	C₄	Q₈×D₁₃₉	1	2	yes	—
133	C₂	—	278	2	no	—
134	C₂	C₂×Q₈×D₁₃₉	1	1	yes	characteristic
135	C₂	#17984888	1	1	yes	characteristic
136	C₂	#17984889	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17984842

C₂

yes

maximal, characteristic, complemented

#17984841

C₂

yes

maximal, characteristic, complemented

#17984841

C₂

yes

maximal, complemented

#17984833

C₂

yes

maximal, characteristic, complemented

#17984832

C₂

yes

maximal, complemented

#17984822

C₂

yes

maximal

#17984820

C₂

yes

maximal

#17984800

C₂

yes

maximal, characteristic

#17984799

C₂

yes

maximal, characteristic

C₄⋊C₄×D₁₃₉

C₂

yes

maximal, characteristic

#125×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₇₈

C₂²

yes

—

C₂×Dic₂₇₈

C₂²

yes

—

C₂×C₄×D₁₃₉

C₂²

yes

—

C₂×C₄×D₁₃₉

C₂²

yes

characteristic

C₄⋊Dic₁₃₉

C₂²

yes

characteristic

C₄⋊Dic₁₃₉

C₂²

yes

characteristic

C₄⋊Dic₁₃₉

C₂²

yes

—

C₄⋊Dic₁₃₉

C₂²

yes

—

Dic₁₃₉⋊C₄

C₂²

yes

—

Dic₁₃₉⋊C₄

C₂²

yes

characteristic

Dic₁₃₉⋊C₄

C₂²

yes

characteristic

Dic₁₃₉⋊C₄

C₂²

yes

—

Dic₁₃₉⋊C₄

C₂²

yes

—

(C₂×D₁₃₉)⋊C₄

C₂²

yes

characteristic

(C₂×D₁₃₉)⋊C₄

C₂²

yes

characteristic

(C₂×D₁₃₉)⋊C₄

C₂²

yes

—

C₄×Dic₁₃₉

C₂²

yes

characteristic

C₄×Dic₁₃₉

C₂²

yes

—

C₄⋊C₄×C₁₃₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₃₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₃₉

C₂²

yes

—

C₄⋊C₄×C₁₃₉

C₂²

yes

—

C₄²×C₁₃₉

C₂²

yes

characteristic

C₂²×D₁₃₉

C₂³

yes

characteristic

Dic₂₇₈

—

Dic₂₇₈

—

C₂×Dic₁₃₉

C₂³

yes

characteristic

C₂×Dic₁₃₉

C₂³

yes

characteristic

C₂×Dic₁₃₉

C₂³

yes

—

C₂×Dic₁₃₉

C₂³

yes

—

C₂×Dic₁₃₉

C₂³

yes

characteristic

C₄×D₁₃₉

—

C₄×D₁₃₉

Q₈

yes

complemented

C₂×C₄×C₁₃₉

C₂³

yes

characteristic

C₂×C₄×C₁₃₉

C₂³

yes

characteristic

C₂×C₄×C₁₃₉

C₂³

yes

—

C₂×C₄×C₁₃₉

C₂³

yes

—

C₂×C₄×C₁₃₉

C₂³

yes

characteristic

C₂²×C₁₃₉

C₂⁴

yes

characteristic

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

C₂×Q₈

yes

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×Q₈

yes

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

C₂×Q₈

yes

—

C₂×C₁₃₉

C₂²×Q₈

yes

characteristic

C₂×C₁₃₉

H(5,𝔽₂)

yes

characteristic

C₂×C₁₃₉

2₋¹⁺⁴

yes

characteristic

D₁₃₉

—

C₁₃₉

#544

yes

characteristic, complemented, Sylow

#544

—

139

maximal, Sylow

C₄⋊Q₈

—

139

—

C₄⋊Q₈

—

139

—

C₄⋊Q₈

—

139

—

#125

—

139

—

#125

—

139

—

#125

D₁₃₉

yes

characteristic, complemented

C₂²⋊Q₈

—

139

—

C₂²⋊Q₈

—

139

—

#117

—

139

—

#117

—

139

—

C₂×C₄⋊C₄

—

139

—

C₂×Q₈

—

139

—

C₂×Q₈

—

139

—

C₂²×C₄

—

139

—

C₂²×C₄

—

139

—

C₄⋊C₄

—

139

—

C₄⋊C₄

—

139

—

C₄⋊C₄

—

139

—

C₄⋊C₄

—

139

—

C₄⋊C₄

—

139

—

C₄⋊C₄

—

139

—

C₄⋊C₄

—

139

—

C₄⋊C₄

—

139

—

C₄⋊C₄

—

139

—

C₄⋊C₄

C₂×D₁₃₉

yes

characteristic

C₄⋊C₄

C₂×D₁₃₉

yes

characteristic

C₄⋊C₄

C₂×D₁₃₉

yes

—

C₄⋊C₄

C₂×D₁₃₉

yes

—

C₂²⋊C₄

—

139

—

C₂²⋊C₄

—

139

—

100

C₂²⋊C₄

—

139

—

101

C₄²

—

139

—

102

C₄²

—

139

—

103

C₄²

C₂×D₁₃₉

yes

characteristic

104

C₂³

—

139

—

105

Q₈

—

556

—

106

Q₈

—

556

—

107

C₂×C₄

—

556

—

108

C₂×C₄

—

139

—

109

C₂×C₄

—

139

—

110

C₂×C₄

—

139

—

111

C₂×C₄

—

139

—

112

C₂×C₄

—

139

—

113

C₂×C₄

—

139

—

114

C₂×C₄

C₂²×D₁₃₉

yes

—

115

C₂×C₄

C₂²×D₁₃₉

yes

characteristic

116

C₂×C₄

C₂²×D₁₃₉

yes

characteristic

117

C₂×C₄

C₂²×D₁₃₉

yes

—

118

C₂×C₄

C₂²×D₁₃₉

yes

characteristic

119

C₂²

—

278

—

120

C₂²

—

278

—

121

C₂²

—

139

—

122

C₂²

C₂³×D₁₃₉

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

278

—

128

C₄

—

278

—

129

C₄

—

278

—

130

C₄

—

278

—

131

C₄

—

139

—

132

C₄

Q₈×D₁₃₉

yes

—

133

C₂

—

278

—

134

C₂

C₂×Q₈×D₁₃₉

yes

characteristic

135

C₂

#17984888

yes

characteristic

136

C₂

#17984889

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₃₉
commutator, socle	C₂²×C₁₃₉	C₂⁴
Fitting	#125×C₁₃₉	C₂
hypercenter	#125	D₁₃₉
hypocenter	C₁₃₉	#544
LCS	G ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ #125

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₃₉

commutator, socle

C₂²×C₁₃₉

C₂⁴

Fitting

#125×C₁₃₉

C₂

D₁₃₉

C₁₃₉

G ▷ C₂²×C₁₃₉ ▷ C₁₃₉

C₁ ◁ C₂² ◁ #125

N	H ≅ G/N	type
1. #17984842	C₂	semidirect
2. #17984841	C₂	semidirect
3. #17984841	C₂	semidirect
4. #17984833	C₂	semidirect
5. #17984832	C₂	semidirect
11. #125×C₁₃₉	C₂	semidirect
44. C₄×D₁₃₉	Q₈	semidirect
68. C₁₃₉	#544	semidirect
75. #125	D₁₃₉	semidirect

H ≅ G/N

type

1. #17984842

C₂

semidirect

2. #17984841

C₂

semidirect

3. #17984841

C₂

semidirect

4. #17984833

C₂

semidirect

5. #17984832

C₂

semidirect

11. #125×C₁₃₉

C₂

semidirect

44. C₄×D₁₃₉

Q₈

semidirect

68. C₁₃₉

#544

semidirect

75. #125

D₁₃₉

semidirect

Subgroups — #22581583

Conjugacy classes of subgroups

Complements