Group #22581505 — subgroups

Subgroups — #22581505

Order 8896.

222 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 109 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	#17984841	C₂	1	1	yes	maximal, characteristic, complemented
2	#17984840	C₂	1	1	yes	maximal, characteristic, complemented
3	#17984832	C₂	1	2	yes	maximal, complemented
4	#17984821	C₂	1	1	yes	maximal, characteristic
5	#17984819	C₂	1	2	yes	maximal
6	Q₈×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
7	#17984807	C₂	1	2	yes	maximal, complemented
8	#17984807	C₂	1	1	yes	maximal, characteristic, complemented
9	#17984798	C₂	1	2	yes	maximal
10	C₂×C₄×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic
11	C₂²⋊Q₈×C₁₃₉	C₂	1	1	yes	maximal, characteristic
12	C₂×Dic₂₇₈	C₂²	1	2	yes	—
13	C₂×Dic₂₇₈	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₃₉	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₃₉	C₂²	1	1	yes	characteristic
16	C₄⋊Dic₁₃₉	C₂²	1	2	yes	—
17	C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
18	Dic₁₃₉⋊C₄	C₂²	1	2	yes	—
19	Dic₁₃₉⋊C₄	C₂²	1	2	yes	—
20	Dic₁₃₉⋊C₄	C₂²	1	2	yes	—
21	Dic₁₃₉⋊C₄	C₂²	1	2	yes	—
22	Dic₁₃₉⋊C₄	C₂²	1	2	yes	—
23	C₂²⋊Dic₁₃₉	C₂²	1	2	yes	—
24	C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
25	C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
26	C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
27	C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
28	C₄×Dic₁₃₉	C₂²	1	2	yes	—
29	C₄×Dic₁₃₉	C₂²	1	2	yes	—
30	C₂×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
31	C₂²×C₄×C₁₃₉	C₂²	1	1	yes	characteristic
32	C₄⋊C₄×C₁₃₉	C₂²	1	2	yes	—
33	C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
34	C₂²⋊C₄×C₁₃₉	C₂²	1	2	yes	—
35	Dic₂₇₈	—	2	2	no	—
36	Dic₂₇₈	—	2	4	no	—
37	C₂×Dic₁₃₉	—	2	2	no	—
38	C₂×Dic₁₃₉	Q₈	1	4	yes	complemented
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	C₂×Dic₁₃₉	C₂³	1	1	yes	characteristic
44	C₂×Dic₁₃₉	C₂³	1	1	yes	characteristic
45	C₂³×C₁₃₉	C₂³	1	1	yes	characteristic
46	Q₈×C₁₃₉	—	2	2	no	—
47	C₂×C₄×C₁₃₉	—	2	2	no	—
48	C₂×C₄×C₁₃₉	C₂³	1	2	yes	—
49	C₂×C₄×C₁₃₉	C₂³	1	2	yes	—
50	C₂×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
51	C₂×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
52	C₂²×C₁₃₉	—	2	1	no	—
53	C₂²×C₁₃₉	—	2	1	no	—
54	C₂²×C₁₃₉	C₂×Q₈	1	2	yes	—
55	C₂²×C₁₃₉	C₂⁴	1	1	yes	characteristic
56	C₄×C₁₃₉	—	2	1	no	—
57	C₄×C₁₃₉	—	2	2	no	—
58	C₄×C₁₃₉	—	2	2	no	—
59	C₄×C₁₃₉	#41	1	2	yes	—
60	Dic₁₃₉	—	2	1	no	—
61	Dic₁₃₉	—	2	2	no	—
62	Dic₁₃₉	—	2	2	no	—
63	Dic₁₃₉	C₂×Q₈	1	2	yes	—
64	Dic₁₃₉	C₂×Q₈	1	2	yes	—
65	Dic₁₃₉	#41	1	2	yes	—
66	C₂×C₁₃₉	—	2	2	no	—
67	C₂×C₁₃₉	C₂²×Q₈	1	1	yes	characteristic
68	C₂×C₁₃₉	C₂×#41	1	1	yes	characteristic
69	C₂×C₁₃₉	C₂×#41	1	1	yes	characteristic
70	C₁₃₉	#533	1	1	yes	characteristic, complemented, Sylow
71	#533	—	139	1	no	maximal, Sylow
72	C₄⋊Q₈	—	139	1	no	—
73	C₄⋊Q₈	—	139	1	no	—
74	#125	—	139	2	no	—
75	C₂²⋊Q₈	—	139	1	no	—
76	C₂²⋊Q₈	—	139	2	no	—
77	C₂²⋊Q₈	D₁₃₉	1	1	yes	characteristic
78	C₄×Q₈	—	139	1	no	—
79	C₄×Q₈	—	139	1	no	—
80	C₄×Q₈	—	139	2	no	—
81	#117	—	139	2	no	—
82	C₂×C₄²	—	139	1	no	—
83	C₂×Q₈	—	139	1	no	—
84	C₂×Q₈	—	139	2	no	—
85	C₂×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
86	C₂²×C₄	—	139	1	no	—
87	C₂²×C₄	—	139	1	no	—
88	C₂²×C₄	C₂×D₁₃₉	1	1	yes	characteristic
89	C₄⋊C₄	—	139	2	no	—
90	C₄⋊C₄	—	139	2	no	—
91	C₄⋊C₄	—	139	2	no	—
92	C₄⋊C₄	—	139	2	no	—
93	C₄⋊C₄	—	139	2	no	—
94	C₄⋊C₄	—	139	2	no	—
95	C₄⋊C₄	—	139	1	no	—
96	C₄⋊C₄	C₂×D₁₃₉	1	2	yes	—
97	C₄⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
98	C₂²⋊C₄	—	139	2	no	—
99	C₂²⋊C₄	C₂×D₁₃₉	1	2	yes	—
100	C₄²	—	139	1	no	—
101	C₄²	—	139	1	no	—
102	C₄²	—	139	1	no	—
103	C₄²	—	139	1	no	—
104	C₄²	—	139	2	no	—
105	C₄²	—	139	2	no	—
106	C₂³	C₂²×D₁₃₉	1	1	yes	characteristic
107	Q₈	—	2	2	no	—
108	Q₈	—	278	4	no	—
109	Q₈	—	278	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	278	2	no	—
112	C₂×C₄	—	139	1	no	—
113	C₂×C₄	—	139	2	no	—
114	C₂×C₄	—	139	2	no	—
115	C₂×C₄	—	139	4	no	—
116	C₂×C₄	—	139	1	no	—
117	C₂×C₄	—	139	1	no	—
118	C₂×C₄	—	139	1	no	—
119	C₂×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
120	C₂×C₄	C₂²×D₁₃₉	1	2	yes	—
121	C₂×C₄	C₂²×D₁₃₉	1	2	yes	—
122	C₂×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
123	C₂²	—	2	1	no	—
124	C₂²	—	2	1	no	—
125	C₂²	Q₈×D₁₃₉	1	2	yes	—
126	C₂²	C₂³×D₁₃₉	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	—	2	1	no	—
129	C₄	—	2	2	no	—
130	C₄	—	278	1	no	—
131	C₄	—	278	2	no	—
132	C₄	—	278	2	no	—
133	C₄	—	139	2	no	—
134	C₄	—	139	2	no	—
135	C₄	—	139	2	no	—
136	C₄	#14555784	1	2	yes	—
137	C₂	—	2	2	no	—
138	C₂	C₂×Q₈×D₁₃₉	1	1	yes	characteristic
139	C₂	#41×D₁₃₉	1	1	yes	characteristic
140	C₂	C₂×#14555784	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17984841

C₂

yes

maximal, characteristic, complemented

#17984840

C₂

yes

maximal, characteristic, complemented

#17984832

C₂

yes

maximal, complemented

#17984821

C₂

yes

maximal, characteristic

#17984819

C₂

yes

maximal

Q₈×Dic₁₃₉

C₂

yes

maximal, characteristic, complemented

#17984807

C₂

yes

maximal, complemented

#17984807

C₂

yes

maximal, characteristic, complemented

#17984798

C₂

yes

maximal

C₂×C₄×Dic₁₃₉

C₂

yes

maximal, characteristic

C₂²⋊Q₈×C₁₃₉

C₂

yes

maximal, characteristic

C₂×Dic₂₇₈

C₂²

yes

—

C₂×Dic₂₇₈

C₂²

yes

characteristic

C₂²×Dic₁₃₉

C₂²

yes

characteristic

C₂²×Dic₁₃₉

C₂²

yes

characteristic

C₄⋊Dic₁₃₉

C₂²

yes

—

C₄⋊Dic₁₃₉

C₂²

yes

characteristic

Dic₁₃₉⋊C₄

C₂²

yes

—

Dic₁₃₉⋊C₄

C₂²

yes

—

Dic₁₃₉⋊C₄

C₂²

yes

—

Dic₁₃₉⋊C₄

C₂²

yes

—

Dic₁₃₉⋊C₄

C₂²

yes

—

C₂²⋊Dic₁₃₉

C₂²

yes

—

C₄×Dic₁₃₉

C₂²

yes

characteristic

C₄×Dic₁₃₉

C₂²

yes

characteristic

C₄×Dic₁₃₉

C₂²

yes

characteristic

C₄×Dic₁₃₉

C₂²

yes

characteristic

C₄×Dic₁₃₉

C₂²

yes

—

C₄×Dic₁₃₉

C₂²

yes

—

C₂×Q₈×C₁₃₉

C₂²

yes

characteristic

C₂²×C₄×C₁₃₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₃₉

C₂²

yes

—

C₄⋊C₄×C₁₃₉

C₂²

yes

characteristic

C₂²⋊C₄×C₁₃₉

C₂²

yes

—

Dic₂₇₈

—

Dic₂₇₈

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

Q₈

yes

complemented

C₂×Dic₁₃₉

C₂³

yes

characteristic

C₂×Dic₁₃₉

C₂³

yes

—

C₂×Dic₁₃₉

C₂³

yes

—

C₂×Dic₁₃₉

C₂³

yes

characteristic

C₂×Dic₁₃₉

C₂³

yes

characteristic

C₂×Dic₁₃₉

C₂³

yes

characteristic

C₂³×C₁₃₉

C₂³

yes

characteristic

Q₈×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂³

yes

—

C₂×C₄×C₁₃₉

C₂³

yes

—

C₂×C₄×C₁₃₉

C₂³

yes

characteristic

C₂×C₄×C₁₃₉

C₂³

yes

characteristic

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

C₂×Q₈

yes

—

C₂²×C₁₃₉

C₂⁴

yes

characteristic

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

#41

yes

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

C₂×Q₈

yes

—

Dic₁₃₉

C₂×Q₈

yes

—

Dic₁₃₉

#41

yes

—

C₂×C₁₃₉

—

C₂×C₁₃₉

C₂²×Q₈

yes

characteristic

C₂×C₁₃₉

C₂×#41

yes

characteristic

C₂×C₁₃₉

C₂×#41

yes

characteristic

C₁₃₉

#533

yes

characteristic, complemented, Sylow

#533

—

139

maximal, Sylow

C₄⋊Q₈

—

139

—

C₄⋊Q₈

—

139

—

#125

—

139

—

C₂²⋊Q₈

—

139

—

C₂²⋊Q₈

—

139

—

C₂²⋊Q₈

D₁₃₉

yes

characteristic

C₄×Q₈

—

139

—

C₄×Q₈

—

139

—

C₄×Q₈

—

139

—

#117

—

139

—

C₂×C₄²

—

139

—

C₂×Q₈

—

139

—

C₂×Q₈

—

139

—

C₂×Q₈

C₂×D₁₃₉

yes

characteristic

C₂²×C₄

—

139

—

C₂²×C₄

—

139

—

C₂²×C₄

C₂×D₁₃₉

yes

characteristic

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

—

C₄⋊C₄

C₂×D₁₃₉

yes

characteristic

C₂²⋊C₄

—

139

—

C₂²⋊C₄

C₂×D₁₃₉

yes

—

100

C₄²

—

139

—

101

C₄²

—

139

—

102

C₄²

—

139

—

103

C₄²

—

139

—

104

C₄²

—

139

—

105

C₄²

—

139

—

106

C₂³

C₂²×D₁₃₉

yes

characteristic

107

Q₈

—

108

Q₈

—

278

—

109

Q₈

—

278

—

110

C₂×C₄

—

111

C₂×C₄

—

278

—

112

C₂×C₄

—

139

—

113

C₂×C₄

—

139

—

114

C₂×C₄

—

139

—

115

C₂×C₄

—

139

—

116

C₂×C₄

—

139

—

117

C₂×C₄

—

139

—

118

C₂×C₄

—

139

—

119

C₂×C₄

C₂²×D₁₃₉

yes

characteristic

120

C₂×C₄

C₂²×D₁₃₉

yes

—

121

C₂×C₄

C₂²×D₁₃₉

yes

—

122

C₂×C₄

C₂²×D₁₃₉

yes

characteristic

123

C₂²

—

124

C₂²

—

125

C₂²

Q₈×D₁₃₉

yes

—

126

C₂²

C₂³×D₁₃₉

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

278

—

131

C₄

—

278

—

132

C₄

—

278

—

133

C₄

—

139

—

134

C₄

—

139

—

135

C₄

—

139

—

136

C₄

#14555784

yes

—

137

C₂

—

138

C₂

C₂×Q₈×D₁₃₉

yes

characteristic

139

C₂

#41×D₁₃₉

yes

characteristic

140

C₂

C₂×#14555784

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₃₉
commutator, socle	C₂²×C₁₃₉	C₂⁴
Fitting	C₂²⋊Q₈×C₁₃₉	C₂
hypercenter	C₂²⋊Q₈	D₁₃₉
hypocenter	C₁₃₉	#533
LCS	G ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₃₉

commutator, socle

C₂²×C₁₃₉

C₂⁴

Fitting

C₂²⋊Q₈×C₁₃₉

C₂

hypercenter

C₂²⋊Q₈

D₁₃₉

C₁₃₉

G ▷ C₂²×C₁₃₉ ▷ C₁₃₉

C₁ ◁ C₂² ◁ C₂²⋊Q₈

N	H ≅ G/N	type
1. #17984841	C₂	semidirect
2. #17984840	C₂	semidirect
3. #17984832	C₂	semidirect
6. Q₈×Dic₁₃₉	C₂	semidirect
7. #17984807	C₂	semidirect
8. #17984807	C₂	semidirect
38. C₂×Dic₁₃₉	Q₈	semidirect
70. C₁₃₉	#533	semidirect

H ≅ G/N

type

1. #17984841

C₂

semidirect

2. #17984840

C₂

semidirect

3. #17984832

C₂

semidirect

6. Q₈×Dic₁₃₉

C₂

semidirect

7. #17984807

C₂

semidirect

8. #17984807

C₂

semidirect

38. C₂×Dic₁₃₉

Q₈

semidirect

70. C₁₃₉

#533

semidirect

Subgroups — #22581505

Conjugacy classes of subgroups

Complements