Group #22581661 — subgroups

Subgroups — #22581661

Order 8896.

228 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 107 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	#17984832	C₂	1	2	yes	maximal, complemented
3	#17984822	C₂	1	2	yes	maximal, complemented
4	#17984819	C₂	1	2	yes	maximal, complemented
5	#17984818	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₈×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₂²⋊D₁₃₉	C₂	1	1	yes	maximal, characteristic
8	(C₂²⋊D₁₃₉)⋊C₄	C₂	1	2	yes	maximal
9	C₂×C₄⋊Dic₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄⋊C₄×D₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²⋊Q₈×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Dic₂₇₈	C₂²	1	2	yes	—
13	C₂×C₂²⋊D₁₃₉	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₃₉	C₂²	1	2	yes	—
15	C₂×C₄×D₁₃₉	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₁₃₉	C₂²	1	2	yes	—
17	C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
18	C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
19	C₄⋊Dic₁₃₉	C₂²	1	2	yes	—
20	C₄⋊Dic₁₃₉	C₂²	1	2	yes	—
21	Dic₁₃₉⋊C₄	C₂²	1	2	yes	—
22	Dic₁₃₉⋊C₄	C₂²	1	2	yes	—
23	Dic₁₃₉⋊C₄	C₂²	1	1	yes	characteristic
24	Dic₁₃₉⋊C₄	C₂²	1	2	yes	—
25	C₂²⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
26	(C₂×D₁₃₉)⋊C₄	C₂²	1	2	yes	—
27	(C₂×D₁₃₉)⋊C₄	C₂²	1	1	yes	characteristic
28	C₄×Dic₁₃₉	C₂²	1	2	yes	—
29	C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
30	C₂×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
31	C₂²×C₄×C₁₃₉	C₂²	1	1	yes	characteristic
32	C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×C₁₃₉	C₂²	1	2	yes	—
34	C₂²⋊C₄×C₁₃₉	C₂²	1	2	yes	—
35	C₂²×D₁₃₉	C₂³	1	1	yes	characteristic
36	Dic₂₇₈	—	4	2	no	—
37	C₂²⋊D₁₃₉	Q₈	1	4	yes	complemented
38	C₂×Dic₁₃₉	—	4	2	no	—
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	C₄×D₁₃₉	—	2	2	no	—
45	C₄×D₁₃₉	—	4	2	no	—
46	C₂³×C₁₃₉	C₂³	1	1	yes	characteristic
47	Q₈×C₁₃₉	—	2	2	no	—
48	C₂×C₄×C₁₃₉	—	2	2	no	—
49	C₂×C₄×C₁₃₉	C₂³	1	2	yes	—
50	C₂×C₄×C₁₃₉	C₂³	1	2	yes	—
51	C₂×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
52	C₂×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
53	C₂²×C₁₃₉	—	2	1	no	—
54	C₂²×C₁₃₉	—	2	1	no	—
55	C₂²×C₁₃₉	C₂×Q₈	1	2	yes	—
56	C₂²×C₁₃₉	C₂⁴	1	1	yes	characteristic
57	C₂×D₁₃₉	—	2	1	no	—
58	C₂×D₁₃₉	—	2	1	no	—
59	C₂×D₁₃₉	C₂×Q₈	1	2	yes	—
60	C₄×C₁₃₉	—	2	1	no	—
61	C₄×C₁₃₉	—	2	2	no	—
62	C₄×C₁₃₉	—	2	2	no	—
63	C₄×C₁₃₉	#41	1	2	yes	—
64	Dic₁₃₉	—	2	1	no	—
65	Dic₁₃₉	—	2	2	no	—
66	Dic₁₃₉	—	2	2	no	—
67	Dic₁₃₉	—	2	1	no	—
68	Dic₁₃₉	C₂×Q₈	1	2	yes	—
69	C₂×C₁₃₉	—	2	2	no	—
70	C₂×C₁₃₉	C₂²×Q₈	1	1	yes	characteristic
71	C₂×C₁₃₉	C₂×#41	1	1	yes	characteristic
72	C₂×C₁₃₉	H(5,𝔽₂)	1	1	yes	characteristic
73	D₁₃₉	—	2	2	no	—
74	C₁₃₉	#554	1	1	yes	characteristic, complemented, Sylow
75	#554	—	139	1	no	maximal, Sylow
76	C₄⋊Q₈	—	139	1	no	—
77	#125	—	139	2	no	—
78	C₂²⋊Q₈	—	139	1	no	—
79	C₂²⋊Q₈	—	139	2	no	—
80	C₂²⋊Q₈	—	139	2	no	—
81	C₂²⋊Q₈	D₁₃₉	1	1	yes	characteristic, complemented
82	C₄×Q₈	—	139	1	no	—
83	C₄×D₄	—	139	1	no	—
84	C₄×D₄	—	139	2	no	—
85	C₂×C₄⋊C₄	—	139	1	no	—
86	C₂×C₄⋊C₄	—	139	1	no	—
87	C₂×Q₈	—	139	2	no	—
88	C₂×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
89	C₂×D₄	—	139	1	no	—
90	C₂²×C₄	—	139	1	no	—
91	C₂²×C₄	—	139	2	no	—
92	C₂²×C₄	—	139	2	no	—
93	C₂²×C₄	C₂×D₁₃₉	1	1	yes	characteristic
94	C₄⋊C₄	—	139	1	no	—
95	C₄⋊C₄	—	139	2	no	—
96	C₄⋊C₄	—	139	2	no	—
97	C₄⋊C₄	—	139	1	no	—
98	C₄⋊C₄	—	139	1	no	—
99	C₄⋊C₄	—	139	2	no	—
100	C₄⋊C₄	—	139	2	no	—
101	C₄⋊C₄	—	139	2	no	—
102	C₄⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
103	C₄⋊C₄	C₂×D₁₃₉	1	2	yes	—
104	C₂²⋊C₄	—	139	1	no	—
105	C₂²⋊C₄	—	139	1	no	—
106	C₂²⋊C₄	—	139	2	no	—
107	C₂²⋊C₄	C₂×D₁₃₉	1	2	yes	—
108	C₄²	—	139	2	no	—
109	C₄²	—	139	1	no	—
110	C₂³	—	139	1	no	—
111	C₂³	C₂²×D₁₃₉	1	1	yes	characteristic
112	Q₈	—	2	2	no	—
113	Q₈	—	556	2	no	—
114	D₄	—	139	4	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	278	2	no	—
117	C₂×C₄	—	556	2	no	—
118	C₂×C₄	—	556	2	no	—
119	C₂×C₄	—	139	1	no	—
120	C₂×C₄	—	139	2	no	—
121	C₂×C₄	—	139	1	no	—
122	C₂×C₄	—	139	2	no	—
123	C₂×C₄	—	139	1	no	—
124	C₂×C₄	C₂²×D₁₃₉	1	2	yes	—
125	C₂×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
126	C₂×C₄	C₂²×D₁₃₉	1	2	yes	—
127	C₂×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
128	C₂²	—	2	1	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	278	1	no	—
131	C₂²	—	278	1	no	—
132	C₂²	—	139	2	no	—
133	C₂²	Q₈×D₁₃₉	1	2	yes	—
134	C₂²	C₂³×D₁₃₉	1	1	yes	characteristic
135	C₄	—	2	1	no	—
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	278	1	no	—
139	C₄	—	278	2	no	—
140	C₄	—	278	2	no	—
141	C₄	—	278	1	no	—
142	C₄	—	139	2	no	—
143	C₄	#14555784	1	2	yes	—
144	C₂	—	2	2	no	—
145	C₂	—	278	2	no	—
146	C₂	C₂×Q₈×D₁₃₉	1	1	yes	characteristic
147	C₂	C₂×#14555784	1	1	yes	characteristic
148	C₂	#17984888	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17984841

C₂

yes

maximal, characteristic, complemented

#17984832

C₂

yes

maximal, complemented

#17984822

C₂

yes

maximal, complemented

#17984819

C₂

yes

maximal, complemented

#17984818

C₂

yes

maximal, characteristic, complemented

Q₈×Dic₁₃₉

C₂

yes

maximal, characteristic, complemented

C₄×C₂²⋊D₁₃₉

C₂

yes

maximal, characteristic

(C₂²⋊D₁₃₉)⋊C₄

C₂

yes

maximal

C₂×C₄⋊Dic₁₃₉

C₂

yes

maximal, characteristic, complemented

C₄⋊C₄×D₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂²⋊Q₈×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₇₈

C₂²

yes

—

C₂×C₂²⋊D₁₃₉

C₂²

yes

characteristic

C₂²×Dic₁₃₉

C₂²

yes

—

C₂×C₄×D₁₃₉

C₂²

yes

characteristic

C₂×C₄×D₁₃₉

C₂²

yes

—

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

—

Dic₁₃₉⋊C₄

C₂²

yes

characteristic

Dic₁₃₉⋊C₄

C₂²

yes

—

C₂²⋊Dic₁₃₉

C₂²

yes

characteristic

(C₂×D₁₃₉)⋊C₄

C₂²

yes

—

(C₂×D₁₃₉)⋊C₄

C₂²

yes

characteristic

C₄×Dic₁₃₉

C₂²

yes

—

C₄×Dic₁₃₉

C₂²

yes

characteristic

C₂×Q₈×C₁₃₉

C₂²

yes

characteristic

C₂²×C₄×C₁₃₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₃₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₃₉

C₂²

yes

—

C₂²⋊C₄×C₁₃₉

C₂²

yes

—

C₂²×D₁₃₉

C₂³

yes

characteristic

Dic₂₇₈

—

C₂²⋊D₁₃₉

Q₈

yes

complemented

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

C₂³

yes

characteristic

C₂×Dic₁₃₉

C₂³

yes

—

C₂×Dic₁₃₉

C₂³

yes

—

C₂×Dic₁₃₉

C₂³

yes

characteristic

C₂×Dic₁₃₉

C₂³

yes

characteristic

C₄×D₁₃₉

—

C₄×D₁₃₉

—

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

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

C₂×Q₈

yes

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

#41

yes

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

C₂×Q₈

yes

—

C₂×C₁₃₉

—

C₂×C₁₃₉

C₂²×Q₈

yes

characteristic

C₂×C₁₃₉

C₂×#41

yes

characteristic

C₂×C₁₃₉

H(5,𝔽₂)

yes

characteristic

D₁₃₉

—

C₁₃₉

#554

yes

characteristic, complemented, Sylow

#554

—

139

maximal, Sylow

C₄⋊Q₈

—

139

—

#125

—

139

—

C₂²⋊Q₈

—

139

—

C₂²⋊Q₈

—

139

—

C₂²⋊Q₈

—

139

—

C₂²⋊Q₈

D₁₃₉

yes

characteristic, complemented

C₄×Q₈

—

139

—

C₄×D₄

—

139

—

C₄×D₄

—

139

—

C₂×C₄⋊C₄

—

139

—

C₂×C₄⋊C₄

—

139

—

C₂×Q₈

—

139

—

C₂×Q₈

C₂×D₁₃₉

yes

characteristic

C₂×D₄

—

139

—

C₂²×C₄

—

139

—

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

—

100

C₄⋊C₄

—

139

—

101

C₄⋊C₄

—

139

—

102

C₄⋊C₄

C₂×D₁₃₉

yes

characteristic

103

C₄⋊C₄

C₂×D₁₃₉

yes

—

104

C₂²⋊C₄

—

139

—

105

C₂²⋊C₄

—

139

—

106

C₂²⋊C₄

—

139

—

107

C₂²⋊C₄

C₂×D₁₃₉

yes

—

108

C₄²

—

139

—

109

C₄²

—

139

—

110

C₂³

—

139

—

111

C₂³

C₂²×D₁₃₉

yes

characteristic

112

Q₈

—

113

Q₈

—

556

—

114

D₄

—

139

—

115

C₂×C₄

—

116

C₂×C₄

—

278

—

117

C₂×C₄

—

556

—

118

C₂×C₄

—

556

—

119

C₂×C₄

—

139

—

120

C₂×C₄

—

139

—

121

C₂×C₄

—

139

—

122

C₂×C₄

—

139

—

123

C₂×C₄

—

139

—

124

C₂×C₄

C₂²×D₁₃₉

yes

—

125

C₂×C₄

C₂²×D₁₃₉

yes

characteristic

126

C₂×C₄

C₂²×D₁₃₉

yes

—

127

C₂×C₄

C₂²×D₁₃₉

yes

characteristic

128

C₂²

—

129

C₂²

—

130

C₂²

—

278

—

131

C₂²

—

278

—

132

C₂²

—

139

—

133

C₂²

Q₈×D₁₃₉

yes

—

134

C₂²

C₂³×D₁₃₉

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

278

—

139

C₄

—

278

—

140

C₄

—

278

—

141

C₄

—

278

—

142

C₄

—

139

—

143

C₄

#14555784

yes

—

144

C₂

—

145

C₂

—

278

—

146

C₂

C₂×Q₈×D₁₃₉

yes

characteristic

147

C₂

C₂×#14555784

yes

characteristic

148

C₂

#17984888

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	#554
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. #17984832	C₂	semidirect
3. #17984822	C₂	semidirect
4. #17984819	C₂	semidirect
5. #17984818	C₂	semidirect
6. Q₈×Dic₁₃₉	C₂	semidirect
9. C₂×C₄⋊Dic₁₃₉	C₂	semidirect
10. C₄⋊C₄×D₁₃₉	C₂	semidirect
11. C₂²⋊Q₈×C₁₃₉	C₂	semidirect
37. C₂²⋊D₁₃₉	Q₈	semidirect
74. C₁₃₉	#554	semidirect
81. C₂²⋊Q₈	D₁₃₉	semidirect

H ≅ G/N

type

1. #17984841

C₂

semidirect

2. #17984832

C₂

semidirect

3. #17984822

C₂

semidirect

4. #17984819

C₂

semidirect

5. #17984818

C₂

semidirect

6. Q₈×Dic₁₃₉

C₂

semidirect

9. C₂×C₄⋊Dic₁₃₉

C₂

semidirect

10. C₄⋊C₄×D₁₃₉

C₂

semidirect

11. C₂²⋊Q₈×C₁₃₉

C₂

semidirect

37. C₂²⋊D₁₃₉

Q₈

semidirect

74. C₁₃₉

#554

semidirect

81. C₂²⋊Q₈

D₁₃₉

semidirect

Subgroups — #22581661

Conjugacy classes of subgroups

Complements