Group #23750346 — subgroups

Subgroups — #23750346

Order 9536.

212 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 97 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	#18011910	C₂	1	1	yes	maximal, characteristic, complemented
2	#18011906	C₂	1	2	yes	maximal, complemented
3	#18011904	C₂	1	2	yes	maximal, complemented
4	#18011897	C₂	1	2	yes	maximal
5	#18011887	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₈×Dic₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
7	#18011876	C₂	1	2	yes	maximal, complemented
8	C₄×C₂²⋊D₁₄₉	C₂	1	1	yes	maximal, characteristic
9	#18011870	C₂	1	1	yes	maximal, characteristic, complemented
10	#18011869	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²⋊Q₈×C₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Dic₂₉₈	C₂²	1	2	yes	—
13	C₂×C₂²⋊D₁₄₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₁₄₉	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₁₄₉	C₂²	1	2	yes	—
16	C₄⋊Dic₁₄₉	C₂²	1	1	yes	characteristic
17	C₄⋊Dic₁₄₉	C₂²	1	1	yes	characteristic
18	#14558715	C₂²	1	2	yes	—
19	#14558715	C₂²	1	2	yes	—
20	#14558715	C₂²	1	1	yes	characteristic
21	C₂²⋊Dic₁₄₉	C₂²	1	1	yes	characteristic
22	C₂²⋊Dic₁₄₉	C₂²	1	2	yes	—
23	#14558713	C₂²	1	1	yes	characteristic
24	#14558713	C₂²	1	2	yes	—
25	#14558713	C₂²	1	2	yes	—
26	C₄×Dic₁₄₉	C₂²	1	2	yes	—
27	C₄×Dic₁₄₉	C₂²	1	2	yes	—
28	C₄×Dic₁₄₉	C₂²	1	1	yes	characteristic
29	C₄×Dic₁₄₉	C₂²	1	2	yes	—
30	C₂×Q₈×C₁₄₉	C₂²	1	1	yes	characteristic
31	C₂²×C₄×C₁₄₉	C₂²	1	1	yes	characteristic
32	C₄⋊C₄×C₁₄₉	C₂²	1	2	yes	—
33	C₄⋊C₄×C₁₄₉	C₂²	1	1	yes	characteristic
34	C₂²⋊C₄×C₁₄₉	C₂²	1	2	yes	—
35	C₂²×D₁₄₉	C₂³	1	1	yes	characteristic
36	Dic₂₉₈	—	2	4	no	—
37	C₂²⋊D₁₄₉	—	2	2	no	—
38	C₂×Dic₁₄₉	C₂³	1	1	yes	characteristic
39	C₂×Dic₁₄₉	C₂³	1	2	yes	—
40	C₂×Dic₁₄₉	C₂³	1	2	yes	—
41	C₂×Dic₁₄₉	C₂³	1	1	yes	characteristic
42	C₂×Dic₁₄₉	C₂³	1	1	yes	characteristic
43	C₄×D₁₄₉	—	2	2	no	—
44	C₂³×C₁₄₉	C₂³	1	1	yes	characteristic
45	Q₈×C₁₄₉	—	2	2	no	—
46	C₂×C₄×C₁₄₉	—	2	2	no	—
47	C₂×C₄×C₁₄₉	C₂³	1	2	yes	—
48	C₂×C₄×C₁₄₉	C₂³	1	2	yes	—
49	C₂×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
50	C₂×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
51	C₂²×C₁₄₉	—	2	1	no	—
52	C₂²×C₁₄₉	—	2	1	no	—
53	C₂²×C₁₄₉	—	2	1	no	—
54	C₂²×C₁₄₉	C₂⁴	1	1	yes	characteristic
55	C₂×D₁₄₉	—	2	1	no	—
56	C₂×D₁₄₉	—	2	1	no	—
57	C₂×D₁₄₉	—	2	1	no	—
58	C₄×C₁₄₉	—	2	1	no	—
59	C₄×C₁₄₉	—	2	2	no	—
60	C₄×C₁₄₉	—	2	2	no	—
61	C₄×C₁₄₉	#41	1	2	yes	—
62	Dic₁₄₉	—	2	1	no	—
63	Dic₁₄₉	—	2	2	no	—
64	Dic₁₄₉	—	2	2	no	—
65	Dic₁₄₉	—	2	1	no	—
66	Dic₁₄₉	#41	1	2	yes	—
67	C₂×C₁₄₉	—	4	1	no	—
68	C₂×C₁₄₉	C₂×#41	1	1	yes	characteristic
69	C₂×C₁₄₉	C₂×#41	1	1	yes	characteristic
70	C₂×C₁₄₉	2₋¹⁺⁴	1	1	yes	characteristic
71	D₁₄₉	—	4	1	no	—
72	C₁₄₉	#556	1	1	yes	characteristic, complemented, Sylow
73	#556	—	149	1	no	maximal, Sylow
74	C₄⋊Q₈	—	149	1	no	—
75	#126	—	149	2	no	—
76	#126	—	149	2	no	—
77	#124	—	149	2	no	—
78	C₂²⋊Q₈	—	149	1	no	—
79	C₂²⋊Q₈	D₁₄₉	1	1	yes	characteristic, complemented
80	C₄×Q₈	—	149	1	no	—
81	C₄×Q₈	—	149	2	no	—
82	C₄×D₄	—	149	1	no	—
83	#117	—	149	1	no	—
84	#117	—	149	1	no	—
85	C₂×Q₈	—	149	2	no	—
86	C₂×Q₈	C₂×D₁₄₉	1	1	yes	characteristic
87	C₂×D₄	—	149	1	no	—
88	C₂²×C₄	—	149	1	no	—
89	C₂²×C₄	C₂×D₁₄₉	1	1	yes	characteristic
90	C₄⋊C₄	—	149	1	no	—
91	C₄⋊C₄	—	149	1	no	—
92	C₄⋊C₄	—	149	1	no	—
93	C₄⋊C₄	—	149	2	no	—
94	C₄⋊C₄	—	149	2	no	—
95	C₄⋊C₄	—	149	2	no	—
96	C₄⋊C₄	C₂×D₁₄₉	1	1	yes	characteristic
97	C₄⋊C₄	C₂×D₁₄₉	1	2	yes	—
98	C₂²⋊C₄	—	149	2	no	—
99	C₂²⋊C₄	—	149	1	no	—
100	C₂²⋊C₄	—	149	2	no	—
101	C₂²⋊C₄	—	149	1	no	—
102	C₂²⋊C₄	—	149	2	no	—
103	C₂²⋊C₄	C₂×D₁₄₉	1	2	yes	—
104	C₄²	—	149	2	no	—
105	C₄²	—	149	2	no	—
106	C₄²	—	149	1	no	—
107	C₄²	—	149	2	no	—
108	C₂³	—	149	1	no	—
109	C₂³	C₂²×D₁₄₉	1	1	yes	characteristic
110	Q₈	—	2	2	no	—
111	Q₈	—	298	4	no	—
112	D₄	—	298	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	298	2	no	—
115	C₂×C₄	—	149	2	no	—
116	C₂×C₄	—	149	1	no	—
117	C₂×C₄	—	149	1	no	—
118	C₂×C₄	—	149	2	no	—
119	C₂×C₄	—	149	1	no	—
120	C₂×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
121	C₂×C₄	C₂²×D₁₄₉	1	2	yes	—
122	C₂×C₄	C₂²×D₁₄₉	1	2	yes	—
123	C₂×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
124	C₂²	—	2	1	no	—
125	C₂²	—	2	1	no	—
126	C₂²	—	2	1	no	—
127	C₂²	—	298	1	no	—
128	C₂²	—	298	1	no	—
129	C₂²	—	298	1	no	—
130	C₂²	C₂³×D₁₄₉	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	2	1	no	—
133	C₄	—	2	2	no	—
134	C₄	—	298	2	no	—
135	C₄	—	298	1	no	—
136	C₄	—	298	2	no	—
137	C₄	—	298	1	no	—
138	C₄	—	149	2	no	—
139	C₄	#14558735	1	2	yes	—
140	C₂	—	4	1	no	—
141	C₂	—	596	1	no	—
142	C₂	#41×D₁₄₉	1	1	yes	characteristic
143	C₂	C₂×#14558735	1	1	yes	characteristic
144	C₂	#18011959	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#18011910

C₂

yes

maximal, characteristic, complemented

#18011906

C₂

yes

maximal, complemented

#18011904

C₂

yes

maximal, complemented

#18011897

C₂

yes

maximal

#18011887

C₂

yes

maximal, characteristic, complemented

Q₈×Dic₁₄₉

C₂

yes

maximal, characteristic, complemented

#18011876

C₂

yes

maximal, complemented

C₄×C₂²⋊D₁₄₉

C₂

yes

maximal, characteristic

#18011870

C₂

yes

maximal, characteristic, complemented

#18011869

C₂

yes

maximal, characteristic, complemented

C₂²⋊Q₈×C₁₄₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₉₈

C₂²

yes

—

C₂×C₂²⋊D₁₄₉

C₂²

yes

characteristic

C₂×C₄×D₁₄₉

C₂²

yes

characteristic

C₄⋊Dic₁₄₉

C₂²

yes

—

C₄⋊Dic₁₄₉

C₂²

yes

characteristic

C₄⋊Dic₁₄₉

C₂²

yes

characteristic

#14558715

C₂²

yes

—

#14558715

C₂²

yes

—

#14558715

C₂²

yes

characteristic

C₂²⋊Dic₁₄₉

C₂²

yes

characteristic

C₂²⋊Dic₁₄₉

C₂²

yes

—

#14558713

C₂²

yes

characteristic

#14558713

C₂²

yes

—

#14558713

C₂²

yes

—

C₄×Dic₁₄₉

C₂²

yes

—

C₄×Dic₁₄₉

C₂²

yes

—

C₄×Dic₁₄₉

C₂²

yes

characteristic

C₄×Dic₁₄₉

C₂²

yes

—

C₂×Q₈×C₁₄₉

C₂²

yes

characteristic

C₂²×C₄×C₁₄₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₄₉

C₂²

yes

—

C₄⋊C₄×C₁₄₉

C₂²

yes

characteristic

C₂²⋊C₄×C₁₄₉

C₂²

yes

—

C₂²×D₁₄₉

C₂³

yes

characteristic

Dic₂₉₈

—

C₂²⋊D₁₄₉

—

C₂×Dic₁₄₉

C₂³

yes

characteristic

C₂×Dic₁₄₉

C₂³

yes

—

C₂×Dic₁₄₉

C₂³

yes

—

C₂×Dic₁₄₉

C₂³

yes

characteristic

C₂×Dic₁₄₉

C₂³

yes

characteristic

C₄×D₁₄₉

—

C₂³×C₁₄₉

C₂³

yes

characteristic

Q₈×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

C₂³

yes

—

C₂×C₄×C₁₄₉

C₂³

yes

—

C₂×C₄×C₁₄₉

C₂³

yes

characteristic

C₂×C₄×C₁₄₉

C₂³

yes

characteristic

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

C₂⁴

yes

characteristic

C₂×D₁₄₉

—

C₂×D₁₄₉

—

C₂×D₁₄₉

—

C₄×C₁₄₉

—

C₄×C₁₄₉

—

C₄×C₁₄₉

—

C₄×C₁₄₉

#41

yes

—

Dic₁₄₉

—

Dic₁₄₉

—

Dic₁₄₉

—

Dic₁₄₉

—

Dic₁₄₉

#41

yes

—

C₂×C₁₄₉

—

C₂×C₁₄₉

C₂×#41

yes

characteristic

C₂×C₁₄₉

C₂×#41

yes

characteristic

C₂×C₁₄₉

2₋¹⁺⁴

yes

characteristic

D₁₄₉

—

C₁₄₉

#556

yes

characteristic, complemented, Sylow

#556

—

149

maximal, Sylow

C₄⋊Q₈

—

149

—

#126

—

149

—

#126

—

149

—

#124

—

149

—

C₂²⋊Q₈

—

149

—

C₂²⋊Q₈

D₁₄₉

yes

characteristic, complemented

C₄×Q₈

—

149

—

C₄×Q₈

—

149

—

C₄×D₄

—

149

—

#117

—

149

—

#117

—

149

—

C₂×Q₈

—

149

—

C₂×Q₈

C₂×D₁₄₉

yes

characteristic

C₂×D₄

—

149

—

C₂²×C₄

—

149

—

C₂²×C₄

C₂×D₁₄₉

yes

characteristic

C₄⋊C₄

—

149

—

C₄⋊C₄

—

149

—

C₄⋊C₄

—

149

—

C₄⋊C₄

—

149

—

C₄⋊C₄

—

149

—

C₄⋊C₄

—

149

—

C₄⋊C₄

C₂×D₁₄₉

yes

characteristic

C₄⋊C₄

C₂×D₁₄₉

yes

—

C₂²⋊C₄

—

149

—

C₂²⋊C₄

—

149

—

100

C₂²⋊C₄

—

149

—

101

C₂²⋊C₄

—

149

—

102

C₂²⋊C₄

—

149

—

103

C₂²⋊C₄

C₂×D₁₄₉

yes

—

104

C₄²

—

149

—

105

C₄²

—

149

—

106

C₄²

—

149

—

107

C₄²

—

149

—

108

C₂³

—

149

—

109

C₂³

C₂²×D₁₄₉

yes

characteristic

110

Q₈

—

111

Q₈

—

298

—

112

D₄

—

298

—

113

C₂×C₄

—

114

C₂×C₄

—

298

—

115

C₂×C₄

—

149

—

116

C₂×C₄

—

149

—

117

C₂×C₄

—

149

—

118

C₂×C₄

—

149

—

119

C₂×C₄

—

149

—

120

C₂×C₄

C₂²×D₁₄₉

yes

characteristic

121

C₂×C₄

C₂²×D₁₄₉

yes

—

122

C₂×C₄

C₂²×D₁₄₉

yes

—

123

C₂×C₄

C₂²×D₁₄₉

yes

characteristic

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

298

—

128

C₂²

—

298

—

129

C₂²

—

298

—

130

C₂²

C₂³×D₁₄₉

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

298

—

135

C₄

—

298

—

136

C₄

—

298

—

137

C₄

—

298

—

138

C₄

—

149

—

139

C₄

#14558735

yes

—

140

C₂

—

141

C₂

—

596

—

142

C₂

#41×D₁₄₉

yes

characteristic

143

C₂

C₂×#14558735

yes

characteristic

144

C₂

#18011959

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₄₉
commutator, socle, cosocle	C₂²×C₁₄₉	C₂⁴
Fitting	C₂²⋊Q₈×C₁₄₉	C₂
prime radical	C₂×C₂²⋊D₁₄₉	C₂²
hypercenter	C₂²⋊Q₈	D₁₄₉
hypocenter	C₁₄₉	#556
LCS	G ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₄₉

commutator, socle, cosocle

C₂²×C₁₄₉

C₂⁴

Fitting

C₂²⋊Q₈×C₁₄₉

C₂

prime radical

C₂×C₂²⋊D₁₄₉

C₂²

hypercenter

C₂²⋊Q₈

D₁₄₉

C₁₄₉

G ▷ C₂²×C₁₄₉ ▷ C₁₄₉

C₁ ◁ C₂² ◁ C₂²⋊Q₈

N	H ≅ G/N	type
1. #18011910	C₂	semidirect
2. #18011906	C₂	semidirect
3. #18011904	C₂	semidirect
5. #18011887	C₂	semidirect
6. Q₈×Dic₁₄₉	C₂	semidirect
7. #18011876	C₂	semidirect
9. #18011870	C₂	semidirect
10. #18011869	C₂	semidirect
11. C₂²⋊Q₈×C₁₄₉	C₂	semidirect
72. C₁₄₉	#556	semidirect
79. C₂²⋊Q₈	D₁₄₉	semidirect

H ≅ G/N

type

1. #18011910

C₂

semidirect

2. #18011906

C₂

semidirect

3. #18011904

C₂

semidirect

5. #18011887

C₂

semidirect

6. Q₈×Dic₁₄₉

C₂

semidirect

7. #18011876

C₂

semidirect

9. #18011870

C₂

semidirect

10. #18011869

C₂

semidirect

11. C₂²⋊Q₈×C₁₄₉

C₂

semidirect

72. C₁₄₉

#556

semidirect

79. C₂²⋊Q₈

D₁₄₉

semidirect

Subgroups — #23750346

Conjugacy classes of subgroups

Complements