Group #34060822 — subgroups

Subgroups — #34060822

Order 19072.

308 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 78 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	C₂×#18011974	C₂	1	4	yes	maximal
2	#23750584	C₂	1	1	yes	maximal, characteristic
3	#23750569	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Q₈×D₁₄₉	C₂	1	1	yes	maximal, characteristic
5	C₂×D₁₄₉⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊#10597010	C₂²	1	2	yes	—
7	#18011974	—	2	8	no	—
8	#18011966	—	2	2	no	—
9	(C₂×#10597010)⋊C₄	—	2	2	no	—
10	C₂×Q₈×D₁₄₉	C₂²	1	2	yes	—
11	#18011911	C₄	1	1	yes	characteristic, complemented
12	#18011910	—	2	2	no	—
13	#18011909	C₄	1	2	yes	complemented
14	C₄⋊C₄×D₁₄₉	—	2	2	no	—
15	C₄²×D₁₄₉	C₂²	1	1	yes	characteristic
16	C₄⋊Q₈×C₁₄₉	C₄	1	1	yes	characteristic, complemented
17	C₂²×#10597010	—	2	2	no	—
18	D₁₄₉⋊C₈	—	2	2	no	—
19	D₁₄₉⋊C₈	D₄	1	4	yes	—
20	C₂×#11924753	—	2	2	no	—
21	C₄⋊#10597010	—	2	4	no	—
22	C₂×Dic₂₉₈	—	2	2	no	—
23	C₂×Dic₂₉₈	C₂×C₄	1	2	yes	—
24	Q₈×D₁₄₉	—	2	4	no	—
25	Q₈×D₁₄₉	—	2	4	no	—
26	C₂×C₄×D₁₄₉	—	2	2	no	—
27	C₂×C₄×D₁₄₉	D₄	1	2	yes	—
28	C₂×C₄×D₁₄₉	C₂³	1	1	yes	characteristic
29	C₄⋊Dic₁₄₉	—	2	2	no	—
30	#14558715	—	2	4	no	—
31	C₄×Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
32	C₄×Dic₁₄₉	D₄	1	2	yes	—
33	C₂×Q₈×C₁₄₉	C₂×C₄	1	2	yes	—
34	C₄⋊C₄×C₁₄₉	—	2	2	no	—
35	C₄²×C₁₄₉	C₂×C₄	1	1	yes	characteristic
36	C₂×#10597010	—	4	2	no	—
37	C₂×#10597010	—	4	4	no	—
38	#11924753	—	2	4	no	—
39	C₂²×D₁₄₉	C₂×D₄	1	1	yes	characteristic
40	Dic₂₉₈	—	2	4	no	—
41	Dic₂₉₈	—	4	4	no	—
42	C₂×Dic₁₄₉	—	2	2	no	—
43	C₂×Dic₁₄₉	C₂²⋊C₄	1	2	yes	—
44	C₂×Dic₁₄₉	C₂×D₄	1	1	yes	characteristic
45	C₄×D₁₄₉	—	2	4	no	—
46	C₄×D₁₄₉	—	4	2	no	—
47	C₄×D₁₄₉	—	4	2	no	—
48	C₄×D₁₄₉	C₂×D₄	1	2	yes	—
49	C₄×D₁₄₉	#41	1	2	yes	—
50	Q₈×C₁₄₉	—	2	4	no	—
51	C₂×C₄×C₁₄₉	—	2	2	no	—
52	C₂×C₄×C₁₄₉	C₂²⋊C₄	1	2	yes	—
53	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₁₄₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂×D₁₄₉	#124	1	1	yes	characteristic
56	C₂×D₁₄₉	#127	1	1	yes	characteristic
57	C₂×D₁₄₉	#137	1	4	yes	—
58	#10597010	—	8	2	no	—
59	C₄×C₁₄₉	—	2	2	no	—
60	C₄×C₁₄₉	—	4	2	no	—
61	C₄×C₁₄₉	C₄×D₄	1	2	yes	—
62	Dic₁₄₉	—	2	2	no	—
63	Dic₁₄₉	—	4	2	no	—
64	Dic₁₄₉	#121	1	2	yes	—
65	C₂×C₁₄₉	#390	1	1	yes	characteristic
66	C₂×C₁₄₉	#419	1	2	yes	—
67	D₁₄₉	#491	1	2	yes	—
68	D₁₄₉	#497	1	2	yes	—
69	C₁₄₉	#1960	1	1	yes	characteristic, complemented, Sylow
70	#1960	—	149	1	no	maximal, Sylow
71	C₂×C₄⋊Q₈	—	149	1	no	—
72	C₂×#103	—	149	4	no	—
73	C₂×#97	—	149	1	no	—
74	#382	—	149	1	no	—
75	C₂²×Q₈	—	149	2	no	—
76	C₂²×C₈	—	149	2	no	—
77	C₄⋊Q₈	—	298	2	no	—
78	C₄⋊Q₈	—	149	2	no	—
79	C₄⋊Q₈	—	149	1	no	—
80	C₄⋊Q₈	#10597010	1	1	yes	characteristic, complemented
81	C₂×C₄⋊C₄	—	298	2	no	—
82	C₂×C₄⋊C₄	—	149	2	no	—
83	C₂×C₄²	—	149	1	no	—
84	#103	—	298	8	no	—
85	#97	—	298	2	no	—
86	#95	—	298	2	no	—
87	C₂×Q₈	—	298	2	no	—
88	C₂×Q₈	—	298	4	no	—
89	C₂×Q₈	—	298	4	no	—
90	C₂×Q₈	—	149	2	no	—
91	C₂×Q₈	C₂×#10597010	1	2	yes	—
92	C₂²×C₄	—	298	2	no	—
93	C₂²×C₄	—	298	2	no	—
94	C₂²×C₄	—	149	2	no	—
95	C₂²×C₄	—	149	1	no	—
96	C₂×C₈	—	298	2	no	—
97	C₂×C₈	—	298	2	no	—
98	C₂×C₈	—	149	4	no	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	298	4	no	—
101	C₄⋊C₄	—	298	2	no	—
102	C₄⋊C₄	—	298	4	no	—
103	C₄²	—	149	2	no	—
104	C₄²	—	149	1	no	—
105	C₄²	C₂×#10597010	1	1	yes	characteristic
106	C₂³	—	149	1	no	—
107	Q₈	—	2	4	no	—
108	Q₈	—	298	4	no	—
109	Q₈	—	596	4	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	298	4	no	—
112	C₂×C₄	—	298	2	no	—
113	C₂×C₄	—	596	2	no	—
114	C₂×C₄	—	596	2	no	—
115	C₂×C₄	—	596	2	no	—
116	C₂×C₄	—	596	4	no	—
117	C₂×C₄	—	149	2	no	—
118	C₂×C₄	—	149	1	no	—
119	C₂×C₄	—	149	2	no	—
120	C₂×C₄	—	149	2	no	—
121	C₂×C₄	C₂²⋊#10597010	1	2	yes	—
122	C₂×C₄	C₂²×#10597010	1	1	yes	characteristic
123	C₈	—	298	4	no	—
124	C₂²	—	149	1	no	—
125	C₂²	—	149	1	no	—
126	C₂²	—	149	4	no	—
127	C₂²	C₂×C₂²⋊#10597010	1	1	yes	characteristic
128	C₄	—	2	2	no	—
129	C₄	—	4	2	no	—
130	C₄	—	298	2	no	—
131	C₄	—	596	2	no	—
132	C₄	—	149	2	no	—
133	C₄	—	1192	2	no	—
134	C₄	D₄×#10597010	1	2	yes	—
135	C₂	—	149	2	no	—
136	C₂	—	149	2	no	—
137	C₂	#23750577	1	1	yes	characteristic
138	C₂	#23750618	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#18011974

C₂

yes

maximal

#23750584

C₂

yes

maximal, characteristic

#23750569

C₂

yes

maximal, characteristic

C₄⋊Q₈×D₁₄₉

C₂

yes

maximal, characteristic

C₂×D₁₄₉⋊C₈

C₂²

yes

—

C₂×C₄⋊#10597010

C₂²

yes

—

#18011974

—

#18011966

—

(C₂×#10597010)⋊C₄

—

C₂×Q₈×D₁₄₉

C₂²

yes

—

#18011911

C₄

yes

characteristic, complemented

#18011910

—

#18011909

C₄

yes

complemented

C₄⋊C₄×D₁₄₉

—

C₄²×D₁₄₉

C₂²

yes

characteristic

C₄⋊Q₈×C₁₄₉

C₄

yes

characteristic, complemented

C₂²×#10597010

—

D₁₄₉⋊C₈

—

D₁₄₉⋊C₈

D₄

yes

—

C₂×#11924753

—

C₄⋊#10597010

—

C₂×Dic₂₉₈

—

C₂×Dic₂₉₈

C₂×C₄

yes

—

Q₈×D₁₄₉

—

Q₈×D₁₄₉

—

C₂×C₄×D₁₄₉

—

C₂×C₄×D₁₄₉

D₄

yes

—

C₂×C₄×D₁₄₉

C₂³

yes

characteristic

C₄⋊Dic₁₄₉

—

#14558715

—

C₄×Dic₁₄₉

C₂×C₄

yes

characteristic

C₄×Dic₁₄₉

D₄

yes

—

C₂×Q₈×C₁₄₉

C₂×C₄

yes

—

C₄⋊C₄×C₁₄₉

—

C₄²×C₁₄₉

C₂×C₄

yes

—

—

—

C₂²×D₁₄₉

C₂×D₄

yes

characteristic

Dic₂₉₈

—

Dic₂₉₈

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

C₂²⋊C₄

yes

—

C₂×Dic₁₄₉

C₂×D₄

yes

characteristic

C₄×D₁₄₉

—

C₄×D₁₄₉

—

C₄×D₁₄₉

—

C₄×D₁₄₉

C₂×D₄

yes

—

C₄×D₁₄₉

#41

yes

—

Q₈×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₄₉

C₂²×C₄

yes

characteristic

C₂²×C₁₄₉

C₂×C₂²⋊C₄

yes

C₂×D₁₄₉

yes

C₂×D₁₄₉

yes

C₂×D₁₄₉

yes

—

#10597010

—

C₄×C₁₄₉

—

C₄×C₁₄₉

—

C₄×C₁₄₉

C₄×D₄

yes

—

Dic₁₄₉

—

Dic₁₄₉

—

Dic₁₄₉

#121

yes

—

C₂×C₁₄₉

#390

yes

characteristic

C₂×C₁₄₉

#419

yes

—

D₁₄₉

#491

yes

—

D₁₄₉

#497

yes

—

C₁₄₉

#1960

yes

characteristic, complemented, Sylow

#1960

—

149

maximal, Sylow

C₂×C₄⋊Q₈

—

149

—

C₂×#103

—

149

—

C₂×#97

—

149

—

#382

—

149

—

C₂²×Q₈

—

149

—

C₂²×C₈

—

149

—

C₄⋊Q₈

—

298

—

C₄⋊Q₈

—

149

—

C₄⋊Q₈

—

149

—

C₄⋊Q₈

#10597010

yes

characteristic, complemented

C₂×C₄⋊C₄

—

298

—

C₂×C₄⋊C₄

—

149

—

C₂×C₄²

—

149

—

#103

—

298

—

#97

—

298

—

#95

—

298

—

C₂×Q₈

—

298

—

C₂×Q₈

—

298

—

C₂×Q₈

—

298

—

C₂×Q₈

—

149

—

C₂×Q₈

C₂×#10597010

yes

—

C₂²×C₄

—

298

—

C₂²×C₄

—

298

—

C₂²×C₄

—

149

—

C₂²×C₄

—

149

—

C₂×C₈

—

298

—

C₂×C₈

—

298

—

C₂×C₈

—

149

—

C₄⋊C₄

—

100

C₄⋊C₄

—

298

—

101

C₄⋊C₄

—

298

—

102

C₄⋊C₄

—

298

—

103

C₄²

—

149

—

104

C₄²

—

149

—

105

C₄²

C₂×#10597010

yes

characteristic

106

C₂³

—

149

—

107

Q₈

—

108

Q₈

—

298

—

109

Q₈

—

596

—

110

C₂×C₄

—

111

C₂×C₄

—

298

—

112

C₂×C₄

—

298

—

113

C₂×C₄

—

596

—

114

C₂×C₄

—

596

—

115

C₂×C₄

—

596

—

116

C₂×C₄

—

596

—

117

C₂×C₄

—

149

—

118

C₂×C₄

—

149

—

119

C₂×C₄

—

149

—

120

C₂×C₄

—

149

—

121

C₂×C₄

C₂²⋊#10597010

yes

—

122

C₂×C₄

C₂²×#10597010

yes

characteristic

123

C₈

—

298

—

124

C₂²

—

149

—

125

C₂²

—

149

—

126

C₂²

—

149

—

127

C₂²

C₂×C₂²⋊#10597010

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

298

—

131

C₄

—

596

—

132

C₄

—

149

—

133

C₄

—

1192

—

134

C₄

D₄×#10597010

yes

—

135

C₂

—

149

—

136

C₂

—

149

—

137

C₂

#23750577

yes

characteristic

138

C₂

#23750618

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#10597010
commutator	C₂×C₄×C₁₄₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×#10597010
Fitting	C₄⋊Q₈×C₁₄₉	C₄
socle	C₂²×C₁₄₉	C₂×C₂²⋊C₄
prime radical	C₂²×D₁₄₉	C₂×D₄
cosocle	C₂×C₄×D₁₄₉	C₂³
hypercenter	C₄⋊Q₈	#10597010
hypocenter	C₁₄₉	#1960
LCS	G ▷ C₂×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#10597010

commutator

C₂×C₄×C₁₄₉

C₂²×C₄

Frattini

C₂×C₄

C₂²×#10597010

Fitting

C₄⋊Q₈×C₁₄₉

C₄

socle

C₂²×C₁₄₉

C₂×C₂²⋊C₄

prime radical

C₂²×D₁₄₉

C₂×D₄

cosocle

C₂×C₄×D₁₄₉

C₂³

C₄⋊Q₈

C₁₄₉

G ▷ C₂×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
11. #18011911	C₄	semidirect
13. #18011909	C₄	semidirect
16. C₄⋊Q₈×C₁₄₉	C₄	semidirect
69. C₁₄₉	#1960	semidirect
80. C₄⋊Q₈	#10597010	semidirect

H ≅ G/N

type

11. #18011911

C₄

semidirect

13. #18011909

C₄

semidirect

16. C₄⋊Q₈×C₁₄₉

C₄

semidirect

69. C₁₄₉

#1960

semidirect

80. C₄⋊Q₈

#10597010

semidirect

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

subgroup	index
C₂×#18011974	2
#23750584	2
#23750569	2
C₄⋊Q₈×D₁₄₉	2
#1960	149

subgroup

index

C₄⋊Q₈×D₁₄₉

149

Subgroups — #34060822

Conjugacy classes of subgroups

Complements

Maximal subgroups