Group #34060923 — subgroups

Subgroups — #34060923

Order 19072.

228 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 82 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×#18011985	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#14558739	C₂	1	1	yes	maximal, characteristic, complemented
3	#23750520	C₂	1	4	yes	maximal, complemented
4	C₂×#18011919	C₂	1	1	yes	maximal, characteristic
5	#18011985	—	2	2	no	—
6	C₂×#14558739	—	2	2	no	—
7	C₂×#14558739	C₂²	1	2	yes	—
8	C₂×#14558739	C₂²	1	2	yes	—
9	C₂×#14558736	C₄	1	1	yes	characteristic
10	#18011919	—	2	2	no	—
11	#18011919	—	2	2	no	—
12	C₂×#14558719	C₄	1	2	yes	—
13	C₂²×#11924746	C₂²	1	1	yes	characteristic
14	C₂×C₈×D₁₄₉	C₂²	1	2	yes	—
15	C₂×M₁₆×C₁₄₉	C₄	1	1	yes	characteristic
16	#14558739	—	2	2	no	—
17	#14558739	—	2	2	no	—
18	#14558736	—	2	2	no	—
19	#14558736	—	2	2	no	—
20	C₂×Dic₂₉₈	C₈	1	1	yes	characteristic
21	C₂×D₅₉₆	C₈	1	1	yes	characteristic
22	C₂×C₂²⋊D₁₄₉	C₈	1	2	yes	—
23	C₂×C₄×D₁₄₉	C₂×C₄	1	2	yes	—
24	#14558719	—	2	4	no	—
25	C₂×#11924746	C₂×C₄	1	1	yes	characteristic
26	C₂×#11924746	D₄	1	2	yes	—
27	C₂×#11924746	D₄	1	2	yes	—
28	C₂×#11924746	C₂³	1	1	yes	characteristic
29	C₈×D₁₄₉	—	2	2	no	—
30	C₈×D₁₄₉	—	2	2	no	—
31	C₂²×C₄×C₁₄₉	C₂×C₄	1	1	yes	characteristic
32	M₁₆×C₁₄₉	—	2	2	no	—
33	C₂×C₈×C₁₄₉	C₂×C₄	1	2	yes	—
34	C₂²×D₁₄₉	C₂×C₈	1	2	yes	—
35	Dic₂₉₈	—	2	2	no	—
36	D₅₉₆	—	2	2	no	—
37	C₂²⋊D₁₄₉	—	2	4	no	—
38	C₂×Dic₁₄₉	C₂×C₈	1	2	yes	—
39	C₄×D₁₄₉	—	2	2	no	—
40	C₄×D₁₄₉	—	2	2	no	—
41	#11924746	C₂²⋊C₄	1	2	yes	—
42	#11924746	C₂×D₄	1	1	yes	characteristic
43	#11924746	C₂×D₄	1	1	yes	characteristic
44	C₂³×C₁₄₉	C₂×C₈	1	1	yes	characteristic
45	C₂×C₄×C₁₄₉	C₂²⋊C₄	1	2	yes	—
46	C₂×C₄×C₁₄₉	C₂×C₈	1	1	yes	characteristic
47	C₂×C₄×C₁₄₉	M₁₆	1	2	yes	—
48	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
49	C₈×C₁₄₉	—	2	2	no	—
50	C₂²×C₁₄₉	—	2	2	no	—
51	C₂²×C₁₄₉	C₂²⋊C₈	1	2	yes	—
52	C₂²×C₁₄₉	C₂²×C₈	1	1	yes	characteristic
53	C₂×D₁₄₉	—	2	2	no	—
54	C₂×D₁₄₉	—	2	4	no	—
55	C₄×C₁₄₉	C₂²⋊C₈	1	2	yes	—
56	C₄×C₁₄₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₄×C₁₄₉	C₂×M₁₆	1	1	yes	characteristic
58	Dic₁₄₉	—	2	2	no	—
59	C₂×C₁₄₉	—	2	2	no	—
60	C₂×C₁₄₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
61	C₂×C₁₄₉	#504	1	2	yes	—
62	D₁₄₉	—	4	2	no	—
63	C₁₄₉	#2113	1	1	yes	characteristic, complemented, Sylow
64	#2113	—	149	1	no	maximal, Sylow
65	C₂×#131	—	149	1	no	—
66	C₂×M₃₂	—	149	1	no	—
67	C₂²×C₁₆	—	149	1	no	—
68	C₂²⋊C₁₆	—	149	4	no	—
69	C₂×#41	—	149	1	no	—
70	#131	—	298	2	no	—
71	#131	—	298	2	no	—
72	C₂×M₁₆	—	149	2	no	—
73	C₂×M₁₆	#10597010	1	1	yes	characteristic
74	C₂²×C₈	—	149	2	no	—
75	C₂²×C₈	—	149	1	no	—
76	M₃₂	—	298	2	no	—
77	C₂×C₁₆	—	298	2	no	—
78	C₂×C₁₆	—	149	2	no	—
79	C₂×C₁₆	—	149	2	no	—
80	#41	—	298	2	no	—
81	#41	—	298	2	no	—
82	C₂×Q₈	—	149	1	no	—
83	C₂×D₄	—	149	2	no	—
84	C₂×D₄	—	149	1	no	—
85	C₂²×C₄	—	149	2	no	—
86	C₂²×C₄	C₂×#10597010	1	1	yes	characteristic
87	M₁₆	—	2	2	no	—
88	M₁₆	—	298	4	no	—
89	C₂×C₈	—	298	2	no	—
90	C₂×C₈	—	298	2	no	—
91	C₂×C₈	—	149	2	no	—
92	C₂×C₈	—	149	2	no	—
93	C₂×C₈	—	149	1	no	—
94	C₂×C₈	—	149	1	no	—
95	C₂×C₈	C₂×#10597010	1	2	yes	—
96	C₁₆	—	298	2	no	—
97	C₁₆	—	298	2	no	—
98	C₂³	—	149	2	no	—
99	C₂³	D₁₄₉⋊C₈	1	1	yes	characteristic
100	Q₈	—	298	2	no	—
101	D₄	—	298	4	no	—
102	D₄	—	298	2	no	—
103	C₂×C₄	—	298	2	no	—
104	C₂×C₄	—	298	2	no	—
105	C₂×C₄	—	149	2	no	—
106	C₂×C₄	C₂²⋊#10597010	1	2	yes	—
107	C₂×C₄	D₁₄₉⋊C₈	1	1	yes	characteristic
108	C₂×C₄	#14558746	1	2	yes	—
109	C₂×C₄	C₂²×#10597010	1	1	yes	characteristic
110	C₈	—	2	2	no	—
111	C₈	—	149	2	no	—
112	C₈	—	149	1	no	—
113	C₈	—	149	1	no	—
114	C₂²	—	2	2	no	—
115	C₂²	—	298	4	no	—
116	C₂²	—	298	2	no	—
117	C₂²	#18011968	1	2	yes	—
118	C₂²	C₂×D₁₄₉⋊C₈	1	1	yes	characteristic
119	C₄	—	298	2	no	—
120	C₄	#18011968	1	2	yes	—
121	C₄	C₂×C₂²⋊#10597010	1	1	yes	characteristic
122	C₄	C₂×#14558746	1	1	yes	characteristic
123	C₂	—	2	2	no	—
124	C₂	—	596	2	no	—
125	C₂	C₂×#18011968	1	1	yes	characteristic
126	C₂	#23750684	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#18011985

C₂

yes

maximal, characteristic, complemented

C₂²×#14558739

C₂

yes

maximal, characteristic, complemented

#23750520

C₂

yes

maximal, complemented

C₂×#18011919

C₂

yes

maximal, characteristic

#18011985

—

C₂×#14558739

—

C₂×#14558739

C₂²

yes

—

C₂×#14558739

C₂²

yes

—

C₂×#14558736

C₄

yes

—

—

C₄

yes

—

C₂²×#11924746

C₂²

yes

characteristic

C₂×C₈×D₁₄₉

C₂²

yes

—

C₂×M₁₆×C₁₄₉

C₄

yes

—

—

—

—

C₂×Dic₂₉₈

C₈

yes

characteristic

C₂×D₅₉₆

C₈

yes

characteristic

C₂×C₂²⋊D₁₄₉

C₈

yes

—

C₂×C₄×D₁₄₉

C₂×C₄

yes

—

#14558719

—

C₂×#11924746

C₂×C₄

yes

characteristic

C₂×#11924746

D₄

yes

—

C₂×#11924746

D₄

yes

—

C₂×#11924746

C₂³

yes

characteristic

C₈×D₁₄₉

—

C₈×D₁₄₉

—

C₂²×C₄×C₁₄₉

C₂×C₄

yes

characteristic

M₁₆×C₁₄₉

—

C₂×C₈×C₁₄₉

C₂×C₄

yes

—

C₂²×D₁₄₉

C₂×C₈

yes

—

Dic₂₉₈

—

D₅₉₆

—

C₂²⋊D₁₄₉

—

C₂×Dic₁₄₉

C₂×C₈

yes

—

C₄×D₁₄₉

—

C₄×D₁₄₉

—

#11924746

C₂²⋊C₄

yes

—

#11924746

C₂×D₄

yes

characteristic

#11924746

C₂×D₄

yes

characteristic

C₂³×C₁₄₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₄₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₄₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₄₉

M₁₆

yes

—

C₂×C₄×C₁₄₉

C₂²×C₄

yes

characteristic

C₈×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

C₂²⋊C₈

yes

—

C₂²×C₁₄₉

C₂²×C₈

yes

characteristic

C₂×D₁₄₉

—

C₂×D₁₄₉

—

C₄×C₁₄₉

C₂²⋊C₈

yes

—

C₄×C₁₄₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₄₉

C₂×M₁₆

yes

characteristic

Dic₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₁₄₉

#504

yes

—

D₁₄₉

—

C₁₄₉

#2113

yes

characteristic, complemented, Sylow

#2113

—

149

maximal, Sylow

C₂×#131

—

149

—

C₂×M₃₂

—

149

—

C₂²×C₁₆

—

149

—

C₂²⋊C₁₆

—

149

—

C₂×#41

—

149

—

#131

—

298

—

#131

—

298

—

C₂×M₁₆

—

149

—

C₂×M₁₆

#10597010

yes

characteristic

C₂²×C₈

—

149

—

C₂²×C₈

—

149

—

M₃₂

—

298

—

C₂×C₁₆

—

298

—

C₂×C₁₆

—

149

—

C₂×C₁₆

—

149

—

#41

—

298

—

#41

—

298

—

C₂×Q₈

—

149

—

C₂×D₄

—

149

—

C₂×D₄

—

149

—

C₂²×C₄

—

149

—

C₂²×C₄

C₂×#10597010

yes

characteristic

M₁₆

—

M₁₆

—

298

—

C₂×C₈

—

298

—

C₂×C₈

—

298

—

C₂×C₈

—

149

—

C₂×C₈

—

149

—

C₂×C₈

—

149

—

C₂×C₈

—

149

—

C₂×C₈

C₂×#10597010

yes

—

C₁₆

—

298

—

C₁₆

—

298

—

C₂³

—

149

—

C₂³

D₁₄₉⋊C₈

yes

characteristic

100

Q₈

—

298

—

101

D₄

—

298

—

102

D₄

—

298

—

103

C₂×C₄

—

298

—

104

C₂×C₄

—

298

—

105

C₂×C₄

—

149

—

106

C₂×C₄

C₂²⋊#10597010

yes

—

107

C₂×C₄

D₁₄₉⋊C₈

yes

characteristic

108

C₂×C₄

#14558746

yes

—

109

C₂×C₄

C₂²×#10597010

yes

characteristic

110

C₈

—

111

C₈

—

149

—

112

C₈

—

149

—

113

C₈

—

149

—

114

C₂²

—

115

C₂²

—

298

—

116

C₂²

—

298

—

117

C₂²

#18011968

yes

—

118

C₂²

C₂×D₁₄₉⋊C₈

yes

characteristic

119

C₄

—

298

—

120

C₄

#18011968

yes

—

121

C₄

C₂×C₂²⋊#10597010

yes

characteristic

122

C₄

C₂×#14558746

yes

characteristic

123

C₂

—

124

C₂

—

596

—

125

C₂

C₂×#18011968

yes

characteristic

126

C₂

#23750684

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#10597010
commutator, socle	C₂²×C₁₄₉	C₂²×C₈
Fitting	C₂×M₁₆×C₁₄₉	C₄
prime radical	C₂×#14558736	C₄
cosocle	C₂×#11924746	C₂³
hypercenter	C₂×M₁₆	#10597010
hypocenter	C₁₄₉	#2113
LCS	G ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#10597010

commutator, socle

C₂²×C₁₄₉

C₂²×C₈

Fitting

C₂×M₁₆×C₁₄₉

C₄

C₄

C₂³

C₂×M₁₆

C₁₄₉

G ▷ C₂²×C₁₄₉ ▷ C₁₄₉

C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

N	H ≅ G/N	type
1. C₂×#18011985	C₂	semidirect
2. C₂²×#14558739	C₂	semidirect
3. #23750520	C₂	semidirect
63. C₁₄₉	#2113	semidirect

H ≅ G/N

type

1. C₂×#18011985

C₂

semidirect

2. C₂²×#14558739

C₂

semidirect

3. #23750520

C₂

semidirect

63. C₁₄₉

#2113

semidirect

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

subgroup	index
C₂×#18011985	2
C₂²×#14558739	2
#23750520	2
C₂×#18011919	2
#2113	149

subgroup

index

149

Subgroups — #34060923

Conjugacy classes of subgroups

Complements

Maximal subgroups