Group #34046137 — subgroups

Subgroups — #34046137

Order 19072.

198 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 63 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	#23749624	C₂	1	2	yes	maximal
2	#23749591	C₂	1	1	yes	maximal, characteristic
3	#23749394	C₂	1	2	yes	maximal, complemented
4	#23749390	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#106×C₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14558736	C₂²	1	1	yes	characteristic
7	C₂×#14558720	C₂²	1	2	yes	—
8	C₂×#14558720	C₂²	1	1	yes	characteristic
9	#18011834	—	2	2	no	—
10	#18011831	—	2	2	no	—
11	#18011823	—	2	2	no	—
12	C₂²×C₈×C₁₄₉	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₁₄₉	C₂²	1	2	yes	—
14	#106×C₁₄₉	—	2	2	no	—
15	#14558736	—	2	2	no	—
16	#14558736	—	2	2	no	—
17	C₂×Dic₂₉₈	D₄	1	1	yes	characteristic
18	C₂×D₅₉₆	D₄	1	1	yes	characteristic
19	C₂×C₂²⋊D₁₄₉	—	2	1	no	—
20	C₂×C₄×D₁₄₉	—	2	1	no	—
21	#14558720	—	2	2	no	—
22	#14558720	—	2	2	no	—
23	#14558720	—	2	2	no	—
24	C₂×#11924746	—	2	1	no	—
25	C₂×#11924746	—	2	2	no	—
26	C₂²×C₄×C₁₄₉	—	2	2	no	—
27	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₁₄₉	—	2	2	no	—
29	C₂×C₈×C₁₄₉	D₄	1	2	yes	—
30	C₄⋊C₄×C₁₄₉	—	2	2	no	—
31	C₄⋊C₄×C₁₄₉	—	2	2	no	—
32	C₂²×D₁₄₉	—	2	1	no	—
33	Dic₂₉₈	—	2	2	no	—
34	D₅₉₆	—	2	2	no	—
35	C₂²⋊D₁₄₉	—	4	2	no	—
36	C₂×Dic₁₄₉	—	2	1	no	—
37	C₄×D₁₄₉	—	4	1	no	—
38	C₄×D₁₄₉	—	4	1	no	—
39	#11924746	—	4	2	no	—
40	#11924746	—	4	1	no	—
41	C₂³×C₁₄₉	#41	1	1	yes	characteristic
42	C₂×C₄×C₁₄₉	—	4	2	no	—
43	C₂×C₄×C₁₄₉	—	8	2	no	—
44	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₄₉	#41	1	2	yes	—
47	C₂×C₄×C₁₄₉	#41	1	2	yes	—
48	C₈×C₁₄₉	—	2	2	no	—
49	C₂²×C₁₄₉	—	2	2	no	—
50	C₂²×C₁₄₉	#121	1	1	yes	characteristic
51	C₂²×C₁₄₉	#126	1	2	yes	—
52	C₂×D₁₄₉	—	4	1	no	—
53	C₂×D₁₄₉	—	4	2	no	—
54	C₄×C₁₄₉	—	8	2	no	—
55	C₄×C₁₄₉	#123	1	1	yes	characteristic
56	C₄×C₁₄₉	#123	1	2	yes	—
57	C₄×C₁₄₉	#124	1	1	yes	characteristic
58	Dic₁₄₉	—	4	1	no	—
59	C₂×C₁₄₉	—	2	2	no	—
60	C₂×C₁₄₉	#397	1	1	yes	characteristic
61	C₂×C₁₄₉	#471	1	2	yes	—
62	D₁₄₉	—	8	1	no	—
63	C₁₄₉	#2079	1	1	yes	characteristic, complemented, Sylow
64	#2079	—	149	1	no	maximal, Sylow
65	C₂×#106	D₁₄₉	1	1	yes	characteristic, complemented
66	#417	—	149	2	no	—
67	#408	—	149	1	no	—
68	#343	—	149	2	no	—
69	#341	—	149	1	no	—
70	C₂×#41	—	149	1	no	—
71	C₂×M₁₆	—	149	2	no	—
72	C₂×M₁₆	—	149	1	no	—
73	C₂²×C₈	C₂×D₁₄₉	1	1	yes	characteristic
74	C₂×C₄⋊C₄	C₂×D₁₄₉	1	2	yes	—
75	#106	—	2	2	no	—
76	#103	—	298	2	no	—
77	#102	—	298	2	no	—
78	C₂²⋊C₈	—	298	2	no	—
79	#41	—	298	2	no	—
80	#41	—	298	2	no	—
81	C₂×Q₈	—	149	1	no	—
82	C₂×D₄	—	298	1	no	—
83	C₂×D₄	—	149	1	no	—
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	—	298	1	no	—
86	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
87	M₁₆	—	298	2	no	—
88	M₁₆	—	298	2	no	—
89	M₁₆	—	298	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	298	2	no	—
92	C₂×C₈	—	298	1	no	—
93	C₂×C₈	C₂²⋊D₁₄₉	1	2	yes	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	2	2	no	—
96	C₂³	—	298	1	no	—
97	C₂³	#14558735	1	1	yes	characteristic
98	Q₈	—	298	2	no	—
99	D₄	—	298	2	no	—
100	D₄	—	596	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	8	2	no	—
103	C₂×C₄	—	298	1	no	—
104	C₂×C₄	—	596	1	no	—
105	C₂×C₄	—	596	1	no	—
106	C₂×C₄	D₄×D₁₄₉	1	1	yes	characteristic
107	C₂×C₄	C₂×C₂²⋊D₁₄₉	1	1	yes	characteristic
108	C₂×C₄	#14558736	1	2	yes	—
109	C₂×C₄	#14558737	1	2	yes	—
110	C₈	—	2	2	no	—
111	C₈	—	596	2	no	—
112	C₈	—	596	1	no	—
113	C₂²	—	2	2	no	—
114	C₂²	—	596	1	no	—
115	C₂²	—	596	2	no	—
116	C₂²	#18011882	1	1	yes	characteristic
117	C₂²	#18011906	1	2	yes	—
118	C₄	—	8	2	no	—
119	C₄	—	596	1	no	—
120	C₄	#18011894	1	1	yes	characteristic
121	C₄	#18011896	1	2	yes	—
122	C₄	#18011900	1	1	yes	characteristic
123	C₂	—	2	2	no	—
124	C₂	—	1192	1	no	—
125	C₂	#23749551	1	1	yes	characteristic
126	C₂	#23749885	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23749624

C₂

yes

maximal

#23749591

C₂

yes

maximal, characteristic

#23749394

C₂

yes

maximal, complemented

#23749390

C₂

yes

maximal, characteristic, complemented

C₂×#106×C₁₄₉

C₂

yes

maximal, characteristic, complemented

C₂×#14558736

C₂²

yes

characteristic

C₂×#14558720

C₂²

yes

—

C₂×#14558720

C₂²

yes

—

—

—

C₂²×C₈×C₁₄₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₄₉

C₂²

yes

—

#106×C₁₄₉

—

#14558736

—

#14558736

—

C₂×Dic₂₉₈

D₄

yes

characteristic

C₂×D₅₉₆

D₄

yes

characteristic

C₂×C₂²⋊D₁₄₉

—

C₂×C₄×D₁₄₉

—

—

—

—

—

—

C₂²×C₄×C₁₄₉

—

C₂²×C₄×C₁₄₉

C₂³

yes

characteristic

C₂×C₈×C₁₄₉

—

C₂×C₈×C₁₄₉

D₄

yes

—

C₄⋊C₄×C₁₄₉

—

C₄⋊C₄×C₁₄₉

—

C₂²×D₁₄₉

—

Dic₂₉₈

—

D₅₉₆

—

C₂²⋊D₁₄₉

—

C₂×Dic₁₄₉

—

C₄×D₁₄₉

—

C₄×D₁₄₉

—

#11924746

—

#11924746

—

C₂³×C₁₄₉

#41

yes

characteristic

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₄₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₄₉

#41

yes

—

C₂×C₄×C₁₄₉

#41

yes

—

C₈×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

#121

yes

characteristic

C₂²×C₁₄₉

#126

yes

—

C₂×D₁₄₉

—

C₂×D₁₄₉

—

C₄×C₁₄₉

—

C₄×C₁₄₉

#123

yes

characteristic

C₄×C₁₄₉

#123

yes

—

C₄×C₁₄₉

#124

yes

characteristic

Dic₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

#397

yes

characteristic

C₂×C₁₄₉

#471

yes

—

D₁₄₉

—

C₁₄₉

#2079

yes

characteristic, complemented, Sylow

#2079

—

149

maximal, Sylow

C₂×#106

D₁₄₉

yes

characteristic, complemented

#417

—

149

—

#408

—

149

—

#343

—

149

—

#341

—

149

—

C₂×#41

—

149

—

C₂×M₁₆

—

149

—

C₂×M₁₆

—

149

—

C₂²×C₈

C₂×D₁₄₉

yes

characteristic

C₂×C₄⋊C₄

C₂×D₁₄₉

yes

—

#106

—

#103

—

298

—

#102

—

298

—

C₂²⋊C₈

—

298

—

#41

—

298

—

#41

—

298

—

C₂×Q₈

—

149

—

C₂×D₄

—

298

—

C₂×D₄

—

149

—

C₂²×C₄

—

C₂²×C₄

—

298

—

C₂²×C₄

C₂²×D₁₄₉

yes

characteristic

M₁₆

—

298

—

M₁₆

—

298

—

M₁₆

—

298

—

C₂×C₈

—

C₂×C₈

—

298

—

C₂×C₈

—

298

—

C₂×C₈

C₂²⋊D₁₄₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂³

—

298

—

C₂³

#14558735

yes

characteristic

Q₈

—

298

—

D₄

—

298

—

100

D₄

—

596

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

298

—

104

C₂×C₄

—

596

—

105

C₂×C₄

—

596

—

106

C₂×C₄

D₄×D₁₄₉

yes

characteristic

107

C₂×C₄

C₂×C₂²⋊D₁₄₉

yes

characteristic

108

C₂×C₄

#14558736

yes

—

109

C₂×C₄

#14558737

yes

—

110

C₈

—

111

C₈

—

596

—

112

C₈

—

596

—

113

C₂²

—

114

C₂²

—

596

—

115

C₂²

—

596

—

116

C₂²

#18011882

yes

characteristic

117

C₂²

#18011906

yes

—

118

C₄

—

119

C₄

—

596

—

120

C₄

#18011894

yes

characteristic

121

C₄

#18011896

yes

—

122

C₄

#18011900

yes

characteristic

123

C₂

—

124

C₂

—

1192

—

125

C₂

#23749551

yes

characteristic

126

C₂

#23749885

yes

—

	subgroup	G/H
center	C₂²	#18011882
commutator, cosocle	C₂²×C₄×C₁₄₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₄₉
Fitting	C₂×#106×C₁₄₉	C₂
socle	C₂²×C₁₄₉	#121
prime radical	C₂×#14558736	C₂²
hypercenter	C₂×#106	D₁₄₉
hypocenter	C₁₄₉	#2079
LCS	G ▷ C₂²×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂²

#18011882

commutator, cosocle

C₂²×C₄×C₁₄₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₄₉

Fitting

C₂×#106×C₁₄₉

C₂

C₂²×C₁₄₉

C₂²

D₁₄₉

C₁₄₉

G ▷ C₂²×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
3. #23749394	C₂	semidirect
4. #23749390	C₂	semidirect
5. C₂×#106×C₁₄₉	C₂	semidirect
63. C₁₄₉	#2079	semidirect
65. C₂×#106	D₁₄₉	semidirect

H ≅ G/N

type

3. #23749394

C₂

semidirect

4. #23749390

C₂

semidirect

5. C₂×#106×C₁₄₉

C₂

semidirect

63. C₁₄₉

#2079

semidirect

65. C₂×#106

D₁₄₉

semidirect

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

subgroup	index
#23749624	2
#23749591	2
#23749394	2
#23749390	2
C₂×#106×C₁₄₉	2
#2079	149

subgroup

index

C₂×#106×C₁₄₉

149

Subgroups — #34046137

Conjugacy classes of subgroups

Complements

Maximal subgroups