Group #34044913 — subgroups

Subgroups — #34044913

Order 19072.

252 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 93 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	#23750392	C₂	1	1	yes	maximal, characteristic
2	#23749674	C₂	1	2	yes	maximal, complemented
3	#23749592	C₂	1	1	yes	maximal, characteristic
4	#23749389	C₂	1	2	yes	maximal, complemented
5	#413×C₁₄₉	C₂	1	1	yes	maximal, characteristic
6	#18011918	—	2	4	no	—
7	C₂×#14558720	—	2	2	no	—
8	C₂×#14558720	C₂²	1	1	yes	characteristic
9	C₂×#14558720	C₂²	1	2	yes	—
10	C₂²×#11924746	C₂²	1	1	yes	characteristic
11	#18011847	—	2	4	no	—
12	C₂²⋊#11924746	—	2	2	no	—
13	C₂×#41×C₁₄₉	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₁₄₉	C₂²	1	2	yes	—
15	#101×C₁₄₉	C₄	1	2	yes	—
16	#100×C₁₄₉	C₄	1	2	yes	—
17	#14558720	—	2	4	no	—
18	#14558720	—	2	2	no	—
19	#14558720	—	2	2	no	—
20	#14558720	D₄	1	4	yes	—
21	C₂×#11924746	—	2	4	no	—
22	C₂×#11924746	—	2	2	no	—
23	C₂×#11924746	—	2	1	no	—
24	C₂×#11924746	—	2	1	no	—
25	C₂×#11924746	—	2	2	no	—
26	#41×C₁₄₉	—	2	2	no	—
27	#41×C₁₄₉	—	2	2	no	—
28	C₂×Q₈×C₁₄₉	C₂×C₄	1	1	yes	characteristic
29	C₂×D₄×C₁₄₉	C₂×C₄	1	1	yes	characteristic
30	C₂×D₄×C₁₄₉	Q₈	1	2	yes	—
31	C₂²×C₄×C₁₄₉	D₄	1	2	yes	—
32	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
33	M₁₆×C₁₄₉	—	2	2	no	—
34	M₁₆×C₁₄₉	C₂×C₄	1	4	yes	—
35	C₂×C₈×C₁₄₉	—	2	2	no	—
36	#11924746	—	2	4	no	—
37	#11924746	—	4	1	no	—
38	#11924746	—	4	1	no	—
39	C₂³×C₁₄₉	C₄⋊C₄	1	2	yes	—
40	C₂³×C₁₄₉	C₂×Q₈	1	1	yes	characteristic
41	Q₈×C₁₄₉	—	2	2	no	—
42	D₄×C₁₄₉	—	2	4	no	—
43	D₄×C₁₄₉	—	2	2	no	—
44	C₂×C₄×C₁₄₉	—	2	2	no	—
45	C₂×C₄×C₁₄₉	—	2	2	no	—
46	C₂×C₄×C₁₄₉	C₄⋊C₄	1	2	yes	—
47	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₄₉	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₄₉	#41	1	2	yes	—
51	C₈×C₁₄₉	—	2	4	no	—
52	C₂²×C₁₄₉	—	2	2	no	—
53	C₂²×C₁₄₉	—	4	1	no	—
54	C₂²×C₁₄₉	—	4	2	no	—
55	C₂²×C₁₄₉	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₁₄₉	C₂²⋊Q₈	1	2	yes	—
57	C₄×C₁₄₉	—	2	2	no	—
58	C₄×C₁₄₉	C₄×D₄	1	2	yes	—
59	C₄×C₁₄₉	#120	1	1	yes	characteristic
60	C₄×C₁₄₉	#123	1	1	yes	characteristic
61	C₂×C₁₄₉	—	2	1	no	—
62	C₂×C₁₄₉	—	2	2	no	—
63	C₂×C₁₄₉	—	4	2	no	—
64	C₂×C₁₄₉	#385	1	1	yes	characteristic
65	C₁₄₉	#1858	1	1	yes	characteristic, complemented, Sylow
66	#1858	—	149	1	no	maximal, Sylow
67	#568	—	149	1	no	—
68	#430	—	149	2	no	—
69	#413	D₁₄₉	1	1	yes	characteristic
70	#408	—	149	1	no	—
71	#341	—	149	2	no	—
72	C₂×#41	C₂×D₁₄₉	1	1	yes	characteristic
73	#131	—	298	4	no	—
74	C₂×M₁₆	—	298	2	no	—
75	C₂×M₁₆	—	149	1	no	—
76	C₂×M₁₆	—	149	2	no	—
77	C₂×M₁₆	C₂×D₁₄₉	1	2	yes	—
78	C₂²×C₈	—	149	1	no	—
79	#108	—	298	4	no	—
80	#101	Dic₁₄₉	1	2	yes	—
81	#100	Dic₁₄₉	1	2	yes	—
82	C₂²⋊C₈	—	298	2	no	—
83	#41	—	2	2	no	—
84	#41	—	2	2	no	—
85	C₂×Q₈	C₂×Dic₁₄₉	1	1	yes	characteristic
86	C₂×D₄	C₂×Dic₁₄₉	1	1	yes	characteristic
87	C₂×D₄	Dic₂₉₈	1	2	yes	—
88	C₂²×C₄	D₅₉₆	1	2	yes	—
89	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
90	M₁₆	—	2	2	no	—
91	M₁₆	—	298	4	no	—
92	M₁₆	—	298	2	no	—
93	M₁₆	—	298	2	no	—
94	M₁₆	—	149	4	no	—
95	M₁₆	C₂×Dic₁₄₉	1	4	yes	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	298	4	no	—
98	C₂×C₈	—	298	2	no	—
99	C₂×C₈	—	298	1	no	—
100	C₂×C₈	—	298	1	no	—
101	C₂×C₈	—	298	2	no	—
102	C₂³	C₄⋊Dic₁₄₉	1	2	yes	—
103	C₂³	C₂×Dic₂₉₈	1	1	yes	characteristic
104	Q₈	—	2	2	no	—
105	D₄	—	2	4	no	—
106	D₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	C₄⋊Dic₁₄₉	1	2	yes	—
110	C₂×C₄	C₂²×Dic₁₄₉	1	1	yes	characteristic
111	C₂×C₄	D₄×D₁₄₉	1	2	yes	—
112	C₂×C₄	C₂×D₅₉₆	1	1	yes	characteristic
113	C₂×C₄	#14558735	1	2	yes	—
114	C₈	—	2	4	no	—
115	C₈	—	298	4	no	—
116	C₈	—	596	1	no	—
117	C₈	—	596	1	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	4	1	no	—
120	C₂²	—	4	2	no	—
121	C₂²	C₂×C₄⋊Dic₁₄₉	1	1	yes	characteristic
122	C₂²	#18011888	1	2	yes	—
123	C₄	—	2	2	no	—
124	C₄	D₄×Dic₁₄₉	1	2	yes	—
125	C₄	#18011881	1	1	yes	characteristic
126	C₄	#18011895	1	1	yes	characteristic
127	C₂	—	2	1	no	—
128	C₂	—	2	2	no	—
129	C₂	—	4	2	no	—
130	C₂	#23749500	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23750392

C₂

yes

maximal, characteristic

#23749674

C₂

yes

maximal, complemented

#23749592

C₂

yes

maximal, characteristic

#23749389

C₂

yes

maximal, complemented

#413×C₁₄₉

C₂

yes

maximal, characteristic

#18011918

—

C₂×#14558720

—

C₂×#14558720

C₂²

yes

characteristic

C₂×#14558720

C₂²

yes

—

C₂²×#11924746

C₂²

yes

characteristic

#18011847

—

C₂²⋊#11924746

—

C₂×#41×C₁₄₉

C₂²

yes

characteristic

C₂×M₁₆×C₁₄₉

C₂²

yes

—

#101×C₁₄₉

C₄

yes

—

#100×C₁₄₉

C₄

yes

—

—

—

—

D₄

yes

—

—

—

—

—

—

#41×C₁₄₉

—

#41×C₁₄₉

—

C₂×Q₈×C₁₄₉

C₂×C₄

yes

characteristic

C₂×D₄×C₁₄₉

C₂×C₄

yes

characteristic

C₂×D₄×C₁₄₉

Q₈

yes

—

C₂²×C₄×C₁₄₉

D₄

yes

—

C₂²×C₄×C₁₄₉

C₂³

yes

characteristic

M₁₆×C₁₄₉

—

M₁₆×C₁₄₉

C₂×C₄

yes

—

C₂×C₈×C₁₄₉

—

#11924746

—

#11924746

—

#11924746

—

C₂³×C₁₄₉

C₄⋊C₄

yes

—

C₂³×C₁₄₉

C₂×Q₈

yes

characteristic

Q₈×C₁₄₉

—

D₄×C₁₄₉

—

D₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₄₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₄₉

C₂×D₄

yes

—

C₂×C₄×C₁₄₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₄₉

#41

yes

—

C₈×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₄₉

C₂²⋊Q₈

yes

—

C₄×C₁₄₉

—

C₄×C₁₄₉

C₄×D₄

yes

—

C₄×C₁₄₉

yes

C₄×C₁₄₉

yes

C₂×C₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

#385

yes

characteristic

C₁₄₉

#1858

yes

characteristic, complemented, Sylow

#1858

—

149

maximal, Sylow

#568

—

149

—

#430

—

149

—

#413

D₁₄₉

yes

characteristic

#408

—

149

—

#341

—

149

—

C₂×#41

C₂×D₁₄₉

yes

characteristic

#131

—

298

—

C₂×M₁₆

—

298

—

C₂×M₁₆

—

149

—

C₂×M₁₆

—

149

—

C₂×M₁₆

C₂×D₁₄₉

yes

—

C₂²×C₈

—

149

—

#108

—

298

—

#101

Dic₁₄₉

yes

—

#100

Dic₁₄₉

yes

—

C₂²⋊C₈

—

298

—

#41

—

#41

—

C₂×Q₈

C₂×Dic₁₄₉

yes

characteristic

C₂×D₄

C₂×Dic₁₄₉

yes

characteristic

C₂×D₄

Dic₂₉₈

yes

—

C₂²×C₄

D₅₉₆

yes

—

C₂²×C₄

C₂²×D₁₄₉

yes

characteristic

M₁₆

—

M₁₆

—

298

—

M₁₆

—

298

—

M₁₆

—

298

—

M₁₆

—

149

—

M₁₆

C₂×Dic₁₄₉

yes

—

C₂×C₈

—

C₂×C₈

—

298

—

C₂×C₈

—

298

—

C₂×C₈

—

298

—

100

C₂×C₈

—

298

—

101

C₂×C₈

—

298

—

102

C₂³

C₄⋊Dic₁₄₉

yes

—

103

C₂³

C₂×Dic₂₉₈

yes

characteristic

104

Q₈

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₄⋊Dic₁₄₉

yes

—

110

C₂×C₄

C₂²×Dic₁₄₉

yes

characteristic

111

C₂×C₄

D₄×D₁₄₉

yes

—

112

C₂×C₄

C₂×D₅₉₆

yes

characteristic

113

C₂×C₄

#14558735

yes

—

114

C₈

—

115

C₈

—

298

—

116

C₈

—

596

—

117

C₈

—

596

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂×C₄⋊Dic₁₄₉

yes

characteristic

122

C₂²

#18011888

yes

—

123

C₄

—

124

C₄

D₄×Dic₁₄₉

yes

—

125

C₄

#18011881

yes

characteristic

126

C₄

#18011895

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#23749500

yes

characteristic

	subgroup	G/H
center	C₄	#18011881
commutator	C₂×C₄×C₁₄₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₄₉
Fitting	#413×C₁₄₉	C₂
socle	C₂×C₁₄₉	#385
prime radical	C₂×#41×C₁₄₉	C₂²
cosocle	C₂²×C₄×C₁₄₉	C₂³
hypercenter	#413	D₁₄₉
hypocenter	C₁₄₉	#1858
LCS	G ▷ C₂×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

subgroup

G/H

center

C₄

#18011881

commutator

C₂×C₄×C₁₄₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₄₉

Fitting

#413×C₁₄₉

C₂

socle

C₂×C₁₄₉

#385

prime radical

C₂×#41×C₁₄₉

C₂²

cosocle

C₂²×C₄×C₁₄₉

C₂³

D₁₄₉

C₁₄₉

G ▷ C₂×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

N	H ≅ G/N	type
2. #23749674	C₂	semidirect
4. #23749389	C₂	semidirect
65. C₁₄₉	#1858	semidirect

H ≅ G/N

type

2. #23749674

C₂

semidirect

4. #23749389

C₂

semidirect

65. C₁₄₉

#1858

semidirect

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

subgroup	index
#23750392	2
#23749674	2
#23749592	2
#23749389	2
#413×C₁₄₉	2
#1858	149

subgroup

index

#413×C₁₄₉

149

Subgroups — #34044913

Conjugacy classes of subgroups

Complements

Maximal subgroups