Group #34044800 — subgroups

Subgroups — #34044800

Order 19072.

272 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 81 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	#23750100	C₂	1	1	yes	maximal, characteristic
2	C₂×#18011828	C₂	1	1	yes	maximal, characteristic
3	#23749397	C₂	1	2	yes	maximal, complemented
4	#23749396	C₂	1	2	yes	maximal, complemented
5	C₂×#100×C₁₄₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14558720	C₂²	1	2	yes	—
7	D₄×Dic₁₄₉	—	2	2	no	—
8	D₄×Dic₁₄₉	—	2	2	no	—
9	#18011870	C₂²	1	1	yes	characteristic
10	#18011870	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊Dic₁₄₉	C₄	1	4	yes	—
12	#18011828	—	2	2	no	—
13	C₂²×D₄×C₁₄₉	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₁₄₉	C₂²	1	2	yes	—
15	#100×C₁₄₉	—	2	2	no	—
16	C₂²×Dic₁₄₉	—	2	4	no	—
17	#14558720	—	2	2	no	—
18	#14558720	—	2	2	no	—
19	C₂×#11924746	—	2	2	no	—
20	C₄⋊Dic₁₄₉	—	2	1	no	—
21	C₄⋊Dic₁₄₉	—	2	1	no	—
22	C₂²⋊Dic₁₄₉	—	2	4	no	—
23	C₂²⋊Dic₁₄₉	C₂×C₄	1	2	yes	—
24	C₂²⋊Dic₁₄₉	C₂×C₄	1	2	yes	—
25	C₄×Dic₁₄₉	—	2	1	no	—
26	C₄×Dic₁₄₉	—	2	1	no	—
27	C₂⁴×C₁₄₉	C₂×C₄	1	2	yes	—
28	C₂×D₄×C₁₄₉	—	2	2	no	—
29	C₂×D₄×C₁₄₉	—	2	2	no	—
30	C₂×D₄×C₁₄₉	D₄	1	2	yes	—
31	C₂×D₄×C₁₄₉	D₄	1	2	yes	—
32	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
33	M₁₆×C₁₄₉	—	2	2	no	—
34	M₁₆×C₁₄₉	—	2	2	no	—
35	C₂×C₈×C₁₄₉	—	2	2	no	—
36	C₂×Dic₁₄₉	—	2	2	no	—
37	C₂×Dic₁₄₉	—	2	2	no	—
38	C₂×Dic₁₄₉	—	8	4	no	—
39	#11924746	—	4	2	no	—
40	C₂³×C₁₄₉	—	2	2	no	—
41	C₂³×C₁₄₉	—	2	2	no	—
42	C₂³×C₁₄₉	—	8	2	no	—
43	C₂³×C₁₄₉	C₂²⋊C₄	1	4	yes	—
44	C₂³×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
45	D₄×C₁₄₉	—	2	4	no	—
46	D₄×C₁₄₉	—	2	4	no	—
47	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₄₉	#41	1	2	yes	—
50	C₂×C₄×C₁₄₉	#41	1	2	yes	—
51	C₈×C₁₄₉	—	4	2	no	—
52	C₂²×C₁₄₉	—	2	4	no	—
53	C₂²×C₁₄₉	—	4	1	no	—
54	C₂²×C₁₄₉	—	4	4	no	—
55	C₂²×C₁₄₉	—	8	2	no	—
56	C₂²×C₁₄₉	—	8	2	no	—
57	C₂²×C₁₄₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₁₄₉	#117	1	1	yes	characteristic
59	C₂²×C₁₄₉	#117	1	1	yes	characteristic
60	C₄×C₁₄₉	#123	1	2	yes	—
61	C₄×C₁₄₉	#123	1	2	yes	—
62	Dic₁₄₉	—	4	2	no	—
63	Dic₁₄₉	—	4	2	no	—
64	C₂×C₁₄₉	—	2	1	no	—
65	C₂×C₁₄₉	—	2	1	no	—
66	C₂×C₁₄₉	—	2	1	no	—
67	C₂×C₁₄₉	—	4	4	no	—
68	C₂×C₁₄₉	#381	1	1	yes	characteristic
69	C₁₄₉	#1829	1	1	yes	characteristic, complemented, Sylow
70	#1829	—	149	1	no	maximal, Sylow
71	#518	—	149	1	no	—
72	C₂×#100	—	149	1	no	—
73	C₂×#100	D₁₄₉	1	1	yes	characteristic
74	#344	—	149	2	no	—
75	#344	—	149	2	no	—
76	C₂²×D₄	C₂×D₁₄₉	1	1	yes	characteristic
77	C₂×M₁₆	—	149	2	no	—
78	C₂×M₁₆	C₂×D₁₄₉	1	2	yes	—
79	C₄×D₄	—	298	2	no	—
80	C₄×D₄	—	298	2	no	—
81	#117	—	149	1	no	—
82	#117	—	149	1	no	—
83	C₂×C₂²⋊C₄	—	149	4	no	—
84	#100	—	2	2	no	—
85	#100	—	298	2	no	—
86	C₂⁴	C₄×D₁₄₉	1	2	yes	—
87	C₂×D₄	—	2	2	no	—
88	C₂×D₄	—	2	2	no	—
89	C₂×D₄	C₂²⋊D₁₄₉	1	2	yes	—
90	C₂×D₄	D₅₉₆	1	2	yes	—
91	C₂²×C₄	—	298	4	no	—
92	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
93	M₁₆	—	2	2	no	—
94	M₁₆	—	2	2	no	—
95	M₁₆	—	298	2	no	—
96	M₁₆	—	298	2	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	298	2	no	—
99	C₄⋊C₄	—	298	1	no	—
100	C₄⋊C₄	—	298	1	no	—
101	C₂²⋊C₄	—	298	4	no	—
102	C₂²⋊C₄	—	149	2	no	—
103	C₂²⋊C₄	—	149	2	no	—
104	C₄²	—	298	1	no	—
105	C₄²	—	298	1	no	—
106	C₂³	—	2	2	no	—
107	C₂³	—	2	2	no	—
108	C₂³	—	8	2	no	—
109	C₂³	#14558713	1	4	yes	—
110	C₂³	C₂×C₄×D₁₄₉	1	1	yes	characteristic
111	D₄	—	2	4	no	—
112	D₄	—	2	4	no	—
113	C₂×C₄	—	298	2	no	—
114	C₂×C₄	—	298	2	no	—
115	C₂×C₄	—	1192	4	no	—
116	C₂×C₄	C₂×C₂²⋊D₁₄₉	1	1	yes	characteristic
117	C₂×C₄	C₂×D₅₉₆	1	1	yes	characteristic
118	C₂×C₄	#14558735	1	2	yes	—
119	C₂×C₄	#14558735	1	2	yes	—
120	C₈	—	4	2	no	—
121	C₈	—	596	2	no	—
122	C₂²	—	2	4	no	—
123	C₂²	—	4	1	no	—
124	C₂²	—	4	4	no	—
125	C₂²	—	8	2	no	—
126	C₂²	—	8	2	no	—
127	C₂²	C₂×#14558713	1	1	yes	characteristic
128	C₂²	#18011867	1	1	yes	characteristic
129	C₂²	#18011867	1	1	yes	characteristic
130	C₄	—	596	2	no	—
131	C₄	—	596	2	no	—
132	C₄	#18011892	1	2	yes	—
133	C₄	#18011895	1	2	yes	—
134	C₂	—	2	1	no	—
135	C₂	—	2	1	no	—
136	C₂	—	2	1	no	—
137	C₂	—	4	4	no	—
138	C₂	#23749486	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23750100

C₂

yes

maximal, characteristic

C₂×#18011828

C₂

yes

maximal, characteristic

#23749397

C₂

yes

maximal, complemented

#23749396

C₂

yes

maximal, complemented

C₂×#100×C₁₄₉

C₂

yes

maximal, characteristic

C₂×#14558720

C₂²

yes

—

D₄×Dic₁₄₉

—

D₄×Dic₁₄₉

—

#18011870

C₂²

yes

characteristic

#18011870

C₂²

yes

characteristic

C₂×C₂²⋊Dic₁₄₉

C₄

yes

—

#18011828

—

C₂²×D₄×C₁₄₉

C₂²

yes

characteristic

C₂×M₁₆×C₁₄₉

C₂²

yes

—

#100×C₁₄₉

—

C₂²×Dic₁₄₉

—

#14558720

—

#14558720

—

C₂×#11924746

—

C₄⋊Dic₁₄₉

—

C₄⋊Dic₁₄₉

—

C₂²⋊Dic₁₄₉

—

C₂²⋊Dic₁₄₉

C₂×C₄

yes

—

C₂²⋊Dic₁₄₉

C₂×C₄

yes

—

C₄×Dic₁₄₉

—

C₄×Dic₁₄₉

—

C₂⁴×C₁₄₉

C₂×C₄

yes

—

C₂×D₄×C₁₄₉

—

C₂×D₄×C₁₄₉

—

C₂×D₄×C₁₄₉

D₄

yes

—

C₂×D₄×C₁₄₉

D₄

yes

—

C₂²×C₄×C₁₄₉

C₂³

yes

characteristic

M₁₆×C₁₄₉

—

M₁₆×C₁₄₉

—

C₂×C₈×C₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

#11924746

—

C₂³×C₁₄₉

—

C₂³×C₁₄₉

—

C₂³×C₁₄₉

—

C₂³×C₁₄₉

C₂²⋊C₄

yes

—

C₂³×C₁₄₉

C₂²×C₄

yes

characteristic

D₄×C₁₄₉

—

D₄×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

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

C₂×C₂²⋊C₄

yes

C₂²×C₁₄₉

yes

C₂²×C₁₄₉

yes

C₄×C₁₄₉

yes

—

C₄×C₁₄₉

#123

yes

—

Dic₁₄₉

—

Dic₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

#381

yes

characteristic

C₁₄₉

#1829

yes

characteristic, complemented, Sylow

#1829

—

149

maximal, Sylow

#518

—

149

—

C₂×#100

—

149

—

C₂×#100

D₁₄₉

yes

characteristic

#344

—

149

—

#344

—

149

—

C₂²×D₄

C₂×D₁₄₉

yes

characteristic

C₂×M₁₆

—

149

—

C₂×M₁₆

C₂×D₁₄₉

yes

—

C₄×D₄

—

298

—

C₄×D₄

—

298

—

#117

—

149

—

#117

—

149

—

C₂×C₂²⋊C₄

—

149

—

#100

—

#100

—

298

—

C₂⁴

C₄×D₁₄₉

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂²⋊D₁₄₉

yes

—

C₂×D₄

D₅₉₆

yes

—

C₂²×C₄

—

298

—

C₂²×C₄

C₂²×D₁₄₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

298

—

M₁₆

—

298

—

C₂×C₈

—

C₂×C₈

—

298

—

C₄⋊C₄

—

298

—

100

C₄⋊C₄

—

298

—

101

C₂²⋊C₄

—

298

—

102

C₂²⋊C₄

—

149

—

103

C₂²⋊C₄

—

149

—

104

C₄²

—

298

—

105

C₄²

—

298

—

106

C₂³

—

107

C₂³

—

108

C₂³

—

109

C₂³

#14558713

yes

—

110

C₂³

C₂×C₄×D₁₄₉

yes

characteristic

111

D₄

—

112

D₄

—

113

C₂×C₄

—

298

—

114

C₂×C₄

—

298

—

115

C₂×C₄

—

1192

—

116

C₂×C₄

C₂×C₂²⋊D₁₄₉

yes

characteristic

117

C₂×C₄

C₂×D₅₉₆

yes

characteristic

118

C₂×C₄

#14558735

yes

—

119

C₂×C₄

#14558735

yes

—

120

C₈

—

121

C₈

—

596

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

C₂×#14558713

yes

characteristic

128

C₂²

#18011867

yes

characteristic

129

C₂²

#18011867

yes

characteristic

130

C₄

—

596

—

131

C₄

—

596

—

132

C₄

#18011892

yes

—

133

C₄

#18011895

yes

—

134

C₂

—

135

C₂

—

136

C₂

—

137

C₂

—

138

C₂

#23749486

yes

characteristic

	subgroup	G/H
center	C₂	#23749486
commutator	C₂³×C₁₄₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₄₉
Fitting	C₂×#100×C₁₄₉	C₂
socle	C₂×C₁₄₉	#381
prime radical	C₂²×D₄×C₁₄₉	C₂²
cosocle	C₂²×C₄×C₁₄₉	C₂³
hypercenter	C₂×#100	D₁₄₉
hypocenter	C₁₄₉	#1829
LCS	G ▷ C₂³×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#100

subgroup

G/H

center

C₂

#23749486

commutator

C₂³×C₁₄₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₄₉

Fitting

C₂×#100×C₁₄₉

C₂

socle

C₂×C₁₄₉

#381

prime radical

C₂²×D₄×C₁₄₉

C₂²

cosocle

C₂²×C₄×C₁₄₉

C₂³

D₁₄₉

C₁₄₉

G ▷ C₂³×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#100

N	H ≅ G/N	type
3. #23749397	C₂	semidirect
4. #23749396	C₂	semidirect
69. C₁₄₉	#1829	semidirect

H ≅ G/N

type

3. #23749397

C₂

semidirect

4. #23749396

C₂

semidirect

69. C₁₄₉

#1829

semidirect

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

subgroup	index
#23750100	2
C₂×#18011828	2
#23749397	2
#23749396	2
C₂×#100×C₁₄₉	2
#1829	149

subgroup

index

C₂×#100×C₁₄₉

149

Subgroups — #34044800

Conjugacy classes of subgroups

Complements

Maximal subgroups