Group #34047410 — subgroups

Subgroups — #34047410

Order 19072.

516 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 205 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×#18011911	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₂₉₈	C₂	1	2	yes	maximal
3	#23749531	C₂	1	2	yes	maximal
4	#23749510	C₂	1	8	yes	maximal
5	#23749491	C₂	1	1	yes	maximal, characteristic
6	#376×C₁₄₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₉₈	C₂²	1	2	yes	—
8	#18011911	C₄	1	8	yes	complemented
9	C₄×Dic₂₉₈	—	2	8	no	—
10	C₂×C₄⋊Dic₁₄₉	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₁₄₉	C₂²	1	4	yes	—
12	C₂×#14558715	C₂²	1	4	yes	—
13	C₂×C₄×Dic₁₄₉	C₂²	1	4	yes	—
14	#18011818	C₂²	1	4	yes	—
15	#18011818	C₂²	1	8	yes	—
16	C₂×C₄²×C₁₄₉	C₂²	1	1	yes	characteristic
17	C₂×C₄²×C₁₄₉	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₁₄₉	C₂²	1	1	yes	characteristic
19	#95×C₁₄₉	C₂²	1	4	yes	—
20	C₂×Dic₂₉₈	—	2	4	no	—
21	C₂×Dic₂₉₈	—	2	4	no	—
22	C₂×Dic₂₉₈	C₂×C₄	1	8	yes	—
23	C₂²×Dic₁₄₉	C₂³	1	4	yes	—
24	C₂²×Dic₁₄₉	C₂³	1	4	yes	—
25	C₄⋊Dic₁₄₉	—	2	4	no	—
26	C₄⋊Dic₁₄₉	C₂×C₄	1	16	yes	—
27	#14558715	—	2	8	no	—
28	C₄×Dic₁₄₉	—	2	8	no	—
29	C₂²×C₄×C₁₄₉	C₂³	1	2	yes	—
30	C₂²×C₄×C₁₄₉	C₂³	1	2	yes	—
31	C₂²×C₄×C₁₄₉	C₂³	1	2	yes	—
32	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
33	C₄²×C₁₄₉	—	2	2	no	—
34	C₄²×C₁₄₉	C₂×C₄	1	4	yes	—
35	C₄²×C₁₄₉	D₄	1	4	yes	—
36	Dic₂₉₈	—	2	16	no	—
37	C₂×Dic₁₄₉	—	2	4	no	—
38	C₂×Dic₁₄₉	—	2	4	no	—
39	C₂×Dic₁₄₉	—	2	4	no	—
40	C₂×Dic₁₄₉	—	2	8	no	—
41	C₂×Dic₁₄₉	C₂²×C₄	1	8	yes	—
42	C₂³×C₁₄₉	C₂⁴	1	1	yes	characteristic
43	C₂×C₄×C₁₄₉	—	2	4	no	—
44	C₂×C₄×C₁₄₉	—	2	4	no	—
45	C₂×C₄×C₁₄₉	—	2	2	no	—
46	C₂×C₄×C₁₄₉	—	2	4	no	—
47	C₂×C₄×C₁₄₉	C₂²×C₄	1	4	yes	—
48	C₂×C₄×C₁₄₉	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₁₄₉	C₂×D₄	1	4	yes	—
50	C₂×C₄×C₁₄₉	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₁₄₉	#41	1	2	yes	—
52	C₂²×C₁₄₉	C₂³×C₄	1	1	yes	characteristic
53	C₂²×C₁₄₉	C₂²×D₄	1	1	yes	characteristic
54	C₂²×C₁₄₉	C₂×#41	1	1	yes	characteristic
55	C₂²×C₁₄₉	2₋¹⁺⁴	1	4	yes	—
56	C₄×C₁₄₉	—	2	4	no	—
57	C₄×C₁₄₉	—	2	4	no	—
58	C₄×C₁₄₉	—	4	2	no	—
59	C₄×C₁₄₉	C₄×D₄	1	4	yes	—
60	Dic₁₄₉	—	2	8	no	—
61	Dic₁₄₉	—	4	4	no	—
62	C₂×C₁₄₉	C₂×C₄×D₄	1	1	yes	characteristic
63	C₂×C₁₄₉	#519	1	2	yes	—
64	C₂×C₁₄₉	#536	1	2	yes	—
65	C₂×C₁₄₉	#541	1	2	yes	—
66	C₁₄₉	#2327	1	1	yes	characteristic, complemented, Sylow
67	#2327	—	149	1	no	maximal, Sylow
68	C₂×C₄⋊Q₈	—	149	1	no	—
69	C₂×C₄×Q₈	—	149	2	no	—
70	#391	—	149	2	no	—
71	#387	—	149	8	no	—
72	#382	—	149	1	no	—
73	#376	D₁₄₉	1	1	yes	characteristic
74	C₂²×Q₈	—	149	2	no	—
75	C₄⋊Q₈	—	149	8	no	—
76	C₄×Q₈	—	298	8	no	—
77	C₂×C₄⋊C₄	—	149	2	no	—
78	C₂×C₄⋊C₄	—	149	4	no	—
79	C₂×C₄⋊C₄	—	149	4	no	—
80	C₂×C₄²	—	149	4	no	—
81	C₂×C₄²	C₂×D₁₄₉	1	1	yes	characteristic
82	C₂×C₄²	C₂×D₁₄₉	1	1	yes	characteristic
83	C₂×C₄²	C₂×D₁₄₉	1	1	yes	characteristic
84	#95	—	149	4	no	—
85	#95	—	149	8	no	—
86	#95	C₂×D₁₄₉	1	4	yes	—
87	C₂×Q₈	—	298	4	no	—
88	C₂×Q₈	—	298	4	no	—
89	C₂×Q₈	—	149	8	no	—
90	C₂²×C₄	—	149	4	no	—
91	C₂²×C₄	—	149	4	no	—
92	C₂²×C₄	C₂²×D₁₄₉	1	2	yes	—
93	C₂²×C₄	C₂²×D₁₄₉	1	2	yes	—
94	C₂²×C₄	C₂²×D₁₄₉	1	2	yes	—
95	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
96	C₄⋊C₄	—	298	8	no	—
97	C₄⋊C₄	—	298	4	no	—
98	C₄⋊C₄	—	149	16	no	—
99	C₄²	—	2	2	no	—
100	C₄²	—	298	8	no	—
101	C₄²	C₄×D₁₄₉	1	4	yes	—
102	C₄²	D₅₉₆	1	4	yes	—
103	C₂³	C₂³×D₁₄₉	1	1	yes	characteristic
104	Q₈	—	298	16	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	298	4	no	—
110	C₂×C₄	—	298	4	no	—
111	C₂×C₄	—	298	4	no	—
112	C₂×C₄	—	298	8	no	—
113	C₂×C₄	—	149	8	no	—
114	C₂×C₄	C₂×C₄×D₁₄₉	1	4	yes	—
115	C₂×C₄	C₂×C₄×D₁₄₉	1	2	yes	—
116	C₂×C₄	C₂×D₅₉₆	1	4	yes	—
117	C₂×C₄	C₂×D₅₉₆	1	2	yes	—
118	C₂×C₄	#14558736	1	2	yes	—
119	C₂²	C₂²×C₄×D₁₄₉	1	1	yes	characteristic
120	C₂²	C₂²×D₅₉₆	1	1	yes	characteristic
121	C₂²	C₂×#14558736	1	1	yes	characteristic
122	C₂²	#18011958	1	4	yes	—
123	C₄	—	2	4	no	—
124	C₄	—	2	4	no	—
125	C₄	—	4	2	no	—
126	C₄	—	298	8	no	—
127	C₄	—	596	4	no	—
128	C₄	C₄×D₅₉₆	1	4	yes	—
129	C₂	C₂×C₄×D₅₉₆	1	1	yes	characteristic
130	C₂	#23750102	1	2	yes	—
131	C₂	#23750201	1	2	yes	—
132	C₂	#23750237	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#18011911

C₂

yes

maximal, characteristic

C₂×C₄×Dic₂₉₈

C₂

yes

maximal

#23749531

C₂

yes

maximal

#23749510

C₂

yes

maximal

#23749491

C₂

yes

maximal, characteristic

#376×C₁₄₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₉₈

C₂²

yes

—

#18011911

C₄

yes

complemented

C₄×Dic₂₉₈

—

C₂×C₄⋊Dic₁₄₉

C₂²

yes

—

C₂×C₄⋊Dic₁₄₉

C₂²

yes

—

C₂×#14558715

C₂²

yes

—

C₂×C₄×Dic₁₄₉

C₂²

yes

—

#18011818

C₂²

yes

—

#18011818

C₂²

yes

—

C₂×C₄²×C₁₄₉

C₂²

yes

characteristic

C₂×C₄²×C₁₄₉

C₂²

yes

characteristic

C₂×C₄²×C₁₄₉

C₂²

yes

characteristic

#95×C₁₄₉

C₂²

yes

—

C₂×Dic₂₉₈

—

C₂×Dic₂₉₈

—

C₂×Dic₂₉₈

C₂×C₄

yes

—

C₂²×Dic₁₄₉

C₂³

yes

—

C₂²×Dic₁₄₉

C₂³

yes

—

C₄⋊Dic₁₄₉

—

C₄⋊Dic₁₄₉

C₂×C₄

yes

—

#14558715

—

C₄×Dic₁₄₉

—

C₂²×C₄×C₁₄₉

C₂³

yes

—

C₂²×C₄×C₁₄₉

C₂³

yes

—

C₂²×C₄×C₁₄₉

C₂³

yes

—

C₂²×C₄×C₁₄₉

C₂³

yes

characteristic

C₄²×C₁₄₉

—

C₄²×C₁₄₉

C₂×C₄

yes

—

C₄²×C₁₄₉

D₄

yes

—

Dic₂₉₈

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

C₂²×C₄

yes

—

C₂³×C₁₄₉

C₂⁴

yes

characteristic

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

C₂²×C₄

yes

—

C₂×C₄×C₁₄₉

C₂²×C₄

yes

—

C₂×C₄×C₁₄₉

C₂×D₄

yes

—

C₂×C₄×C₁₄₉

C₂×D₄

yes

—

C₂×C₄×C₁₄₉

#41

yes

—

C₂²×C₁₄₉

C₂³×C₄

yes

characteristic

C₂²×C₁₄₉

C₂²×D₄

yes

characteristic

C₂²×C₁₄₉

C₂×#41

yes

characteristic

C₂²×C₁₄₉

2₋¹⁺⁴

yes

—

C₄×C₁₄₉

—

C₄×C₁₄₉

—

C₄×C₁₄₉

—

C₄×C₁₄₉

C₄×D₄

yes

—

Dic₁₄₉

—

Dic₁₄₉

—

C₂×C₁₄₉

C₂×C₄×D₄

yes

characteristic

C₂×C₁₄₉

#519

yes

—

C₂×C₁₄₉

#536

yes

—

C₂×C₁₄₉

#541

yes

—

C₁₄₉

#2327

yes

characteristic, complemented, Sylow

#2327

—

149

maximal, Sylow

C₂×C₄⋊Q₈

—

149

—

C₂×C₄×Q₈

—

149

—

#391

—

149

—

#387

—

149

—

#382

—

149

—

#376

D₁₄₉

yes

characteristic

C₂²×Q₈

—

149

—

C₄⋊Q₈

—

149

—

C₄×Q₈

—

298

—

C₂×C₄⋊C₄

—

149

—

C₂×C₄⋊C₄

—

149

—

C₂×C₄⋊C₄

—

149

—

C₂×C₄²

—

149

—

C₂×C₄²

C₂×D₁₄₉

yes

characteristic

C₂×C₄²

C₂×D₁₄₉

yes

characteristic

C₂×C₄²

C₂×D₁₄₉

yes

characteristic

#95

—

149

—

#95

—

149

—

#95

C₂×D₁₄₉

yes

—

C₂×Q₈

—

298

—

C₂×Q₈

—

298

—

C₂×Q₈

—

149

—

C₂²×C₄

—

149

—

C₂²×C₄

—

149

—

C₂²×C₄

C₂²×D₁₄₉

yes

—

C₂²×C₄

C₂²×D₁₄₉

yes

—

C₂²×C₄

C₂²×D₁₄₉

yes

—

C₂²×C₄

C₂²×D₁₄₉

yes

characteristic

C₄⋊C₄

—

298

—

C₄⋊C₄

—

298

—

C₄⋊C₄

—

149

—

C₄²

—

100

C₄²

—

298

—

101

C₄²

C₄×D₁₄₉

yes

—

102

C₄²

D₅₉₆

yes

—

103

C₂³

C₂³×D₁₄₉

yes

characteristic

104

Q₈

—

298

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

298

—

110

C₂×C₄

—

298

—

111

C₂×C₄

—

298

—

112

C₂×C₄

—

298

—

113

C₂×C₄

—

149

—

114

C₂×C₄

C₂×C₄×D₁₄₉

yes

—

115

C₂×C₄

C₂×C₄×D₁₄₉

yes

—

116

C₂×C₄

C₂×D₅₉₆

yes

—

117

C₂×C₄

C₂×D₅₉₆

yes

—

118

C₂×C₄

#14558736

yes

—

119

C₂²

C₂²×C₄×D₁₄₉

yes

characteristic

120

C₂²

C₂²×D₅₉₆

yes

characteristic

121

C₂²

C₂×#14558736

yes

characteristic

122

C₂²

#18011958

yes

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

298

—

127

C₄

—

596

—

128

C₄

C₄×D₅₉₆

yes

—

129

C₂

C₂×C₄×D₅₉₆

yes

characteristic

130

C₂

#23750102

yes

—

131

C₂

#23750201

yes

—

132

C₂

#23750237

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₄₉
commutator	C₂²×C₁₄₉	C₂³×C₄
Fitting	#376×C₁₄₉	C₂
socle, prime radical, cosocle	C₂³×C₁₄₉	C₂⁴
hypercenter	#376	D₁₄₉
hypocenter	C₁₄₉	#2327
LCS	G ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂³ ◁ #376

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₄₉

commutator

C₂²×C₁₄₉

C₂³×C₄

Fitting

#376×C₁₄₉

C₂

socle, prime radical, cosocle

C₂³×C₁₄₉

C₂⁴

D₁₄₉

C₁₄₉

G ▷ C₂²×C₁₄₉ ▷ C₁₄₉

C₁ ◁ C₂³ ◁ #376

N	H ≅ G/N	type
8. #18011911	C₄	semidirect
66. C₁₄₉	#2327	semidirect

H ≅ G/N

type

8. #18011911

C₄

semidirect

66. C₁₄₉

#2327

semidirect

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

subgroup	index
C₂×#18011911	2
C₂×C₄×Dic₂₉₈	2
#23749531	2
#23749510	2
#23749491	2
#376×C₁₄₉	2
#2327	149

subgroup

index

C₂×C₄×Dic₂₉₈

#376×C₁₄₉

149

Subgroups — #34047410

Conjugacy classes of subgroups

Complements

Maximal subgroups