Group #34061108 — subgroups

Subgroups — #34061108

Order 19072.

126 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 64 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	#23750546	C₂	1	1	yes	maximal, characteristic
2	#23750546	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#14558739	C₂	1	1	yes	maximal, characteristic
4	#23750514	C₂	1	1	yes	maximal, characteristic
5	#23750513	C₂	1	1	yes	maximal, characteristic
6	#23750511	C₂	1	1	yes	maximal, characteristic
7	#23749749	C₂	1	1	yes	maximal, characteristic
8	C₂×#14558739	C₂²	1	1	yes	characteristic
9	C₂×#14558739	C₂²	1	1	yes	characteristic
10	C₂×#14558739	C₂²	1	1	yes	characteristic
11	C₂×#14558739	C₂²	1	1	yes	characteristic
12	C₄×Dic₂₉₈	C₄	1	1	yes	characteristic
13	#18011840	C₄	1	1	yes	characteristic
14	#18011840	C₄	1	1	yes	characteristic
15	C₄×#11924746	C₂²	1	1	yes	characteristic
16	C₈×Dic₁₄₉	C₂²	1	1	yes	characteristic
17	C₈×Dic₁₄₉	C₂²	1	1	yes	characteristic
18	C₄⋊C₈×C₁₄₉	C₄	1	1	yes	characteristic
19	#14558739	—	2	1	no	—
20	#14558739	—	2	1	no	—
21	#14558739	—	2	1	no	—
22	#14558739	Q₈	1	2	yes	complemented
23	C₂×Dic₂₉₈	C₈	1	1	yes	characteristic
24	C₂×#11924746	C₂×C₄	1	1	yes	characteristic
25	C₂×#11924746	C₂³	1	1	yes	characteristic
26	C₄⋊Dic₁₄₉	C₈	1	1	yes	characteristic
27	#14558715	C₈	1	1	yes	characteristic
28	#14558715	C₈	1	1	yes	characteristic
29	C₄×Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
30	C₄×Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×C₁₄₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₈×C₁₄₉	C₂×C₄	1	1	yes	characteristic
33	C₄²×C₁₄₉	C₂×C₄	1	1	yes	characteristic
34	Dic₂₉₈	—	2	2	no	—
35	C₂×Dic₁₄₉	C₂×C₈	1	1	yes	characteristic
36	C₂×Dic₁₄₉	C₂×C₈	1	1	yes	characteristic
37	C₂×Dic₁₄₉	C₂×C₈	1	1	yes	characteristic
38	C₂×Dic₁₄₉	C₂×C₈	1	1	yes	characteristic
39	#11924746	—	2	1	no	—
40	#11924746	C₂×Q₈	1	1	yes	characteristic
41	#11924746	#41	1	1	yes	characteristic
42	C₂×C₄×C₁₄₉	C₂×C₈	1	1	yes	characteristic
43	C₂×C₄×C₁₄₉	C₂×C₈	1	1	yes	characteristic
44	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
45	C₈×C₁₄₉	—	2	1	no	—
46	C₈×C₁₄₉	—	2	1	no	—
47	C₂²×C₁₄₉	C₂²×C₈	1	1	yes	characteristic
48	C₄×C₁₄₉	—	2	1	no	—
49	C₄×C₁₄₉	—	2	1	no	—
50	C₄×C₁₄₉	C₄×Q₈	1	1	yes	characteristic
51	C₄×C₁₄₉	#131	1	1	yes	characteristic
52	Dic₁₄₉	—	2	1	no	—
53	Dic₁₄₉	—	2	1	no	—
54	Dic₁₄₉	—	2	1	no	—
55	Dic₁₄₉	M₃₂	1	2	yes	—
56	C₂×C₁₄₉	C₈×Q₈	1	1	yes	characteristic
57	C₂×C₁₄₉	C₂×M₃₂	1	1	yes	characteristic
58	C₂×C₁₄₉	#504	1	1	yes	characteristic
59	C₁₄₉	#2183	1	1	yes	characteristic, complemented, Sylow
60	#2183	—	149	1	no	maximal, Sylow
61	C₈×Q₈	—	149	1	no	—
62	C₄⋊C₁₆	—	149	1	no	—
63	C₄⋊C₁₆	—	149	1	no	—
64	C₄⋊C₁₆	—	149	1	no	—
65	#346	—	149	1	no	—
66	#346	—	149	1	no	—
67	C₄×C₁₆	—	149	1	no	—
68	C₄×Q₈	—	149	1	no	—
69	C₂×C₁₆	—	149	1	no	—
70	C₂×C₁₆	—	149	1	no	—
71	C₂×C₁₆	—	149	1	no	—
72	C₂×C₁₆	—	149	1	no	—
73	C₄⋊C₈	—	149	1	no	—
74	C₄⋊C₈	—	149	1	no	—
75	C₄⋊C₈	#10597010	1	1	yes	characteristic
76	C₄×C₈	—	149	1	no	—
77	C₄×C₈	—	149	1	no	—
78	C₄×C₈	—	149	1	no	—
79	C₂×Q₈	—	149	1	no	—
80	C₂×C₈	—	149	1	no	—
81	C₂×C₈	—	149	1	no	—
82	C₂×C₈	C₂×#10597010	1	1	yes	characteristic
83	C₂×C₈	C₂×#10597010	1	1	yes	characteristic
84	C₄⋊C₄	—	149	1	no	—
85	C₄⋊C₄	—	149	1	no	—
86	C₄⋊C₄	—	149	1	no	—
87	C₄²	—	149	1	no	—
88	C₄²	—	149	1	no	—
89	C₄²	C₂×#10597010	1	1	yes	characteristic
90	C₁₆	—	298	1	no	—
91	C₁₆	—	298	1	no	—
92	C₁₆	—	298	1	no	—
93	C₁₆	—	149	2	no	—
94	Q₈	—	298	2	no	—
95	C₂×C₄	—	149	1	no	—
96	C₂×C₄	—	149	1	no	—
97	C₂×C₄	—	149	1	no	—
98	C₂×C₄	—	149	1	no	—
99	C₂×C₄	D₁₄₉⋊C₈	1	1	yes	characteristic
100	C₂×C₄	D₁₄₉⋊C₈	1	1	yes	characteristic
101	C₂×C₄	C₂²×#10597010	1	1	yes	characteristic
102	C₈	—	2	1	no	—
103	C₈	—	2	1	no	—
104	C₈	—	298	1	no	—
105	C₈	—	149	1	no	—
106	C₈	—	149	1	no	—
107	C₂²	C₂×D₁₄₉⋊C₈	1	1	yes	characteristic
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	—	298	1	no	—
111	C₄	—	298	1	no	—
112	C₄	—	298	1	no	—
113	C₄	—	149	2	no	—
114	C₄	Q₈×#10597010	1	1	yes	characteristic
115	C₄	#18011998	1	1	yes	characteristic
116	C₂	#23750674	1	1	yes	characteristic
117	C₂	D₁₄₉⋊M₃₂	1	1	yes	characteristic
118	C₂	#23750684	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23750546

C₂

yes

maximal, characteristic

#23750546

C₂

yes

maximal, characteristic

C₄⋊#14558739

C₂

yes

maximal, characteristic

#23750514

C₂

yes

maximal, characteristic

#23750513

C₂

yes

maximal, characteristic

#23750511

C₂

yes

maximal, characteristic

#23749749

C₂

yes

maximal, characteristic

C₂×#14558739

C₂²

yes

characteristic

C₂×#14558739

C₂²

yes

characteristic

C₂×#14558739

C₂²

yes

characteristic

C₂×#14558739

C₂²

yes

characteristic

C₄×Dic₂₉₈

C₄

yes

characteristic

#18011840

C₄

yes

characteristic

#18011840

C₄

yes

characteristic

C₄×#11924746

C₂²

yes

characteristic

C₈×Dic₁₄₉

C₂²

yes

characteristic

C₈×Dic₁₄₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₄₉

C₄

yes

—

—

—

Q₈

yes

complemented

C₂×Dic₂₉₈

C₈

yes

characteristic

C₂×#11924746

C₂×C₄

yes

characteristic

C₂×#11924746

C₂³

yes

characteristic

C₄⋊Dic₁₄₉

C₈

yes

characteristic

#14558715

C₈

yes

characteristic

#14558715

C₈

yes

characteristic

C₄×Dic₁₄₉

C₂×C₄

yes

characteristic

C₄×Dic₁₄₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₄₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₄₉

C₂×C₄

yes

characteristic

C₄²×C₁₄₉

C₂×C₄

yes

characteristic

Dic₂₉₈

—

C₂×Dic₁₄₉

C₂×C₈

yes

characteristic

C₂×Dic₁₄₉

C₂×C₈

yes

characteristic

C₂×Dic₁₄₉

C₂×C₈

yes

characteristic

C₂×Dic₁₄₉

C₂×C₈

yes

characteristic

#11924746

—

#11924746

C₂×Q₈

yes

yes

C₂×C₄×C₁₄₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₄₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₄₉

C₂²×C₄

yes

characteristic

C₈×C₁₄₉

—

C₈×C₁₄₉

—

C₂²×C₁₄₉

C₂²×C₈

yes

characteristic

C₄×C₁₄₉

—

C₄×C₁₄₉

—

C₄×C₁₄₉

C₄×Q₈

yes

characteristic

C₄×C₁₄₉

#131

yes

characteristic

Dic₁₄₉

—

Dic₁₄₉

—

Dic₁₄₉

—

Dic₁₄₉

M₃₂

yes

—

C₂×C₁₄₉

C₈×Q₈

yes

characteristic

C₂×C₁₄₉

C₂×M₃₂

yes

C₂×C₁₄₉

yes

C₁₄₉

yes

characteristic, complemented, Sylow

#2183

—

149

maximal, Sylow

C₈×Q₈

—

149

—

C₄⋊C₁₆

—

149

—

C₄⋊C₁₆

—

149

—

C₄⋊C₁₆

—

149

—

#346

—

149

—

#346

—

149

—

C₄×C₁₆

—

149

—

C₄×Q₈

—

149

—

C₂×C₁₆

—

149

—

C₂×C₁₆

—

149

—

C₂×C₁₆

—

149

—

C₂×C₁₆

—

149

—

C₄⋊C₈

—

149

—

C₄⋊C₈

—

149

—

C₄⋊C₈

#10597010

yes

characteristic

C₄×C₈

—

149

—

C₄×C₈

—

149

—

C₄×C₈

—

149

—

C₂×Q₈

—

149

—

C₂×C₈

—

149

—

C₂×C₈

—

149

—

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

149

—

C₄⋊C₄

—

149

—

C₄⋊C₄

—

149

—

C₄²

—

149

—

C₄²

—

149

—

C₄²

C₂×#10597010

yes

characteristic

C₁₆

—

298

—

C₁₆

—

298

—

C₁₆

—

298

—

C₁₆

—

149

—

Q₈

—

298

—

C₂×C₄

—

149

—

C₂×C₄

—

149

—

C₂×C₄

—

149

—

C₂×C₄

—

149

—

C₂×C₄

D₁₄₉⋊C₈

yes

characteristic

100

C₂×C₄

D₁₄₉⋊C₈

yes

characteristic

101

C₂×C₄

C₂²×#10597010

yes

characteristic

102

C₈

—

103

C₈

—

104

C₈

—

298

—

105

C₈

—

149

—

106

C₈

—

149

—

107

C₂²

C₂×D₁₄₉⋊C₈

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

298

—

111

C₄

—

298

—

112

C₄

—

298

—

113

C₄

—

149

—

114

C₄

Q₈×#10597010

yes

characteristic

115

C₄

#18011998

yes

characteristic

116

C₂

#23750674

yes

characteristic

117

C₂

D₁₄₉⋊M₃₂

yes

characteristic

118

C₂

#23750684

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#10597010
commutator, socle, prime radical	C₂²×C₁₄₉	C₂²×C₈
Fitting	C₄⋊C₈×C₁₄₉	C₄
cosocle	C₂×#11924746	C₂³
hypercenter	C₄⋊C₈	#10597010
hypocenter	C₁₄₉	#2183
LCS	G ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#10597010

commutator, socle, prime radical

C₂²×C₁₄₉

C₂²×C₈

Fitting

C₄⋊C₈×C₁₄₉

C₄

C₂³

C₄⋊C₈

C₁₄₉

G ▷ C₂²×C₁₄₉ ▷ C₁₄₉

C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

N	H ≅ G/N	type
22. #14558739	Q₈	semidirect
59. C₁₄₉	#2183	semidirect

H ≅ G/N

type

22. #14558739

Q₈

semidirect

59. C₁₄₉

#2183

semidirect

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

subgroup	index
#23750546	2
C₄⋊#14558739	2
#23750514	2
#23750513	2
#23750511	2
#23749749	2
#2183	149

subgroup

index

149

Subgroups — #34061108

Conjugacy classes of subgroups

Complements

Maximal subgroups