Group #34060927 — subgroups

Subgroups — #34060927

Order 19072.

220 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 86 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×#18011985	C₂	1	2	yes	maximal, complemented
2	#23750521	C₂	1	4	yes	maximal, complemented
3	C₂×#18011918	C₂	1	1	yes	maximal, characteristic
4	#18011985	—	2	2	no	—
5	#18011985	C₂²	1	4	yes	—
6	C₂×#14558739	—	2	2	no	—
7	#18011918	—	2	2	no	—
8	#18011918	—	2	2	no	—
9	C₂×#14558720	C₄	1	1	yes	characteristic
10	C₂×#14558720	C₂²	1	2	yes	—
11	C₂²×#11924746	C₄	1	2	yes	—
12	C₂²×#11924746	C₂²	1	1	yes	characteristic
13	C₂×#41×C₁₄₉	C₄	1	1	yes	characteristic
14	#14558739	—	2	4	no	—
15	#14558720	—	2	4	no	—
16	#14558720	—	2	2	no	—
17	C₂×#11924746	—	2	2	no	—
18	C₂×#11924746	—	2	2	no	—
19	C₂×#11924746	C₂×C₄	1	2	yes	—
20	C₂×#11924746	C₂×C₄	1	1	yes	characteristic
21	C₂×#11924746	D₄	1	2	yes	—
22	C₂×#11924746	D₄	1	2	yes	—
23	C₂×#11924746	C₂³	1	1	yes	characteristic
24	#41×C₁₄₉	—	2	2	no	—
25	#41×C₁₄₉	—	2	2	no	—
26	C₂×Q₈×C₁₄₉	C₈	1	1	yes	characteristic
27	C₂×D₄×C₁₄₉	C₈	1	2	yes	—
28	C₂×D₄×C₁₄₉	C₈	1	1	yes	characteristic
29	C₂²×C₄×C₁₄₉	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₁₄₉	C₂×C₄	1	1	yes	characteristic
31	#11924746	—	2	2	no	—
32	#11924746	C₂²⋊C₄	1	1	yes	characteristic
33	#11924746	C₂²⋊C₄	1	1	yes	characteristic
34	#11924746	C₂×D₄	1	2	yes	—
35	C₂³×C₁₄₉	C₂×C₈	1	2	yes	—
36	C₂³×C₁₄₉	C₂×C₈	1	1	yes	characteristic
37	Q₈×C₁₄₉	—	2	2	no	—
38	D₄×C₁₄₉	—	2	4	no	—
39	D₄×C₁₄₉	—	2	2	no	—
40	C₂×C₄×C₁₄₉	—	2	2	no	—
41	C₂×C₄×C₁₄₉	—	2	2	no	—
42	C₂×C₄×C₁₄₉	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₁₄₉	C₂×C₈	1	2	yes	—
44	C₂×C₄×C₁₄₉	C₂×C₈	1	1	yes	characteristic
45	C₂×C₄×C₁₄₉	M₁₆	1	2	yes	—
46	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₁₄₉	—	2	2	no	—
48	C₂²×C₁₄₉	—	4	1	no	—
49	C₂²×C₁₄₉	—	4	2	no	—
50	C₂²×C₁₄₉	C₂²⋊C₈	1	2	yes	—
51	C₂²×C₁₄₉	C₂²×C₈	1	1	yes	characteristic
52	C₄×C₁₄₉	—	2	2	no	—
53	C₄×C₁₄₉	C₂²⋊C₈	1	2	yes	—
54	C₄×C₁₄₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₄×C₁₄₉	C₂×M₁₆	1	1	yes	characteristic
56	C₂×C₁₄₉	—	2	1	no	—
57	C₂×C₁₄₉	—	2	2	no	—
58	C₂×C₁₄₉	—	4	2	no	—
59	C₂×C₁₄₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
60	C₁₄₉	#2115	1	1	yes	characteristic, complemented, Sylow
61	#2115	—	149	1	no	maximal, Sylow
62	C₂×#131	—	149	1	no	—
63	C₂×M₃₂	—	149	2	no	—
64	#349	—	149	4	no	—
65	C₂×#41	#10597010	1	1	yes	characteristic
66	#131	—	298	2	no	—
67	#131	—	298	2	no	—
68	C₂×M₁₆	—	149	2	no	—
69	C₂×M₁₆	—	149	1	no	—
70	C₂²×C₈	—	149	2	no	—
71	C₂²×C₈	—	149	1	no	—
72	M₃₂	—	298	2	no	—
73	M₃₂	—	149	4	no	—
74	C₂×C₁₆	—	298	2	no	—
75	#41	—	2	2	no	—
76	#41	—	2	2	no	—
77	C₂×Q₈	#11924753	1	1	yes	characteristic
78	C₂×D₄	#11924753	1	2	yes	—
79	C₂×D₄	#11924753	1	1	yes	characteristic
80	C₂²×C₄	C₂×#10597010	1	2	yes	—
81	C₂²×C₄	C₂×#10597010	1	1	yes	characteristic
82	M₁₆	—	298	2	no	—
83	M₁₆	—	298	4	no	—
84	C₂×C₈	—	298	2	no	—
85	C₂×C₈	—	298	2	no	—
86	C₂×C₈	—	149	2	no	—
87	C₂×C₈	—	149	2	no	—
88	C₂×C₈	—	149	2	no	—
89	C₂×C₈	—	149	1	no	—
90	C₂×C₈	—	149	1	no	—
91	C₁₆	—	298	4	no	—
92	C₂³	C₂×#11924753	1	2	yes	—
93	C₂³	C₂×#11924753	1	1	yes	characteristic
94	Q₈	—	2	2	no	—
95	D₄	—	2	4	no	—
96	D₄	—	2	2	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	C₂²⋊#10597010	1	2	yes	—
100	C₂×C₄	C₂×#11924753	1	2	yes	—
101	C₂×C₄	C₂×#11924753	1	1	yes	characteristic
102	C₂×C₄	#14558745	1	2	yes	—
103	C₂×C₄	C₂²×#10597010	1	1	yes	characteristic
104	C₈	—	298	2	no	—
105	C₈	—	149	1	no	—
106	C₈	—	149	1	no	—
107	C₈	—	149	2	no	—
108	C₂²	—	2	2	no	—
109	C₂²	—	4	1	no	—
110	C₂²	—	4	2	no	—
111	C₂²	C₂²⋊#11924753	1	2	yes	—
112	C₂²	C₂²×#11924753	1	1	yes	characteristic
113	C₄	—	2	2	no	—
114	C₄	C₂²⋊#11924753	1	2	yes	—
115	C₄	C₂×C₂²⋊#10597010	1	1	yes	characteristic
116	C₄	C₂×#14558745	1	1	yes	characteristic
117	C₂	—	2	1	no	—
118	C₂	—	2	2	no	—
119	C₂	—	4	2	no	—
120	C₂	C₂×C₂²⋊#11924753	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#18011985

C₂

yes

maximal, complemented

#23750521

C₂

yes

maximal, complemented

C₂×#18011918

C₂

yes

maximal, characteristic

#18011985

—

#18011985

C₂²

yes

—

—

—

—

C₄

yes

characteristic

C₂×#14558720

C₂²

yes

—

C₂²×#11924746

C₄

yes

—

C₂²×#11924746

C₂²

yes

characteristic

C₂×#41×C₁₄₉

C₄

yes

—

—

—

—

—

C₂×C₄

yes

—

C₂×#11924746

C₂×C₄

yes

characteristic

C₂×#11924746

D₄

yes

—

C₂×#11924746

D₄

yes

—

C₂×#11924746

C₂³

yes

characteristic

#41×C₁₄₉

—

#41×C₁₄₉

—

C₂×Q₈×C₁₄₉

C₈

yes

characteristic

C₂×D₄×C₁₄₉

C₈

yes

—

C₂×D₄×C₁₄₉

C₈

yes

characteristic

C₂²×C₄×C₁₄₉

C₂×C₄

yes

—

C₂²×C₄×C₁₄₉

C₂×C₄

yes

characteristic

#11924746

—

#11924746

C₂²⋊C₄

yes

characteristic

#11924746

C₂²⋊C₄

yes

characteristic

#11924746

C₂×D₄

yes

—

C₂³×C₁₄₉

C₂×C₈

yes

—

C₂³×C₁₄₉

C₂×C₈

yes

characteristic

Q₈×C₁₄₉

—

D₄×C₁₄₉

—

D₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₄₉

C₂×C₈

yes

—

C₂×C₄×C₁₄₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₄₉

M₁₆

yes

—

C₂×C₄×C₁₄₉

C₂²×C₄

yes

characteristic

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

C₂²⋊C₈

yes

—

C₂²×C₁₄₉

C₂²×C₈

yes

characteristic

C₄×C₁₄₉

—

C₄×C₁₄₉

C₂²⋊C₈

yes

—

C₄×C₁₄₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₄₉

C₂×M₁₆

yes

characteristic

C₂×C₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

C₂×C₂²⋊C₈

yes

characteristic

C₁₄₉

#2115

yes

characteristic, complemented, Sylow

#2115

—

149

maximal, Sylow

C₂×#131

—

149

—

C₂×M₃₂

—

149

—

#349

—

149

—

C₂×#41

#10597010

yes

characteristic

#131

—

298

—

#131

—

298

—

C₂×M₁₆

—

149

—

C₂×M₁₆

—

149

—

C₂²×C₈

—

149

—

C₂²×C₈

—

149

—

M₃₂

—

298

—

M₃₂

—

149

—

C₂×C₁₆

—

298

—

#41

—

#41

—

C₂×Q₈

#11924753

yes

characteristic

C₂×D₄

#11924753

yes

—

C₂×D₄

#11924753

yes

characteristic

C₂²×C₄

C₂×#10597010

yes

—

C₂²×C₄

C₂×#10597010

yes

characteristic

M₁₆

—

298

—

M₁₆

—

298

—

C₂×C₈

—

298

—

C₂×C₈

—

298

—

C₂×C₈

—

149

—

C₂×C₈

—

149

—

C₂×C₈

—

149

—

C₂×C₈

—

149

—

C₂×C₈

—

149

—

C₁₆

—

298

—

C₂³

C₂×#11924753

yes

—

C₂³

C₂×#11924753

yes

characteristic

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊#10597010

yes

—

100

C₂×C₄

C₂×#11924753

yes

—

101

C₂×C₄

C₂×#11924753

yes

characteristic

102

C₂×C₄

#14558745

yes

—

103

C₂×C₄

C₂²×#10597010

yes

characteristic

104

C₈

—

298

—

105

C₈

—

149

—

106

C₈

—

149

—

107

C₈

—

149

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

C₂²⋊#11924753

yes

—

112

C₂²

C₂²×#11924753

yes

characteristic

113

C₄

—

114

C₄

C₂²⋊#11924753

yes

—

115

C₄

C₂×C₂²⋊#10597010

yes

characteristic

116

C₄

C₂×#14558745

yes

characteristic

117

C₂

—

118

C₂

—

119

C₂

—

120

C₂

C₂×C₂²⋊#11924753

yes

characteristic

	subgroup	G/H
center	C₄	C₂×C₂²⋊#10597010
commutator	C₂²×C₁₄₉	C₂²×C₈
Frattini	C₂×C₄	C₂²×#10597010
Fitting, prime radical	C₂×#41×C₁₄₉	C₄
socle	C₂×C₁₄₉	C₂×C₂²⋊C₈
cosocle	C₂×#11924746	C₂³
hypercenter	C₂×#41	#10597010
hypocenter	C₁₄₉	#2115
LCS	G ▷ C₂²×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#41

subgroup

G/H

center

C₄

C₂×C₂²⋊#10597010

commutator

C₂²×C₁₄₉

C₂²×C₈

Frattini

C₂×C₄

C₂²×#10597010

Fitting, prime radical

C₂×#41×C₁₄₉

C₄

socle

C₂×C₁₄₉

C₂×C₂²⋊C₈

C₂³

C₂×#41

C₁₄₉

G ▷ C₂²×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#41

N	H ≅ G/N	type
1. C₂×#18011985	C₂	semidirect
2. #23750521	C₂	semidirect
60. C₁₄₉	#2115	semidirect

H ≅ G/N

type

1. C₂×#18011985

C₂

semidirect

2. #23750521

C₂

semidirect

60. C₁₄₉

#2115

semidirect

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

subgroup	index
C₂×#18011985	2
#23750521	2
C₂×#18011918	2
#2115	149

subgroup

index

149

Subgroups — #34060927

Conjugacy classes of subgroups

Complements

Maximal subgroups