Group #34061601 — subgroups

Subgroups — #34061601

Order 19072.

330 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 148 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	#23750677	C₂	1	4	yes	maximal, complemented
2	#23750674	C₂	1	4	yes	maximal, complemented
3	#23750650	C₂	1	4	yes	maximal
4	#23750643	C₂	1	2	yes	maximal
5	#23750181	C₂	1	1	yes	maximal, characteristic
6	#18011978	C₂²	1	2	yes	—
7	#18011978	C₂²	1	4	yes	—
8	#18011978	C₂²	1	2	yes	—
9	C₄⋊#11924753	C₂²	1	4	yes	—
10	#18011968	C₂²	1	4	yes	—
11	#18011966	C₂²	1	4	yes	—
12	C₄×#11924753	C₂²	1	4	yes	—
13	C₄×#11924753	C₂²	1	4	yes	—
14	#18011910	C₄	1	2	yes	—
15	#18011903	C₄	1	1	yes	characteristic
16	#18011889	C₄	1	4	yes	—
17	#18011876	C₂²	1	4	yes	—
18	#18011869	C₂²	1	2	yes	—
19	C₄²×D₁₄₉	C₂²	1	1	yes	characteristic
20	#125×C₁₄₉	C₄	1	1	yes	characteristic
21	C₂×#11924753	C₂³	1	4	yes	—
22	C₂×#11924753	C₂³	1	4	yes	—
23	C₂×Dic₂₉₈	C₂×C₄	1	4	yes	—
24	C₂×C₄×D₁₄₉	C₂×C₄	1	2	yes	—
25	C₂×C₄×D₁₄₉	C₂³	1	1	yes	characteristic
26	C₄⋊Dic₁₄₉	C₂×C₄	1	2	yes	—
27	#14558715	C₂×C₄	1	4	yes	—
28	#14558715	C₂×C₄	1	4	yes	—
29	#14558713	C₂×C₄	1	4	yes	—
30	C₄×Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
31	C₄×Dic₁₄₉	C₂³	1	2	yes	—
32	C₄×Dic₁₄₉	C₂³	1	4	yes	—
33	C₄⋊C₄×C₁₄₉	C₂×C₄	1	2	yes	—
34	C₄⋊C₄×C₁₄₉	C₂×C₄	1	4	yes	—
35	C₄²×C₁₄₉	C₂×C₄	1	1	yes	characteristic
36	#11924753	—	2	4	no	—
37	#11924753	—	2	4	no	—
38	C₂²×D₁₄₉	C₂²×C₄	1	1	yes	characteristic
39	Dic₂₉₈	—	2	8	no	—
40	C₂×Dic₁₄₉	C₂²×C₄	1	4	yes	—
41	C₂×Dic₁₄₉	C₂²×C₄	1	2	yes	—
42	C₂×Dic₁₄₉	C₂⁴	1	1	yes	characteristic
43	C₄×D₁₄₉	—	2	4	no	—
44	C₄×D₁₄₉	—	2	2	no	—
45	C₂×C₄×C₁₄₉	C₂²×C₄	1	2	yes	—
46	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₄₉	C₂²×C₄	1	4	yes	—
48	C₂²×C₁₄₉	C₂³×C₄	1	1	yes	characteristic
49	C₂×D₁₄₉	—	2	1	no	—
50	C₂×D₁₄₉	—	2	2	no	—
51	C₄×C₁₄₉	—	2	1	no	—
52	C₄×C₁₄₉	—	2	2	no	—
53	C₄×C₁₄₉	—	2	4	no	—
54	Dic₁₄₉	—	2	4	no	—
55	Dic₁₄₉	#131	1	4	yes	—
56	Dic₁₄₉	2₋¹⁺⁴	1	2	yes	—
57	C₂×C₁₄₉	#519	1	1	yes	characteristic
58	C₂×C₁₄₉	C₂×#131	1	2	yes	—
59	D₁₄₉	—	4	1	no	—
60	C₁₄₉	#2986	1	1	yes	characteristic, complemented, Sylow
61	#2986	—	149	1	no	maximal, Sylow
62	#533	—	149	1	no	—
63	#446	—	149	4	no	—
64	C₈×Q₈	—	149	4	no	—
65	#433	—	149	4	no	—
66	#431	—	149	2	no	—
67	C₄⋊Q₈	—	149	2	no	—
68	#125	—	149	1	no	—
69	#125	#10597010	1	1	yes	characteristic
70	C₂²⋊Q₈	—	149	4	no	—
71	C₄×Q₈	—	149	4	no	—
72	#117	—	149	2	no	—
73	C₂×C₄²	—	149	1	no	—
74	C₄⋊C₈	—	149	4	no	—
75	C₄⋊C₈	—	149	2	no	—
76	C₄⋊C₈	—	149	2	no	—
77	C₄⋊C₈	—	149	4	no	—
78	C₂²⋊C₈	—	149	4	no	—
79	#97	—	149	4	no	—
80	C₄×C₈	—	149	4	no	—
81	C₄×C₈	—	149	4	no	—
82	C₂×Q₈	—	149	4	no	—
83	C₂²×C₄	—	149	2	no	—
84	C₂²×C₄	—	149	1	no	—
85	C₂×C₈	—	149	4	no	—
86	C₂×C₈	—	149	4	no	—
87	C₄⋊C₄	—	149	2	no	—
88	C₄⋊C₄	—	149	4	no	—
89	C₄⋊C₄	—	149	4	no	—
90	C₄⋊C₄	C₂×#10597010	1	2	yes	—
91	C₄⋊C₄	C₂×#10597010	1	4	yes	—
92	C₂²⋊C₄	—	149	4	no	—
93	C₄²	—	149	2	no	—
94	C₄²	—	149	4	no	—
95	C₄²	—	149	1	no	—
96	C₄²	C₂×#10597010	1	1	yes	characteristic
97	C₂³	—	149	1	no	—
98	Q₈	—	298	8	no	—
99	C₂×C₄	—	298	2	no	—
100	C₂×C₄	—	298	4	no	—
101	C₂×C₄	—	149	4	no	—
102	C₂×C₄	—	149	2	no	—
103	C₂×C₄	—	149	1	no	—
104	C₂×C₄	C₂²×#10597010	1	1	yes	characteristic
105	C₂×C₄	C₂²×#10597010	1	2	yes	—
106	C₂×C₄	C₂²×#10597010	1	4	yes	—
107	C₈	—	298	4	no	—
108	C₈	—	298	4	no	—
109	C₂²	—	298	1	no	—
110	C₂²	—	298	2	no	—
111	C₂²	C₂³×#10597010	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	2	2	no	—
114	C₄	—	2	4	no	—
115	C₄	—	298	4	no	—
116	C₄	—	149	2	no	—
117	C₄	—	149	4	no	—
118	C₂	—	596	1	no	—
119	C₂	#23750693	1	1	yes	characteristic
120	C₂	D₁₄₉⋊#131	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23750677

C₂

yes

maximal, complemented

#23750674

C₂

yes

maximal, complemented

#23750650

C₂

yes

maximal

#23750643

C₂

yes

maximal

#23750181

C₂

yes

maximal, characteristic

#18011978

C₂²

yes

—

#18011978

C₂²

yes

—

#18011978

C₂²

yes

—

C₄⋊#11924753

C₂²

yes

—

#18011968

C₂²

yes

—

#18011966

C₂²

yes

—

C₄×#11924753

C₂²

yes

—

C₄×#11924753

C₂²

yes

—

#18011910

C₄

yes

—

#18011903

C₄

yes

characteristic

#18011889

C₄

yes

—

#18011876

C₂²

yes

—

#18011869

C₂²

yes

—

C₄²×D₁₄₉

C₂²

yes

characteristic

#125×C₁₄₉

C₄

yes

characteristic

C₂×#11924753

C₂³

yes

—

C₂×#11924753

C₂³

yes

—

C₂×Dic₂₉₈

C₂×C₄

yes

—

C₂×C₄×D₁₄₉

C₂×C₄

yes

—

C₂×C₄×D₁₄₉

C₂³

yes

characteristic

C₄⋊Dic₁₄₉

C₂×C₄

yes

—

#14558715

C₂×C₄

yes

—

#14558715

C₂×C₄

yes

—

#14558713

C₂×C₄

yes

—

C₄×Dic₁₄₉

C₂×C₄

yes

characteristic

C₄×Dic₁₄₉

C₂³

yes

—

C₄×Dic₁₄₉

C₂³

yes

—

C₄⋊C₄×C₁₄₉

C₂×C₄

yes

—

C₄⋊C₄×C₁₄₉

C₂×C₄

yes

—

C₄²×C₁₄₉

C₂×C₄

yes

characteristic

#11924753

—

#11924753

—

C₂²×D₁₄₉

C₂²×C₄

yes

characteristic

Dic₂₉₈

—

C₂×Dic₁₄₉

C₂²×C₄

yes

—

C₂×Dic₁₄₉

C₂²×C₄

yes

—

C₂×Dic₁₄₉

C₂⁴

yes

characteristic

C₄×D₁₄₉

—

C₄×D₁₄₉

—

C₂×C₄×C₁₄₉

C₂²×C₄

yes

—

C₂×C₄×C₁₄₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₄₉

C₂²×C₄

yes

—

C₂²×C₁₄₉

C₂³×C₄

yes

characteristic

C₂×D₁₄₉

—

C₂×D₁₄₉

—

C₄×C₁₄₉

—

C₄×C₁₄₉

—

C₄×C₁₄₉

—

Dic₁₄₉

—

Dic₁₄₉

#131

yes

—

Dic₁₄₉

2₋¹⁺⁴

yes

—

C₂×C₁₄₉

#519

yes

characteristic

C₂×C₁₄₉

C₂×#131

yes

—

D₁₄₉

—

C₁₄₉

#2986

yes

characteristic, complemented, Sylow

#2986

—

149

maximal, Sylow

#533

—

149

—

#446

—

149

—

C₈×Q₈

—

149

—

#433

—

149

—

#431

—

149

—

C₄⋊Q₈

—

149

—

#125

—

149

—

#125

#10597010

yes

characteristic

C₂²⋊Q₈

—

149

—

C₄×Q₈

—

149

—

#117

—

149

—

C₂×C₄²

—

149

—

C₄⋊C₈

—

149

—

C₄⋊C₈

—

149

—

C₄⋊C₈

—

149

—

C₄⋊C₈

—

149

—

C₂²⋊C₈

—

149

—

#97

—

149

—

C₄×C₈

—

149

—

C₄×C₈

—

149

—

C₂×Q₈

—

149

—

C₂²×C₄

—

149

—

C₂²×C₄

—

149

—

C₂×C₈

—

149

—

C₂×C₈

—

149

—

C₄⋊C₄

—

149

—

C₄⋊C₄

—

149

—

C₄⋊C₄

—

149

—

C₄⋊C₄

C₂×#10597010

yes

—

C₄⋊C₄

C₂×#10597010

yes

—

C₂²⋊C₄

—

149

—

C₄²

—

149

—

C₄²

—

149

—

C₄²

—

149

—

C₄²

C₂×#10597010

yes

characteristic

C₂³

—

149

—

Q₈

—

298

—

C₂×C₄

—

298

—

100

C₂×C₄

—

298

—

101

C₂×C₄

—

149

—

102

C₂×C₄

—

149

—

103

C₂×C₄

—

149

—

104

C₂×C₄

C₂²×#10597010

yes

characteristic

105

C₂×C₄

C₂²×#10597010

yes

—

106

C₂×C₄

C₂²×#10597010

yes

—

107

C₈

—

298

—

108

C₈

—

298

—

109

C₂²

—

298

—

110

C₂²

—

298

—

111

C₂²

C₂³×#10597010

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

298

—

116

C₄

—

149

—

117

C₄

—

149

—

118

C₂

—

596

—

119

C₂

#23750693

yes

characteristic

120

C₂

D₁₄₉⋊#131

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×#10597010
commutator, socle	C₂²×C₁₄₉	C₂³×C₄
Fitting	#125×C₁₄₉	C₄
prime radical	C₂²×D₁₄₉	C₂²×C₄
cosocle	C₂×Dic₁₄₉	C₂⁴
hypercenter	#125	#10597010
hypocenter	C₁₄₉	#2986
LCS	G ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂² ◁ #125

subgroup

G/H

center, Frattini

C₂²

C₂³×#10597010

commutator, socle

C₂²×C₁₄₉

C₂³×C₄

Fitting

#125×C₁₄₉

C₄

prime radical

C₂²×D₁₄₉

C₂²×C₄

cosocle

C₂×Dic₁₄₉

C₂⁴

C₁₄₉

G ▷ C₂²×C₁₄₉ ▷ C₁₄₉

C₁ ◁ C₂² ◁ #125

N	H ≅ G/N	type
1. #23750677	C₂	semidirect
2. #23750674	C₂	semidirect
60. C₁₄₉	#2986	semidirect

H ≅ G/N

type

1. #23750677

C₂

semidirect

2. #23750674

C₂

semidirect

60. C₁₄₉

#2986

semidirect

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

subgroup	index
#23750677	2
#23750674	2
#23750650	2
#23750643	2
#23750181	2
#2986	149

subgroup

index

149

Subgroups — #34061601

Conjugacy classes of subgroups

Complements

Maximal subgroups