Group #34056495 — subgroups

Subgroups — #34056495

Order 19072.

444 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 117 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	#23750438	C₂	1	4	yes	maximal, complemented
2	#23749897	C₂	1	4	yes	maximal, complemented
3	#23749807	C₂	1	4	yes	maximal, complemented
4	#23749743	C₂	1	2	yes	maximal, complemented
5	#494×C₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#18011959	C₂²	1	4	yes	—
7	#18011943	—	2	8	no	—
8	#18011942	C₂²	1	8	yes	—
9	#18011934	—	2	4	no	—
10	#18011933	C₂²	1	2	yes	—
11	Q₁₆×D₁₄₉	—	2	4	no	—
12	#18011917	C₂²	1	4	yes	—
13	#18011847	C₂²	1	2	yes	—
14	#18011839	C₂²	1	4	yes	—
15	#18011830	C₂²	1	4	yes	—
16	C₂×Q₁₆×C₁₄₉	C₂²	1	4	yes	—
17	C₄⋊Q₈×C₁₄₉	C₂²	1	2	yes	—
18	C₄×C₈×C₁₄₉	C₂²	1	1	yes	characteristic
19	#14558737	—	2	8	no	—
20	#14558736	—	4	4	no	—
21	#14558736	C₂³	1	4	yes	—
22	Q₈×D₁₄₉	—	2	8	no	—
23	Dic₅₉₆	D₄	1	2	yes	—
24	#14558726	—	2	8	no	—
25	#14558725	D₄	1	4	yes	—
26	#14558723	—	2	8	no	—
27	D₁₁₉₂	D₄	1	2	yes	—
28	#14558720	C₂³	1	4	yes	—
29	#14558719	—	2	4	no	—
30	C₈×D₁₄₉	—	2	4	no	—
31	C₂×Q₈×C₁₄₉	C₂³	1	4	yes	—
32	Q₁₆×C₁₄₉	—	2	4	no	—
33	Q₁₆×C₁₄₉	—	2	4	no	—
34	C₂×C₈×C₁₄₉	C₂³	1	2	yes	—
35	C₄⋊C₄×C₁₄₉	—	4	2	no	—
36	C₄²×C₁₄₉	C₂³	1	1	yes	characteristic
37	Dic₂₉₈	—	4	4	no	—
38	Dic₂₉₈	C₂×D₄	1	4	yes	—
39	D₅₉₆	—	4	4	no	—
40	D₅₉₆	C₂×D₄	1	4	yes	—
41	C₂²⋊D₁₄₉	—	2	4	no	—
42	C₄×D₁₄₉	—	2	4	no	—
43	C₄×D₁₄₉	—	4	8	no	—
44	#11924746	—	2	4	no	—
45	Q₈×C₁₄₉	—	2	8	no	—
46	C₂×C₄×C₁₄₉	—	2	1	no	—
47	C₂×C₄×C₁₄₉	—	2	4	no	—
48	C₂×C₄×C₁₄₉	C₂⁴	1	1	yes	characteristic
49	C₈×C₁₄₉	C₂×D₄	1	2	yes	—
50	C₈×C₁₄₉	C₂×D₄	1	2	yes	—
51	C₂²×C₁₄₉	H(5,𝔽₂)	1	1	yes	characteristic
52	C₂×D₁₄₉	—	2	4	no	—
53	C₄×C₁₄₉	—	2	2	no	—
54	C₄×C₁₄₉	—	4	4	no	—
55	C₄×C₁₄₉	C₂²×D₄	1	2	yes	—
56	Dic₁₄₉	—	2	4	no	—
57	C₂×C₁₄₉	—	2	1	no	—
58	C₂×C₁₄₉	D₄²	1	1	yes	characteristic
59	D₁₄₉	—	4	4	no	—
60	C₁₄₉	#3293	1	1	yes	characteristic, complemented, Sylow
61	#3293	—	149	1	no	maximal, Sylow
62	#578	—	149	4	no	—
63	#494	D₁₄₉	1	1	yes	characteristic, complemented
64	#473	—	149	4	no	—
65	#456	—	149	4	no	—
66	#443	—	149	2	no	—
67	2₋¹⁺⁴	—	149	4	no	—
68	#137	—	298	8	no	—
69	#137	—	149	8	no	—
70	#135	—	298	4	no	—
71	#135	—	149	2	no	—
72	C₂×Q₁₆	—	298	4	no	—
73	C₂×Q₁₆	C₂×D₁₄₉	1	4	yes	—
74	#131	—	149	4	no	—
75	C₄⋊Q₈	C₂×D₁₄₉	1	2	yes	—
76	#108	—	149	2	no	—
77	#104	—	149	4	no	—
78	#101	—	149	4	no	—
79	C₄×C₈	C₂×D₁₄₉	1	1	yes	characteristic
80	#41	—	298	8	no	—
81	#41	—	596	4	no	—
82	#41	—	149	4	no	—
83	C₂×Q₈	—	298	8	no	—
84	C₂×Q₈	C₂²×D₁₄₉	1	4	yes	—
85	Q₁₆	—	2	4	no	—
86	Q₁₆	—	2	4	no	—
87	Q₁₆	—	298	8	no	—
88	Q₁₆	—	149	2	no	—
89	SD₁₆	—	298	8	no	—
90	SD₁₆	—	149	4	no	—
91	D₈	—	149	2	no	—
92	M₁₆	—	298	4	no	—
93	M₁₆	—	149	4	no	—
94	C₂×C₈	—	298	4	no	—
95	C₂×C₈	C₂²×D₁₄₉	1	2	yes	—
96	C₄⋊C₄	—	4	2	no	—
97	C₄²	C₂²×D₁₄₉	1	1	yes	characteristic
98	Q₈	—	2	8	no	—
99	Q₈	—	596	4	no	—
100	Q₈	—	149	4	no	—
101	D₄	—	298	4	no	—
102	D₄	—	596	4	no	—
103	D₄	—	149	4	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	298	4	no	—
107	C₂×C₄	—	596	8	no	—
108	C₂×C₄	C₂³×D₁₄₉	1	1	yes	characteristic
109	C₈	—	298	4	no	—
110	C₈	D₄×D₁₄₉	1	2	yes	—
111	C₈	D₄×D₁₄₉	1	2	yes	—
112	C₂²	—	298	4	no	—
113	C₂²	#18011956	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	—	4	4	no	—
116	C₄	—	298	4	no	—
117	C₄	C₂×D₄×D₁₄₉	1	2	yes	—
118	C₂	—	2	1	no	—
119	C₂	—	596	4	no	—
120	C₂	#23750260	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23750438

C₂

yes

maximal, complemented

#23749897

C₂

yes

maximal, complemented

#23749807

C₂

yes

maximal, complemented

#23749743

C₂

yes

maximal, complemented

#494×C₁₄₉

C₂

yes

maximal, characteristic, complemented

#18011959

C₂²

yes

—

#18011943

—

#18011942

C₂²

yes

—

#18011934

—

#18011933

C₂²

yes

—

Q₁₆×D₁₄₉

—

#18011917

C₂²

yes

—

#18011847

C₂²

yes

—

#18011839

C₂²

yes

—

#18011830

C₂²

yes

—

C₂×Q₁₆×C₁₄₉

C₂²

yes

—

C₄⋊Q₈×C₁₄₉

C₂²

yes

—

C₄×C₈×C₁₄₉

C₂²

yes

—

—

C₂³

yes

—

Q₈×D₁₄₉

—

Dic₅₉₆

D₄

yes

—

#14558726

—

#14558725

D₄

yes

—

#14558723

—

D₁₁₉₂

D₄

yes

—

#14558720

C₂³

yes

—

#14558719

—

C₈×D₁₄₉

—

C₂×Q₈×C₁₄₉

C₂³

yes

—

Q₁₆×C₁₄₉

—

Q₁₆×C₁₄₉

—

C₂×C₈×C₁₄₉

C₂³

yes

—

C₄⋊C₄×C₁₄₉

—

C₄²×C₁₄₉

C₂³

yes

characteristic

Dic₂₉₈

—

Dic₂₉₈

C₂×D₄

yes

—

D₅₉₆

—

D₅₉₆

C₂×D₄

yes

—

C₂²⋊D₁₄₉

—

C₄×D₁₄₉

—

C₄×D₁₄₉

—

#11924746

—

Q₈×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

C₂⁴

yes

characteristic

C₈×C₁₄₉

C₂×D₄

yes

—

C₈×C₁₄₉

C₂×D₄

yes

—

C₂²×C₁₄₉

H(5,𝔽₂)

yes

characteristic

C₂×D₁₄₉

—

C₄×C₁₄₉

—

C₄×C₁₄₉

—

C₄×C₁₄₉

C₂²×D₄

yes

—

Dic₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

D₄²

yes

characteristic

D₁₄₉

—

C₁₄₉

#3293

yes

characteristic, complemented, Sylow

#3293

—

149

maximal, Sylow

#578

—

149

—

#494

D₁₄₉

yes

characteristic, complemented

#473

—

149

—

#456

—

149

—

#443

—

149

—

2₋¹⁺⁴

—

149

—

#137

—

298

—

#137

—

149

—

#135

—

298

—

#135

—

149

—

C₂×Q₁₆

—

298

—

C₂×Q₁₆

C₂×D₁₄₉

yes

—

#131

—

149

—

C₄⋊Q₈

C₂×D₁₄₉

yes

—

#108

—

149

—

#104

—

149

—

#101

—

149

—

C₄×C₈

C₂×D₁₄₉

yes

characteristic

#41

—

298

—

#41

—

596

—

#41

—

149

—

C₂×Q₈

—

298

—

C₂×Q₈

C₂²×D₁₄₉

yes

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

298

—

Q₁₆

—

149

—

SD₁₆

—

298

—

SD₁₆

—

149

—

D₈

—

149

—

M₁₆

—

298

—

M₁₆

—

149

—

C₂×C₈

—

298

—

C₂×C₈

C₂²×D₁₄₉

yes

—

C₄⋊C₄

—

C₄²

C₂²×D₁₄₉

yes

characteristic

Q₈

—

Q₈

—

596

—

100

Q₈

—

149

—

101

D₄

—

298

—

102

D₄

—

596

—

103

D₄

—

149

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

298

—

107

C₂×C₄

—

596

—

108

C₂×C₄

C₂³×D₁₄₉

yes

characteristic

109

C₈

—

298

—

110

C₈

D₄×D₁₄₉

yes

—

111

C₈

D₄×D₁₄₉

yes

—

112

C₂²

—

298

—

113

C₂²

#18011956

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

298

—

117

C₄

C₂×D₄×D₁₄₉

yes

—

118

C₂

—

119

C₂

—

596

—

120

C₂

#23750260

yes

characteristic

	subgroup	G/H
center	C₂	#23750260
commutator, cosocle	C₂×C₄×C₁₄₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₄₉
Fitting	#494×C₁₄₉	C₂
socle	C₂×C₁₄₉	D₄²
hypercenter	#494	D₁₄₉
hypocenter	C₁₄₉	#3293
LCS	G ▷ C₂×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

subgroup

G/H

center

C₂

#23750260

commutator, cosocle

C₂×C₄×C₁₄₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₄₉

Fitting

#494×C₁₄₉

C₂

socle

C₂×C₁₄₉

D₄²

D₁₄₉

C₁₄₉

G ▷ C₂×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

N	H ≅ G/N	type
1. #23750438	C₂	semidirect
2. #23749897	C₂	semidirect
3. #23749807	C₂	semidirect
4. #23749743	C₂	semidirect
5. #494×C₁₄₉	C₂	semidirect
60. C₁₄₉	#3293	semidirect
63. #494	D₁₄₉	semidirect

H ≅ G/N

type

1. #23750438

C₂

semidirect

2. #23749897

C₂

semidirect

3. #23749807

C₂

semidirect

4. #23749743

C₂

semidirect

5. #494×C₁₄₉

C₂

semidirect

60. C₁₄₉

#3293

semidirect

63. #494

D₁₄₉

semidirect

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

subgroup	index
#23750438	2
#23749897	2
#23749807	2
#23749743	2
#494×C₁₄₉	2
#3293	149

subgroup

index

#494×C₁₄₉

149

Subgroups — #34056495

Conjugacy classes of subgroups

Complements

Maximal subgroups