Group #34060324 — subgroups

Subgroups — #34060324

Order 19072.

180 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 58 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	#23750660	C₂	1	2	yes	maximal, complemented
2	#23750488	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄⋊#11924753	C₂	1	2	yes	maximal, complemented
4	C₈×#11924753	C₂	1	1	yes	maximal, characteristic, complemented
5	D₈×Dic₁₄₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14558745	—	2	2	no	—
7	C₄⋊#11924753	C₂²	1	2	yes	—
8	C₂²⋊#11924753	—	2	2	no	—
9	C₄×#11924753	C₂²	1	1	yes	characteristic
10	C₄×#11924753	C₂²	1	1	yes	characteristic
11	D₄×Dic₁₄₉	C₂²	1	2	yes	—
12	#18011845	C₄	1	1	yes	characteristic
13	#18011832	C₄	1	2	yes	—
14	C₈×Dic₁₄₉	C₂²	1	1	yes	characteristic
15	C₂×D₈×C₁₄₉	C₄	1	1	yes	characteristic
16	#14558745	—	4	4	no	—
17	C₂×#11924753	—	2	2	no	—
18	C₂×#11924753	—	2	1	no	—
19	C₂×#11924753	D₄	1	1	yes	characteristic
20	C₂×#11924753	D₄	1	1	yes	characteristic
21	C₂²×Dic₁₄₉	—	2	2	no	—
22	C₂×#11924746	C₂×C₄	1	1	yes	characteristic
23	C₄⋊Dic₁₄₉	C₂×C₄	1	2	yes	—
24	C₂²⋊Dic₁₄₉	—	2	2	no	—
25	C₄×Dic₁₄₉	C₂³	1	1	yes	characteristic
26	C₂×D₄×C₁₄₉	C₂×C₄	1	2	yes	—
27	D₈×C₁₄₉	—	2	2	no	—
28	C₂×C₈×C₁₄₉	C₂×C₄	1	1	yes	characteristic
29	#11924753	—	2	2	no	—
30	#11924753	—	2	1	no	—
31	#11924753	—	4	2	no	—
32	#11924753	D₈	1	2	yes	complemented
33	C₂×Dic₁₄₉	—	2	2	no	—
34	C₂×Dic₁₄₉	—	4	2	no	—
35	C₂×Dic₁₄₉	—	4	2	no	—
36	C₂×Dic₁₄₉	C₂×D₄	1	1	yes	characteristic
37	C₂×Dic₁₄₉	#41	1	1	yes	characteristic
38	#11924746	—	2	1	no	—
39	C₂³×C₁₄₉	—	2	2	no	—
40	D₄×C₁₄₉	—	2	2	no	—
41	D₄×C₁₄₉	—	2	2	no	—
42	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
43	C₈×C₁₄₉	M₁₆	1	2	yes	complemented
44	C₂²×C₁₄₉	—	4	2	no	—
45	C₂²×C₁₄₉	—	4	2	no	—
46	C₂²×C₁₄₉	—	4	2	no	—
47	C₂²×C₁₄₉	C₄×D₄	1	1	yes	characteristic
48	C₄×C₁₄₉	C₂×M₁₆	1	1	yes	characteristic
49	C₄×C₁₄₉	#131	1	1	yes	characteristic
50	Dic₁₄₉	—	2	1	no	—
51	Dic₁₄₉	—	4	2	no	—
52	Dic₁₄₉	C₂×D₈	1	1	yes	characteristic
53	Dic₁₄₉	#135	1	1	yes	characteristic
54	C₂×C₁₄₉	—	8	2	no	—
55	C₂×C₁₄₉	#436	1	1	yes	characteristic
56	C₂×C₁₄₉	C₄×D₈	1	1	yes	characteristic
57	C₂×C₁₄₉	#443	1	1	yes	characteristic
58	C₁₄₉	#1589	1	1	yes	characteristic, complemented, Sylow
59	#1589	—	149	1	no	maximal, Sylow
60	C₄×D₈	—	149	1	no	—
61	#436	—	149	2	no	—
62	#335	—	149	1	no	—
63	D₄⋊C₈	—	149	2	no	—
64	C₈²	—	149	1	no	—
65	C₂×D₈	#10597010	1	1	yes	characteristic
66	C₂×M₁₆	—	298	2	no	—
67	C₄×D₄	—	149	2	no	—
68	#107	—	149	1	no	—
69	C₄⋊C₈	—	149	2	no	—
70	#102	—	149	2	no	—
71	C₂²⋊C₈	—	298	2	no	—
72	C₄×C₈	—	149	1	no	—
73	C₄×C₈	—	149	1	no	—
74	C₄×C₈	—	149	1	no	—
75	C₂×D₄	C₂×#10597010	1	2	yes	—
76	C₂²×C₄	—	298	2	no	—
77	D₈	—	2	2	no	—
78	M₁₆	—	596	4	no	—
79	C₂×C₈	—	298	1	no	—
80	C₂×C₈	—	298	2	no	—
81	C₂×C₈	—	149	1	no	—
82	C₂×C₈	—	149	1	no	—
83	C₂×C₈	—	149	1	no	—
84	C₂×C₈	C₂×#10597010	1	1	yes	characteristic
85	C₄⋊C₄	—	149	2	no	—
86	C₂²⋊C₄	—	298	2	no	—
87	C₄²	—	149	1	no	—
88	C₂³	—	2	2	no	—
89	D₄	—	2	2	no	—
90	D₄	—	2	2	no	—
91	C₂×C₄	—	298	2	no	—
92	C₂×C₄	—	596	2	no	—
93	C₂×C₄	—	596	2	no	—
94	C₂×C₄	—	149	1	no	—
95	C₂×C₄	—	149	1	no	—
96	C₂×C₄	C₂²×#10597010	1	1	yes	characteristic
97	C₈	—	298	2	no	—
98	C₈	—	298	1	no	—
99	C₈	—	298	1	no	—
100	C₈	—	596	2	no	—
101	C₈	—	149	2	no	—
102	C₈	#14558745	1	2	yes	complemented
103	C₂²	—	4	2	no	—
104	C₂²	—	4	2	no	—
105	C₂²	—	4	2	no	—
106	C₂²	D₄×#10597010	1	1	yes	characteristic
107	C₄	—	298	1	no	—
108	C₄	—	596	2	no	—
109	C₄	—	149	1	no	—
110	C₄	—	149	1	no	—
111	C₄	C₂×#14558745	1	1	yes	characteristic
112	C₄	#18011998	1	1	yes	characteristic
113	C₂	—	8	2	no	—
114	C₂	#23750660	1	1	yes	characteristic
115	C₂	D₈×#10597010	1	1	yes	characteristic
116	C₂	#23750667	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23750660

C₂

yes

maximal, complemented

#23750488

C₂

yes

maximal, characteristic, complemented

D₄⋊#11924753

C₂

yes

maximal, complemented

C₈×#11924753

C₂

yes

maximal, characteristic, complemented

D₈×Dic₁₄₉

C₂

yes

maximal, characteristic

C₂×#14558745

—

C₄⋊#11924753

C₂²

yes

—

C₂²⋊#11924753

—

C₄×#11924753

C₂²

yes

characteristic

C₄×#11924753

C₂²

yes

characteristic

D₄×Dic₁₄₉

C₂²

yes

—

#18011845

C₄

yes

characteristic

#18011832

C₄

yes

—

C₈×Dic₁₄₉

C₂²

yes

characteristic

C₂×D₈×C₁₄₉

C₄

yes

—

—

—

D₄

yes

characteristic

C₂×#11924753

D₄

yes

characteristic

C₂²×Dic₁₄₉

—

C₂×#11924746

C₂×C₄

yes

characteristic

C₄⋊Dic₁₄₉

C₂×C₄

yes

—

C₂²⋊Dic₁₄₉

—

C₄×Dic₁₄₉

C₂³

yes

characteristic

C₂×D₄×C₁₄₉

C₂×C₄

yes

—

D₈×C₁₄₉

—

C₂×C₈×C₁₄₉

C₂×C₄

yes

—

—

—

D₈

yes

complemented

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

C₂×D₄

yes

C₂×Dic₁₄₉

yes

—

C₂³×C₁₄₉

—

D₄×C₁₄₉

—

D₄×C₁₄₉

—

C₂×C₄×C₁₄₉

C₂²×C₄

yes

characteristic

C₈×C₁₄₉

M₁₆

yes

complemented

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

C₄×D₄

yes

characteristic

C₄×C₁₄₉

C₂×M₁₆

yes

characteristic

C₄×C₁₄₉

#131

yes

characteristic

Dic₁₄₉

—

Dic₁₄₉

—

Dic₁₄₉

C₂×D₈

yes

characteristic

Dic₁₄₉

#135

yes

characteristic

C₂×C₁₄₉

—

C₂×C₁₄₉

#436

yes

characteristic

C₂×C₁₄₉

C₄×D₈

yes

C₂×C₁₄₉

yes

C₁₄₉

yes

characteristic, complemented, Sylow

#1589

—

149

maximal, Sylow

C₄×D₈

—

149

—

#436

—

149

—

#335

—

149

—

D₄⋊C₈

—

149

—

C₈²

—

149

—

C₂×D₈

#10597010

yes

characteristic

C₂×M₁₆

—

298

—

C₄×D₄

—

149

—

#107

—

149

—

C₄⋊C₈

—

149

—

#102

—

149

—

C₂²⋊C₈

—

298

—

C₄×C₈

—

149

—

C₄×C₈

—

149

—

C₄×C₈

—

149

—

C₂×D₄

C₂×#10597010

yes

—

C₂²×C₄

—

298

—

D₈

—

M₁₆

—

596

—

C₂×C₈

—

298

—

C₂×C₈

—

298

—

C₂×C₈

—

149

—

C₂×C₈

—

149

—

C₂×C₈

—

149

—

C₂×C₈

C₂×#10597010

yes

characteristic

C₄⋊C₄

—

149

—

C₂²⋊C₄

—

298

—

C₄²

—

149

—

C₂³

—

D₄

—

D₄

—

C₂×C₄

—

298

—

C₂×C₄

—

596

—

C₂×C₄

—

596

—

C₂×C₄

—

149

—

C₂×C₄

—

149

—

C₂×C₄

C₂²×#10597010

yes

characteristic

C₈

—

298

—

C₈

—

298

—

C₈

—

298

—

100

C₈

—

596

—

101

C₈

—

149

—

102

C₈

#14558745

yes

complemented

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

D₄×#10597010

yes

characteristic

107

C₄

—

298

—

108

C₄

—

596

—

109

C₄

—

149

—

110

C₄

—

149

—

111

C₄

C₂×#14558745

yes

characteristic

112

C₄

#18011998

yes

characteristic

113

C₂

—

114

C₂

#23750660

yes

characteristic

115

C₂

D₈×#10597010

yes

characteristic

116

C₂

#23750667

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×#10597010
commutator	C₂×C₄×C₁₄₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×#10597010
Fitting, prime radical	C₂×D₈×C₁₄₉	C₄
socle	C₂²×C₁₄₉	C₄×D₄
cosocle	C₄×Dic₁₄₉	C₂³
hypercenter	C₂×D₈	#10597010
hypocenter	C₁₄₉	#1589
LCS	G ▷ C₂×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

subgroup

G/H

center

C₂²

D₄×#10597010

commutator

C₂×C₄×C₁₄₉

C₂²×C₄

Frattini

C₂×C₄

C₂²×#10597010

Fitting, prime radical

C₂×D₈×C₁₄₉

C₄

socle

C₂²×C₁₄₉

C₄×D₄

cosocle

C₄×Dic₁₄₉

C₂³

C₂×D₈

C₁₄₉

G ▷ C₂×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

N	H ≅ G/N	type
1. #23750660	C₂	semidirect
2. #23750488	C₂	semidirect
3. D₄⋊#11924753	C₂	semidirect
4. C₈×#11924753	C₂	semidirect
32. #11924753	D₈	semidirect
43. C₈×C₁₄₉	M₁₆	semidirect
58. C₁₄₉	#1589	semidirect
102. C₈	#14558745	semidirect

H ≅ G/N

type

1. #23750660

C₂

semidirect

2. #23750488

C₂

semidirect

3. D₄⋊#11924753

C₂

semidirect

4. C₈×#11924753

C₂

semidirect

32. #11924753

D₈

semidirect

43. C₈×C₁₄₉

M₁₆

semidirect

58. C₁₄₉

#1589

semidirect

102. C₈

#14558745

semidirect

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

subgroup	index
#23750660	2
#23750488	2
D₄⋊#11924753	2
C₈×#11924753	2
D₈×Dic₁₄₉	2
#1589	149

subgroup

index

D₈×Dic₁₄₉

149

Subgroups — #34060324

Conjugacy classes of subgroups

Complements

Maximal subgroups