Group #34045668 — subgroups

Subgroups — #34045668

Order 19072.

284 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 101 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	#23750182	C₂	1	1	yes	maximal, characteristic
2	#23749649	C₂	1	1	yes	maximal, characteristic
3	#23749385	C₂	1	4	yes	maximal
4	C₄³×C₁₄₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#14558720	C₂²	1	2	yes	—
6	#18011911	C₄	1	2	yes	complemented
7	#18011903	C₄	1	2	yes	complemented
8	#18011890	—	2	2	no	—
9	C₄×Dic₂₉₈	—	2	2	no	—
10	#18011870	C₂²	1	2	yes	—
11	C₄⋊#11924746	—	2	2	no	—
12	C₂×C₄²×C₁₄₉	—	2	2	no	—
13	C₂×C₄²×C₁₄₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₄₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₉₈	—	2	2	no	—
16	#14558720	—	2	4	no	—
17	C₂×#11924746	—	2	2	no	—
18	C₄⋊Dic₁₄₉	—	2	2	no	—
19	C₄⋊Dic₁₄₉	C₂×C₄	1	4	yes	—
20	#14558715	—	2	4	no	—
21	C₂²⋊Dic₁₄₉	—	2	2	no	—
22	C₄×Dic₁₄₉	—	2	2	no	—
23	C₂²×C₄×C₁₄₉	—	2	2	no	—
24	C₂²×C₄×C₁₄₉	D₄	1	2	yes	—
25	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
26	C₄²×C₁₄₉	—	2	8	no	—
27	C₄²×C₁₄₉	—	2	2	no	—
28	C₄²×C₁₄₉	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₁₄₉	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₁₄₉	D₄	1	2	yes	—
31	C₄²×C₁₄₉	Q₈	1	4	yes	complemented
32	Dic₂₉₈	—	4	4	no	—
33	C₂×Dic₁₄₉	—	2	4	no	—
34	#11924746	—	4	2	no	—
35	C₂³×C₁₄₉	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₄₉	—	2	8	no	—
37	C₂×C₄×C₁₄₉	—	2	4	no	—
38	C₂×C₄×C₁₄₉	—	2	4	no	—
39	C₂×C₄×C₁₄₉	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₁₄₉	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₁₄₉	C₂×Q₈	1	2	yes	—
44	C₂×C₄×C₁₄₉	#41	1	2	yes	—
45	C₂²×C₁₄₉	—	2	2	no	—
46	C₂²×C₁₄₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₁₄₉	C₂²⋊Q₈	1	2	yes	—
48	C₄×C₁₄₉	—	2	8	no	—
49	C₄×C₁₄₉	—	2	2	no	—
50	C₄×C₁₄₉	#104	1	4	yes	—
51	C₄×C₁₄₉	C₄×Q₈	1	2	yes	—
52	C₄×C₁₄₉	#124	1	1	yes	characteristic
53	C₄×C₁₄₉	C₄⋊Q₈	1	1	yes	characteristic
54	Dic₁₄₉	—	4	4	no	—
55	C₂×C₁₄₉	—	2	2	no	—
56	C₂×C₁₄₉	#391	1	1	yes	characteristic
57	C₂×C₁₄₉	C₂×#104	1	2	yes	—
58	C₁₄₉	#1994	1	1	yes	characteristic, complemented, Sylow
59	#1994	—	149	1	no	maximal, Sylow
60	#533	—	149	1	no	—
61	#423	—	149	1	no	—
62	C₄³	D₁₄₉	1	1	yes	characteristic
63	#339	—	149	4	no	—
64	C₂×M₁₆	—	149	2	no	—
65	C₄⋊Q₈	—	149	2	no	—
66	#125	—	149	2	no	—
67	C₂²⋊Q₈	—	298	2	no	—
68	C₄×Q₈	—	298	2	no	—
69	#117	—	149	2	no	—
70	C₂×C₄²	—	2	2	no	—
71	C₂×C₄²	C₂×D₁₄₉	1	2	yes	—
72	C₂×C₄²	C₂×D₁₄₉	1	1	yes	characteristic
73	C₄⋊C₈	—	298	2	no	—
74	C₂×Q₈	—	298	2	no	—
75	C₂²×C₄	—	2	2	no	—
76	C₂²×C₄	C₂²⋊D₁₄₉	1	2	yes	—
77	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
78	M₁₆	—	298	4	no	—
79	C₂×C₈	—	298	2	no	—
80	C₄⋊C₄	—	298	4	no	—
81	C₄⋊C₄	—	298	2	no	—
82	C₄⋊C₄	—	149	4	no	—
83	C₂²⋊C₄	—	298	2	no	—
84	C₄²	—	2	8	no	—
85	C₄²	—	2	2	no	—
86	C₄²	—	298	2	no	—
87	C₄²	C₄×D₁₄₉	1	1	yes	characteristic
88	C₄²	C₄×D₁₄₉	1	1	yes	characteristic
89	C₄²	D₅₉₆	1	2	yes	—
90	C₄²	Dic₂₉₈	1	4	yes	complemented
91	C₂³	C₂×C₂²⋊D₁₄₉	1	1	yes	characteristic
92	Q₈	—	596	4	no	—
93	C₂×C₄	—	2	8	no	—
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	298	4	no	—
97	C₂×C₄	#14558713	1	2	yes	—
98	C₂×C₄	#14558713	1	2	yes	—
99	C₂×C₄	C₂×C₄×D₁₄₉	1	1	yes	characteristic
100	C₂×C₄	C₂×D₅₉₆	1	1	yes	characteristic
101	C₂×C₄	C₂×Dic₂₉₈	1	2	yes	—
102	C₂×C₄	#14558736	1	2	yes	—
103	C₈	—	596	2	no	—
104	C₂²	—	2	2	no	—
105	C₂²	C₂×#14558713	1	1	yes	characteristic
106	C₂²	#18011890	1	2	yes	—
107	C₄	—	2	8	no	—
108	C₄	—	2	2	no	—
109	C₄	—	596	4	no	—
110	C₄	#18011839	1	4	yes	—
111	C₄	C₄×Dic₂₉₈	1	2	yes	—
112	C₄	#18011899	1	1	yes	characteristic
113	C₄	#18011911	1	1	yes	characteristic
114	C₂	—	2	2	no	—
115	C₂	#23749531	1	1	yes	characteristic
116	C₂	C₂×#18011839	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23750182

C₂

yes

maximal, characteristic

#23749649

C₂

yes

maximal, characteristic

#23749385

C₂

yes

maximal

C₄³×C₁₄₉

C₂

yes

maximal, characteristic

C₂×#14558720

C₂²

yes

—

#18011911

C₄

yes

complemented

#18011903

C₄

yes

complemented

#18011890

—

C₄×Dic₂₉₈

—

#18011870

C₂²

yes

—

C₄⋊#11924746

—

C₂×C₄²×C₁₄₉

—

C₂×C₄²×C₁₄₉

C₂²

yes

—

C₂×C₄²×C₁₄₉

C₂²

yes

characteristic

C₂×Dic₂₉₈

—

#14558720

—

C₂×#11924746

—

C₄⋊Dic₁₄₉

—

C₄⋊Dic₁₄₉

C₂×C₄

yes

—

#14558715

—

C₂²⋊Dic₁₄₉

—

C₄×Dic₁₄₉

—

C₂²×C₄×C₁₄₉

—

C₂²×C₄×C₁₄₉

D₄

yes

—

C₂²×C₄×C₁₄₉

C₂³

yes

characteristic

C₄²×C₁₄₉

—

C₄²×C₁₄₉

—

C₄²×C₁₄₉

C₂×C₄

yes

characteristic

C₄²×C₁₄₉

C₂×C₄

yes

characteristic

C₄²×C₁₄₉

D₄

yes

—

C₄²×C₁₄₉

Q₈

yes

complemented

Dic₂₉₈

—

C₂×Dic₁₄₉

—

#11924746

—

C₂³×C₁₄₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₄₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₄₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₄₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₄₉

C₂×Q₈

yes

—

C₂×C₄×C₁₄₉

#41

yes

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₄₉

C₂²⋊Q₈

yes

—

C₄×C₁₄₉

—

C₄×C₁₄₉

—

C₄×C₁₄₉

#104

yes

—

C₄×C₁₄₉

C₄×Q₈

yes

—

C₄×C₁₄₉

#124

yes

characteristic

C₄×C₁₄₉

C₄⋊Q₈

yes

characteristic

Dic₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

#391

yes

characteristic

C₂×C₁₄₉

C₂×#104

yes

—

C₁₄₉

#1994

yes

characteristic, complemented, Sylow

#1994

—

149

maximal, Sylow

#533

—

149

—

#423

—

149

—

C₄³

D₁₄₉

yes

characteristic

#339

—

149

—

C₂×M₁₆

—

149

—

C₄⋊Q₈

—

149

—

#125

—

149

—

C₂²⋊Q₈

—

298

—

C₄×Q₈

—

298

—

#117

—

149

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₄₉

yes

—

C₂×C₄²

C₂×D₁₄₉

yes

characteristic

C₄⋊C₈

—

298

—

C₂×Q₈

—

298

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₄₉

yes

—

C₂²×C₄

C₂²×D₁₄₉

yes

characteristic

M₁₆

—

298

—

C₂×C₈

—

298

—

C₄⋊C₄

—

298

—

C₄⋊C₄

—

298

—

C₄⋊C₄

—

149

—

C₂²⋊C₄

—

298

—

C₄²

—

C₄²

—

C₄²

—

298

—

C₄²

C₄×D₁₄₉

yes

characteristic

C₄²

C₄×D₁₄₉

yes

characteristic

C₄²

D₅₉₆

yes

—

C₄²

Dic₂₉₈

yes

complemented

C₂³

C₂×C₂²⋊D₁₄₉

yes

characteristic

Q₈

—

596

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

298

—

C₂×C₄

#14558713

yes

—

C₂×C₄

#14558713

yes

—

C₂×C₄

C₂×C₄×D₁₄₉

yes

characteristic

100

C₂×C₄

C₂×D₅₉₆

yes

characteristic

101

C₂×C₄

C₂×Dic₂₉₈

yes

—

102

C₂×C₄

#14558736

yes

—

103

C₈

—

596

—

104

C₂²

—

105

C₂²

C₂×#14558713

yes

characteristic

106

C₂²

#18011890

yes

—

107

C₄

—

108

C₄

—

109

C₄

—

596

—

110

C₄

#18011839

yes

—

111

C₄

C₄×Dic₂₉₈

yes

—

112

C₄

#18011899

yes

characteristic

113

C₄

#18011911

yes

characteristic

114

C₂

—

115

C₂

#23749531

yes

characteristic

116

C₂

C₂×#18011839

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₅₉₆
commutator	C₂×C₄×C₁₄₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₄₉
Fitting	C₄³×C₁₄₉	C₂
socle	C₂²×C₁₄₉	C₂×C₂²⋊C₄
prime radical	C₂³×C₁₄₉	C₂×D₄
cosocle	C₂²×C₄×C₁₄₉	C₂³
hypercenter	C₄³	D₁₄₉
hypocenter	C₁₄₉	#1994
LCS	G ▷ C₂×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄³

subgroup

G/H

center

C₂×C₄

C₂×D₅₉₆

commutator

C₂×C₄×C₁₄₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₄₉

Fitting

C₄³×C₁₄₉

C₂

socle

C₂²×C₁₄₉

C₂×C₂²⋊C₄

prime radical

C₂³×C₁₄₉

C₂×D₄

cosocle

C₂²×C₄×C₁₄₉

C₂³

hypercenter

C₄³

D₁₄₉

hypocenter

C₁₄₉

#1994

LCS

G ▷ C₂×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄³

N	H ≅ G/N	type
6. #18011911	C₄	semidirect
7. #18011903	C₄	semidirect
31. C₄²×C₁₄₉	Q₈	semidirect
58. C₁₄₉	#1994	semidirect
90. C₄²	Dic₂₉₈	semidirect

H ≅ G/N

type

6. #18011911

C₄

semidirect

7. #18011903

C₄

semidirect

31. C₄²×C₁₄₉

Q₈

semidirect

58. C₁₄₉

#1994

semidirect

90. C₄²

Dic₂₉₈

semidirect

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

subgroup	index
#23750182	2
#23749649	2
#23749385	2
C₄³×C₁₄₉	2
#1994	149

subgroup

index

C₄³×C₁₄₉

149

Subgroups — #34045668

Conjugacy classes of subgroups

Complements

Maximal subgroups