Group #34045671 — subgroups

Subgroups — #34045671

Order 19072.

244 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 89 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	#23749471	C₂	1	1	yes	maximal, characteristic
3	#23749386	C₂	1	4	yes	maximal
4	#423×C₁₄₉	C₂	1	1	yes	maximal, characteristic
5	#18011911	C₄	1	2	yes	complemented
6	#18011903	C₄	1	2	yes	complemented
7	#18011890	—	2	2	no	—
8	C₄×Dic₂₉₈	—	2	2	no	—
9	#18011870	C₂²	1	2	yes	—
10	C₂×C₄×Dic₁₄₉	C₂²	1	2	yes	—
11	#18011818	—	2	2	no	—
12	C₂×M₁₆×C₁₄₉	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₄₉	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₁₄₉	—	2	2	no	—
15	C₂×Dic₂₉₈	—	2	2	no	—
16	C₂²×Dic₁₄₉	—	2	2	no	—
17	C₄⋊Dic₁₄₉	—	2	2	no	—
18	C₄⋊Dic₁₄₉	C₂×C₄	1	4	yes	—
19	#14558715	—	2	4	no	—
20	C₂²⋊Dic₁₄₉	—	2	2	no	—
21	C₄×Dic₁₄₉	—	2	2	no	—
22	C₄×Dic₁₄₉	—	2	2	no	—
23	C₄×Dic₁₄₉	Q₈	1	4	yes	complemented
24	C₂²×C₄×C₁₄₉	D₄	1	2	yes	—
25	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
26	M₁₆×C₁₄₉	—	2	4	no	—
27	C₂×C₈×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	Dic₂₉₈	—	4	4	no	—
32	C₂×Dic₁₄₉	—	2	4	no	—
33	C₂×Dic₁₄₉	—	2	4	no	—
34	C₂×Dic₁₄₉	—	4	4	no	—
35	C₂³×C₁₄₉	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₄₉	—	4	2	no	—
37	C₂×C₄×C₁₄₉	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₁₄₉	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₄₉	C₂×Q₈	1	2	yes	—
42	C₂×C₄×C₁₄₉	#41	1	2	yes	—
43	C₈×C₁₄₉	—	4	2	no	—
44	C₂²×C₁₄₉	—	2	2	no	—
45	C₂²×C₁₄₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₁₄₉	C₂²⋊Q₈	1	2	yes	—
47	C₄×C₁₄₉	—	2	2	no	—
48	C₄×C₁₄₉	—	2	2	no	—
49	C₄×C₁₄₉	C₄×Q₈	1	2	yes	—
50	C₄×C₁₄₉	#124	1	1	yes	characteristic
51	C₄×C₁₄₉	C₄⋊Q₈	1	1	yes	characteristic
52	Dic₁₄₉	—	4	4	no	—
53	Dic₁₄₉	—	4	4	no	—
54	C₂×C₁₄₉	—	2	2	no	—
55	C₂×C₁₄₉	#391	1	1	yes	characteristic
56	C₂×C₁₄₉	#421	1	2	yes	—
57	C₁₄₉	#1995	1	1	yes	characteristic, complemented, Sylow
58	#1995	—	149	1	no	maximal, Sylow
59	#533	—	149	1	no	—
60	#423	D₁₄₉	1	1	yes	characteristic
61	#376	—	149	1	no	—
62	#339	—	149	4	no	—
63	C₂×M₁₆	C₂×D₁₄₉	1	2	yes	—
64	C₄⋊Q₈	—	149	2	no	—
65	#125	—	149	2	no	—
66	C₂²⋊Q₈	—	298	2	no	—
67	C₄×Q₈	—	298	2	no	—
68	#117	—	149	2	no	—
69	C₂×C₄²	—	149	2	no	—
70	C₂×C₄²	C₂×D₁₄₉	1	1	yes	characteristic
71	C₄⋊C₈	—	2	2	no	—
72	#95	—	298	2	no	—
73	C₂×Q₈	—	298	2	no	—
74	C₂²×C₄	—	298	2	no	—
75	C₂²×C₄	D₅₉₆	1	2	yes	—
76	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
77	M₁₆	—	2	4	no	—
78	C₂×C₈	—	2	2	no	—
79	C₄⋊C₄	—	298	4	no	—
80	C₄⋊C₄	—	298	2	no	—
81	C₄⋊C₄	—	149	4	no	—
82	C₂²⋊C₄	—	298	2	no	—
83	C₄²	—	298	2	no	—
84	C₄²	—	298	2	no	—
85	C₄²	—	149	4	no	—
86	C₄²	C₄×D₁₄₉	1	1	yes	characteristic
87	C₄²	C₄×D₁₄₉	1	1	yes	characteristic
88	C₄²	C₂²⋊D₁₄₉	1	2	yes	—
89	C₂³	C₂×D₅₉₆	1	1	yes	characteristic
90	Q₈	—	596	4	no	—
91	C₂×C₄	—	4	2	no	—
92	C₂×C₄	—	298	4	no	—
93	C₂×C₄	—	298	4	no	—
94	C₂×C₄	—	596	4	no	—
95	C₂×C₄	#14558713	1	2	yes	—
96	C₂×C₄	#14558713	1	2	yes	—
97	C₂×C₄	C₂×C₄×D₁₄₉	1	1	yes	characteristic
98	C₂×C₄	C₂×C₂²⋊D₁₄₉	1	1	yes	characteristic
99	C₂×C₄	Q₈×D₁₄₉	1	2	yes	—
100	C₂×C₄	#14558735	1	2	yes	—
101	C₈	—	4	2	no	—
102	C₂²	—	2	2	no	—
103	C₂²	C₂×#14558713	1	1	yes	characteristic
104	C₂²	#18011891	1	2	yes	—
105	C₄	—	2	2	no	—
106	C₄	—	2	2	no	—
107	C₄	—	596	4	no	—
108	C₄	—	596	4	no	—
109	C₄	#18011876	1	2	yes	—
110	C₄	#18011898	1	1	yes	characteristic
111	C₄	#18011909	1	1	yes	characteristic
112	C₂	—	2	2	no	—
113	C₂	#23749530	1	1	yes	characteristic
114	C₂	#23749641	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23750182

C₂

yes

maximal, characteristic

#23749471

C₂

yes

maximal, characteristic

#23749386

C₂

yes

maximal

#423×C₁₄₉

C₂

yes

maximal, characteristic

#18011911

C₄

yes

complemented

#18011903

C₄

yes

complemented

#18011890

—

C₄×Dic₂₉₈

—

#18011870

C₂²

yes

—

C₂×C₄×Dic₁₄₉

C₂²

yes

—

#18011818

—

C₂×M₁₆×C₁₄₉

C₂²

yes

—

C₂×C₄²×C₁₄₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₄₉

—

C₂×Dic₂₉₈

—

C₂²×Dic₁₄₉

—

C₄⋊Dic₁₄₉

—

C₄⋊Dic₁₄₉

C₂×C₄

yes

—

#14558715

—

C₂²⋊Dic₁₄₉

—

C₄×Dic₁₄₉

—

C₄×Dic₁₄₉

—

C₄×Dic₁₄₉

Q₈

yes

complemented

C₂²×C₄×C₁₄₉

D₄

yes

—

C₂²×C₄×C₁₄₉

C₂³

yes

characteristic

M₁₆×C₁₄₉

—

C₂×C₈×C₁₄₉

—

C₄²×C₁₄₉

C₂×C₄

yes

characteristic

C₄²×C₁₄₉

C₂×C₄

yes

characteristic

C₄²×C₁₄₉

D₄

yes

—

Dic₂₉₈

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂³×C₁₄₉

C₂×D₄

yes

characteristic

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

—

C₈×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₄₉

C₂²⋊Q₈

yes

—

C₄×C₁₄₉

—

C₄×C₁₄₉

—

C₄×C₁₄₉

C₄×Q₈

yes

—

C₄×C₁₄₉

#124

yes

characteristic

C₄×C₁₄₉

C₄⋊Q₈

yes

characteristic

Dic₁₄₉

—

Dic₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

#391

yes

characteristic

C₂×C₁₄₉

#421

yes

—

C₁₄₉

#1995

yes

characteristic, complemented, Sylow

#1995

—

149

maximal, Sylow

#533

—

149

—

#423

D₁₄₉

yes

characteristic

#376

—

149

—

#339

—

149

—

C₂×M₁₆

C₂×D₁₄₉

yes

—

C₄⋊Q₈

—

149

—

#125

—

149

—

C₂²⋊Q₈

—

298

—

C₄×Q₈

—

298

—

#117

—

149

—

C₂×C₄²

—

149

—

C₂×C₄²

C₂×D₁₄₉

yes

characteristic

C₄⋊C₈

—

#95

—

298

—

C₂×Q₈

—

298

—

C₂²×C₄

—

298

—

C₂²×C₄

D₅₉₆

yes

—

C₂²×C₄

C₂²×D₁₄₉

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

298

—

C₄⋊C₄

—

298

—

C₄⋊C₄

—

149

—

C₂²⋊C₄

—

298

—

C₄²

—

298

—

C₄²

—

298

—

C₄²

—

149

—

C₄²

C₄×D₁₄₉

yes

characteristic

C₄²

C₄×D₁₄₉

yes

characteristic

C₄²

C₂²⋊D₁₄₉

yes

—

C₂³

C₂×D₅₉₆

yes

characteristic

Q₈

—

596

—

C₂×C₄

—

C₂×C₄

—

298

—

C₂×C₄

—

298

—

C₂×C₄

—

596

—

C₂×C₄

#14558713

yes

—

C₂×C₄

#14558713

yes

—

C₂×C₄

C₂×C₄×D₁₄₉

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₄₉

yes

characteristic

C₂×C₄

Q₈×D₁₄₉

yes

—

100

C₂×C₄

#14558735

yes

—

101

C₈

—

102

C₂²

—

103

C₂²

C₂×#14558713

yes

characteristic

104

C₂²

#18011891

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

596

—

108

C₄

—

596

—

109

C₄

#18011876

yes

—

110

C₄

#18011898

yes

characteristic

111

C₄

#18011909

yes

characteristic

112

C₂

—

113

C₂

#23749530

yes

characteristic

114

C₂

#23749641

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₄₉
commutator	C₂×C₄×C₁₄₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₄₉
Fitting	#423×C₁₄₉	C₂
socle	C₂²×C₁₄₉	C₂×C₂²⋊C₄
prime radical	C₂³×C₁₄₉	C₂×D₄
cosocle	C₂²×C₄×C₁₄₉	C₂³
hypercenter	#423	D₁₄₉
hypocenter	C₁₄₉	#1995
LCS	G ▷ C₂×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₄₉

commutator

C₂×C₄×C₁₄₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₄₉

Fitting

#423×C₁₄₉

C₂

socle

C₂²×C₁₄₉

C₂×C₂²⋊C₄

prime radical

C₂³×C₁₄₉

C₂×D₄

cosocle

C₂²×C₄×C₁₄₉

C₂³

D₁₄₉

C₁₄₉

G ▷ C₂×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
5. #18011911	C₄	semidirect
6. #18011903	C₄	semidirect
23. C₄×Dic₁₄₉	Q₈	semidirect
57. C₁₄₉	#1995	semidirect

H ≅ G/N

type

5. #18011911

C₄

semidirect

6. #18011903

C₄

semidirect

23. C₄×Dic₁₄₉

Q₈

semidirect

57. C₁₄₉

#1995

semidirect

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

subgroup	index
#23750182	2
#23749471	2
#23749386	2
#423×C₁₄₉	2
#1995	149

subgroup

index

#423×C₁₄₉

149

Subgroups — #34045671

Conjugacy classes of subgroups

Complements

Maximal subgroups