Group #34044357 — subgroups

Subgroups — #34044357

Order 19072.

242 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 69 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	#23750002	C₂	1	1	yes	maximal, characteristic, complemented
2	#23750002	C₂	1	2	yes	maximal, complemented
3	#23749998	C₂	1	1	yes	maximal, characteristic, complemented
4	#23749972	C₂	1	2	yes	maximal, complemented
5	#322×C₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14558725	—	2	2	no	—
7	C₂×D₁₁₉₂	—	2	4	no	—
8	C₂×D₁₁₉₂	—	2	2	no	—
9	#18011911	C₂²	1	1	yes	characteristic
10	#18011908	C₂²	1	2	yes	—
11	#18011908	C₂²	1	1	yes	characteristic
12	#18011833	—	2	4	no	—
13	C₄×C₈×C₁₄₉	C₂²	1	2	yes	—
14	C₄×C₈×C₁₄₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₉₈	—	2	1	no	—
16	C₂×D₅₉₆	—	2	2	no	—
17	C₂×D₅₉₆	—	2	2	no	—
18	C₂×D₅₉₆	—	2	2	no	—
19	C₂×D₅₉₆	—	2	1	no	—
20	C₂×D₅₉₆	—	2	2	no	—
21	#14558725	—	4	4	no	—
22	D₁₁₉₂	—	4	8	no	—
23	D₁₁₉₂	—	4	4	no	—
24	C₄⋊Dic₁₄₉	—	2	2	no	—
25	C₂×C₈×C₁₄₉	D₄	1	4	yes	—
26	C₂×C₈×C₁₄₉	D₄	1	2	yes	—
27	C₄²×C₁₄₉	C₂³	1	1	yes	characteristic
28	C₂²×D₁₄₉	—	4	1	no	—
29	C₂²×D₁₄₉	—	4	2	no	—
30	Dic₂₉₈	—	4	2	no	—
31	D₅₉₆	—	4	4	no	—
32	D₅₉₆	—	4	2	no	—
33	D₅₉₆	—	4	4	no	—
34	D₅₉₆	—	4	4	no	—
35	D₅₉₆	—	4	4	no	—
36	C₂×Dic₁₄₉	—	4	1	no	—
37	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₄₉	C₂×D₄	1	2	yes	—
39	C₈×C₁₄₉	—	2	4	no	—
40	C₈×C₁₄₉	D₈	1	4	yes	complemented
41	C₂²×C₁₄₉	#127	1	1	yes	characteristic
42	C₂×D₁₄₉	—	8	2	no	—
43	C₂×D₁₄₉	—	8	1	no	—
44	C₂×D₁₄₉	—	8	2	no	—
45	C₂×D₁₄₉	—	8	2	no	—
46	C₂×D₁₄₉	—	8	2	no	—
47	C₄×C₁₄₉	C₂×D₈	1	2	yes	—
48	C₄×C₁₄₉	AGL(1,ℤ/8)	1	4	yes	—
49	Dic₁₄₉	—	8	1	no	—
50	C₂×C₁₄₉	#493	1	1	yes	characteristic
51	C₂×C₁₄₉	#496	1	2	yes	—
52	D₁₄₉	—	16	2	no	—
53	D₁₄₉	—	16	1	no	—
54	C₁₄₉	#1721	1	1	yes	characteristic, complemented, Sylow
55	#1721	—	149	1	no	maximal, Sylow
56	#493	—	149	2	no	—
57	#493	—	149	1	no	—
58	#492	—	149	1	no	—
59	#486	—	149	2	no	—
60	#322	D₁₄₉	1	1	yes	characteristic, complemented
61	C₂×SD₁₆	—	298	2	no	—
62	C₂×D₈	—	298	4	no	—
63	C₂×D₈	—	298	2	no	—
64	C₄⋊Q₈	—	149	1	no	—
65	#127	—	149	2	no	—
66	#127	—	149	1	no	—
67	#102	—	298	4	no	—
68	C₄×C₈	C₂×D₁₄₉	1	2	yes	—
69	C₄×C₈	C₂×D₁₄₉	1	1	yes	characteristic
70	C₂×Q₈	—	298	1	no	—
71	C₂×D₄	—	298	2	no	—
72	C₂×D₄	—	298	2	no	—
73	C₂×D₄	—	298	2	no	—
74	C₂×D₄	—	298	2	no	—
75	C₂×D₄	—	298	1	no	—
76	SD₁₆	—	596	4	no	—
77	D₈	—	596	8	no	—
78	D₈	—	596	4	no	—
79	C₂×C₈	D₅₉₆	1	4	yes	—
80	C₂×C₈	D₅₉₆	1	2	yes	—
81	C₄⋊C₄	—	298	2	no	—
82	C₄²	C₂²×D₁₄₉	1	1	yes	characteristic
83	C₂³	—	596	2	no	—
84	C₂³	—	596	1	no	—
85	Q₈	—	596	2	no	—
86	D₄	—	596	4	no	—
87	D₄	—	596	2	no	—
88	D₄	—	596	4	no	—
89	D₄	—	596	4	no	—
90	D₄	—	596	4	no	—
91	C₂×C₄	—	596	1	no	—
92	C₂×C₄	C₂×D₅₉₆	1	1	yes	characteristic
93	C₂×C₄	C₂×D₅₉₆	1	2	yes	—
94	C₈	—	2	4	no	—
95	C₈	D₁₁₉₂	1	4	yes	complemented
96	C₂²	—	1192	2	no	—
97	C₂²	—	1192	1	no	—
98	C₂²	—	1192	2	no	—
99	C₂²	—	1192	2	no	—
100	C₂²	—	1192	2	no	—
101	C₂²	#18011908	1	1	yes	characteristic
102	C₄	—	1192	1	no	—
103	C₄	C₂×D₁₁₉₂	1	2	yes	—
104	C₄	#18011939	1	4	yes	—
105	C₂	—	2384	2	no	—
106	C₂	—	2384	1	no	—
107	C₂	#23750002	1	1	yes	characteristic
108	C₂	#23750014	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23750002

C₂

yes

maximal, characteristic, complemented

#23750002

C₂

yes

maximal, complemented

#23749998

C₂

yes

maximal, characteristic, complemented

#23749972

C₂

yes

maximal, complemented

#322×C₁₄₉

C₂

yes

maximal, characteristic, complemented

C₂×#14558725

—

C₂×D₁₁₉₂

—

C₂×D₁₁₉₂

—

#18011911

C₂²

yes

characteristic

#18011908

C₂²

yes

—

#18011908

C₂²

yes

characteristic

#18011833

—

C₄×C₈×C₁₄₉

C₂²

yes

—

C₄×C₈×C₁₄₉

C₂²

yes

characteristic

C₂×Dic₂₉₈

—

C₂×D₅₉₆

—

C₂×D₅₉₆

—

C₂×D₅₉₆

—

C₂×D₅₉₆

—

C₂×D₅₉₆

—

#14558725

—

D₁₁₉₂

—

D₁₁₉₂

—

C₄⋊Dic₁₄₉

—

C₂×C₈×C₁₄₉

D₄

yes

—

C₂×C₈×C₁₄₉

D₄

yes

—

C₄²×C₁₄₉

C₂³

yes

characteristic

C₂²×D₁₄₉

—

C₂²×D₁₄₉

—

Dic₂₉₈

—

D₅₉₆

—

D₅₉₆

—

D₅₉₆

—

D₅₉₆

—

D₅₉₆

—

C₂×Dic₁₄₉

—

C₂×C₄×C₁₄₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₄₉

C₂×D₄

yes

—

C₈×C₁₄₉

—

C₈×C₁₄₉

D₈

yes

complemented

C₂²×C₁₄₉

#127

yes

characteristic

C₂×D₁₄₉

—

C₂×D₁₄₉

—

C₂×D₁₄₉

—

C₂×D₁₄₉

—

C₂×D₁₄₉

—

C₄×C₁₄₉

C₂×D₈

yes

—

C₄×C₁₄₉

AGL(1,ℤ/8)

yes

—

Dic₁₄₉

—

C₂×C₁₄₉

#493

yes

characteristic

C₂×C₁₄₉

#496

yes

—

D₁₄₉

—

D₁₄₉

—

C₁₄₉

#1721

yes

characteristic, complemented, Sylow

#1721

—

149

maximal, Sylow

#493

—

149

—

#493

—

149

—

#492

—

149

—

#486

—

149

—

#322

D₁₄₉

yes

characteristic, complemented

C₂×SD₁₆

—

298

—

C₂×D₈

—

298

—

C₂×D₈

—

298

—

C₄⋊Q₈

—

149

—

#127

—

149

—

#127

—

149

—

#102

—

298

—

C₄×C₈

C₂×D₁₄₉

yes

—

C₄×C₈

C₂×D₁₄₉

yes

characteristic

C₂×Q₈

—

298

—

C₂×D₄

—

298

—

C₂×D₄

—

298

—

C₂×D₄

—

298

—

C₂×D₄

—

298

—

C₂×D₄

—

298

—

SD₁₆

—

596

—

D₈

—

596

—

D₈

—

596

—

C₂×C₈

D₅₉₆

yes

—

C₂×C₈

D₅₉₆

yes

—

C₄⋊C₄

—

298

—

C₄²

C₂²×D₁₄₉

yes

characteristic

C₂³

—

596

—

C₂³

—

596

—

Q₈

—

596

—

D₄

—

596

—

D₄

—

596

—

D₄

—

596

—

D₄

—

596

—

D₄

—

596

—

C₂×C₄

—

596

—

C₂×C₄

C₂×D₅₉₆

yes

characteristic

C₂×C₄

C₂×D₅₉₆

yes

—

C₈

—

C₈

D₁₁₉₂

yes

complemented

C₂²

—

1192

—

C₂²

—

1192

—

C₂²

—

1192

—

C₂²

—

1192

—

100

C₂²

—

1192

—

101

C₂²

#18011908

yes

characteristic

102

C₄

—

1192

—

103

C₄

C₂×D₁₁₉₂

yes

—

104

C₄

#18011939

yes

—

105

C₂

—

2384

—

106

C₂

—

2384

—

107

C₂

#23750002

yes

characteristic

108

C₂

#23750014

yes

—

	subgroup	G/H
center	C₂²	#18011908
commutator, cosocle	C₄²×C₁₄₉	C₂³
Frattini	C₄²	C₂²×D₁₄₉
Fitting	#322×C₁₄₉	C₂
socle	C₂²×C₁₄₉	#127
hypercenter	#322	D₁₄₉
hypocenter	C₁₄₉	#1721
LCS	G ▷ C₄²×C₁₄₉ ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#18011908

commutator, cosocle

C₄²×C₁₄₉

C₂³

Frattini

C₄²

C₂²×D₁₄₉

Fitting

#322×C₁₄₉

C₂

C₂²×C₁₄₉

D₁₄₉

C₁₄₉

G ▷ C₄²×C₁₄₉ ▷ C₂²×C₁₄₉ ▷ C₁₄₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
1. #23750002	C₂	semidirect
2. #23750002	C₂	semidirect
3. #23749998	C₂	semidirect
4. #23749972	C₂	semidirect
5. #322×C₁₄₉	C₂	semidirect
40. C₈×C₁₄₉	D₈	semidirect
54. C₁₄₉	#1721	semidirect
60. #322	D₁₄₉	semidirect
95. C₈	D₁₁₉₂	semidirect

H ≅ G/N

type

1. #23750002

C₂

semidirect

2. #23750002

C₂

semidirect

3. #23749998

C₂

semidirect

4. #23749972

C₂

semidirect

5. #322×C₁₄₉

C₂

semidirect

40. C₈×C₁₄₉

D₈

semidirect

54. C₁₄₉

#1721

semidirect

60. #322

D₁₄₉

semidirect

95. C₈

D₁₁₉₂

semidirect

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

subgroup	index
#23750002	2
#23749998	2
#23749972	2
#322×C₁₄₉	2
#1721	149

subgroup

index

#322×C₁₄₉

149

Subgroups — #34044357

Conjugacy classes of subgroups

Complements

Maximal subgroups