Group #34044598 — subgroups

Subgroups — #34044598

Order 19072.

220 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 113 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	C₂×#18011846	C₂	1	2	yes	maximal
2	C₄×#14558720	C₂	1	1	yes	maximal, characteristic
3	C₂×#18011822	C₂	1	1	yes	maximal, characteristic
4	#23749389	C₂	1	2	yes	maximal
5	C₄×M₁₆×C₁₄₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14558720	C₂²	1	2	yes	—
7	C₂²×#11924746	C₂²	1	1	yes	characteristic
8	C₂²×#11924746	C₂²	1	1	yes	characteristic
9	#18011846	C₄	1	8	yes	complemented
10	#18011822	—	2	1	no	—
11	#18011822	C₄	1	4	yes	—
12	C₄×#11924746	—	2	1	no	—
13	C₂×M₁₆×C₁₄₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₄₉	C₂²	1	1	yes	characteristic
15	#97×C₁₄₉	—	2	1	no	—
16	C₄×C₈×C₁₄₉	—	2	1	no	—
17	#14558720	—	2	2	no	—
18	#14558720	C₂×C₄	1	4	yes	—
19	C₂×#11924746	—	2	2	no	—
20	C₂×#11924746	—	2	2	no	—
21	C₂×#11924746	C₂×C₄	1	2	yes	—
22	C₂×#11924746	C₂×C₄	1	2	yes	—
23	C₂×#11924746	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₁₄₉	Q₈	1	2	yes	—
25	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
26	M₁₆×C₁₄₉	—	2	2	no	—
27	M₁₆×C₁₄₉	C₂×C₄	1	4	yes	—
28	C₂×C₈×C₁₄₉	—	2	2	no	—
29	C₄²×C₁₄₉	C₂×C₄	1	2	yes	—
30	C₄²×C₁₄₉	D₄	1	1	yes	characteristic
31	C₄²×C₁₄₉	D₄	1	1	yes	characteristic
32	#11924746	—	2	2	no	—
33	#11924746	—	2	4	no	—
34	#11924746	C₄²	1	4	yes	—
35	C₂³×C₁₄₉	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₄×C₁₄₉	—	4	2	no	—
37	C₂×C₄×C₁₄₉	C₄⋊C₄	1	2	yes	—
38	C₂×C₄×C₁₄₉	C₄⋊C₄	1	2	yes	—
39	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₄₉	C₂²×C₄	1	2	yes	—
41	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₄₉	#41	1	2	yes	—
43	C₈×C₁₄₉	—	2	4	no	—
44	C₂²×C₁₄₉	—	2	2	no	—
45	C₂²×C₁₄₉	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₁₄₉	C₄×Q₈	1	2	yes	—
47	C₄×C₁₄₉	—	2	2	no	—
48	C₄×C₁₄₉	—	2	2	no	—
49	C₄×C₁₄₉	C₂×C₄²	1	1	yes	characteristic
50	C₄×C₁₄₉	#117	1	1	yes	characteristic
51	C₄×C₁₄₉	C₄×D₄	1	2	yes	—
52	C₂×C₁₄₉	—	2	2	no	—
53	C₂×C₁₄₉	C₄×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₁₄₉	#430	1	2	yes	—
55	C₁₄₉	#1778	1	1	yes	characteristic, complemented, Sylow
56	#1778	—	149	1	no	maximal, Sylow
57	C₂×#108	—	149	2	no	—
58	C₄×M₁₆	—	149	1	no	—
59	C₄×M₁₆	D₁₄₉	1	1	yes	characteristic
60	C₂×#97	—	149	1	no	—
61	#341	—	149	2	no	—
62	C₂×M₁₆	—	149	2	no	—
63	C₂×M₁₆	C₂×D₁₄₉	1	2	yes	—
64	C₂²×C₈	—	149	1	no	—
65	C₂²×C₈	—	149	1	no	—
66	C₂×C₄²	C₂×D₁₄₉	1	1	yes	characteristic
67	#108	—	149	8	no	—
68	#97	—	2	1	no	—
69	#97	—	298	1	no	—
70	#97	—	149	4	no	—
71	C₄×C₈	—	2	1	no	—
72	C₄×C₈	—	298	1	no	—
73	C₂²×C₄	Dic₂₉₈	1	2	yes	—
74	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
75	M₁₆	—	2	2	no	—
76	M₁₆	—	298	2	no	—
77	M₁₆	—	149	4	no	—
78	M₁₆	C₄×D₁₄₉	1	4	yes	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	298	2	no	—
81	C₂×C₈	—	298	2	no	—
82	C₂×C₈	—	149	2	no	—
83	C₂×C₈	—	149	2	no	—
84	C₂×C₈	—	149	4	no	—
85	C₄²	C₄×D₁₄₉	1	2	yes	—
86	C₄²	C₂²⋊D₁₄₉	1	1	yes	characteristic
87	C₄²	C₂²⋊D₁₄₉	1	1	yes	characteristic
88	C₂³	C₂×Dic₂₉₈	1	1	yes	characteristic
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	#14558715	1	2	yes	—
91	C₂×C₄	#14558715	1	2	yes	—
92	C₂×C₄	C₂×C₄×D₁₄₉	1	1	yes	characteristic
93	C₂×C₄	C₂×C₄×D₁₄₉	1	2	yes	—
94	C₂×C₄	C₂×C₂²⋊D₁₄₉	1	1	yes	characteristic
95	C₂×C₄	#14558736	1	2	yes	—
96	C₈	—	2	4	no	—
97	C₈	—	298	2	no	—
98	C₈	—	298	4	no	—
99	C₈	—	149	4	no	—
100	C₂²	—	2	2	no	—
101	C₂²	C₂×#14558715	1	1	yes	characteristic
102	C₂²	C₄×Dic₂₉₈	1	2	yes	—
103	C₄	—	2	2	no	—
104	C₄	—	2	2	no	—
105	C₄	C₄²×D₁₄₉	1	1	yes	characteristic
106	C₄	#18011868	1	1	yes	characteristic
107	C₄	C₄×C₂²⋊D₁₄₉	1	2	yes	—
108	C₂	—	2	2	no	—
109	C₂	C₄×#14558715	1	1	yes	characteristic
110	C₂	#23749673	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#18011846

C₂

yes

maximal

C₄×#14558720

C₂

yes

maximal, characteristic

C₂×#18011822

C₂

yes

maximal, characteristic

#23749389

C₂

yes

maximal

C₄×M₁₆×C₁₄₉

C₂

yes

maximal, characteristic

C₂×#14558720

C₂²

yes

—

C₂²×#11924746

C₂²

yes

characteristic

C₂²×#11924746

C₂²

yes

characteristic

#18011846

C₄

yes

complemented

#18011822

—

#18011822

C₄

yes

—

C₄×#11924746

—

C₂×M₁₆×C₁₄₉

C₂²

yes

—

C₂×C₄²×C₁₄₉

C₂²

yes

characteristic

#97×C₁₄₉

—

C₄×C₈×C₁₄₉

—

#14558720

—

#14558720

C₂×C₄

yes

—

C₂×#11924746

—

C₂×#11924746

—

C₂×#11924746

C₂×C₄

yes

—

C₂×#11924746

C₂×C₄

yes

—

C₂×#11924746

C₂×C₄

yes

—

C₂²×C₄×C₁₄₉

Q₈

yes

—

C₂²×C₄×C₁₄₉

C₂³

yes

characteristic

M₁₆×C₁₄₉

—

M₁₆×C₁₄₉

C₂×C₄

yes

—

C₂×C₈×C₁₄₉

—

C₄²×C₁₄₉

C₂×C₄

yes

—

C₄²×C₁₄₉

D₄

yes

characteristic

C₄²×C₁₄₉

D₄

yes

—

—

C₄²

yes

—

C₂³×C₁₄₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₄₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₄₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₄₉

C₂²×C₄

yes

—

C₂×C₄×C₁₄₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₄₉

#41

yes

—

C₈×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₄₉

C₄×Q₈

yes

—

C₄×C₁₄₉

—

C₄×C₁₄₉

—

C₄×C₁₄₉

C₂×C₄²

yes

characteristic

C₄×C₁₄₉

#117

yes

characteristic

C₄×C₁₄₉

C₄×D₄

yes

—

C₂×C₁₄₉

—

C₂×C₁₄₉

C₄×C₄⋊C₄

yes

characteristic

C₂×C₁₄₉

#430

yes

—

C₁₄₉

#1778

yes

characteristic, complemented, Sylow

#1778

—

149

maximal, Sylow

C₂×#108

—

149

—

C₄×M₁₆

—

149

—

C₄×M₁₆

D₁₄₉

yes

characteristic

C₂×#97

—

149

—

#341

—

149

—

C₂×M₁₆

—

149

—

C₂×M₁₆

C₂×D₁₄₉

yes

—

C₂²×C₈

—

149

—

C₂²×C₈

—

149

—

C₂×C₄²

C₂×D₁₄₉

yes

characteristic

#108

—

149

—

#97

—

#97

—

298

—

#97

—

149

—

C₄×C₈

—

C₄×C₈

—

298

—

C₂²×C₄

Dic₂₉₈

yes

—

C₂²×C₄

C₂²×D₁₄₉

yes

characteristic

M₁₆

—

M₁₆

—

298

—

M₁₆

—

149

—

M₁₆

C₄×D₁₄₉

yes

—

C₂×C₈

—

C₂×C₈

—

298

—

C₂×C₈

—

298

—

C₂×C₈

—

149

—

C₂×C₈

—

149

—

C₂×C₈

—

149

—

C₄²

C₄×D₁₄₉

yes

—

C₄²

C₂²⋊D₁₄₉

yes

characteristic

C₄²

C₂²⋊D₁₄₉

yes

characteristic

C₂³

C₂×Dic₂₉₈

yes

characteristic

C₂×C₄

—

C₂×C₄

#14558715

yes

—

C₂×C₄

#14558715

yes

—

C₂×C₄

C₂×C₄×D₁₄₉

yes

characteristic

C₂×C₄

C₂×C₄×D₁₄₉

yes

—

C₂×C₄

C₂×C₂²⋊D₁₄₉

yes

characteristic

C₂×C₄

#14558736

yes

—

C₈

—

C₈

—

298

—

C₈

—

298

—

C₈

—

149

—

100

C₂²

—

101

C₂²

C₂×#14558715

yes

characteristic

102

C₂²

C₄×Dic₂₉₈

yes

—

103

C₄

—

104

C₄

—

105

C₄

C₄²×D₁₄₉

yes

characteristic

106

C₄

#18011868

yes

characteristic

107

C₄

C₄×C₂²⋊D₁₄₉

yes

—

108

C₂

—

109

C₂

C₄×#14558715

yes

characteristic

110

C₂

#23749673

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₄₉
commutator	C₄×C₁₄₉	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₄₉
Fitting	C₄×M₁₆×C₁₄₉	C₂
socle	C₂²×C₁₄₉	C₂×C₄⋊C₄
prime radical	C₂³×C₁₄₉	C₂×Q₈
cosocle	C₂²×C₄×C₁₄₉	C₂³
hypercenter	C₄×M₁₆	D₁₄₉
hypocenter	C₁₄₉	#1778
LCS	G ▷ C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₄₉

commutator

C₄×C₁₄₉

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₄₉

Fitting

C₄×M₁₆×C₁₄₉

C₂

socle

C₂²×C₁₄₉

C₂×C₄⋊C₄

prime radical

C₂³×C₁₄₉

C₂×Q₈

cosocle

C₂²×C₄×C₁₄₉

C₂³

hypercenter

C₄×M₁₆

D₁₄₉

hypocenter

C₁₄₉

#1778

LCS

G ▷ C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
9. #18011846	C₄	semidirect
55. C₁₄₉	#1778	semidirect

H ≅ G/N

type

9. #18011846

C₄

semidirect

55. C₁₄₉

#1778

semidirect

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

subgroup	index
C₂×#18011846	2
C₄×#14558720	2
C₂×#18011822	2
#23749389	2
C₄×M₁₆×C₁₄₉	2
#1778	149

subgroup

index

C₄×M₁₆×C₁₄₉

149

Subgroups — #34044598

Conjugacy classes of subgroups

Complements

Maximal subgroups