Group #34060617 — subgroups

Subgroups — #34060617

Order 19072.

180 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 66 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	#23750630	C₂	1	1	yes	maximal, characteristic
2	#23750629	C₂	1	1	yes	maximal, characteristic
3	#23750501	C₂	1	4	yes	maximal
4	#97×D₁₄₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#14558746	C₂²	1	2	yes	—
6	C₂×#14558746	C₂²	1	2	yes	—
7	#18011978	—	2	2	no	—
8	C₄⋊#11924753	—	2	2	no	—
9	C₂×C₈×D₁₄₉	C₂²	1	2	yes	—
10	C₄²×D₁₄₉	C₂²	1	1	yes	characteristic
11	#18011822	C₄	1	1	yes	characteristic
12	#18011821	C₄	1	2	yes	—
13	#97×C₁₄₉	C₄	1	1	yes	characteristic
14	#14558746	—	2	4	no	—
15	#14558746	—	2	4	no	—
16	C₂×#11924753	—	2	2	no	—
17	C₂×#11924753	—	2	2	no	—
18	C₂×C₄×D₁₄₉	Q₈	1	2	yes	—
19	C₂×C₄×D₁₄₉	C₂³	1	1	yes	characteristic
20	C₂×#11924746	C₂×C₄	1	2	yes	—
21	C₈×D₁₄₉	—	2	2	no	—
22	C₈×D₁₄₉	—	2	2	no	—
23	C₄×Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
24	C₄×Dic₁₄₉	D₄	1	2	yes	—
25	C₂×C₈×C₁₄₉	C₂×C₄	1	2	yes	—
26	C₄²×C₁₄₉	C₂×C₄	1	1	yes	characteristic
27	#11924753	—	4	2	no	—
28	#11924753	—	4	2	no	—
29	C₂²×D₁₄₉	C₂×Q₈	1	1	yes	characteristic
30	C₂×Dic₁₄₉	C₄⋊C₄	1	2	yes	—
31	C₂×Dic₁₄₉	C₂×D₄	1	1	yes	characteristic
32	C₄×D₁₄₉	—	4	2	no	—
33	C₄×D₁₄₉	#41	1	2	yes	—
34	C₄×D₁₄₉	#41	1	2	yes	—
35	#11924746	—	2	2	no	—
36	C₂×C₄×C₁₄₉	C₄⋊C₄	1	2	yes	—
37	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
38	C₈×C₁₄₉	—	2	2	no	—
39	C₂²×C₁₄₉	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂×D₁₄₉	—	2	2	no	—
41	C₂×D₁₄₉	#125	1	2	yes	—
42	C₄×C₁₄₉	—	2	2	no	—
43	C₄×C₁₄₉	#117	1	2	yes	—
44	Dic₁₄₉	—	2	2	no	—
45	Dic₁₄₉	#124	1	1	yes	characteristic
46	Dic₁₄₉	#124	1	1	yes	characteristic
47	C₂×C₁₄₉	#382	1	1	yes	characteristic
48	C₂×C₁₄₉	#430	1	2	yes	—
49	D₁₄₉	—	2	2	no	—
50	C₁₄₉	#1838	1	1	yes	characteristic, complemented, Sylow
51	#1838	—	149	1	no	maximal, Sylow
52	#423	—	149	1	no	—
53	#423	—	149	1	no	—
54	C₂×#97	—	149	1	no	—
55	#341	—	149	4	no	—
56	C₂×M₁₆	—	149	2	no	—
57	C₂×M₁₆	—	149	2	no	—
58	C₂²×C₈	—	149	2	no	—
59	C₂×C₄²	—	149	1	no	—
60	C₄⋊C₈	—	298	2	no	—
61	C₄⋊C₈	—	298	2	no	—
62	#97	—	149	2	no	—
63	#97	—	149	1	no	—
64	#97	#10597010	1	1	yes	characteristic
65	C₂²×C₄	—	149	2	no	—
66	C₂²×C₄	—	149	1	no	—
67	M₁₆	—	298	4	no	—
68	M₁₆	—	298	4	no	—
69	C₂×C₈	—	298	2	no	—
70	C₂×C₈	—	298	2	no	—
71	C₂×C₈	—	298	2	no	—
72	C₂×C₈	—	298	2	no	—
73	C₂×C₈	—	149	2	no	—
74	C₂×C₈	C₂×#10597010	1	2	yes	—
75	C₄²	—	149	2	no	—
76	C₄²	—	149	1	no	—
77	C₄²	C₂×#10597010	1	1	yes	characteristic
78	C₂³	—	149	1	no	—
79	C₂×C₄	—	596	2	no	—
80	C₂×C₄	—	149	2	no	—
81	C₂×C₄	—	149	1	no	—
82	C₂×C₄	—	149	2	no	—
83	C₂×C₄	—	149	2	no	—
84	C₂×C₄	C₄⋊#10597010	1	2	yes	—
85	C₂×C₄	C₂²×#10597010	1	1	yes	characteristic
86	C₈	—	2	2	no	—
87	C₈	—	298	2	no	—
88	C₈	—	596	2	no	—
89	C₈	—	596	2	no	—
90	C₂²	—	298	2	no	—
91	C₂²	—	149	2	no	—
92	C₂²	C₂×C₄⋊#10597010	1	1	yes	characteristic
93	C₄	—	2	2	no	—
94	C₄	—	298	2	no	—
95	C₄	—	149	1	no	—
96	C₄	—	149	1	no	—
97	C₄	#18011990	1	2	yes	—
98	C₂	—	298	2	no	—
99	C₂	#23750569	1	1	yes	characteristic
100	C₂	#23750640	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23750630

C₂

yes

maximal, characteristic

#23750629

C₂

yes

maximal, characteristic

#23750501

C₂

yes

maximal

#97×D₁₄₉

C₂

yes

maximal, characteristic

C₂×#14558746

C₂²

yes

—

C₂×#14558746

C₂²

yes

—

#18011978

—

C₄⋊#11924753

—

C₂×C₈×D₁₄₉

C₂²

yes

—

C₄²×D₁₄₉

C₂²

yes

characteristic

#18011822

C₄

yes

characteristic

#18011821

C₄

yes

—

#97×C₁₄₉

C₄

yes

—

—

—

—

C₂×C₄×D₁₄₉

Q₈

yes

—

C₂×C₄×D₁₄₉

C₂³

yes

characteristic

C₂×#11924746

C₂×C₄

yes

—

C₈×D₁₄₉

—

C₈×D₁₄₉

—

C₄×Dic₁₄₉

C₂×C₄

yes

characteristic

C₄×Dic₁₄₉

D₄

yes

—

C₂×C₈×C₁₄₉

C₂×C₄

yes

—

C₄²×C₁₄₉

C₂×C₄

yes

characteristic

#11924753

—

#11924753

—

C₂²×D₁₄₉

C₂×Q₈

yes

characteristic

C₂×Dic₁₄₉

C₄⋊C₄

yes

—

C₂×Dic₁₄₉

C₂×D₄

yes

characteristic

C₄×D₁₄₉

—

C₄×D₁₄₉

#41

yes

—

C₄×D₁₄₉

#41

yes

—

#11924746

—

C₂×C₄×C₁₄₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₄₉

C₂²×C₄

yes

characteristic

C₈×C₁₄₉

—

C₂²×C₁₄₉

C₂×C₄⋊C₄

yes

characteristic

C₂×D₁₄₉

—

C₂×D₁₄₉

#125

yes

—

C₄×C₁₄₉

—

C₄×C₁₄₉

#117

yes

—

Dic₁₄₉

—

Dic₁₄₉

yes

Dic₁₄₉

yes

C₂×C₁₄₉

yes

C₂×C₁₄₉

yes

—

D₁₄₉

—

C₁₄₉

#1838

yes

characteristic, complemented, Sylow

#1838

—

149

maximal, Sylow

#423

—

149

—

#423

—

149

—

C₂×#97

—

149

—

#341

—

149

—

C₂×M₁₆

—

149

—

C₂×M₁₆

—

149

—

C₂²×C₈

—

149

—

C₂×C₄²

—

149

—

C₄⋊C₈

—

298

—

C₄⋊C₈

—

298

—

#97

—

149

—

#97

—

149

—

#97

#10597010

yes

characteristic

C₂²×C₄

—

149

—

C₂²×C₄

—

149

—

M₁₆

—

298

—

M₁₆

—

298

—

C₂×C₈

—

298

—

C₂×C₈

—

298

—

C₂×C₈

—

298

—

C₂×C₈

—

298

—

C₂×C₈

—

149

—

C₂×C₈

C₂×#10597010

yes

—

C₄²

—

149

—

C₄²

—

149

—

C₄²

C₂×#10597010

yes

characteristic

C₂³

—

149

—

C₂×C₄

—

596

—

C₂×C₄

—

149

—

C₂×C₄

—

149

—

C₂×C₄

—

149

—

C₂×C₄

—

149

—

C₂×C₄

C₄⋊#10597010

yes

—

C₂×C₄

C₂²×#10597010

yes

characteristic

C₈

—

C₈

—

298

—

C₈

—

596

—

C₈

—

596

—

C₂²

—

298

—

C₂²

—

149

—

C₂²

C₂×C₄⋊#10597010

yes

characteristic

C₄

—

C₄

—

298

—

C₄

—

149

—

C₄

—

149

—

C₄

#18011990

yes

—

C₂

—

298

—

C₂

#23750569

yes

characteristic

100

C₂

#23750640

yes

—

	subgroup	G/H
center	C₂²	C₂×C₄⋊#10597010
commutator	C₂×C₄×C₁₄₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×#10597010
Fitting	#97×C₁₄₉	C₄
socle	C₂²×C₁₄₉	C₂×C₄⋊C₄
prime radical	C₂²×D₁₄₉	C₂×Q₈
cosocle	C₂×C₄×D₁₄₉	C₂³
hypercenter	#97	#10597010
hypocenter	C₁₄₉	#1838
LCS	G ▷ C₂×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

subgroup

G/H

center

C₂²

C₂×C₄⋊#10597010

commutator

C₂×C₄×C₁₄₉

C₂²×C₄

Frattini

C₂×C₄

C₂²×#10597010

Fitting

#97×C₁₄₉

C₄

socle

C₂²×C₁₄₉

C₂×C₄⋊C₄

prime radical

C₂²×D₁₄₉

C₂×Q₈

cosocle

C₂×C₄×D₁₄₉

C₂³

C₁₄₉

G ▷ C₂×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #97

N	H ≅ G/N	type
50. C₁₄₉	#1838	semidirect

H ≅ G/N

type

50. C₁₄₉

#1838

semidirect

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

subgroup	index
#23750630	2
#23750629	2
#23750501	2
#97×D₁₄₉	2
#1838	149

subgroup

index

#97×D₁₄₉

149

Subgroups — #34060617

Conjugacy classes of subgroups

Complements

Maximal subgroups