Group #34060407 — subgroups

Subgroups — #34060407

Order 19072.

272 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 98 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	#23750584	C₂	1	1	yes	maximal, characteristic
2	C₄²×#10597010	C₂	1	1	yes	maximal, characteristic
3	(C₂×#10597010)⋊C₈	C₂	1	4	yes	maximal
4	#97×D₁₄₉	C₂	1	1	yes	maximal, characteristic
5	C₂×D₁₄₉⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄×#10597010	C₄	1	4	yes	—
7	C₂×C₄×#10597010	C₂²	1	2	yes	—
8	#18011966	C₄	1	4	yes	complemented
9	C₂×C₈×D₁₄₉	C₂²	1	2	yes	—
10	C₄²×D₁₄₉	C₂²	1	1	yes	characteristic
11	#18011822	C₄	1	1	yes	characteristic, complemented
12	#18011821	C₄	1	2	yes	complemented
13	#97×C₁₄₉	C₄	1	1	yes	characteristic, complemented
14	C₂²×#10597010	C₂×C₄	1	4	yes	—
15	D₁₄₉⋊C₈	—	2	4	no	—
16	C₂×#11924753	C₂×C₄	1	4	yes	—
17	C₄×#10597010	—	2	8	no	—
18	C₄×#10597010	—	2	4	no	—
19	C₂×C₄×D₁₄₉	C₂×C₄	1	2	yes	—
20	C₂×C₄×D₁₄₉	C₂³	1	1	yes	characteristic
21	C₂×#11924746	C₂×C₄	1	2	yes	—
22	C₈×D₁₄₉	—	2	4	no	—
23	C₄×Dic₁₄₉	C₂×C₄	1	2	yes	—
24	C₄×Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₁₄₉	C₂×C₄	1	2	yes	—
26	C₄²×C₁₄₉	C₂×C₄	1	1	yes	characteristic
27	C₂×#10597010	—	2	4	no	—
28	C₂×#10597010	—	2	4	no	—
29	C₂×#10597010	M₁₆	1	8	yes	—
30	#11924753	—	2	4	no	—
31	C₂²×D₁₄₉	C₂²×C₄	1	1	yes	characteristic
32	C₂×Dic₁₄₉	C₄²	1	2	yes	—
33	C₂×Dic₁₄₉	C₂²×C₄	1	1	yes	characteristic
34	C₄×D₁₄₉	—	2	4	no	—
35	C₄×D₁₄₉	#41	1	4	yes	—
36	#11924746	—	2	2	no	—
37	C₂×C₄×C₁₄₉	C₄²	1	2	yes	—
38	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
39	C₈×C₁₄₉	—	2	2	no	—
40	C₂²×C₁₄₉	C₂×C₄²	1	1	yes	characteristic
41	C₂×D₁₄₉	#117	1	2	yes	—
42	C₂×D₁₄₉	C₂×M₁₆	1	4	yes	—
43	#10597010	—	2	8	no	—
44	C₄×C₁₄₉	—	2	2	no	—
45	C₄×C₁₄₉	#117	1	2	yes	—
46	Dic₁₄₉	—	2	2	no	—
47	Dic₁₄₉	#117	1	2	yes	—
48	C₂×C₁₄₉	#376	1	1	yes	characteristic
49	C₂×C₁₄₉	C₄×M₁₆	1	2	yes	—
50	D₁₄₉	#432	1	4	yes	—
51	C₁₄₉	#1745	1	1	yes	characteristic, complemented, Sylow
52	#1745	—	149	1	no	maximal, Sylow
53	C₂×#97	—	149	1	no	—
54	C₂×#97	—	149	1	no	—
55	C₄³	—	149	1	no	—
56	#336	—	149	4	no	—
57	C₂²×C₈	—	149	2	no	—
58	C₂²×C₈	—	149	2	no	—
59	C₂×C₄²	—	149	2	no	—
60	C₂×C₄²	—	149	4	no	—
61	C₂×C₄²	—	149	1	no	—
62	#97	—	149	4	no	—
63	#97	—	149	2	no	—
64	#97	—	149	1	no	—
65	#97	#10597010	1	1	yes	characteristic, complemented
66	C₂²×C₄	—	149	4	no	—
67	C₂²×C₄	—	149	2	no	—
68	C₂²×C₄	—	149	1	no	—
69	C₂×C₈	—	298	4	no	—
70	C₂×C₈	—	298	4	no	—
71	C₂×C₈	—	149	4	no	—
72	C₂×C₈	—	149	2	no	—
73	C₂×C₈	C₂×#10597010	1	2	yes	—
74	C₄²	—	298	8	no	—
75	C₄²	—	298	4	no	—
76	C₄²	—	149	2	no	—
77	C₄²	—	149	1	no	—
78	C₄²	C₂×#10597010	1	1	yes	characteristic
79	C₂³	—	149	1	no	—
80	C₂×C₄	—	298	4	no	—
81	C₂×C₄	—	298	4	no	—
82	C₂×C₄	—	298	4	no	—
83	C₂×C₄	—	149	8	no	—
84	C₂×C₄	—	149	2	no	—
85	C₂×C₄	—	149	1	no	—
86	C₂×C₄	—	149	4	no	—
87	C₂×C₄	C₄×#10597010	1	2	yes	—
88	C₂×C₄	C₂²×#10597010	1	1	yes	characteristic
89	C₈	—	2	2	no	—
90	C₈	—	298	4	no	—
91	C₈	—	298	2	no	—
92	C₂²	—	149	2	no	—
93	C₂²	—	149	4	no	—
94	C₂²	C₂×C₄×#10597010	1	1	yes	characteristic
95	C₄	—	2	2	no	—
96	C₄	—	298	8	no	—
97	C₄	—	298	2	no	—
98	C₄	—	149	2	no	—
99	C₄	#18011990	1	2	yes	—
100	C₂	—	149	4	no	—
101	C₂	#23750561	1	1	yes	characteristic
102	C₂	M₁₆×#10597010	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23750584

C₂

yes

maximal, characteristic

C₄²×#10597010

C₂

yes

maximal, characteristic

(C₂×#10597010)⋊C₈

C₂

yes

maximal

#97×D₁₄₉

C₂

yes

maximal, characteristic

C₂×D₁₄₉⋊C₈

C₂²

yes

—

C₂×C₄×#10597010

C₄

yes

—

C₂×C₄×#10597010

C₂²

yes

—

#18011966

C₄

yes

complemented

C₂×C₈×D₁₄₉

C₂²

yes

—

C₄²×D₁₄₉

C₂²

yes

characteristic

#18011822

C₄

yes

characteristic, complemented

#18011821

C₄

yes

complemented

#97×C₁₄₉

C₄

yes

characteristic, complemented

C₂²×#10597010

C₂×C₄

yes

—

D₁₄₉⋊C₈

—

C₂×#11924753

C₂×C₄

yes

—

C₄×#10597010

—

C₄×#10597010

—

C₂×C₄×D₁₄₉

C₂×C₄

yes

—

C₂×C₄×D₁₄₉

C₂³

yes

characteristic

C₂×#11924746

C₂×C₄

yes

—

C₈×D₁₄₉

—

C₄×Dic₁₄₉

C₂×C₄

yes

—

C₄×Dic₁₄₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₄₉

C₂×C₄

yes

—

C₄²×C₁₄₉

C₂×C₄

yes

—

—

M₁₆

yes

—

#11924753

—

C₂²×D₁₄₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₄₉

C₄²

yes

—

C₂×Dic₁₄₉

C₂²×C₄

yes

characteristic

C₄×D₁₄₉

—

C₄×D₁₄₉

#41

yes

—

#11924746

—

C₂×C₄×C₁₄₉

C₄²

yes

—

C₂×C₄×C₁₄₉

C₂²×C₄

yes

characteristic

C₈×C₁₄₉

—

C₂²×C₁₄₉

C₂×C₄²

yes

characteristic

C₂×D₁₄₉

#117

yes

—

C₂×D₁₄₉

C₂×M₁₆

yes

—

#10597010

—

C₄×C₁₄₉

—

C₄×C₁₄₉

#117

yes

—

Dic₁₄₉

—

Dic₁₄₉

#117

yes

—

C₂×C₁₄₉

#376

yes

characteristic

C₂×C₁₄₉

C₄×M₁₆

yes

—

D₁₄₉

#432

yes

—

C₁₄₉

#1745

yes

characteristic, complemented, Sylow

#1745

—

149

maximal, Sylow

C₂×#97

—

149

—

C₂×#97

—

149

—

C₄³

—

149

—

#336

—

149

—

C₂²×C₈

—

149

—

C₂²×C₈

—

149

—

C₂×C₄²

—

149

—

C₂×C₄²

—

149

—

C₂×C₄²

—

149

—

#97

—

149

—

#97

—

149

—

#97

—

149

—

#97

#10597010

yes

characteristic, complemented

C₂²×C₄

—

149

—

C₂²×C₄

—

149

—

C₂²×C₄

—

149

—

C₂×C₈

—

298

—

C₂×C₈

—

298

—

C₂×C₈

—

149

—

C₂×C₈

—

149

—

C₂×C₈

C₂×#10597010

yes

—

C₄²

—

298

—

C₄²

—

298

—

C₄²

—

149

—

C₄²

—

149

—

C₄²

C₂×#10597010

yes

characteristic

C₂³

—

149

—

C₂×C₄

—

298

—

C₂×C₄

—

298

—

C₂×C₄

—

298

—

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

—

298

—

C₂²

—

149

—

C₂²

—

149

—

C₂²

C₂×C₄×#10597010

yes

characteristic

C₄

—

C₄

—

298

—

C₄

—

298

—

C₄

—

149

—

C₄

#18011990

yes

—

100

C₂

—

149

—

101

C₂

#23750561

yes

characteristic

102

C₂

M₁₆×#10597010

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#10597010
commutator, socle	C₂²×C₁₄₉	C₂×C₄²
Fitting	#97×C₁₄₉	C₄
prime radical	C₂²×D₁₄₉	C₂²×C₄
cosocle	C₂×C₄×D₁₄₉	C₂³
hypercenter	#97	#10597010
hypocenter	C₁₄₉	#1745
LCS	G ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂×C₄ ◁ #97

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#10597010

commutator, socle

C₂²×C₁₄₉

C₂×C₄²

Fitting

#97×C₁₄₉

C₄

prime radical

C₂²×D₁₄₉

C₂²×C₄

cosocle

C₂×C₄×D₁₄₉

C₂³

C₁₄₉

G ▷ C₂²×C₁₄₉ ▷ C₁₄₉

C₁ ◁ C₂×C₄ ◁ #97

N	H ≅ G/N	type
8. #18011966	C₄	semidirect
11. #18011822	C₄	semidirect
12. #18011821	C₄	semidirect
13. #97×C₁₄₉	C₄	semidirect
51. C₁₄₉	#1745	semidirect
65. #97	#10597010	semidirect

H ≅ G/N

type

8. #18011966

C₄

semidirect

11. #18011822

C₄

semidirect

12. #18011821

C₄

semidirect

13. #97×C₁₄₉

C₄

semidirect

51. C₁₄₉

#1745

semidirect

65. #97

#10597010

semidirect

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

subgroup	index
#23750584	2
C₄²×#10597010	2
(C₂×#10597010)⋊C₈	2
#97×D₁₄₉	2
#1745	149

subgroup

index

#97×D₁₄₉

149

Subgroups — #34060407

Conjugacy classes of subgroups

Complements

Maximal subgroups