Group #11927113 — subgroups

Subgroups — #11927113

Order 1216.

334 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 111 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×#64541	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₇₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×#117	C₂	1	1	yes	maximal, characteristic, complemented
4	#10597341	C₂	1	4	yes	maximal, complemented
5	#10597328	C₂	1	4	yes	maximal, complemented
6	#10597321	C₂	1	2	yes	maximal, complemented
7	#10597320	C₂	1	2	yes	maximal, complemented
8	C₂³×D₁₉	C₂²	1	2	yes	—
9	C₂²×C₄×C₁₉	C₂²	1	1	yes	characteristic
10	C₂×C₂²⋊D₁₉	C₂²	1	2	yes	—
11	#64541	—	2	4	no	—
12	C₂×D₇₆	—	2	4	no	—
13	C₂×D₇₆	C₂²	1	2	yes	—
14	C₂×D₇₆	C₂²	1	2	yes	—
15	C₂×D₇₆	C₂²	1	1	yes	characteristic, complemented
16	C₂×D₇₆	C₂²	1	8	yes	complemented
17	C₂×C₄×D₁₉	C₂²	1	2	yes	complemented
18	C₂×Dic₃₈	C₂²	1	1	yes	characteristic, complemented
19	C₄⋊C₄×C₁₉	C₂²	1	2	yes	complemented
20	C₂²⋊C₄×C₁₉	C₂²	1	2	yes	—
21	C₄²×C₁₉	C₂²	1	2	yes	complemented
22	(C₂×D₁₉)⋊C₄	C₂²	1	8	yes	complemented
23	C₂³×C₁₉	C₂³	1	1	yes	characteristic
24	C₂²×D₁₉	—	2	4	no	—
25	C₂²×D₁₉	—	4	4	no	—
26	C₂²×D₁₉	C₂³	1	4	yes	—
27	C₂²×D₁₉	C₂³	1	2	yes	—
28	C₂×C₄×C₁₉	D₄	1	4	yes	complemented
29	C₂×C₄×C₁₉	C₂³	1	4	yes	—
30	C₂×C₄×C₁₉	C₂³	1	1	yes	characteristic
31	C₂×C₄×C₁₉	C₂³	1	1	yes	characteristic
32	C₂²⋊D₁₉	—	2	4	no	—
33	C₂×Dic₁₉	C₂³	1	2	yes	—
34	D₇₆	—	2	4	no	—
35	D₇₆	—	2	4	no	—
36	D₇₆	—	2	2	no	—
37	D₇₆	—	4	8	no	—
38	C₄×D₁₉	—	2	4	no	—
39	Dic₃₈	—	2	2	no	—
40	C₂²×C₁₉	—	2	2	no	—
41	C₂²×C₁₉	C₂×D₄	1	2	yes	—
42	C₂²×C₁₉	C₂⁴	1	1	yes	characteristic
43	C₂×D₁₉	—	2	2	no	—
44	C₂×D₁₉	—	2	4	no	—
45	C₂×D₁₉	—	2	4	no	—
46	C₂×D₁₉	—	2	8	no	—
47	C₂×D₁₉	—	4	8	no	—
48	C₄×C₁₉	—	2	4	no	—
49	C₄×C₁₉	C₂×D₄	1	2	yes	—
50	C₄×C₁₉	C₂×D₄	1	2	yes	—
51	Dic₁₉	—	2	2	no	—
52	#535	—	19	1	no	maximal, Sylow
53	C₂×C₁₉	—	2	2	no	—
54	C₂×C₁₉	C₂²×D₄	1	1	yes	characteristic
55	C₂×C₁₉	H(5,𝔽₂)	1	2	yes	—
56	D₁₉	—	4	2	no	—
57	D₁₉	—	4	4	no	—
58	C₂×#41	—	19	1	no	—
59	C₂²×D₄	—	19	1	no	—
60	#127	—	19	2	no	—
61	#124	—	19	2	no	—
62	#121	—	19	4	no	—
63	#120	—	19	4	no	—
64	#117	D₁₉	1	1	yes	characteristic, complemented
65	C₁₉	#535	1	1	yes	characteristic, complemented, Sylow
66	C₂⁴	—	19	2	no	—
67	#41	—	38	4	no	—
68	C₂×Q₈	—	19	1	no	—
69	C₂×D₄	—	19	2	no	—
70	C₂×D₄	—	19	2	no	—
71	C₂×D₄	—	19	2	no	—
72	C₂×D₄	—	19	1	no	—
73	C₂×D₄	—	19	8	no	—
74	C₂×D₄	—	38	4	no	—
75	C₂²×C₄	—	19	2	no	—
76	C₂²×C₄	C₂×D₁₉	1	1	yes	characteristic
77	C₄⋊C₄	C₂×D₁₉	1	2	yes	complemented
78	C₂²⋊C₄	—	19	8	no	—
79	C₂²⋊C₄	C₂×D₁₉	1	2	yes	—
80	C₄²	C₂×D₁₉	1	2	yes	complemented
81	C₂³	—	19	4	no	—
82	C₂³	—	19	2	no	—
83	C₂³	—	38	4	no	—
84	C₂³	—	76	4	no	—
85	C₂³	C₂²×D₁₉	1	1	yes	characteristic
86	Q₈	—	38	2	no	—
87	D₄	—	38	4	no	—
88	D₄	—	38	4	no	—
89	D₄	—	38	4	no	—
90	D₄	—	38	2	no	—
91	D₄	—	76	8	no	—
92	C₂×C₄	—	19	2	no	—
93	C₂×C₄	—	38	4	no	—
94	C₂×C₄	D₇₆	1	4	yes	complemented
95	C₂×C₄	C₂²×D₁₉	1	4	yes	—
96	C₂×C₄	C₂²×D₁₉	1	1	yes	characteristic
97	C₂×C₄	C₂²×D₁₉	1	1	yes	characteristic
98	C₂²	—	2	2	no	—
99	C₂²	—	38	2	no	—
100	C₂²	—	38	4	no	—
101	C₂²	—	38	4	no	—
102	C₂²	—	38	8	no	—
103	C₂²	—	76	8	no	—
104	C₂²	C₂×D₇₆	1	2	yes	—
105	C₂²	C₂³×D₁₉	1	1	yes	characteristic
106	C₄	—	2	4	no	—
107	C₄	—	38	2	no	—
108	C₄	C₂×D₇₆	1	2	yes	—
109	C₄	C₂×D₇₆	1	2	yes	—
110	C₂	—	2	2	no	—
111	C₂	—	76	4	no	—
112	C₂	—	76	2	no	—
113	C₂	C₂²×D₇₆	1	1	yes	characteristic
114	C₂	#10597435	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#64541

C₂

yes

maximal, characteristic, complemented

C₂²×D₇₆

C₂

yes

maximal, characteristic, complemented

C₁₉×#117

C₂

yes

maximal, characteristic, complemented

#10597341

C₂

yes

maximal, complemented

#10597328

C₂

yes

maximal, complemented

#10597321

C₂

yes

maximal, complemented

#10597320

C₂

yes

maximal, complemented

C₂³×D₁₉

C₂²

yes

—

C₂²×C₄×C₁₉

C₂²

yes

characteristic

C₂×C₂²⋊D₁₉

C₂²

yes

—

#64541

—

C₂×D₇₆

—

C₂×D₇₆

C₂²

yes

—

C₂×D₇₆

C₂²

yes

—

C₂×D₇₆

C₂²

yes

characteristic, complemented

C₂×D₇₆

C₂²

yes

complemented

C₂×C₄×D₁₉

C₂²

yes

complemented

C₂×Dic₃₈

C₂²

yes

characteristic, complemented

C₄⋊C₄×C₁₉

C₂²

yes

complemented

C₂²⋊C₄×C₁₉

C₂²

yes

—

C₄²×C₁₉

C₂²

yes

complemented

(C₂×D₁₉)⋊C₄

C₂²

yes

complemented

C₂³×C₁₉

C₂³

yes

characteristic

C₂²×D₁₉

—

C₂²×D₁₉

—

C₂²×D₁₉

C₂³

yes

—

C₂²×D₁₉

C₂³

yes

—

C₂×C₄×C₁₉

D₄

yes

complemented

C₂×C₄×C₁₉

C₂³

yes

—

C₂×C₄×C₁₉

C₂³

yes

characteristic

C₂×C₄×C₁₉

C₂³

yes

characteristic

C₂²⋊D₁₉

—

C₂×Dic₁₉

C₂³

yes

—

D₇₆

—

D₇₆

—

D₇₆

—

D₇₆

—

C₄×D₁₉

—

Dic₃₈

—

C₂²×C₁₉

—

C₂²×C₁₉

C₂×D₄

yes

—

C₂²×C₁₉

C₂⁴

yes

characteristic

C₂×D₁₉

—

C₂×D₁₉

—

C₂×D₁₉

—

C₂×D₁₉

—

C₂×D₁₉

—

C₄×C₁₉

—

C₄×C₁₉

C₂×D₄

yes

—

C₄×C₁₉

C₂×D₄

yes

—

Dic₁₉

—

#535

—

maximal, Sylow

C₂×C₁₉

—

C₂×C₁₉

C₂²×D₄

yes

characteristic

C₂×C₁₉

H(5,𝔽₂)

yes

—

D₁₉

—

D₁₉

—

C₂×#41

—

C₂²×D₄

—

—

—

—

—

D₁₉

yes

characteristic, complemented

C₁₉

#535

yes

characteristic, complemented, Sylow

C₂⁴

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₁₉

yes

characteristic

C₄⋊C₄

C₂×D₁₉

yes

complemented

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₁₉

yes

—

C₄²

C₂×D₁₉

yes

complemented

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²×D₁₉

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₇₆

yes

complemented

C₂×C₄

C₂²×D₁₉

yes

—

C₂×C₄

C₂²×D₁₉

yes

characteristic

C₂×C₄

C₂²×D₁₉

yes

characteristic

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

—

102

C₂²

—

103

C₂²

—

104

C₂²

C₂×D₇₆

yes

—

105

C₂²

C₂³×D₁₉

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

C₂×D₇₆

yes

—

109

C₄

C₂×D₇₆

yes

—

110

C₂

—

111

C₂

—

112

C₂

—

113

C₂

C₂²×D₇₆

yes

characteristic

114

C₂

#10597435

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₉
commutator, socle, cosocle	C₂²×C₁₉	C₂⁴
Fitting	C₁₉×#117	C₂
hypercenter	#117	D₁₉
hypocenter	C₁₉	#535
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ #117

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₉

commutator, socle, cosocle

C₂²×C₁₉

C₂⁴

Fitting

C₁₉×#117

C₂

D₁₉

C₁₉

G ▷ C₂²×C₁₉ ▷ C₁₉

C₁ ◁ C₂² ◁ #117

N	H ≅ G/N	type
1. C₂×#64541	C₂	semidirect
2. C₂²×D₇₆	C₂	semidirect
3. C₁₉×#117	C₂	semidirect
4. #10597341	C₂	semidirect
5. #10597328	C₂	semidirect
6. #10597321	C₂	semidirect
7. #10597320	C₂	semidirect
15. C₂×D₇₆	C₂²	semidirect
16. C₂×D₇₆	C₂²	semidirect
17. C₂×C₄×D₁₉	C₂²	semidirect
18. C₂×Dic₃₈	C₂²	semidirect
19. C₄⋊C₄×C₁₉	C₂²	semidirect
21. C₄²×C₁₉	C₂²	semidirect
22. (C₂×D₁₉)⋊C₄	C₂²	semidirect
28. C₂×C₄×C₁₉	D₄	semidirect
64. #117	D₁₉	semidirect
65. C₁₉	#535	semidirect
77. C₄⋊C₄	C₂×D₁₉	semidirect
80. C₄²	C₂×D₁₉	semidirect
94. C₂×C₄	D₇₆	semidirect

H ≅ G/N

type

1. C₂×#64541

C₂

semidirect

2. C₂²×D₇₆

C₂

semidirect

3. C₁₉×#117

C₂

semidirect

4. #10597341

C₂

semidirect

5. #10597328

C₂

semidirect

6. #10597321

C₂

semidirect

7. #10597320

C₂

semidirect

15. C₂×D₇₆

C₂²

semidirect

16. C₂×D₇₆

C₂²

semidirect

17. C₂×C₄×D₁₉

C₂²

semidirect

18. C₂×Dic₃₈

C₂²

semidirect

19. C₄⋊C₄×C₁₉

C₂²

semidirect

21. C₄²×C₁₉

C₂²

semidirect

22. (C₂×D₁₉)⋊C₄

C₂²

semidirect

28. C₂×C₄×C₁₉

D₄

semidirect

64. #117

D₁₉

semidirect

65. C₁₉

#535

semidirect

77. C₄⋊C₄

C₂×D₁₉

semidirect

80. C₄²

C₂×D₁₉

semidirect

94. C₂×C₄

D₇₆

semidirect

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

subgroup	index
C₂×#64541	2
C₂²×D₇₆	2
C₁₉×#117	2
#10597341	2
#10597328	2
#10597321	2
#10597320	2
#535	19

subgroup

index

C₂×#64541

C₂²×D₇₆

C₁₉×#117

Subgroups — #11927113

Conjugacy classes of subgroups

Complements

Maximal subgroups