Group #11927198 — subgroups

Subgroups — #11927198

Order 1216.

292 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 115 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×#64543	C₂	1	2	yes	maximal, complemented
2	C₄×D₄×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#10597376	C₂	1	2	yes	maximal, complemented
4	C₂×C₄⋊Dic₁₉	C₂	1	2	yes	maximal, complemented
5	#10597343	C₂	1	2	yes	maximal, complemented
6	#10597332	C₂	1	4	yes	maximal, complemented
7	C₄×D₇₆	C₂	1	1	yes	maximal, characteristic, complemented
8	#10597315	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×D₄×C₁₉	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₁₉	C₂²	1	2	yes	—
11	C₂×C₂²⋊D₁₉	C₂²	1	4	yes	—
12	C₂²×Dic₁₉	C₂²	1	4	yes	—
13	#64543	—	2	8	no	—
14	C₂×D₇₆	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₁₉	C₂²	1	2	yes	—
16	C₂×Dic₃₈	C₂²	1	2	yes	—
17	C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
18	C₂²⋊C₄×C₁₉	C₂²	1	2	yes	—
19	C₄²×C₁₉	C₂²	1	1	yes	characteristic
20	(C₂×D₁₉)⋊C₄	C₂²	1	4	yes	—
21	(C₂×D₁₉)⋊C₄	C₂²	1	2	yes	—
22	C₄⋊Dic₁₉	C₂²	1	2	yes	—
23	C₄⋊Dic₁₉	C₂²	1	2	yes	—
24	C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
25	C₄⋊Dic₁₉	C₂²	1	4	yes	—
26	C₂³×C₁₉	C₂³	1	2	yes	—
27	C₂²×D₁₉	C₂³	1	2	yes	—
28	D₄×C₁₉	D₄	1	4	yes	complemented
29	C₂×C₄×C₁₉	—	2	4	no	—
30	C₂×C₄×C₁₉	C₂³	1	1	yes	characteristic
31	C₂×C₄×C₁₉	C₂³	1	2	yes	—
32	C₂×C₄×C₁₉	C₂³	1	1	yes	characteristic
33	C₂×C₄×C₁₉	C₂³	1	1	yes	characteristic
34	C₂²⋊D₁₉	—	2	8	no	—
35	C₂×Dic₁₉	—	2	8	no	—
36	C₂×Dic₁₉	C₂³	1	4	yes	—
37	C₂×Dic₁₉	C₂³	1	2	yes	—
38	D₇₆	—	2	2	no	—
39	C₄×D₁₉	—	2	4	no	—
40	Dic₃₈	—	2	4	no	—
41	C₂²×C₁₉	—	2	2	no	—
42	C₂²×C₁₉	—	2	2	no	—
43	C₂²×C₁₉	C₂×D₄	1	4	yes	—
44	C₂²×C₁₉	C₂⁴	1	1	yes	characteristic
45	C₂×D₁₉	—	2	2	no	—
46	C₂×D₁₉	—	2	2	no	—
47	C₂×D₁₉	—	2	2	no	—
48	C₄×C₁₉	—	2	2	no	—
49	C₄×C₁₉	—	2	1	no	—
50	C₄×C₁₉	C₂×D₄	1	2	yes	—
51	C₄×C₁₉	#41	1	2	yes	complemented
52	Dic₁₉	—	2	4	no	—
53	Dic₁₉	—	2	2	no	—
54	#547	—	19	1	no	maximal, Sylow
55	C₂×C₁₉	—	2	4	no	—
56	C₂×C₁₉	C₂²×D₄	1	1	yes	characteristic
57	C₂×C₁₉	C₂×#41	1	1	yes	characteristic
58	C₂×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
59	D₁₉	—	4	2	no	—
60	C₂×#41	—	19	2	no	—
61	C₄⋊Q₈	—	19	1	no	—
62	#123	—	19	4	no	—
63	C₂²⋊Q₈	—	19	2	no	—
64	#121	—	19	2	no	—
65	C₄×D₄	—	19	1	no	—
66	C₄×D₄	D₁₉	1	1	yes	characteristic, complemented
67	C₂×C₄⋊C₄	—	19	2	no	—
68	C₁₉	#547	1	1	yes	characteristic, complemented, Sylow
69	#41	—	38	8	no	—
70	C₂×Q₈	—	19	2	no	—
71	C₂×D₄	—	19	4	no	—
72	C₂×D₄	—	19	1	no	—
73	C₂×D₄	C₂×D₁₉	1	1	yes	characteristic
74	C₂²×C₄	—	19	4	no	—
75	C₂²×C₄	—	19	2	no	—
76	C₂²×C₄	C₂×D₁₉	1	2	yes	—
77	C₄⋊C₄	—	19	1	no	—
78	C₄⋊C₄	—	19	2	no	—
79	C₄⋊C₄	—	19	4	no	—
80	C₄⋊C₄	—	19	2	no	—
81	C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
82	C₂²⋊C₄	—	19	2	no	—
83	C₂²⋊C₄	—	19	4	no	—
84	C₂²⋊C₄	C₂×D₁₉	1	2	yes	—
85	C₄²	C₂×D₁₉	1	1	yes	characteristic
86	C₂³	—	19	2	no	—
87	C₂³	C₂²×D₁₉	1	2	yes	—
88	Q₈	—	38	4	no	—
89	D₄	—	38	8	no	—
90	D₄	—	38	2	no	—
91	D₄	D₇₆	1	4	yes	complemented
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	19	2	no	—
94	C₂×C₄	—	19	4	no	—
95	C₂×C₄	—	38	8	no	—
96	C₂×C₄	—	38	4	no	—
97	C₂×C₄	C₂²×D₁₉	1	1	yes	characteristic
98	C₂×C₄	C₂²×D₁₉	1	2	yes	—
99	C₂×C₄	C₂²×D₁₉	1	1	yes	characteristic
100	C₂×C₄	C₂²×D₁₉	1	1	yes	characteristic
101	C₂²	—	2	2	no	—
102	C₂²	—	2	2	no	—
103	C₂²	—	38	2	no	—
104	C₂²	—	38	2	no	—
105	C₂²	—	38	2	no	—
106	C₂²	C₂×D₇₆	1	4	yes	—
107	C₂²	C₂³×D₁₉	1	1	yes	characteristic
108	C₄	—	2	2	no	—
109	C₄	—	2	1	no	—
110	C₄	—	38	2	no	—
111	C₄	—	38	4	no	—
112	C₄	C₂×D₇₆	1	2	yes	—
113	C₄	#64543	1	2	yes	complemented
114	C₂	—	2	4	no	—
115	C₂	—	76	2	no	—
116	C₂	C₂²×D₇₆	1	1	yes	characteristic
117	C₂	C₂×#64543	1	1	yes	characteristic
118	C₂	#10597436	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#64543

C₂

yes

maximal, complemented

C₄×D₄×C₁₉

C₂

yes

maximal, characteristic, complemented

#10597376

C₂

yes

maximal, complemented

C₂×C₄⋊Dic₁₉

C₂

yes

maximal, complemented

#10597343

C₂

yes

maximal, complemented

#10597332

C₂

yes

maximal, complemented

C₄×D₇₆

C₂

yes

maximal, characteristic, complemented

#10597315

C₂

yes

maximal, characteristic, complemented

C₂×D₄×C₁₉

C₂²

yes

characteristic

C₂²×C₄×C₁₉

C₂²

yes

—

C₂×C₂²⋊D₁₉

C₂²

yes

—

C₂²×Dic₁₉

C₂²

yes

—

#64543

—

C₂×D₇₆

C₂²

yes

characteristic

C₂×C₄×D₁₉

C₂²

yes

—

C₂×Dic₃₈

C₂²

yes

—

C₄⋊C₄×C₁₉

C₂²

yes

characteristic

C₂²⋊C₄×C₁₉

C₂²

yes

—

C₄²×C₁₉

C₂²

yes

characteristic

(C₂×D₁₉)⋊C₄

C₂²

yes

—

(C₂×D₁₉)⋊C₄

C₂²

yes

—

C₄⋊Dic₁₉

C₂²

yes

—

C₄⋊Dic₁₉

C₂²

yes

—

C₄⋊Dic₁₉

C₂²

yes

characteristic

C₄⋊Dic₁₉

C₂²

yes

—

C₂³×C₁₉

C₂³

yes

—

C₂²×D₁₉

C₂³

yes

—

D₄×C₁₉

D₄

yes

complemented

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₂³

yes

characteristic

C₂×C₄×C₁₉

C₂³

yes

—

C₂×C₄×C₁₉

C₂³

yes

characteristic

C₂×C₄×C₁₉

C₂³

yes

characteristic

C₂²⋊D₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

C₂³

yes

—

C₂×Dic₁₉

C₂³

yes

—

D₇₆

—

C₄×D₁₉

—

Dic₃₈

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

C₂×D₄

yes

—

C₂²×C₁₉

C₂⁴

yes

characteristic

C₂×D₁₉

—

C₂×D₁₉

—

C₂×D₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

C₂×D₄

yes

—

C₄×C₁₉

#41

yes

complemented

Dic₁₉

—

Dic₁₉

—

#547

—

maximal, Sylow

C₂×C₁₉

—

C₂×C₁₉

C₂²×D₄

yes

characteristic

C₂×C₁₉

C₂×#41

yes

characteristic

C₂×C₁₉

2₋¹⁺⁴

yes

characteristic

D₁₉

—

C₂×#41

—

C₄⋊Q₈

—

#123

—

C₂²⋊Q₈

—

#121

—

C₄×D₄

—

C₄×D₄

D₁₉

yes

characteristic, complemented

C₂×C₄⋊C₄

—

C₁₉

#547

yes

characteristic, complemented, Sylow

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₁₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₁₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₁₉

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₁₉

yes

—

C₄²

C₂×D₁₉

yes

characteristic

C₂³

—

C₂³

C₂²×D₁₉

yes

—

Q₈

—

D₄

—

D₄

—

D₄

D₇₆

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×D₁₉

yes

characteristic

C₂×C₄

C₂²×D₁₉

yes

—

C₂×C₄

C₂²×D₁₉

yes

characteristic

100

C₂×C₄

C₂²×D₁₉

yes

characteristic

101

C₂²

—

102

C₂²

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

C₂×D₇₆

yes

—

107

C₂²

C₂³×D₁₉

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

C₂×D₇₆

yes

—

113

C₄

#64543

yes

complemented

114

C₂

—

115

C₂

—

116

C₂

C₂²×D₇₆

yes

characteristic

117

C₂

C₂×#64543

yes

characteristic

118

C₂

#10597436

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₉
commutator, socle, cosocle	C₂²×C₁₉	C₂⁴
Fitting	C₄×D₄×C₁₉	C₂
hypercenter	C₄×D₄	D₁₉
hypocenter	C₁₉	#547
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₉

commutator, socle, cosocle

C₂²×C₁₉

C₂⁴

Fitting

C₄×D₄×C₁₉

C₂

hypercenter

C₄×D₄

D₁₉

C₁₉

G ▷ C₂²×C₁₉ ▷ C₁₉

C₁ ◁ C₂² ◁ C₄×D₄

N	H ≅ G/N	type
1. C₂×#64543	C₂	semidirect
2. C₄×D₄×C₁₉	C₂	semidirect
3. #10597376	C₂	semidirect
4. C₂×C₄⋊Dic₁₉	C₂	semidirect
5. #10597343	C₂	semidirect
6. #10597332	C₂	semidirect
7. C₄×D₇₆	C₂	semidirect
8. #10597315	C₂	semidirect
28. D₄×C₁₉	D₄	semidirect
51. C₄×C₁₉	#41	semidirect
66. C₄×D₄	D₁₉	semidirect
68. C₁₉	#547	semidirect
91. D₄	D₇₆	semidirect
113. C₄	#64543	semidirect

H ≅ G/N

type

1. C₂×#64543

C₂

semidirect

2. C₄×D₄×C₁₉

C₂

semidirect

3. #10597376

C₂

semidirect

4. C₂×C₄⋊Dic₁₉

C₂

semidirect

5. #10597343

C₂

semidirect

6. #10597332

C₂

semidirect

7. C₄×D₇₆

C₂

semidirect

8. #10597315

C₂

semidirect

28. D₄×C₁₉

D₄

semidirect

51. C₄×C₁₉

#41

semidirect

66. C₄×D₄

D₁₉

semidirect

68. C₁₉

#547

semidirect

91. D₄

D₇₆

semidirect

113. C₄

#64543

semidirect

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

subgroup	index
C₂×#64543	2
C₄×D₄×C₁₉	2
#10597376	2
C₂×C₄⋊Dic₁₉	2
#10597343	2
#10597332	2
C₄×D₇₆	2
#10597315	2
#547	19

subgroup

index

C₄×D₄×C₁₉

C₂×C₄⋊Dic₁₉

C₄×D₇₆

Subgroups — #11927198

Conjugacy classes of subgroups

Complements

Maximal subgroups