Group #38770569 — subgroups

Subgroups — #38770569

Order 25472.

220 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 85 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	C₂²×Dic₇₉₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#20263895	C₂	1	1	yes	maximal, characteristic
3	#27163821	C₂	1	4	yes	maximal, complemented
4	#427×C₁₉₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₃₉₈	—	2	1	no	—
6	C₂×Dic₇₉₆	—	2	1	no	—
7	C₂×Dic₇₉₆	—	2	2	no	—
8	C₂×Dic₇₉₆	C₄	1	4	yes	complemented
9	C₂×Dic₇₉₆	C₂²	1	2	yes	—
10	#20263895	—	2	2	no	—
11	C₂×C₁₉₉⋊C₁₆	C₂²	1	2	yes	—
12	C₂²×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
13	#117×C₁₉₉	—	2	1	no	—
14	#107×C₁₉₉	C₂²	1	2	yes	—
15	#106×C₁₉₉	—	2	1	no	—
16	C₂×Dic₃₉₈	—	2	4	no	—
17	C₂×Dic₃₉₈	—	2	2	no	—
18	C₂×Dic₃₉₈	—	4	1	no	—
19	C₂×Dic₃₉₈	—	4	2	no	—
20	C₂²×Dic₁₉₉	—	4	1	no	—
21	Dic₇₉₆	—	2	4	no	—
22	Dic₇₉₆	—	2	2	no	—
23	Dic₇₉₆	C₂×C₄	1	4	yes	—
24	C₁₉₉⋊C₁₆	—	2	2	no	—
25	C₂²×C₄×C₁₉₉	D₄	1	1	yes	characteristic
26	C₂×C₈×C₁₉₉	C₂×C₄	1	2	yes	—
27	C₂×C₈×C₁₉₉	D₄	1	1	yes	characteristic
28	C₂×C₈×C₁₉₉	D₄	1	2	yes	—
29	C₂×C₈×C₁₉₉	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₁₉₉	—	2	2	no	—
31	C₂²⋊C₄×C₁₉₉	—	4	1	no	—
32	C₄²×C₁₉₉	—	4	1	no	—
33	Dic₃₉₈	—	2	4	no	—
34	Dic₃₉₈	—	4	4	no	—
35	Dic₃₉₈	—	4	2	no	—
36	C₂×Dic₁₉₉	—	4	4	no	—
37	C₂×Dic₁₉₉	—	4	2	no	—
38	C₂³×C₁₉₉	D₈	1	1	yes	characteristic
39	C₂×C₄×C₁₉₉	—	4	2	no	—
40	C₂×C₄×C₁₉₉	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₁₉₉	D₈	1	1	yes	characteristic
42	C₂×C₄×C₁₉₉	SD₁₆	1	2	yes	—
43	C₂×C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
44	C₈×C₁₉₉	C₂²⋊C₄	1	2	yes	—
45	C₈×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
46	C₈×C₁₉₉	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₁₉₉	—	2	2	no	—
48	C₂²×C₁₉₉	#102	1	2	yes	—
49	C₂²×C₁₉₉	C₂×D₈	1	1	yes	characteristic
50	C₄×C₁₉₉	—	8	2	no	—
51	C₄×C₁₉₉	#102	1	2	yes	—
52	C₄×C₁₉₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₄×C₁₉₉	C₂×SD₁₆	1	1	yes	characteristic
54	Dic₁₉₉	—	4	4	no	—
55	C₂×C₁₉₉	—	2	2	no	—
56	C₂×C₁₉₉	C₂×#102	1	1	yes	characteristic
57	C₂×C₁₉₉	#510	1	2	yes	—
58	C₁₉₉	#2141	1	1	yes	characteristic, complemented, Sylow
59	#2141	—	199	1	no	maximal, Sylow
60	C₂²×Q₁₆	—	199	1	no	—
61	C₂×M₃₂	—	199	1	no	—
62	#427	D₁₉₉	1	1	yes	characteristic
63	#358	—	199	4	no	—
64	C₂²×Q₈	—	398	1	no	—
65	C₂×Q₁₆	—	398	1	no	—
66	C₂×Q₁₆	—	398	2	no	—
67	C₂×Q₁₆	—	199	4	no	—
68	C₂×Q₁₆	—	199	2	no	—
69	C₂²×C₈	C₂×D₁₉₉	1	1	yes	characteristic
70	#117	—	2	1	no	—
71	M₃₂	—	398	2	no	—
72	C₂×C₁₆	—	199	2	no	—
73	#107	C₂×D₁₉₉	1	2	yes	—
74	#106	—	2	1	no	—
75	C₂×Q₈	—	796	1	no	—
76	C₂×Q₈	—	796	2	no	—
77	C₂×Q₈	—	398	4	no	—
78	C₂×Q₈	—	398	2	no	—
79	C₂²×C₄	—	796	1	no	—
80	C₂²×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
81	Q₁₆	—	398	4	no	—
82	Q₁₆	—	398	2	no	—
83	Q₁₆	—	199	4	no	—
84	C₂×C₈	C₄×D₁₉₉	1	2	yes	—
85	C₂×C₈	C₂²⋊D₁₉₉	1	1	yes	characteristic
86	C₂×C₈	D₇₉₆	1	2	yes	—
87	C₂×C₈	C₂²×D₁₉₉	1	1	yes	characteristic
88	C₄⋊C₄	—	2	2	no	—
89	C₂²⋊C₄	—	4	1	no	—
90	C₄²	—	4	1	no	—
91	C₁₆	—	398	2	no	—
92	C₂³	#15691871	1	1	yes	characteristic
93	Q₈	—	796	2	no	—
94	Q₈	—	796	4	no	—
95	Q₈	—	398	4	no	—
96	C₂×C₄	—	4	2	no	—
97	C₂×C₄	—	796	2	no	—
98	C₂×C₄	—	796	4	no	—
99	C₂×C₄	(C₂×D₁₉₉)⋊C₄	1	2	yes	—
100	C₂×C₄	#15691871	1	1	yes	characteristic
101	C₂×C₄	#15691873	1	2	yes	—
102	C₂×C₄	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
103	C₈	(C₂×D₁₉₉)⋊C₄	1	2	yes	—
104	C₈	C₂×C₄×D₁₉₉	1	1	yes	characteristic
105	C₈	C₂×D₇₉₆	1	1	yes	characteristic
106	C₂²	—	2	2	no	—
107	C₂²	#20263875	1	2	yes	—
108	C₂²	C₂×#15691871	1	1	yes	characteristic
109	C₄	—	8	2	no	—
110	C₄	—	796	4	no	—
111	C₄	#20263875	1	2	yes	—
112	C₄	C₂×(C₂×D₁₉₉)⋊C₄	1	1	yes	characteristic
113	C₄	C₂×#15691873	1	1	yes	characteristic
114	C₂	—	2	2	no	—
115	C₂	C₂×#20263875	1	1	yes	characteristic
116	C₂	#27164458	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₇₉₆

C₂

yes

maximal, characteristic

C₂×#20263895

C₂

yes

maximal, characteristic

#27163821

C₂

yes

maximal, complemented

#427×C₁₉₉

C₂

yes

maximal, characteristic

C₂²×Dic₃₉₈

—

C₂×Dic₇₉₆

—

C₂×Dic₇₉₆

—

C₂×Dic₇₉₆

C₄

yes

complemented

C₂×Dic₇₉₆

C₂²

yes

—

#20263895

—

C₂×C₁₉₉⋊C₁₆

C₂²

yes

—

C₂²×C₈×C₁₉₉

C₂²

yes

characteristic

#117×C₁₉₉

—

#107×C₁₉₉

C₂²

yes

—

#106×C₁₉₉

—

C₂×Dic₃₉₈

—

C₂×Dic₃₉₈

—

C₂×Dic₃₉₈

—

C₂×Dic₃₉₈

—

C₂²×Dic₁₉₉

—

Dic₇₉₆

—

Dic₇₉₆

—

Dic₇₉₆

C₂×C₄

yes

—

C₁₉₉⋊C₁₆

—

C₂²×C₄×C₁₉₉

D₄

yes

characteristic

C₂×C₈×C₁₉₉

C₂×C₄

yes

—

C₂×C₈×C₁₉₉

D₄

yes

characteristic

C₂×C₈×C₁₉₉

D₄

yes

—

C₂×C₈×C₁₉₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₉

—

C₂²⋊C₄×C₁₉₉

—

C₄²×C₁₉₉

—

Dic₃₉₈

—

Dic₃₉₈

—

Dic₃₉₈

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₂³×C₁₉₉

D₈

yes

characteristic

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉₉

D₈

yes

characteristic

C₂×C₄×C₁₉₉

SD₁₆

yes

—

C₂×C₄×C₁₉₉

C₂×D₄

yes

characteristic

C₈×C₁₉₉

C₂²⋊C₄

yes

—

C₈×C₁₉₉

C₂²×C₄

yes

characteristic

C₈×C₁₉₉

C₂×D₄

yes

characteristic

C₂²×C₁₉₉

—

C₂²×C₁₉₉

#102

yes

—

C₂²×C₁₉₉

C₂×D₈

yes

characteristic

C₄×C₁₉₉

—

C₄×C₁₉₉

#102

yes

—

C₄×C₁₉₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₉₉

C₂×SD₁₆

yes

characteristic

Dic₁₉₉

—

C₂×C₁₉₉

—

C₂×C₁₉₉

C₂×#102

yes

characteristic

C₂×C₁₉₉

#510

yes

—

C₁₉₉

#2141

yes

characteristic, complemented, Sylow

#2141

—

199

maximal, Sylow

C₂²×Q₁₆

—

199

—

C₂×M₃₂

—

199

—

#427

D₁₉₉

yes

characteristic

#358

—

199

—

C₂²×Q₈

—

398

—

C₂×Q₁₆

—

398

—

C₂×Q₁₆

—

398

—

C₂×Q₁₆

—

199

—

C₂×Q₁₆

—

199

—

C₂²×C₈

C₂×D₁₉₉

yes

characteristic

#117

—

M₃₂

—

398

—

C₂×C₁₆

—

199

—

#107

C₂×D₁₉₉

yes

—

#106

—

C₂×Q₈

—

796

—

C₂×Q₈

—

796

—

C₂×Q₈

—

398

—

C₂×Q₈

—

398

—

C₂²×C₄

—

796

—

C₂²×C₄

C₂²⋊D₁₉₉

yes

characteristic

Q₁₆

—

398

—

Q₁₆

—

398

—

Q₁₆

—

199

—

C₂×C₈

C₄×D₁₉₉

yes

—

C₂×C₈

C₂²⋊D₁₉₉

yes

characteristic

C₂×C₈

D₇₉₆

yes

—

C₂×C₈

C₂²×D₁₉₉

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

398

—

C₂³

#15691871

yes

characteristic

Q₈

—

796

—

Q₈

—

796

—

Q₈

—

398

—

C₂×C₄

—

C₂×C₄

—

796

—

C₂×C₄

—

796

—

C₂×C₄

(C₂×D₁₉₉)⋊C₄

yes

—

100

C₂×C₄

#15691871

yes

characteristic

101

C₂×C₄

#15691873

yes

—

102

C₂×C₄

C₂×C₂²⋊D₁₉₉

yes

characteristic

103

C₈

(C₂×D₁₉₉)⋊C₄

yes

—

104

C₈

C₂×C₄×D₁₉₉

yes

characteristic

105

C₈

C₂×D₇₉₆

yes

characteristic

106

C₂²

—

107

C₂²

#20263875

yes

—

108

C₂²

C₂×#15691871

yes

characteristic

109

C₄

—

110

C₄

—

796

—

111

C₄

#20263875

yes

—

112

C₄

C₂×(C₂×D₁₉₉)⋊C₄

yes

characteristic

113

C₄

C₂×#15691873

yes

characteristic

114

C₂

—

115

C₂

C₂×#20263875

yes

characteristic

116

C₂

#27164458

yes

—

	subgroup	G/H
center	C₂²	C₂×#15691871
commutator	C₈×C₁₉₉	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₉₉
Fitting	#427×C₁₉₉	C₂
socle	C₂²×C₁₉₉	C₂×D₈
prime radical	C₂³×C₁₉₉	D₈
cosocle	C₂×C₈×C₁₉₉	C₂³
hypercenter	#427	D₁₉₉
hypocenter	C₁₉₉	#2141
LCS	G ▷ C₈×C₁₉₉ ▷ C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #427

subgroup

G/H

center

C₂²

C₂×#15691871

commutator

C₈×C₁₉₉

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₁₉₉

Fitting

#427×C₁₉₉

C₂

socle

C₂²×C₁₉₉

C₂×D₈

prime radical

C₂³×C₁₉₉

D₈

cosocle

C₂×C₈×C₁₉₉

C₂³

D₁₉₉

C₁₉₉

G ▷ C₈×C₁₉₉ ▷ C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #427

N	H ≅ G/N	type
3. #27163821	C₂	semidirect
8. C₂×Dic₇₉₆	C₄	semidirect
58. C₁₉₉	#2141	semidirect

H ≅ G/N

type

3. #27163821

C₂

semidirect

8. C₂×Dic₇₉₆

C₄

semidirect

58. C₁₉₉

#2141

semidirect

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

subgroup	index
C₂²×Dic₇₉₆	2
C₂×#20263895	2
#27163821	2
#427×C₁₉₉	2
#2141	199

subgroup

index

C₂²×Dic₇₉₆

C₂×#20263895

#27163821

#427×C₁₉₉

#2141

199

Subgroups — #38770569

Conjugacy classes of subgroups

Complements

Maximal subgroups