Group #44449693 — subgroups

Subgroups — #44449693

Order 29824.

188 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 105 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×C₄⋊#14272338	C₂	1	2	yes	maximal
2	#384×C₂₃₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×#14272338	—	2	2	no	—
4	C₄⋊#14272338	—	2	2	no	—
5	C₄⋊#14272338	C₄	1	4	yes	complemented
6	C₂×C₄⋊C₄×C₂₃₃	—	2	2	no	—
7	C₂×C₄⋊C₄×C₂₃₃	C₄	1	2	yes	—
8	C₂×C₄²×C₂₃₃	C₂²	1	1	yes	characteristic
9	C₂×#14272338	—	2	4	no	—
10	C₂×#14272338	—	4	2	no	—
11	C₂×#14272338	—	4	2	no	—
12	C₂²×C₄×C₂₃₃	—	2	2	no	—
13	C₂²×C₄×C₂₃₃	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄×C₂₃₃	D₄	1	2	yes	—
15	C₄⋊C₄×C₂₃₃	—	4	2	no	—
16	C₄⋊C₄×C₂₃₃	—	4	2	no	—
17	C₄⋊C₄×C₂₃₃	—	4	2	no	—
18	C₄²×C₂₃₃	C₂×C₄	1	2	yes	—
19	C₄²×C₂₃₃	D₄	1	1	yes	characteristic
20	C₄²×C₂₃₃	Q₈	1	1	yes	characteristic
21	#14272338	—	4	4	no	—
22	C₂³×C₂₃₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×C₄×C₂₃₃	—	2	1	no	—
24	C₂×C₄×C₂₃₃	—	2	1	no	—
25	C₂×C₄×C₂₃₃	—	4	2	no	—
26	C₂×C₄×C₂₃₃	—	4	4	no	—
27	C₂×C₄×C₂₃₃	C₄²	1	1	yes	characteristic
28	C₂×C₄×C₂₃₃	C₂²⋊C₄	1	2	yes	—
29	C₂×C₄×C₂₃₃	C₄⋊C₄	1	2	yes	—
30	C₂×C₄×C₂₃₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₂₃₃	D₈	1	2	yes	—
32	C₂×C₄×C₂₃₃	SD₁₆	1	2	yes	—
33	C₂×C₄×C₂₃₃	SD₁₆	1	2	yes	—
34	C₂×C₄×C₂₃₃	Q₁₆	1	2	yes	—
35	C₂²×C₂₃₃	#95	1	1	yes	characteristic
36	C₂²×C₂₃₃	#100	1	1	yes	characteristic
37	C₂²×C₂₃₃	#101	1	1	yes	characteristic
38	C₂²×C₂₃₃	#102	1	2	yes	—
39	C₂²×C₂₃₃	#103	1	2	yes	—
40	C₄×C₂₃₃	—	2	1	no	—
41	C₄×C₂₃₃	—	2	1	no	—
42	C₄×C₂₃₃	—	8	2	no	—
43	C₄×C₂₃₃	#102	1	2	yes	—
44	C₄×C₂₃₃	#103	1	2	yes	—
45	C₄×C₂₃₃	#106	1	2	yes	—
46	C₄×C₂₃₃	#107	1	2	yes	—
47	C₂×C₂₃₃	#331	1	1	yes	characteristic
48	C₂×C₂₃₃	#332	1	2	yes	—
49	C₂×C₂₃₃	#333	1	1	yes	characteristic
50	C₂×C₂₃₃	#340	1	2	yes	—
51	C₂×C₂₃₃	#343	1	1	yes	characteristic
52	C₂₃₃	#1296	1	1	yes	characteristic, complemented, Sylow
53	#1296	—	233	1	no	maximal, Sylow
54	C₂×C₄⋊C₈	—	233	2	no	—
55	#384	D₂₃₃	1	1	yes	characteristic
56	C₂²×C₈	—	466	2	no	—
57	C₂×C₄⋊C₄	—	2	2	no	—
58	C₂×C₄⋊C₄	Dic₂₃₃	1	2	yes	—
59	C₂×C₄²	C₂×D₂₃₃	1	1	yes	characteristic
60	C₄⋊C₈	—	466	2	no	—
61	C₄⋊C₈	—	233	4	no	—
62	C₂²×C₄	—	2	2	no	—
63	C₂²×C₄	C₂×Dic₂₃₃	1	1	yes	characteristic
64	C₂²×C₄	C₂²⋊D₂₃₃	1	2	yes	—
65	C₂×C₈	—	932	2	no	—
66	C₂×C₈	—	932	2	no	—
67	C₂×C₈	—	466	4	no	—
68	C₄⋊C₄	—	4	2	no	—
69	C₄⋊C₄	—	4	2	no	—
70	C₄⋊C₄	—	4	2	no	—
71	C₄²	C₄×D₂₃₃	1	2	yes	—
72	C₄²	D₉₃₂	1	1	yes	characteristic
73	C₄²	Dic₄₆₆	1	1	yes	characteristic
74	C₂³	C₂²⋊Dic₂₃₃	1	1	yes	characteristic
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	4	2	no	—
78	C₂×C₄	—	4	4	no	—
79	C₂×C₄	C₄×Dic₂₃₃	1	1	yes	characteristic
80	C₂×C₄	#16836571	1	2	yes	—
81	C₂×C₄	#16836573	1	2	yes	—
82	C₂×C₄	C₄⋊Dic₂₃₃	1	1	yes	characteristic
83	C₂×C₄	#16836579	1	2	yes	—
84	C₂×C₄	#16836581	1	2	yes	—
85	C₂×C₄	#16836582	1	2	yes	—
86	C₂×C₄	#16836584	1	2	yes	—
87	C₈	—	932	4	no	—
88	C₂²	#21418442	1	1	yes	characteristic
89	C₂²	#21418452	1	1	yes	characteristic
90	C₂²	#21418454	1	1	yes	characteristic
91	C₂²	#21418456	1	2	yes	—
92	C₂²	#21418459	1	2	yes	—
93	C₄	—	2	1	no	—
94	C₄	—	2	1	no	—
95	C₄	—	8	2	no	—
96	C₄	#21418455	1	2	yes	—
97	C₄	#21418458	1	2	yes	—
98	C₄	#21418466	1	2	yes	—
99	C₄	#21418468	1	2	yes	—
100	C₂	#28416179	1	1	yes	characteristic
101	C₂	#28416181	1	2	yes	—
102	C₂	#28416184	1	1	yes	characteristic
103	C₂	#28416197	1	2	yes	—
104	C₂	#28416204	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#14272338

C₂

yes

maximal

#384×C₂₃₃

C₂

yes

maximal, characteristic

C₂²×#14272338

—

C₄⋊#14272338

—

C₄⋊#14272338

C₄

yes

complemented

C₂×C₄⋊C₄×C₂₃₃

—

C₂×C₄⋊C₄×C₂₃₃

C₄

yes

—

C₂×C₄²×C₂₃₃

C₂²

yes

—

—

—

C₂²×C₄×C₂₃₃

—

C₂²×C₄×C₂₃₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃₃

D₄

yes

—

C₄⋊C₄×C₂₃₃

—

C₄⋊C₄×C₂₃₃

—

C₄⋊C₄×C₂₃₃

—

C₄²×C₂₃₃

C₂×C₄

yes

—

C₄²×C₂₃₃

D₄

yes

characteristic

C₄²×C₂₃₃

Q₈

yes

characteristic

#14272338

—

C₂³×C₂₃₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₃₃

—

C₂×C₄×C₂₃₃

—

C₂×C₄×C₂₃₃

—

C₂×C₄×C₂₃₃

—

C₂×C₄×C₂₃₃

C₄²

yes

characteristic

C₂×C₄×C₂₃₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃₃

C₄⋊C₄

yes

—

C₂×C₄×C₂₃₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₃₃

D₈

yes

—

C₂×C₄×C₂₃₃

SD₁₆

yes

—

C₂×C₄×C₂₃₃

SD₁₆

yes

—

C₂×C₄×C₂₃₃

Q₁₆

yes

—

C₂²×C₂₃₃

yes

C₂²×C₂₃₃

yes

C₂²×C₂₃₃

yes

C₂²×C₂₃₃

yes

—

C₂²×C₂₃₃

#103

yes

—

C₄×C₂₃₃

—

C₄×C₂₃₃

—

C₄×C₂₃₃

—

C₄×C₂₃₃

#102

yes

—

C₄×C₂₃₃

#103

yes

—

C₄×C₂₃₃

#106

yes

—

C₄×C₂₃₃

#107

yes

—

C₂×C₂₃₃

#331

yes

characteristic

C₂×C₂₃₃

#332

yes

—

C₂×C₂₃₃

#333

yes

characteristic

C₂×C₂₃₃

#340

yes

—

C₂×C₂₃₃

#343

yes

characteristic

C₂₃₃

#1296

yes

characteristic, complemented, Sylow

#1296

—

233

maximal, Sylow

C₂×C₄⋊C₈

—

233

—

#384

D₂₃₃

yes

characteristic

C₂²×C₈

—

466

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Dic₂₃₃

yes

—

C₂×C₄²

C₂×D₂₃₃

yes

characteristic

C₄⋊C₈

—

466

—

C₄⋊C₈

—

233

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₂₃₃

yes

characteristic

C₂²×C₄

C₂²⋊D₂₃₃

yes

—

C₂×C₈

—

932

—

C₂×C₈

—

932

—

C₂×C₈

—

466

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₄×D₂₃₃

yes

—

C₄²

D₉₃₂

yes

characteristic

C₄²

Dic₄₆₆

yes

characteristic

C₂³

C₂²⋊Dic₂₃₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₂₃₃

yes

characteristic

C₂×C₄

#16836571

yes

—

C₂×C₄

#16836573

yes

—

C₂×C₄

C₄⋊Dic₂₃₃

yes

characteristic

C₂×C₄

#16836579

yes

—

C₂×C₄

#16836581

yes

—

C₂×C₄

#16836582

yes

—

C₂×C₄

#16836584

yes

—

C₈

—

932

—

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#21418459

yes

—

C₄

—

C₄

—

C₄

—

C₄

#21418455

yes

—

C₄

#21418458

yes

—

C₄

#21418466

yes

—

C₄

#21418468

yes

—

100

C₂

#28416179

yes

characteristic

101

C₂

#28416181

yes

—

102

C₂

#28416184

yes

characteristic

103

C₂

#28416197

yes

—

104

C₂

#28416204

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²⋊Dic₂₃₃
commutator	C₂×C₄×C₂₃₃	C₄²
Frattini	C₂×C₄²	C₂×D₂₃₃
Fitting	#384×C₂₃₃	C₂
socle, prime radical	C₂³×C₂₃₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₂₃₃	C₂²
hypercenter	#384	D₂₃₃
hypocenter	C₂₃₃	#1296
LCS	G ▷ C₂×C₄×C₂₃₃ ▷ C₂²×C₂₃₃ ▷ C₂₃₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂²⋊Dic₂₃₃

commutator

C₂×C₄×C₂₃₃

C₄²

Frattini

C₂×C₄²

C₂×D₂₃₃

Fitting

#384×C₂₃₃

C₂

socle, prime radical

C₂³×C₂₃₃

C₂²⋊C₄

cosocle

C₂×C₄²×C₂₃₃

C₂²

D₂₃₃

C₂₃₃

G ▷ C₂×C₄×C₂₃₃ ▷ C₂²×C₂₃₃ ▷ C₂₃₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
5. C₄⋊#14272338	C₄	semidirect
52. C₂₃₃	#1296	semidirect

H ≅ G/N

type

5. C₄⋊#14272338

C₄

semidirect

52. C₂₃₃

#1296

semidirect

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

subgroup	index
C₂×C₄⋊#14272338	2
#384×C₂₃₃	2
#1296	233

subgroup

index

C₂×C₄⋊#14272338

#384×C₂₃₃

#1296

233

Subgroups — #44449693

Conjugacy classes of subgroups

Complements

Maximal subgroups