Subgroups — C₂×#20297237 #28383613

Order 14272.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₂₂₃	C₂	1	1	yes	maximal, characteristic
3	#20297237	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄⋊C₄×C₂₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₂₃	—	2	1	no	—
6	C₂×C₂₂₃⋊C₈	C₄	1	4	yes	complemented
7	C₂×C₂₂₃⋊C₈	C₂²	1	2	yes	—
8	C₄⋊Dic₂₂₃	—	2	1	no	—
9	C₄⋊Dic₂₂₃	C₂²	1	2	yes	—
10	C₂²×C₄×C₂₂₃	—	2	1	no	—
11	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₂₂₃	—	2	1	no	—
13	C₄⋊C₄×C₂₂₃	C₂²	1	2	yes	—
14	C₂×Dic₂₂₃	—	2	2	no	—
15	C₂×Dic₂₂₃	—	4	2	no	—
16	C₂₂₃⋊C₈	C₂×C₄	1	4	yes	complemented
17	C₂³×C₂₂₃	D₄	1	1	yes	characteristic
18	C₂×C₄×C₂₂₃	—	2	2	no	—
19	C₂×C₄×C₂₂₃	—	4	2	no	—
20	C₂×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
21	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
22	C₂×C₄×C₂₂₃	Q₈	1	2	yes	—
23	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
24	C₂²×C₂₂₃	C₄⋊C₄	1	2	yes	—
25	C₂²×C₂₂₃	D₈	1	2	yes	—
26	C₂²×C₂₂₃	Q₁₆	1	2	yes	—
27	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
28	C₄×C₂₂₃	—	4	2	no	—
29	C₄×C₂₂₃	C₄⋊C₄	1	2	yes	—
30	C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
31	C₄×C₂₂₃	C₂×Q₈	1	1	yes	characteristic
32	Dic₂₂₃	—	4	2	no	—
33	C₂×C₂₂₃	#107	1	4	yes	complemented
34	C₂×C₂₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₂₂₃	C₂×D₈	1	1	yes	characteristic
36	C₂×C₂₂₃	C₂×Q₁₆	1	1	yes	characteristic
37	C₂₂₃	C₂×#107	1	1	yes	characteristic, complemented, Sylow
38	C₂×#107	—	223	1	no	maximal, Sylow
39	C₂²×C₈	—	223	1	no	—
40	C₂×C₄⋊C₄	—	223	1	no	—
41	C₂×C₄⋊C₄	D₂₂₃	1	1	yes	characteristic
42	#107	—	223	4	no	—
43	C₂²×C₄	—	2	1	no	—
44	C₂²×C₄	—	446	1	no	—
45	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
46	C₂×C₈	—	223	4	no	—
47	C₂×C₈	—	223	2	no	—
48	C₄⋊C₄	—	2	1	no	—
49	C₄⋊C₄	—	446	1	no	—
50	C₄⋊C₄	—	223	2	no	—
51	C₄⋊C₄	C₂×D₂₂₃	1	2	yes	—
52	C₂³	C₂²⋊D₂₂₃	1	1	yes	characteristic
53	C₂×C₄	—	2	2	no	—
54	C₂×C₄	—	4	2	no	—
55	C₂×C₄	—	892	2	no	—
56	C₂×C₄	—	446	2	no	—
57	C₂×C₄	C₄×D₂₂₃	1	2	yes	—
58	C₂×C₄	C₂²⋊D₂₂₃	1	1	yes	characteristic
59	C₂×C₄	Dic₄₄₆	1	2	yes	—
60	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
61	C₈	—	223	4	no	—
62	C₂²	Dic₂₂₃⋊C₄	1	2	yes	—
63	C₂²	#16813155	1	2	yes	—
64	C₂²	#16813160	1	2	yes	—
65	C₂²	C₂×C₂²⋊D₂₂₃	1	1	yes	characteristic
66	C₄	—	4	2	no	—
67	C₄	—	892	2	no	—
68	C₄	Dic₂₂₃⋊C₄	1	2	yes	—
69	C₄	C₂×C₄×D₂₂₃	1	1	yes	characteristic
70	C₄	C₂×Dic₄₄₆	1	1	yes	characteristic
71	C₂	#20297237	1	4	yes	direct product factor
72	C₂	C₂×Dic₂₂₃⋊C₄	1	1	yes	characteristic
73	C₂	C₂×#16813155	1	1	yes	characteristic
74	C₂	C₂×#16813160	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊D₂₂₃
commutator	C₄×C₂₂₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₂₃
Fitting	C₂×C₄⋊C₄×C₂₂₃	C₂
socle, prime radical	C₂³×C₂₂₃	D₄
cosocle	C₂×C₄×C₂₂₃	C₂³
hypercenter	C₂×C₄⋊C₄	D₂₂₃
hypocenter	C₂₂₃	C₂×#107
LCS	G ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. #20297237	C₂	direct (N × H)
6. C₂×C₂₂₃⋊C₈	C₄	semidirect
16. C₂₂₃⋊C₈	C₂×C₄	semidirect
33. C₂×C₂₂₃	#107	semidirect
37. C₂₂₃	C₂×#107	semidirect
71. C₂	#20297237	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₂₂₃⋊C₈	2
C₂×C₄⋊Dic₂₂₃	2
#20297237	2
C₂×C₄⋊C₄×C₂₂₃	2
C₂×#107	223

Subgroups — C2×#20297237 #28383613

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20297237 #28383613