Subgroups — #28383785

Order 14272.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16813158	C₂	1	2	yes	maximal
2	#20297304	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊(C₂₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	#20297227	C₂	1	2	yes	maximal, complemented
5	C₄×D₄×C₂₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₄₆	C₂²	1	2	yes	—
7	#16813158	—	2	4	no	—
8	C₂×C₂₂₃⋊C₈	C₂²	1	2	yes	—
9	C₄⋊Dic₂₂₃	—	2	1	no	—
10	C₄⋊Dic₂₂₃	—	2	1	no	—
11	C₂×D₄×C₂₂₃	C₂²	1	1	yes	characteristic
12	C₂²×C₄×C₂₂₃	—	2	1	no	—
13	C₄⋊C₄×C₂₂₃	C₂²	1	1	yes	characteristic
14	C₂²⋊C₄×C₂₂₃	—	2	1	no	—
15	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
16	Dic₄₄₆	—	2	2	no	—
17	Dic₄₄₆	—	2	2	no	—
18	C₂×Dic₂₂₃	—	2	2	no	—
19	C₂₂₃⋊C₈	—	2	2	no	—
20	C₂³×C₂₂₃	—	2	1	no	—
21	D₄×C₂₂₃	—	2	1	no	—
22	D₄×C₂₂₃	D₄	1	2	yes	—
23	C₂×C₄×C₂₂₃	—	2	1	no	—
24	C₂×C₄×C₂₂₃	—	4	2	no	—
25	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
26	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
27	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
28	C₂²×C₂₂₃	—	2	2	no	—
29	C₂²×C₂₂₃	—	4	1	no	—
30	C₂²×C₂₂₃	—	4	1	no	—
31	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
32	C₄×C₂₂₃	—	2	1	no	—
33	C₄×C₂₂₃	—	4	1	no	—
34	C₄×C₂₂₃	SD₁₆	1	2	yes	complemented
35	C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
36	C₄×C₂₂₃	#41	1	1	yes	characteristic
37	Dic₂₂₃	—	4	2	no	—
38	C₂×C₂₂₃	—	4	2	no	—
39	C₂×C₂₂₃	#121	1	1	yes	characteristic
40	C₂×C₂₂₃	C₂×SD₁₆	1	1	yes	characteristic
41	C₂×C₂₂₃	#137	1	1	yes	characteristic
42	C₂₂₃	#461	1	1	yes	characteristic, complemented, Sylow
43	#461	—	223	1	no	maximal, Sylow
44	C₂×SD₁₆	—	223	2	no	—
45	C₄⋊Q₈	—	223	1	no	—
46	C₄×D₄	D₂₂₃	1	1	yes	characteristic
47	C₄⋊C₈	—	223	1	no	—
48	#103	—	223	2	no	—
49	C₂×Q₈	—	223	2	no	—
50	C₂×D₄	C₂×D₂₂₃	1	1	yes	characteristic
51	C₂²×C₄	—	2	1	no	—
52	SD₁₆	—	446	4	no	—
53	C₂×C₈	—	223	2	no	—
54	C₄⋊C₄	—	446	1	no	—
55	C₄⋊C₄	—	446	1	no	—
56	C₄⋊C₄	C₂×D₂₂₃	1	1	yes	characteristic
57	C₂²⋊C₄	—	2	1	no	—
58	C₄²	C₂×D₂₂₃	1	1	yes	characteristic
59	C₂³	—	2	1	no	—
60	Q₈	—	446	2	no	—
61	Q₈	—	446	2	no	—
62	D₄	—	2	1	no	—
63	D₄	D₈₉₂	1	2	yes	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	—	446	2	no	—
67	C₂×C₄	C₂²⋊D₂₂₃	1	1	yes	characteristic
68	C₂×C₄	C₂²⋊D₂₂₃	1	1	yes	characteristic
69	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
70	C₈	—	446	2	no	—
71	C₂²	—	2	2	no	—
72	C₂²	—	4	1	no	—
73	C₂²	—	4	1	no	—
74	C₂²	C₂×C₂²⋊D₂₂₃	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	4	1	no	—
77	C₄	—	892	2	no	—
78	C₄	#16813158	1	2	yes	complemented
79	C₄	C₂×D₈₉₂	1	1	yes	characteristic
80	C₄	#16813170	1	1	yes	characteristic
81	C₂	—	4	2	no	—
82	C₂	#20297278	1	1	yes	characteristic
83	C₂	C₂×#16813158	1	1	yes	characteristic
84	C₂	#20297334	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₂₃
commutator, cosocle	C₂×C₄×C₂₂₃	C₂³
Frattini	C₂×C₄	C₂²×D₂₂₃
Fitting	C₄×D₄×C₂₂₃	C₂
socle	C₂²×C₂₂₃	C₂×D₄
prime radical	C₂×D₄×C₂₂₃	C₂²
hypercenter	C₄×D₄	D₂₂₃
hypocenter	C₂₂₃	#461
LCS	G ▷ C₂×C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
2. #20297304	C₂	semidirect
3. C₄⋊(C₂₂₃⋊C₈)	C₂	semidirect
4. #20297227	C₂	semidirect
34. C₄×C₂₂₃	SD₁₆	semidirect
42. C₂₂₃	#461	semidirect
78. C₄	#16813158	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16813158	2
#20297304	2
C₄⋊(C₂₂₃⋊C₈)	2
#20297227	2
C₄×D₄×C₂₂₃	2
#461	223