Subgroups — #28383634

Order 14272.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₂₂₃	C₂	1	1	yes	maximal, characteristic
2	Dic₂₂₃⋊C₈	C₂	1	2	yes	maximal, complemented
3	C₂²⋊(C₂₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#20297214	C₂	1	2	yes	maximal, complemented
5	C₂²⋊C₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₂₃	C₄	1	1	yes	characteristic
7	C₂²×Dic₂₂₃	C₄	1	1	yes	characteristic
8	C₂×C₂₂₃⋊C₈	C₂²	1	2	yes	—
9	C₄×Dic₂₂₃	C₄	1	1	yes	characteristic
10	C₄×Dic₂₂₃	C₄	1	1	yes	characteristic
11	C₄×Dic₂₂₃	C₂²	1	1	yes	characteristic
12	C₄×Dic₂₂₃	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₂₂₃	C₂²	1	2	yes	—
15	C₂×Dic₂₂₃	—	2	2	no	—
16	C₂×Dic₂₂₃	—	2	2	no	—
17	C₂×Dic₂₂₃	C₂×C₄	1	1	yes	characteristic
18	C₂×Dic₂₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂×Dic₂₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₂₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂₂₃⋊C₈	—	2	2	no	—
22	C₂³×C₂₂₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₂₂₃	—	2	2	no	—
24	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
26	C₈×C₂₂₃	—	2	2	no	—
27	C₂²×C₂₂₃	—	2	1	no	—
28	C₂²×C₂₂₃	—	2	1	no	—
29	C₂²×C₂₂₃	M₁₆	1	2	yes	—
30	C₂²×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
31	C₄×C₂₂₃	—	2	1	no	—
32	C₄×C₂₂₃	#41	1	2	yes	—
33	Dic₂₂₃	—	2	1	no	—
34	Dic₂₂₃	—	2	1	no	—
35	Dic₂₂₃	—	2	1	no	—
36	Dic₂₂₃	M₁₆	1	2	yes	—
37	C₂×C₂₂₃	—	2	2	no	—
38	C₂×C₂₂₃	#117	1	1	yes	characteristic
39	C₂×C₂₂₃	C₂×M₁₆	1	1	yes	characteristic
40	C₂×C₂₂₃	C₂×M₁₆	1	1	yes	characteristic
41	C₂₂₃	#432	1	1	yes	characteristic, complemented, Sylow
42	#432	—	223	1	no	maximal, Sylow
43	C₂×C₄²	—	223	1	no	—
44	C₄⋊C₈	—	223	2	no	—
45	C₂²⋊C₈	—	223	1	no	—
46	C₂²⋊C₈	D₂₂₃	1	1	yes	characteristic
47	#97	—	223	2	no	—
48	C₂²×C₄	—	223	1	no	—
49	C₂²×C₄	—	223	1	no	—
50	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
51	C₂×C₈	—	223	2	no	—
52	C₂×C₈	C₂×D₂₂₃	1	2	yes	—
53	C₄²	—	223	1	no	—
54	C₄²	—	223	1	no	—
55	C₄²	—	223	1	no	—
56	C₄²	—	223	1	no	—
57	C₂³	C₄×D₂₂₃	1	1	yes	characteristic
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	446	2	no	—
60	C₂×C₄	—	446	2	no	—
61	C₂×C₄	—	223	1	no	—
62	C₂×C₄	—	223	1	no	—
63	C₂×C₄	—	223	1	no	—
64	C₂×C₄	—	223	1	no	—
65	C₂×C₄	C₄×D₂₂₃	1	1	yes	characteristic
66	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
67	C₈	—	2	2	no	—
68	C₈	—	446	2	no	—
69	C₂²	—	2	1	no	—
70	C₂²	—	2	1	no	—
71	C₂²	#16813153	1	2	yes	—
72	C₂²	C₂×C₄×D₂₂₃	1	1	yes	characteristic
73	C₄	—	2	1	no	—
74	C₄	—	446	1	no	—
75	C₄	—	446	1	no	—
76	C₄	—	446	1	no	—
77	C₄	—	223	2	no	—
78	C₄	#16813169	1	2	yes	—
79	C₂	—	2	2	no	—
80	C₂	#20297260	1	1	yes	characteristic
81	C₂	M₁₆×D₂₂₃	1	1	yes	characteristic
82	C₂	C₂×#16813153	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₂₂₃
commutator, socle	C₂²×C₂₂₃	C₂²×C₄
Fitting	C₂²⋊C₈×C₂₂₃	C₂
prime radical	C₂³×C₂₂₃	C₂×C₄
cosocle	C₂×C₄×C₂₂₃	C₂³
hypercenter	C₂²⋊C₈	D₂₂₃
hypocenter	C₂₂₃	#432
LCS	G ▷ C₂²×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
2. Dic₂₂₃⋊C₈	C₂	semidirect
4. #20297214	C₂	semidirect
41. C₂₂₃	#432	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₂₂₃	2
Dic₂₂₃⋊C₈	2
C₂²⋊(C₂₂₃⋊C₈)	2
#20297214	2
C₂²⋊C₈×C₂₂₃	2
#432	223