Subgroups — #28383602

Order 14272.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16813153	C₂	1	2	yes	maximal
2	C₄²×D₂₂₃	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₂₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	Dic₂₂₃⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₄⋊C₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₂₂₃	C₄	1	1	yes	characteristic
7	C₂×C₄×D₂₂₃	C₄	1	1	yes	characteristic
8	C₂×C₄×D₂₂₃	C₂²	1	1	yes	characteristic
9	#16813153	—	2	4	no	—
10	C₂×C₂₂₃⋊C₈	C₂²	1	2	yes	—
11	C₄×Dic₂₂₃	C₄	1	1	yes	characteristic
12	C₄×Dic₂₂₃	C₄	1	1	yes	characteristic
13	C₄×Dic₂₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₂₂₃	C₂²	1	2	yes	—
15	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
16	C₂²×D₂₂₃	C₂×C₄	1	1	yes	characteristic
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	C₄×D₂₂₃	—	2	2	no	—
21	C₄×D₂₂₃	—	2	2	no	—
22	C₄×D₂₂₃	D₄	1	2	yes	—
23	C₄×D₂₂₃	Q₈	1	2	yes	—
24	C₂₂₃⋊C₈	—	2	2	no	—
25	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
28	C₈×C₂₂₃	—	2	2	no	—
29	C₂²×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
30	C₂×D₂₂₃	—	2	1	no	—
31	C₂×D₂₂₃	—	2	1	no	—
32	C₂×D₂₂₃	C₄⋊C₄	1	2	yes	—
33	C₄×C₂₂₃	—	2	1	no	—
34	C₄×C₂₂₃	M₁₆	1	2	yes	complemented
35	C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
36	C₄×C₂₂₃	C₂×Q₈	1	1	yes	characteristic
37	Dic₂₂₃	—	2	1	no	—
38	Dic₂₂₃	C₄⋊C₄	1	2	yes	—
39	Dic₂₂₃	M₁₆	1	2	yes	complemented
40	C₂×C₂₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₂₂₃	C₂×M₁₆	1	1	yes	characteristic
42	C₂×C₂₂₃	C₂×M₁₆	1	1	yes	characteristic
43	D₂₂₃	—	2	2	no	—
44	C₂₂₃	#423	1	1	yes	characteristic, complemented, Sylow
45	#423	—	223	1	no	maximal, Sylow
46	C₂×M₁₆	—	223	2	no	—
47	C₂×C₄²	—	223	1	no	—
48	C₄⋊C₈	—	223	2	no	—
49	C₄⋊C₈	—	223	1	no	—
50	C₄⋊C₈	D₂₂₃	1	1	yes	characteristic, complemented
51	C₂²×C₄	—	223	1	no	—
52	C₂²×C₄	—	223	1	no	—
53	C₂²×C₄	—	223	1	no	—
54	M₁₆	—	446	4	no	—
55	C₂×C₈	—	223	2	no	—
56	C₂×C₈	C₂×D₂₂₃	1	2	yes	—
57	C₄²	—	223	1	no	—
58	C₄²	—	223	1	no	—
59	C₄²	—	223	1	no	—
60	C₄²	C₂×D₂₂₃	1	1	yes	characteristic
61	C₂³	—	223	1	no	—
62	C₂×C₄	—	446	2	no	—
63	C₂×C₄	—	446	2	no	—
64	C₂×C₄	—	223	1	no	—
65	C₂×C₄	—	223	1	no	—
66	C₂×C₄	—	223	1	no	—
67	C₂×C₄	—	223	2	no	—
68	C₂×C₄	—	223	2	no	—
69	C₂×C₄	C₄×D₂₂₃	1	1	yes	characteristic
70	C₂×C₄	C₄×D₂₂₃	1	1	yes	characteristic
71	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
72	C₈	—	2	2	no	—
73	C₈	—	446	2	no	—
74	C₂²	—	446	1	no	—
75	C₂²	—	446	1	no	—
76	C₂²	—	223	2	no	—
77	C₂²	C₂×C₄×D₂₂₃	1	1	yes	characteristic
78	C₄	—	2	1	no	—
79	C₄	—	446	1	no	—
80	C₄	—	223	2	no	—
81	C₄	—	223	2	no	—
82	C₄	#16813153	1	2	yes	complemented
83	C₄	D₄×D₂₂₃	1	1	yes	characteristic
84	C₄	Q₈×D₂₂₃	1	1	yes	characteristic
85	C₂	—	446	2	no	—
86	C₂	C₄⋊C₄×D₂₂₃	1	1	yes	characteristic
87	C₂	M₁₆×D₂₂₃	1	1	yes	characteristic
88	C₂	C₂×#16813153	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₂₂₃
commutator, socle	C₂²×C₂₂₃	C₂²×C₄
Fitting	C₄⋊C₈×C₂₂₃	C₂
prime radical	C₂²×D₂₂₃	C₂×C₄
cosocle	C₂×C₄×C₂₂₃	C₂³
hypercenter	C₄⋊C₈	D₂₂₃
hypocenter	C₂₂₃	#423
LCS	G ▷ C₂²×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
3. C₄⋊(C₂₂₃⋊C₈)	C₂	semidirect
4. Dic₂₂₃⋊C₈	C₂	semidirect
5. C₄⋊C₈×C₂₂₃	C₂	semidirect
34. C₄×C₂₂₃	M₁₆	semidirect
39. Dic₂₂₃	M₁₆	semidirect
44. C₂₂₃	#423	semidirect
50. C₄⋊C₈	D₂₂₃	semidirect
82. C₄	#16813153	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16813153	2
C₄²×D₂₂₃	2
C₄⋊(C₂₂₃⋊C₈)	2
Dic₂₂₃⋊C₈	2
C₄⋊C₈×C₂₂₃	2
#423	223