Subgroups — #28384056

Order 14272.

266 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×D₂₂₃	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×Dic₄₄₆	C₂	1	3	yes	maximal, complemented
4	#20297269	C₂	1	3	yes	maximal, complemented
5	#20297262	C₂	1	3	yes	maximal
6	#20297261	C₂	1	3	yes	maximal
7	C₄×Q₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₄₄₆	C₂²	1	3	yes	—
9	Q₈×D₂₂₃	C₄	1	8	yes	complemented
10	C₂×C₄×D₂₂₃	C₂²	1	3	yes	—
11	C₄⋊Dic₂₂₃	C₂²	1	3	yes	—
12	Dic₂₂₃⋊C₄	C₂²	1	3	yes	—
13	Dic₂₂₃⋊C₄	C₂²	1	3	yes	—
14	(C₂×D₂₂₃)⋊C₄	C₂²	1	1	yes	characteristic
15	(C₂×D₂₂₃)⋊C₄	C₂²	1	3	yes	—
16	C₄×Dic₂₂₃	C₂²	1	3	yes	—
17	C₄×Dic₂₂₃	C₂²	1	6	yes	—
18	C₂×Q₈×C₂₂₃	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₂₂₃	C₂²	1	3	yes	—
20	C₄²×C₂₂₃	C₂²	1	3	yes	—
21	C₂²×D₂₂₃	C₂³	1	1	yes	characteristic
22	Dic₄₄₆	C₂×C₄	1	12	yes	—
23	C₂×Dic₂₂₃	C₂³	1	1	yes	characteristic
24	C₂×Dic₂₂₃	C₂³	1	3	yes	—
25	C₂×Dic₂₂₃	C₂³	1	3	yes	—
26	C₄×D₂₂₃	C₂×C₄	1	12	yes	—
27	Q₈×C₂₂₃	C₂×C₄	1	4	yes	—
28	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
29	C₂×C₄×C₂₂₃	C₂³	1	3	yes	—
30	C₂×C₄×C₂₂₃	C₂³	1	3	yes	—
31	C₂²×C₂₂₃	C₂⁴	1	1	yes	characteristic
32	C₂×D₂₂₃	—	2	1	no	—
33	C₂×D₂₂₃	—	2	1	no	—
34	C₂×D₂₂₃	C₂²×C₄	1	2	yes	—
35	C₄×C₂₂₃	—	2	3	no	—
36	C₄×C₂₂₃	—	2	1	no	—
37	C₄×C₂₂₃	C₂²×C₄	1	6	yes	—
38	Dic₂₂₃	—	2	1	no	—
39	Dic₂₂₃	—	2	3	no	—
40	Dic₂₂₃	C₂²×C₄	1	6	yes	—
41	C₂×C₂₂₃	C₂³×C₄	1	1	yes	characteristic
42	C₂×C₂₂₃	2₋¹⁺⁴	1	1	yes	characteristic
43	C₂×C₂₂₃	2₋¹⁺⁴	1	1	yes	characteristic
44	D₂₂₃	—	2	2	no	—
45	C₂₂₃	#519	1	1	yes	characteristic, complemented, Sylow
46	#519	—	223	1	no	maximal, Sylow
47	C₂²×Q₈	—	223	1	no	—
48	C₄×Q₈	—	223	1	no	—
49	C₄×Q₈	—	223	3	no	—
50	C₄×Q₈	—	223	3	no	—
51	C₄×Q₈	D₂₂₃	1	1	yes	characteristic, complemented
52	#117	—	223	3	no	—
53	#117	—	223	3	no	—
54	C₂×Q₈	—	223	8	no	—
55	C₂×Q₈	—	223	3	no	—
56	C₂×Q₈	C₂×D₂₂₃	1	1	yes	characteristic
57	C₂²×C₄	—	223	3	no	—
58	C₄⋊C₄	—	223	3	no	—
59	C₄⋊C₄	—	223	3	no	—
60	C₄⋊C₄	—	223	3	no	—
61	C₄⋊C₄	C₂×D₂₂₃	1	3	yes	—
62	C₂²⋊C₄	—	223	3	no	—
63	C₂²⋊C₄	—	223	1	no	—
64	C₄²	—	223	3	no	—
65	C₄²	—	223	6	no	—
66	C₄²	C₂×D₂₂₃	1	3	yes	—
67	C₂³	—	223	1	no	—
68	Q₈	—	223	12	no	—
69	Q₈	C₄×D₂₂₃	1	4	yes	—
70	C₂×C₄	—	223	1	no	—
71	C₂×C₄	—	223	3	no	—
72	C₂×C₄	—	223	3	no	—
73	C₂×C₄	—	223	12	no	—
74	C₂×C₄	C₂²×D₂₂₃	1	3	yes	—
75	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
76	C₂×C₄	C₂²×D₂₂₃	1	3	yes	—
77	C₂²	—	446	1	no	—
78	C₂²	—	446	1	no	—
79	C₂²	—	223	2	no	—
80	C₂²	C₂³×D₂₂₃	1	1	yes	characteristic
81	C₄	—	2	3	no	—
82	C₄	—	2	1	no	—
83	C₄	—	446	1	no	—
84	C₄	—	446	3	no	—
85	C₄	—	223	6	no	—
86	C₄	C₂×C₄×D₂₂₃	1	6	yes	—
87	C₂	—	446	2	no	—
88	C₂	C₂²×C₄×D₂₂₃	1	1	yes	characteristic
89	C₂	#20297351	1	1	yes	characteristic
90	C₂	#20297352	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₂₃
commutator	C₂×C₂₂₃	C₂³×C₄
Fitting	C₄×Q₈×C₂₂₃	C₂
socle, cosocle	C₂²×C₂₂₃	C₂⁴
prime radical	C₂²×D₂₂₃	C₂³
hypercenter	C₄×Q₈	D₂₂₃
hypocenter	C₂₂₃	#519
LCS	G ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. Q₈×Dic₂₂₃	C₂	semidirect
3. C₄×Dic₄₄₆	C₂	semidirect
4. #20297269	C₂	semidirect
7. C₄×Q₈×C₂₂₃	C₂	semidirect
9. Q₈×D₂₂₃	C₄	semidirect
45. C₂₂₃	#519	semidirect
51. C₄×Q₈	D₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×D₂₂₃	2
Q₈×Dic₂₂₃	2
C₄×Dic₄₄₆	2
#20297269	2
#20297262	2
#20297261	2
C₄×Q₈×C₂₂₃	2
#519	223