Subgroups — #28384062

Order 14272.

294 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16813171	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×D₈₉₂	C₂	1	3	yes	maximal, complemented
4	#20297266	C₂	1	3	yes	maximal, complemented
5	#20297261	C₂	1	3	yes	maximal, complemented
6	C₄⋊C₄×D₂₂₃	C₂	1	3	yes	maximal, complemented
7	C₄×Q₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#16813171	C₄	1	8	yes	complemented
9	C₂×D₈₉₂	C₂²	1	3	yes	—
10	C₂×C₄×D₂₂₃	C₂²	1	6	yes	—
11	C₂×C₄×D₂₂₃	C₂²	1	3	yes	—
12	C₄⋊Dic₂₂₃	C₂²	1	3	yes	—
13	Dic₂₂₃⋊C₄	C₂²	1	3	yes	—
14	Dic₂₂₃⋊C₄	C₂²	1	1	yes	characteristic
15	(C₂×D₂₂₃)⋊C₄	C₂²	1	3	yes	—
16	(C₂×D₂₂₃)⋊C₄	C₂²	1	3	yes	—
17	C₄×Dic₂₂₃	C₂²	1	3	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	3	yes	—
22	D₈₉₂	C₂×C₄	1	12	yes	complemented
23	C₂×Dic₂₂₃	C₂³	1	1	yes	characteristic
24	C₂×Dic₂₂₃	C₂³	1	3	yes	—
25	C₂×Dic₂₂₃	C₂³	1	1	yes	characteristic
26	C₄×D₂₂₃	—	4	6	no	—
27	C₄×D₂₂₃	C₂×C₄	1	12	yes	complemented
28	Q₈×C₂₂₃	C₂×C₄	1	4	yes	complemented
29	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
30	C₂×C₄×C₂₂₃	C₂³	1	3	yes	—
31	C₂×C₄×C₂₂₃	C₂³	1	3	yes	—
32	C₂²×C₂₂₃	C₂⁴	1	1	yes	characteristic
33	C₂×D₂₂₃	—	2	3	no	—
34	C₂×D₂₂₃	—	2	3	no	—
35	C₂×D₂₂₃	C₂²×C₄	1	6	yes	—
36	C₄×C₂₂₃	—	2	3	no	—
37	C₄×C₂₂₃	—	2	1	no	—
38	C₄×C₂₂₃	C₂²×C₄	1	6	yes	—
39	Dic₂₂₃	—	2	3	no	—
40	Dic₂₂₃	—	2	1	no	—
41	Dic₂₂₃	C₂²×C₄	1	2	yes	—
42	C₂×C₂₂₃	C₂³×C₄	1	1	yes	characteristic
43	C₂×C₂₂₃	H(5,𝔽₂)	1	1	yes	characteristic
44	C₂×C₂₂₃	2₋¹⁺⁴	1	1	yes	characteristic
45	D₂₂₃	—	2	6	no	—
46	C₂₂₃	#520	1	1	yes	characteristic, complemented, Sylow
47	#520	—	223	1	no	maximal, Sylow
48	C₂×#41	—	223	1	no	—
49	C₄×Q₈	—	223	1	no	—
50	C₄×Q₈	D₂₂₃	1	1	yes	characteristic, complemented
51	C₄×D₄	—	223	3	no	—
52	C₄×D₄	—	223	3	no	—
53	#117	—	223	3	no	—
54	C₂×C₄⋊C₄	—	223	3	no	—
55	#41	—	223	8	no	—
56	C₂×Q₈	C₂×D₂₂₃	1	1	yes	characteristic
57	C₂×D₄	—	223	3	no	—
58	C₂²×C₄	—	223	6	no	—
59	C₂²×C₄	—	223	3	no	—
60	C₄⋊C₄	—	223	3	no	—
61	C₄⋊C₄	—	223	1	no	—
62	C₄⋊C₄	—	223	3	no	—
63	C₄⋊C₄	C₂×D₂₂₃	1	3	yes	—
64	C₂²⋊C₄	—	223	3	no	—
65	C₂²⋊C₄	—	223	3	no	—
66	C₄²	—	223	3	no	—
67	C₄²	C₂×D₂₂₃	1	3	yes	—
68	C₂³	—	223	3	no	—
69	Q₈	C₄×D₂₂₃	1	4	yes	complemented
70	D₄	—	223	12	no	—
71	C₂×C₄	—	892	6	no	—
72	C₂×C₄	—	223	1	no	—
73	C₂×C₄	—	223	3	no	—
74	C₂×C₄	—	223	1	no	—
75	C₂×C₄	—	223	12	no	—
76	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
77	C₂×C₄	C₂²×D₂₂₃	1	3	yes	—
78	C₂×C₄	C₂²×D₂₂₃	1	3	yes	—
79	C₂²	—	446	3	no	—
80	C₂²	—	446	3	no	—
81	C₂²	—	223	6	no	—
82	C₂²	C₂³×D₂₂₃	1	1	yes	characteristic
83	C₄	—	2	3	no	—
84	C₄	—	2	1	no	—
85	C₄	—	446	3	no	—
86	C₄	—	446	1	no	—
87	C₄	—	223	2	no	—
88	C₄	C₂×C₄×D₂₂₃	1	6	yes	—
89	C₂	—	446	6	no	—
90	C₂	C₂²×C₄×D₂₂₃	1	1	yes	characteristic
91	C₂	#20297350	1	1	yes	characteristic
92	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	C₂×#16813171	C₂
hypercenter	C₄×Q₈	D₂₂₃
hypocenter	C₂₂₃	#520
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×D₈₉₂	C₂	semidirect
4. #20297266	C₂	semidirect
5. #20297261	C₂	semidirect
6. C₄⋊C₄×D₂₂₃	C₂	semidirect
7. C₄×Q₈×C₂₂₃	C₂	semidirect
8. #16813171	C₄	semidirect
22. D₈₉₂	C₂×C₄	semidirect
27. C₄×D₂₂₃	C₂×C₄	semidirect
28. Q₈×C₂₂₃	C₂×C₄	semidirect
46. C₂₂₃	#520	semidirect
50. C₄×Q₈	D₂₂₃	semidirect
69. Q₈	C₄×D₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16813171	2
Q₈×Dic₂₂₃	2
C₄×D₈₉₂	2
#20297266	2
#20297261	2
C₄⋊C₄×D₂₂₃	2
C₄×Q₈×C₂₂₃	2
#520	223