Subgroups — #2177×C₂₂₃ #43251192

Order 28544.

172 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₆×C₂₂₃	C₂	1	1	yes	maximal, characteristic
2	C₄×D₈×C₂₂₃	C₂	1	2	yes	maximal, complemented
3	#367×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#357×C₂₂₃	C₂	1	2	yes	maximal, complemented
5	#346×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₈×C₂₂₃	C₂²	1	2	yes	—
7	C₄×D₄×C₂₂₃	—	2	2	no	—
8	D₁₆×C₂₂₃	C₄	1	4	yes	complemented
9	C₂×C₁₆×C₂₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₁₆×C₂₂₃	C₂²	1	1	yes	characteristic
11	#107×C₂₂₃	C₂²	1	2	yes	—
12	#102×C₂₂₃	—	2	2	no	—
13	C₄×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
14	C₂×D₄×C₂₂₃	—	2	2	no	—
15	C₂²×C₄×C₂₂₃	—	4	2	no	—
16	D₈×C₂₂₃	—	2	2	no	—
17	D₈×C₂₂₃	C₂×C₄	1	4	yes	complemented
18	C₂×C₈×C₂₂₃	D₄	1	1	yes	characteristic
19	C₂×C₈×C₂₂₃	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₂₂₃	—	2	2	no	—
21	C₂²⋊C₄×C₂₂₃	—	4	2	no	—
22	C₄²×C₂₂₃	D₄	1	1	yes	characteristic
23	C₁₆×C₂₂₃	—	2	1	no	—
24	C₁₆×C₂₂₃	C₂×C₄	1	2	yes	complemented
25	C₂³×C₂₂₃	—	4	2	no	—
26	D₄×C₂₂₃	—	2	4	no	—
27	D₄×C₂₂₃	—	4	2	no	—
28	C₂×C₄×C₂₂₃	—	4	2	no	—
29	C₂×C₄×C₂₂₃	—	8	4	no	—
30	C₂×C₄×C₂₂₃	D₈	1	2	yes	—
31	C₂×C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
32	C₈×C₂₂₃	—	2	1	no	—
33	C₈×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
34	C₈×C₂₂₃	#41	1	1	yes	characteristic
35	C₂²×C₂₂₃	—	4	4	no	—
36	C₂²×C₂₂₃	—	8	2	no	—
37	C₂²×C₂₂₃	—	8	2	no	—
38	C₂²×C₂₂₃	C₂×D₈	1	1	yes	characteristic
39	C₄×C₂₂₃	—	2	2	no	—
40	C₄×C₂₂₃	—	8	2	no	—
41	C₄×C₂₂₃	C₄×D₄	1	1	yes	characteristic
42	C₄×C₂₂₃	#135	1	1	yes	characteristic
43	C₂×C₂₂₃	—	8	4	no	—
44	C₂×C₂₂₃	C₄×D₈	1	1	yes	characteristic
45	C₂×C₂₂₃	#509	1	2	yes	—
46	C₂₂₃	#2177	1	1	yes	characteristic, direct product factor, Sylow
47	#2177	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	C₂×D₁₆	C₂×C₂₂₃	1	1	yes	characteristic
49	C₄×D₈	C₂×C₂₂₃	1	2	yes	complemented
50	#367	C₂×C₂₂₃	1	1	yes	characteristic, complemented
51	#357	C₂×C₂₂₃	1	2	yes	complemented
52	#346	C₂×C₂₂₃	1	1	yes	characteristic, complemented
53	C₂×D₈	C₂²×C₂₂₃	1	2	yes	—
54	C₄×D₄	—	2	2	no	—
55	D₁₆	C₄×C₂₂₃	1	4	yes	complemented
56	C₂×C₁₆	C₂²×C₂₂₃	1	1	yes	characteristic
57	C₂×C₁₆	C₂²×C₂₂₃	1	1	yes	characteristic
58	#107	C₂²×C₂₂₃	1	2	yes	—
59	#102	—	2	2	no	—
60	C₄×C₈	C₂²×C₂₂₃	1	1	yes	characteristic
61	C₂×D₄	—	2	2	no	—
62	C₂²×C₄	—	4	2	no	—
63	D₈	—	2	2	no	—
64	D₈	C₂×C₄×C₂₂₃	1	4	yes	complemented
65	C₂×C₈	D₄×C₂₂₃	1	1	yes	characteristic
66	C₂×C₈	C₂³×C₂₂₃	1	1	yes	characteristic
67	C₄⋊C₄	—	2	2	no	—
68	C₂²⋊C₄	—	4	2	no	—
69	C₄²	D₄×C₂₂₃	1	1	yes	characteristic
70	C₁₆	—	2	1	no	—
71	C₁₆	C₂×C₄×C₂₂₃	1	2	yes	complemented
72	C₂³	—	4	2	no	—
73	D₄	—	2	4	no	—
74	D₄	—	4	2	no	—
75	C₂×C₄	—	4	2	no	—
76	C₂×C₄	—	8	4	no	—
77	C₂×C₄	D₈×C₂₂₃	1	2	yes	—
78	C₂×C₄	C₂×D₄×C₂₂₃	1	1	yes	characteristic
79	C₈	—	2	1	no	—
80	C₈	C₂²×C₄×C₂₂₃	1	1	yes	characteristic
81	C₈	#41×C₂₂₃	1	1	yes	characteristic
82	C₂²	—	4	4	no	—
83	C₂²	—	8	2	no	—
84	C₂²	—	8	2	no	—
85	C₂²	C₂×D₈×C₂₂₃	1	1	yes	characteristic
86	C₄	—	2	2	no	—
87	C₄	—	8	2	no	—
88	C₄	C₄×D₄×C₂₂₃	1	1	yes	characteristic
89	C₄	#135×C₂₂₃	1	1	yes	characteristic
90	C₂	—	8	4	no	—
91	C₂	C₄×D₈×C₂₂₃	1	1	yes	characteristic
92	C₂	#509×C₂₂₃	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₂₃	C₂×D₈
commutator	C₈	C₂²×C₄×C₂₂₃
Frattini, cosocle	C₂×C₈	C₂³×C₂₂₃
prime radical	C₂×D₁₆×C₂₂₃	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₂₃ ◁ C₄²×C₂₂₃ ◁ C₄×C₈×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×D₈×C₂₂₃	C₂	semidirect
3. #367×C₂₂₃	C₂	semidirect
4. #357×C₂₂₃	C₂	semidirect
5. #346×C₂₂₃	C₂	semidirect
8. D₁₆×C₂₂₃	C₄	semidirect
17. D₈×C₂₂₃	C₂×C₄	semidirect
24. C₁₆×C₂₂₃	C₂×C₄	semidirect
46. C₂₂₃	#2177	direct (N × H)
47. #2177	C₂₂₃	direct (N × H)
49. C₄×D₈	C₂×C₂₂₃	semidirect
50. #367	C₂×C₂₂₃	semidirect
51. #357	C₂×C₂₂₃	semidirect
52. #346	C₂×C₂₂₃	semidirect
55. D₁₆	C₄×C₂₂₃	semidirect
64. D₈	C₂×C₄×C₂₂₃	semidirect
71. C₁₆	C₂×C₄×C₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₆×C₂₂₃	2
C₄×D₈×C₂₂₃	2
#367×C₂₂₃	2
#357×C₂₂₃	2
#346×C₂₂₃	2
#2177	223

Subgroups — #2177×C223 #43251192

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2177×C₂₂₃ #43251192