Subgroups — #2176×C₂₂₃ #43251191

Order 28544.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₃₂×C₂₂₃	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₁₆×C₂₂₃	C₂	1	2	yes	maximal
3	#367×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	#358×C₂₂₃	C₂	1	2	yes	maximal
5	#346×C₂₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₁₆×C₂₂₃	C₂²	1	2	yes	—
7	C₄×Q₈×C₂₂₃	—	2	2	no	—
8	Q₃₂×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	#103×C₂₂₃	—	2	2	no	—
13	C₄×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
14	C₂×Q₈×C₂₂₃	—	2	2	no	—
15	Q₁₆×C₂₂₃	—	2	2	no	—
16	Q₁₆×C₂₂₃	C₂×C₄	1	4	yes	—
17	C₂×C₈×C₂₂₃	D₄	1	1	yes	characteristic
18	C₂×C₈×C₂₂₃	C₂³	1	1	yes	characteristic
19	C₄⋊C₄×C₂₂₃	—	2	2	no	—
20	C₄⋊C₄×C₂₂₃	—	4	2	no	—
21	C₄²×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	—
25	Q₈×C₂₂₃	—	2	4	no	—
26	Q₈×C₂₂₃	—	4	2	no	—
27	C₂×C₄×C₂₂₃	—	4	2	no	—
28	C₂×C₄×C₂₂₃	—	4	2	no	—
29	C₂×C₄×C₂₂₃	D₈	1	2	yes	—
30	C₂×C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
31	C₈×C₂₂₃	—	2	1	no	—
32	C₈×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
33	C₈×C₂₂₃	#41	1	1	yes	characteristic
34	C₂²×C₂₂₃	C₂×D₈	1	1	yes	characteristic
35	C₄×C₂₂₃	—	2	2	no	—
36	C₄×C₂₂₃	—	4	4	no	—
37	C₄×C₂₂₃	—	8	2	no	—
38	C₄×C₂₂₃	C₄×D₄	1	1	yes	characteristic
39	C₄×C₂₂₃	#135	1	1	yes	characteristic
40	C₂×C₂₂₃	C₄×D₈	1	1	yes	characteristic
41	C₂×C₂₂₃	#510	1	2	yes	—
42	C₂₂₃	#2176	1	1	yes	characteristic, direct product factor, Sylow
43	#2176	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂×Q₃₂	C₂×C₂₂₃	1	1	yes	characteristic
45	C₄×Q₁₆	C₂×C₂₂₃	1	2	yes	—
46	#367	C₂×C₂₂₃	1	1	yes	characteristic
47	#358	C₂×C₂₂₃	1	2	yes	—
48	#346	C₂×C₂₂₃	1	1	yes	characteristic
49	C₂×Q₁₆	C₂²×C₂₂₃	1	2	yes	—
50	C₄×Q₈	—	2	2	no	—
51	Q₃₂	C₄×C₂₂₃	1	4	yes	complemented
52	C₂×C₁₆	C₂²×C₂₂₃	1	1	yes	characteristic
53	C₂×C₁₆	C₂²×C₂₂₃	1	1	yes	characteristic
54	#107	C₂²×C₂₂₃	1	2	yes	—
55	#103	—	2	2	no	—
56	C₄×C₈	C₂²×C₂₂₃	1	1	yes	characteristic
57	C₂×Q₈	—	2	2	no	—
58	Q₁₆	—	2	2	no	—
59	Q₁₆	C₂×C₄×C₂₂₃	1	4	yes	—
60	C₂×C₈	D₄×C₂₂₃	1	1	yes	characteristic
61	C₂×C₈	C₂³×C₂₂₃	1	1	yes	characteristic
62	C₄⋊C₄	—	2	2	no	—
63	C₄⋊C₄	—	4	2	no	—
64	C₄²	—	4	2	no	—
65	C₄²	D₄×C₂₂₃	1	1	yes	characteristic
66	C₁₆	—	2	1	no	—
67	C₁₆	C₂×C₄×C₂₂₃	1	2	yes	—
68	Q₈	—	2	4	no	—
69	Q₈	—	4	2	no	—
70	C₂×C₄	—	4	2	no	—
71	C₂×C₄	—	4	2	no	—
72	C₂×C₄	D₈×C₂₂₃	1	2	yes	—
73	C₂×C₄	C₂×D₄×C₂₂₃	1	1	yes	characteristic
74	C₈	—	2	1	no	—
75	C₈	C₂²×C₄×C₂₂₃	1	1	yes	characteristic
76	C₈	#41×C₂₂₃	1	1	yes	characteristic
77	C₂²	C₂×D₈×C₂₂₃	1	1	yes	characteristic
78	C₄	—	2	2	no	—
79	C₄	—	4	4	no	—
80	C₄	—	8	2	no	—
81	C₄	C₄×D₄×C₂₂₃	1	1	yes	characteristic
82	C₄	#135×C₂₂₃	1	1	yes	characteristic
83	C₂	C₄×D₈×C₂₂₃	1	1	yes	characteristic
84	C₂	#510×C₂₂₃	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂²×C₂₂₃	C₂×D₈
commutator	C₈	C₂²×C₄×C₂₂₃
Frattini, cosocle	C₂×C₈	C₂³×C₂₂₃
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
8. Q₃₂×C₂₂₃	C₄	semidirect
42. C₂₂₃	#2176	direct (N × H)
43. #2176	C₂₂₃	direct (N × H)
51. Q₃₂	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×Q₃₂×C₂₂₃	2
C₄×Q₁₆×C₂₂₃	2
#367×C₂₂₃	2
#358×C₂₂₃	2
#346×C₂₂₃	2
#2176	223

Subgroups — #2176×C223 #43251191

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2176×C₂₂₃ #43251191