Subgroups — #2906×C₂₂₃ #43251921

Order 28544.

484 conjugacy classes of subgroups.

The subgroup lattice has 138 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#131×C₂₂₃	C₂	1	2	yes	maximal
2	C₄×#41×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	#424×C₂₂₃	C₂	1	6	yes	maximal, complemented
4	#423×C₂₂₃	C₂	1	3	yes	maximal, complemented
5	C₂×C₄⋊C₈×C₂₂₃	C₂	1	3	yes	maximal, complemented
6	C₂×#41×C₂₂₃	C₂²	1	1	yes	characteristic
7	#131×C₂₂₃	—	2	8	no	—
8	C₂×M₁₆×C₂₂₃	C₂²	1	6	yes	—
9	C₂²×C₈×C₂₂₃	C₂²	1	6	yes	—
10	C₄×Q₈×C₂₂₃	C₄	1	1	yes	characteristic
11	C₄×Q₈×C₂₂₃	C₄	1	1	yes	characteristic
12	C₄×D₄×C₂₂₃	C₄	1	3	yes	—
13	C₄×D₄×C₂₂₃	C₄	1	3	yes	—
14	#117×C₂₂₃	C₂²	1	3	yes	—
15	C₂×C₄²×C₂₂₃	C₂²	1	3	yes	—
16	C₄⋊C₈×C₂₂₃	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₂₂₃	C₂²	1	3	yes	—
18	C₄⋊C₈×C₂₂₃	C₂²	1	6	yes	—
19	C₄⋊C₈×C₂₂₃	C₂²	1	6	yes	—
20	#41×C₂₂₃	D₄	1	4	yes	—
21	#41×C₂₂₃	Q₈	1	4	yes	—
22	C₂×Q₈×C₂₂₃	C₂×C₄	1	1	yes	characteristic
23	C₂×D₄×C₂₂₃	C₂×C₄	1	3	yes	—
24	C₂²×C₄×C₂₂₃	C₂×C₄	1	3	yes	—
25	C₂²×C₄×C₂₂₃	C₂×C₄	1	3	yes	—
26	C₂²×C₄×C₂₂₃	C₂³	1	3	yes	—
27	M₁₆×C₂₂₃	—	2	12	no	—
28	C₂×C₈×C₂₂₃	—	2	12	no	—
29	C₂×C₈×C₂₂₃	C₂³	1	2	yes	—
30	C₂×C₈×C₂₂₃	C₂³	1	6	yes	—
31	C₄⋊C₄×C₂₂₃	C₂×C₄	1	3	yes	—
32	C₄⋊C₄×C₂₂₃	C₂×C₄	1	3	yes	—
33	C₂²⋊C₄×C₂₂₃	C₂×C₄	1	6	yes	—
34	C₄²×C₂₂₃	C₂×C₄	1	3	yes	—
35	C₄²×C₂₂₃	C₂×C₄	1	3	yes	—
36	C₄²×C₂₂₃	C₂³	1	1	yes	characteristic
37	C₄²×C₂₂₃	C₂³	1	3	yes	—
38	C₂³×C₂₂₃	C₂²×C₄	1	3	yes	—
39	Q₈×C₂₂₃	C₄⋊C₄	1	4	yes	—
40	D₄×C₂₂₃	C₄⋊C₄	1	12	yes	—
41	C₂×C₄×C₂₂₃	—	2	6	no	—
42	C₂×C₄×C₂₂₃	—	4	3	no	—
43	C₂×C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₂₂₃	C₂²×C₄	1	3	yes	—
45	C₂×C₄×C₂₂₃	C₂²×C₄	1	3	yes	—
46	C₂×C₄×C₂₂₃	C₂²×C₄	1	3	yes	—
47	C₂×C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₂₂₃	C₂×D₄	1	6	yes	—
49	C₂×C₄×C₂₂₃	C₂×Q₈	1	6	yes	—
50	C₂×C₄×C₂₂₃	C₂⁴	1	1	yes	characteristic
51	C₈×C₂₂₃	—	2	2	no	—
52	C₈×C₂₂₃	—	2	6	no	—
53	C₂²×C₂₂₃	—	2	3	no	—
54	C₂²×C₂₂₃	—	2	3	no	—
55	C₂²×C₂₂₃	C₂×C₄⋊C₄	1	6	yes	—
56	C₂²×C₂₂₃	C₂³×C₄	1	1	yes	characteristic
57	C₄×C₂₂₃	—	2	3	no	—
58	C₄×C₂₂₃	—	2	1	no	—
59	C₄×C₂₂₃	—	2	3	no	—
60	C₄×C₂₂₃	C₂×C₄⋊C₄	1	6	yes	—
61	C₄×C₂₂₃	#131	1	2	yes	complemented
62	C₄×C₂₂₃	C₂²×D₄	1	1	yes	characteristic
63	C₄×C₂₂₃	C₂²×Q₈	1	1	yes	characteristic
64	C₂×C₂₂₃	—	2	6	no	—
65	C₂×C₂₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₂₂₃	C₂×#131	1	1	yes	characteristic
67	C₂×C₂₂₃	#568	1	1	yes	characteristic
68	C₂₂₃	#2906	1	1	yes	characteristic, direct product factor, Sylow
69	#2906	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂×#131	C₂×C₂₂₃	1	2	yes	—
71	C₄×#41	C₂×C₂₂₃	1	1	yes	characteristic
72	#424	C₂×C₂₂₃	1	6	yes	complemented
73	#423	C₂×C₂₂₃	1	3	yes	complemented
74	C₂×C₄⋊C₈	C₂×C₂₂₃	1	3	yes	complemented
75	C₂×#41	C₂²×C₂₂₃	1	1	yes	characteristic
76	#131	—	2	8	no	—
77	C₂×M₁₆	C₂²×C₂₂₃	1	6	yes	—
78	C₂²×C₈	C₂²×C₂₂₃	1	6	yes	—
79	C₄×Q₈	C₄×C₂₂₃	1	1	yes	characteristic
80	C₄×Q₈	C₄×C₂₂₃	1	1	yes	characteristic
81	C₄×D₄	C₄×C₂₂₃	1	3	yes	—
82	C₄×D₄	C₄×C₂₂₃	1	3	yes	—
83	#117	C₂²×C₂₂₃	1	3	yes	—
84	C₂×C₄²	C₂²×C₂₂₃	1	3	yes	—
85	C₄⋊C₈	C₂²×C₂₂₃	1	1	yes	characteristic
86	C₄⋊C₈	C₂²×C₂₂₃	1	3	yes	—
87	C₄⋊C₈	C₂²×C₂₂₃	1	6	yes	—
88	C₄⋊C₈	C₂²×C₂₂₃	1	6	yes	—
89	#41	D₄×C₂₂₃	1	4	yes	—
90	#41	Q₈×C₂₂₃	1	4	yes	—
91	C₂×Q₈	C₂×C₄×C₂₂₃	1	1	yes	characteristic
92	C₂×D₄	C₂×C₄×C₂₂₃	1	3	yes	—
93	C₂²×C₄	C₂×C₄×C₂₂₃	1	3	yes	—
94	C₂²×C₄	C₂×C₄×C₂₂₃	1	3	yes	—
95	C₂²×C₄	C₂³×C₂₂₃	1	3	yes	—
96	M₁₆	—	2	12	no	—
97	C₂×C₈	—	2	12	no	—
98	C₂×C₈	C₂³×C₂₂₃	1	2	yes	—
99	C₂×C₈	C₂³×C₂₂₃	1	6	yes	—
100	C₄⋊C₄	C₂×C₄×C₂₂₃	1	3	yes	—
101	C₄⋊C₄	C₂×C₄×C₂₂₃	1	3	yes	—
102	C₂²⋊C₄	C₂×C₄×C₂₂₃	1	6	yes	—
103	C₄²	C₂×C₄×C₂₂₃	1	3	yes	—
104	C₄²	C₂×C₄×C₂₂₃	1	3	yes	—
105	C₄²	C₂³×C₂₂₃	1	1	yes	characteristic
106	C₄²	C₂³×C₂₂₃	1	3	yes	—
107	C₂³	C₂²×C₄×C₂₂₃	1	3	yes	—
108	Q₈	C₄⋊C₄×C₂₂₃	1	4	yes	—
109	D₄	C₄⋊C₄×C₂₂₃	1	12	yes	—
110	C₂×C₄	—	2	6	no	—
111	C₂×C₄	—	4	3	no	—
112	C₂×C₄	C₂²×C₄×C₂₂₃	1	1	yes	characteristic
113	C₂×C₄	C₂²×C₄×C₂₂₃	1	3	yes	—
114	C₂×C₄	C₂²×C₄×C₂₂₃	1	3	yes	—
115	C₂×C₄	C₂²×C₄×C₂₂₃	1	3	yes	—
116	C₂×C₄	C₂²×C₄×C₂₂₃	1	1	yes	characteristic
117	C₂×C₄	C₂×D₄×C₂₂₃	1	6	yes	—
118	C₂×C₄	C₂×Q₈×C₂₂₃	1	6	yes	—
119	C₂×C₄	C₂⁴×C₂₂₃	1	1	yes	characteristic
120	C₈	—	2	2	no	—
121	C₈	—	2	6	no	—
122	C₂²	—	2	3	no	—
123	C₂²	—	2	3	no	—
124	C₂²	C₂×C₄⋊C₄×C₂₂₃	1	6	yes	—
125	C₂²	C₂³×C₄×C₂₂₃	1	1	yes	characteristic
126	C₄	—	2	3	no	—
127	C₄	—	2	1	no	—
128	C₄	—	2	3	no	—
129	C₄	C₂×C₄⋊C₄×C₂₂₃	1	6	yes	—
130	C₄	#131×C₂₂₃	1	2	yes	complemented
131	C₄	C₂²×D₄×C₂₂₃	1	1	yes	characteristic
132	C₄	C₂²×Q₈×C₂₂₃	1	1	yes	characteristic
133	C₂	—	2	6	no	—
134	C₂	C₂²×C₄⋊C₄×C₂₂₃	1	1	yes	characteristic
135	C₂	C₂×#131×C₂₂₃	1	1	yes	characteristic
136	C₂	#568×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₂₃	C₂⁴
commutator	C₂²	C₂³×C₄×C₂₂₃
Frattini, cosocle	C₂×C₄	C₂⁴×C₂₂₃
socle	C₂²×C₂₂₃	C₂³×C₄
prime radical	C₂×#41×C₂₂₃	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. #424×C₂₂₃	C₂	semidirect
4. #423×C₂₂₃	C₂	semidirect
5. C₂×C₄⋊C₈×C₂₂₃	C₂	semidirect
61. C₄×C₂₂₃	#131	semidirect
68. C₂₂₃	#2906	direct (N × H)
69. #2906	C₂₂₃	direct (N × H)
72. #424	C₂×C₂₂₃	semidirect
73. #423	C₂×C₂₂₃	semidirect
74. C₂×C₄⋊C₈	C₂×C₂₂₃	semidirect
130. C₄	#131×C₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#131×C₂₂₃	2
C₄×#41×C₂₂₃	2
#424×C₂₂₃	2
#423×C₂₂₃	2
C₂×C₄⋊C₈×C₂₂₃	2
#2906	223

Subgroups — #2906×C223 #43251921

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2906×C₂₂₃ #43251921