Subgroups — #2956×C₂₂₃ #43251971

Order 28544.

458 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#575×C₂₂₃	C₂	1	1	yes	maximal, characteristic
2	#568×C₂₂₃	C₂	1	2	yes	maximal, complemented
3	#444×C₂₂₃	C₂	1	4	yes	maximal, complemented
4	#443×C₂₂₃	C₂	1	4	yes	maximal, complemented
5	#430×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#104×C₂₂₃	C₂	1	2	yes	maximal, complemented
7	C₄×M₁₆×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#41×C₂₂₃	C₂²	1	2	yes	—
9	#137×C₂₂₃	C₄	1	4	yes	complemented
10	AGL(1,ℤ/8)×C₂₂₃	C₄	1	4	yes	complemented
11	#135×C₂₂₃	C₂²	1	4	yes	—
12	#131×C₂₂₃	—	2	4	no	—
13	#131×C₂₂₃	C₂²	1	8	yes	complemented
14	C₂×M₁₆×C₂₂₃	—	2	4	no	—
15	C₂×M₁₆×C₂₂₃	C₂²	1	2	yes	—
16	C₂×M₁₆×C₂₂₃	C₂²	1	1	yes	characteristic
17	C₂×M₁₆×C₂₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
19	#108×C₂₂₃	C₂²	1	4	yes	complemented
20	#104×C₂₂₃	C₂²	1	4	yes	complemented
21	#104×C₂₂₃	C₂²	1	4	yes	complemented
22	#97×C₂₂₃	C₂²	1	2	yes	complemented
23	C₄×C₈×C₂₂₃	C₂²	1	2	yes	complemented
24	#41×C₂₂₃	—	2	2	no	—
25	#41×C₂₂₃	—	2	2	no	—
26	#41×C₂₂₃	C₂×C₄	1	4	yes	—
27	#41×C₂₂₃	C₂³	1	4	yes	—
28	C₂×Q₈×C₂₂₃	C₂×C₄	1	2	yes	—
29	C₂×D₄×C₂₂₃	—	2	2	no	—
30	C₂×D₄×C₂₂₃	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₂₂₃	—	2	2	no	—
32	C₂²×C₄×C₂₂₃	—	2	1	no	—
33	C₂²×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
34	Q₁₆×C₂₂₃	C₂×C₄	1	4	yes	complemented
35	SD₁₆×C₂₂₃	C₂×C₄	1	8	yes	complemented
36	D₈×C₂₂₃	C₂×C₄	1	4	yes	complemented
37	M₁₆×C₂₂₃	—	2	2	no	—
38	M₁₆×C₂₂₃	—	2	8	no	—
39	M₁₆×C₂₂₃	C₂×C₄	1	4	yes	—
40	M₁₆×C₂₂₃	D₄	1	4	yes	—
41	M₁₆×C₂₂₃	C₂³	1	4	yes	—
42	C₂×C₈×C₂₂₃	—	2	2	no	—
43	C₂×C₈×C₂₂₃	—	2	8	no	—
44	C₂×C₈×C₂₂₃	C₂³	1	2	yes	—
45	C₂×C₈×C₂₂₃	C₂³	1	2	yes	—
46	C₄²×C₂₂₃	—	2	1	no	—
47	C₄²×C₂₂₃	C₂³	1	1	yes	characteristic
48	C₄²×C₂₂₃	C₂³	1	1	yes	characteristic
49	C₂³×C₂₂₃	—	2	2	no	—
50	C₂³×C₂₂₃	#41	1	1	yes	characteristic
51	Q₈×C₂₂₃	—	2	2	no	—
52	Q₈×C₂₂₃	C₂²×C₄	1	4	yes	—
53	D₄×C₂₂₃	—	2	4	no	—
54	D₄×C₂₂₃	—	2	4	no	—
55	D₄×C₂₂₃	—	2	2	no	—
56	D₄×C₂₂₃	C₂²×C₄	1	4	yes	—
57	C₂×C₄×C₂₂₃	—	2	2	no	—
58	C₂×C₄×C₂₂₃	—	2	1	no	—
59	C₂×C₄×C₂₂₃	—	2	4	no	—
60	C₂×C₄×C₂₂₃	—	2	4	no	—
61	C₂×C₄×C₂₂₃	—	2	1	no	—
62	C₂×C₄×C₂₂₃	—	4	2	no	—
63	C₂×C₄×C₂₂₃	C₂²×C₄	1	2	yes	—
64	C₂×C₄×C₂₂₃	C₂×D₄	1	2	yes	—
65	C₂×C₄×C₂₂₃	#41	1	1	yes	characteristic
66	C₂×C₄×C₂₂₃	C₂⁴	1	1	yes	characteristic
67	C₈×C₂₂₃	—	2	4	no	—
68	C₈×C₂₂₃	C₂²×C₄	1	4	yes	—
69	C₈×C₂₂₃	C₂×D₄	1	4	yes	—
70	C₂²×C₂₂₃	—	2	4	no	—
71	C₂²×C₂₂₃	—	4	1	no	—
72	C₂²×C₂₂₃	—	8	2	no	—
73	C₂²×C₂₂₃	C₄×D₄	1	2	yes	—
74	C₂²×C₂₂₃	C₂×#41	1	1	yes	characteristic
75	C₄×C₂₂₃	—	2	2	no	—
76	C₄×C₂₂₃	—	2	4	no	—
77	C₄×C₂₂₃	—	4	1	no	—
78	C₄×C₂₂₃	C₄×D₄	1	2	yes	—
79	C₄×C₂₂₃	C₂³×C₄	1	1	yes	characteristic
80	C₄×C₂₂₃	C₂²×D₄	1	1	yes	characteristic
81	C₂×C₂₂₃	—	2	1	no	—
82	C₂×C₂₂₃	—	2	2	no	—
83	C₂×C₂₂₃	—	4	4	no	—
84	C₂×C₂₂₃	C₂×C₄×D₄	1	1	yes	characteristic
85	C₂₂₃	#2956	1	1	yes	characteristic, direct product factor, Sylow
86	#2956	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	#575	C₂×C₂₂₃	1	1	yes	characteristic
88	#568	C₂×C₂₂₃	1	2	yes	complemented
89	#444	C₂×C₂₂₃	1	4	yes	complemented
90	#443	C₂×C₂₂₃	1	4	yes	complemented
91	#430	C₂×C₂₂₃	1	1	yes	characteristic, complemented
92	C₂×#104	C₂×C₂₂₃	1	2	yes	complemented
93	C₄×M₁₆	C₂×C₂₂₃	1	1	yes	characteristic, complemented
94	C₂×#41	C₂²×C₂₂₃	1	2	yes	—
95	#137	C₄×C₂₂₃	1	4	yes	complemented
96	AGL(1,ℤ/8)	C₄×C₂₂₃	1	4	yes	complemented
97	#135	C₂²×C₂₂₃	1	4	yes	—
98	#131	—	2	4	no	—
99	#131	C₂²×C₂₂₃	1	8	yes	complemented
100	C₂×M₁₆	—	2	4	no	—
101	C₂×M₁₆	C₂²×C₂₂₃	1	2	yes	—
102	C₂×M₁₆	C₂²×C₂₂₃	1	1	yes	characteristic
103	C₂×M₁₆	C₂²×C₂₂₃	1	1	yes	characteristic
104	C₂×C₄²	C₂²×C₂₂₃	1	1	yes	characteristic
105	#108	C₂²×C₂₂₃	1	4	yes	complemented
106	#104	C₂²×C₂₂₃	1	4	yes	complemented
107	#104	C₂²×C₂₂₃	1	4	yes	complemented
108	#97	C₂²×C₂₂₃	1	2	yes	complemented
109	C₄×C₈	C₂²×C₂₂₃	1	2	yes	complemented
110	#41	—	2	2	no	—
111	#41	—	2	2	no	—
112	#41	C₂×C₄×C₂₂₃	1	4	yes	—
113	#41	C₂³×C₂₂₃	1	4	yes	—
114	C₂×Q₈	C₂×C₄×C₂₂₃	1	2	yes	—
115	C₂×D₄	—	2	2	no	—
116	C₂×D₄	C₂×C₄×C₂₂₃	1	2	yes	—
117	C₂²×C₄	—	2	2	no	—
118	C₂²×C₄	—	2	1	no	—
119	C₂²×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
120	Q₁₆	C₂×C₄×C₂₂₃	1	4	yes	complemented
121	SD₁₆	C₂×C₄×C₂₂₃	1	8	yes	complemented
122	D₈	C₂×C₄×C₂₂₃	1	4	yes	complemented
123	M₁₆	—	2	2	no	—
124	M₁₆	—	2	8	no	—
125	M₁₆	C₂×C₄×C₂₂₃	1	4	yes	—
126	M₁₆	D₄×C₂₂₃	1	4	yes	—
127	M₁₆	C₂³×C₂₂₃	1	4	yes	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	2	8	no	—
130	C₂×C₈	C₂³×C₂₂₃	1	2	yes	—
131	C₂×C₈	C₂³×C₂₂₃	1	2	yes	—
132	C₄²	—	2	1	no	—
133	C₄²	C₂³×C₂₂₃	1	1	yes	characteristic
134	C₄²	C₂³×C₂₂₃	1	1	yes	characteristic
135	C₂³	—	2	2	no	—
136	C₂³	#41×C₂₂₃	1	1	yes	characteristic
137	Q₈	—	2	2	no	—
138	Q₈	C₂²×C₄×C₂₂₃	1	4	yes	—
139	D₄	—	2	4	no	—
140	D₄	—	2	4	no	—
141	D₄	—	2	2	no	—
142	D₄	C₂²×C₄×C₂₂₃	1	4	yes	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	C₂²×C₄×C₂₂₃	1	2	yes	—
150	C₂×C₄	C₂×D₄×C₂₂₃	1	2	yes	—
151	C₂×C₄	#41×C₂₂₃	1	1	yes	characteristic
152	C₂×C₄	C₂⁴×C₂₂₃	1	1	yes	characteristic
153	C₈	—	2	4	no	—
154	C₈	C₂²×C₄×C₂₂₃	1	4	yes	—
155	C₈	C₂×D₄×C₂₂₃	1	4	yes	—
156	C₂²	—	2	4	no	—
157	C₂²	—	4	1	no	—
158	C₂²	—	8	2	no	—
159	C₂²	C₄×D₄×C₂₂₃	1	2	yes	—
160	C₂²	C₂×#41×C₂₂₃	1	1	yes	characteristic
161	C₄	—	2	2	no	—
162	C₄	—	2	4	no	—
163	C₄	—	4	1	no	—
164	C₄	C₄×D₄×C₂₂₃	1	2	yes	—
165	C₄	C₂³×C₄×C₂₂₃	1	1	yes	characteristic
166	C₄	C₂²×D₄×C₂₂₃	1	1	yes	characteristic
167	C₂	—	2	1	no	—
168	C₂	—	2	2	no	—
169	C₂	—	4	4	no	—
170	C₂	C₂×C₄×D₄×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₂₂₃	C₂²×D₄
commutator	C₄	C₂³×C₄×C₂₂₃
Frattini, cosocle	C₂×C₄	C₂⁴×C₂₂₃
socle	C₂×C₂₂₃	C₂×C₄×D₄
prime radical	#575×C₂₂₃	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₂₃ ◁ C₂×M₁₆×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. #568×C₂₂₃	C₂	semidirect
3. #444×C₂₂₃	C₂	semidirect
4. #443×C₂₂₃	C₂	semidirect
5. #430×C₂₂₃	C₂	semidirect
6. C₂×#104×C₂₂₃	C₂	semidirect
7. C₄×M₁₆×C₂₂₃	C₂	semidirect
9. #137×C₂₂₃	C₄	semidirect
10. AGL(1,ℤ/8)×C₂₂₃	C₄	semidirect
13. #131×C₂₂₃	C₂²	semidirect
19. #108×C₂₂₃	C₂²	semidirect
20. #104×C₂₂₃	C₂²	semidirect
21. #104×C₂₂₃	C₂²	semidirect
22. #97×C₂₂₃	C₂²	semidirect
23. C₄×C₈×C₂₂₃	C₂²	semidirect
34. Q₁₆×C₂₂₃	C₂×C₄	semidirect
35. SD₁₆×C₂₂₃	C₂×C₄	semidirect
36. D₈×C₂₂₃	C₂×C₄	semidirect
85. C₂₂₃	#2956	direct (N × H)
86. #2956	C₂₂₃	direct (N × H)
88. #568	C₂×C₂₂₃	semidirect
89. #444	C₂×C₂₂₃	semidirect
90. #443	C₂×C₂₂₃	semidirect
91. #430	C₂×C₂₂₃	semidirect
92. C₂×#104	C₂×C₂₂₃	semidirect
93. C₄×M₁₆	C₂×C₂₂₃	semidirect
95. #137	C₄×C₂₂₃	semidirect
96. AGL(1,ℤ/8)	C₄×C₂₂₃	semidirect
99. #131	C₂²×C₂₂₃	semidirect
105. #108	C₂²×C₂₂₃	semidirect
106. #104	C₂²×C₂₂₃	semidirect
107. #104	C₂²×C₂₂₃	semidirect
108. #97	C₂²×C₂₂₃	semidirect
109. C₄×C₈	C₂²×C₂₂₃	semidirect
120. Q₁₆	C₂×C₄×C₂₂₃	semidirect
121. SD₁₆	C₂×C₄×C₂₂₃	semidirect
122. D₈	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
#575×C₂₂₃	2
#568×C₂₂₃	2
#444×C₂₂₃	2
#443×C₂₂₃	2
#430×C₂₂₃	2
C₂×#104×C₂₂₃	2
C₄×M₁₆×C₂₂₃	2
#2956	223

Subgroups — #2956×C223 #43251971

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2956×C₂₂₃ #43251971