Subgroups — C₂₃×#2920 #15670539

Order 2944.

484 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄²×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#432	C₂	1	4	yes	maximal, complemented
3	C₂₃×#431	C₂	1	4	yes	maximal, complemented
4	C₂₃×#423	C₂	1	2	yes	maximal, complemented
5	C₂₃×#407	C₂	1	2	yes	maximal
6	C₄×M₁₆×C₂₃	C₂	1	2	yes	maximal, complemented
7	C₂³×C₄×C₂₃	C₄	1	2	yes	—
8	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×M₁₆×C₂₃	C₂²	1	4	yes	—
10	C₂×C₄²×C₂₃	C₄	1	2	yes	—
11	C₂×C₄²×C₂₃	C₄	1	2	yes	—
12	C₂×C₄²×C₂₃	C₄	1	2	yes	—
13	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
15	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
17	C₂₃×C₄⋊C₈	C₂²	1	8	yes	complemented
18	C₂₃×C₂²⋊C₈	C₂²	1	8	yes	—
19	C₂₃×#97	C₂²	1	4	yes	complemented
20	C₄×C₈×C₂₃	C₂²	1	4	yes	complemented
21	C₂⁴×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₂₃	—	2	8	no	—
23	C₂²×C₄×C₂₃	—	2	4	no	—
24	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
28	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
33	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
34	M₁₆×C₂₃	—	2	8	no	—
35	C₂×C₈×C₂₃	C₂³	1	8	yes	—
36	C₄²×C₂₃	C₂×C₄	1	2	yes	—
37	C₄²×C₂₃	C₂×C₄	1	2	yes	—
38	C₄²×C₂₃	C₂×C₄	1	2	yes	—
39	C₄²×C₂₃	C₂×C₄	1	2	yes	—
40	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
41	C₄²×C₂₃	C₂×C₄	1	2	yes	—
42	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
43	C₄²×C₂₃	C₂³	1	2	yes	—
44	C₄²×C₂₃	C₂³	1	1	yes	characteristic
45	C₄²×C₂₃	C₂³	1	1	yes	characteristic
46	C₂³×C₂₃	—	2	2	no	—
47	C₂³×C₂₃	—	2	4	no	—
48	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
49	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
50	C₂×C₄×C₂₃	—	2	2	no	—
51	C₂×C₄×C₂₃	—	2	4	no	—
52	C₂×C₄×C₂₃	—	2	4	no	—
53	C₂×C₄×C₂₃	—	2	4	no	—
54	C₂×C₄×C₂₃	—	2	4	no	—
55	C₂×C₄×C₂₃	—	2	4	no	—
56	C₂×C₄×C₂₃	—	2	4	no	—
57	C₂×C₄×C₂₃	—	2	4	no	—
58	C₂×C₄×C₂₃	M₁₆	1	8	yes	complemented
59	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
60	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
61	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
62	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
64	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
65	C₂×C₄×C₂₃	#41	1	4	yes	—
66	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
67	C₈×C₂₃	—	2	8	no	—
68	#2920	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂²×C₂₃	—	2	8	no	—
70	C₂²×C₂₃	—	2	2	no	—
71	C₂²×C₂₃	—	2	2	no	—
72	C₂²×C₂₃	—	2	2	no	—
73	C₂²×C₂₃	#117	1	2	yes	—
74	C₂²×C₂₃	C₂×M₁₆	1	4	yes	—
75	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
76	C₄×C₂₃	—	2	2	no	—
77	C₄×C₂₃	—	2	2	no	—
78	C₄×C₂₃	—	2	2	no	—
79	C₄×C₂₃	#117	1	2	yes	—
80	C₄×C₂₃	C₂×M₁₆	1	4	yes	—
81	C₄×C₂₃	C₂×M₁₆	1	4	yes	—
82	C₄×C₂₃	C₂×#41	1	2	yes	—
83	C₂²×C₄²	C₂×C₂₃	1	1	yes	characteristic
84	#432	C₂×C₂₃	1	4	yes	complemented
85	#431	C₂×C₂₃	1	4	yes	complemented
86	#423	C₂×C₂₃	1	2	yes	complemented
87	#407	C₂×C₂₃	1	2	yes	—
88	C₄×M₁₆	C₂×C₂₃	1	2	yes	complemented
89	C₂×C₂₃	—	2	2	no	—
90	C₂×C₂₃	—	2	4	no	—
91	C₂×C₂₃	C₂×#117	1	1	yes	characteristic
92	C₂×C₂₃	C₂²×M₁₆	1	2	yes	—
93	C₂³×C₄	C₄×C₂₃	1	2	yes	—
94	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
95	C₂×M₁₆	C₂²×C₂₃	1	4	yes	—
96	C₂×C₄²	C₄×C₂₃	1	2	yes	—
97	C₂×C₄²	C₄×C₂₃	1	2	yes	—
98	C₂×C₄²	C₄×C₂₃	1	2	yes	—
99	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
100	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
101	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
102	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
103	C₄⋊C₈	C₂²×C₂₃	1	8	yes	complemented
104	C₂²⋊C₈	C₂²×C₂₃	1	8	yes	—
105	#97	C₂²×C₂₃	1	4	yes	complemented
106	C₄×C₈	C₂²×C₂₃	1	4	yes	complemented
107	C₂₃	#2920	1	1	yes	characteristic, direct product factor, Sylow
108	C₂⁴	C₂×C₄×C₂₃	1	1	yes	characteristic
109	C₂²×C₄	—	2	8	no	—
110	C₂²×C₄	—	2	4	no	—
111	C₂²×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
112	C₂²×C₄	C₂×C₄×C₂₃	1	2	yes	—
113	C₂²×C₄	C₂×C₄×C₂₃	1	2	yes	—
114	C₂²×C₄	C₂×C₄×C₂₃	1	2	yes	—
115	C₂²×C₄	C₂×C₄×C₂₃	1	2	yes	—
116	C₂²×C₄	C₂×C₄×C₂₃	1	2	yes	—
117	C₂²×C₄	C₂×C₄×C₂₃	1	2	yes	—
118	C₂²×C₄	C₂×C₄×C₂₃	1	2	yes	—
119	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
120	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
121	M₁₆	—	2	8	no	—
122	C₂×C₈	C₂³×C₂₃	1	8	yes	—
123	C₄²	C₂×C₄×C₂₃	1	2	yes	—
124	C₄²	C₂×C₄×C₂₃	1	2	yes	—
125	C₄²	C₂×C₄×C₂₃	1	2	yes	—
126	C₄²	C₂×C₄×C₂₃	1	2	yes	—
127	C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
128	C₄²	C₂×C₄×C₂₃	1	2	yes	—
129	C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
130	C₄²	C₂³×C₂₃	1	1	yes	characteristic
131	C₄²	C₂³×C₂₃	1	1	yes	characteristic
132	C₄²	C₂³×C₂₃	1	2	yes	—
133	C₂³	—	2	2	no	—
134	C₂³	—	2	4	no	—
135	C₂³	C₂²×C₄×C₂₃	1	1	yes	characteristic
136	C₂³	C₂²×C₄×C₂₃	1	2	yes	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	M₁₆×C₂₃	1	8	yes	complemented
146	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
147	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
148	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
149	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
150	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
151	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
152	C₂×C₄	#41×C₂₃	1	4	yes	—
153	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
154	C₈	—	2	8	no	—
155	C₂²	—	2	8	no	—
156	C₂²	—	2	2	no	—
157	C₂²	—	2	2	no	—
158	C₂²	—	2	2	no	—
159	C₂²	C₂₃×#117	1	2	yes	—
160	C₂²	C₂×M₁₆×C₂₃	1	4	yes	—
161	C₂²	C₂³×C₄×C₂₃	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	2	2	no	—
164	C₄	—	2	2	no	—
165	C₄	C₂₃×#117	1	2	yes	—
166	C₄	C₂×M₁₆×C₂₃	1	4	yes	—
167	C₄	C₂×M₁₆×C₂₃	1	4	yes	—
168	C₄	C₂×#41×C₂₃	1	2	yes	—
169	C₂	—	2	2	no	—
170	C₂	—	2	4	no	—
171	C₂	C₂×C₂₃×#117	1	1	yes	characteristic
172	C₂	C₂²×M₁₆×C₂₃	1	2	yes	—

	subgroup	G/H
center	C₄²×C₂₃	C₂³
commutator	C₂²	C₂³×C₄×C₂₃
Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂³×C₄
prime radical	C₂⁴×C₂₃	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#432	C₂	semidirect
3. C₂₃×#431	C₂	semidirect
4. C₂₃×#423	C₂	semidirect
6. C₄×M₁₆×C₂₃	C₂	semidirect
17. C₂₃×C₄⋊C₈	C₂²	semidirect
19. C₂₃×#97	C₂²	semidirect
20. C₄×C₈×C₂₃	C₂²	semidirect
58. C₂×C₄×C₂₃	M₁₆	semidirect
68. #2920	C₂₃	direct (N × H)
84. #432	C₂×C₂₃	semidirect
85. #431	C₂×C₂₃	semidirect
86. #423	C₂×C₂₃	semidirect
88. C₄×M₁₆	C₂×C₂₃	semidirect
103. C₄⋊C₈	C₂²×C₂₃	semidirect
105. #97	C₂²×C₂₃	semidirect
106. C₄×C₈	C₂²×C₂₃	semidirect
107. C₂₃	#2920	direct (N × H)
145. C₂×C₄	M₁₆×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄²×C₂₃	2
C₂₃×#432	2
C₂₃×#431	2
C₂₃×#423	2
C₂₃×#407	2
C₄×M₁₆×C₂₃	2
#2920	23

Subgroups — C23×#2920 #15670539

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2920 #15670539