Subgroups — C₂₃×#2929 #15670548

Order 2944.

520 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#131	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#525	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#435	C₂	1	4	yes	maximal, complemented
5	C₈×D₄×C₂₃	C₂	1	4	yes	maximal, complemented
6	C₂₃×#424	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#408	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#407	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
12	C₂₃×#131	—	2	4	no	—
13	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×M₁₆×C₂₃	C₂²	1	2	yes	complemented
16	C₂²×C₈×C₂₃	—	2	4	no	—
17	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
19	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
20	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
21	C₂²×C₈×C₂₃	C₂²	1	2	yes	complemented
22	C₂₃×#123	C₄	1	2	yes	—
23	C₂₃×C₂²⋊Q₈	C₄	1	2	yes	—
24	C₂₃×#121	C₄	1	2	yes	—
25	C₂₃×#120	C₄	1	2	yes	—
26	C₄×D₄×C₂₃	C₂²	1	4	yes	—
27	C₂₃×#117	C₂²	1	1	yes	characteristic
28	C₂₃×C₄⋊C₈	C₂²	1	4	yes	complemented
29	C₂₃×C₂²⋊C₈	C₂²	1	4	yes	—
30	C₂₃×C₂²⋊C₈	C₂²	1	4	yes	complemented
31	C₂₃×#97	C₂²	1	2	yes	complemented
32	C₄×C₈×C₂₃	C₂²	1	2	yes	complemented
33	C₂⁴×C₂₃	C₂×C₄	1	1	yes	characteristic
34	#41×C₂₃	—	2	2	no	—
35	#41×C₂₃	—	2	2	no	—
36	C₂×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
37	C₂×D₄×C₂₃	C₂×C₄	1	4	yes	—
38	C₂×D₄×C₂₃	C₂×C₄	1	2	yes	—
39	C₂×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₂₃	—	2	2	no	—
41	C₂²×C₄×C₂₃	—	2	2	no	—
42	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
43	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
45	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
46	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
47	M₁₆×C₂₃	—	2	2	no	—
48	M₁₆×C₂₃	—	2	2	no	—
49	M₁₆×C₂₃	—	2	4	no	—
50	C₂×C₈×C₂₃	—	2	2	no	—
51	C₂×C₈×C₂₃	—	2	4	no	—
52	C₂×C₈×C₂₃	—	2	4	no	—
53	C₂×C₈×C₂₃	—	2	4	no	—
54	C₂×C₈×C₂₃	D₄	1	4	yes	complemented
55	C₂×C₈×C₂₃	C₂³	1	2	yes	—
56	C₂×C₈×C₂₃	C₂³	1	2	yes	—
57	C₂×C₈×C₂₃	C₂³	1	2	yes	—
58	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
59	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
60	C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
61	C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
62	C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
63	C₂²⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
64	C₂²⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
65	C₄²×C₂₃	C₂³	1	2	yes	—
66	C₂³×C₂₃	—	2	2	no	—
67	C₂³×C₂₃	—	2	4	no	—
68	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
69	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
70	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
71	Q₈×C₂₃	—	2	2	no	—
72	D₄×C₂₃	—	2	4	no	—
73	D₄×C₂₃	—	2	8	no	—
74	D₄×C₂₃	—	2	2	no	—
75	C₂×C₄×C₂₃	—	2	2	no	—
76	C₂×C₄×C₂₃	—	2	4	no	—
77	C₂×C₄×C₂₃	—	2	2	no	—
78	C₂×C₄×C₂₃	—	2	2	no	—
79	C₂×C₄×C₂₃	—	2	2	no	—
80	C₂×C₄×C₂₃	—	2	2	no	—
81	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
82	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
86	C₂×C₄×C₂₃	#41	1	2	yes	—
87	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
88	C₈×C₂₃	—	2	2	no	—
89	C₈×C₂₃	—	2	2	no	—
90	C₈×C₂₃	—	2	2	no	—
91	C₈×C₂₃	C₂×D₄	1	2	yes	—
92	C₈×C₂₃	C₂×D₄	1	2	yes	—
93	#2929	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂²×C₂₃	—	2	8	no	—
95	C₂²×C₂₃	—	2	2	no	—
96	C₂²×C₂₃	—	2	2	no	—
97	C₂²×C₂₃	—	2	2	no	—
98	C₂²×C₂₃	—	2	2	no	—
99	C₂²×C₂₃	—	2	4	no	—
100	C₂²×C₂₃	C₄×D₄	1	2	yes	—
101	C₂²×C₂₃	#131	1	4	yes	complemented
102	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
103	C₄×C₂₃	—	2	2	no	—
104	C₄×C₂₃	—	2	2	no	—
105	C₄×C₂₃	—	2	4	no	—
106	C₄×C₂₃	C₄×D₄	1	2	yes	—
107	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
108	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
109	C₂×#131	C₂×C₂₃	1	1	yes	characteristic, complemented
110	C₂³×C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
111	#525	C₂×C₂₃	1	1	yes	characteristic
112	#435	C₂×C₂₃	1	4	yes	complemented
113	C₈×D₄	C₂×C₂₃	1	4	yes	complemented
114	#424	C₂×C₂₃	1	1	yes	characteristic, complemented
115	#408	C₂×C₂₃	1	1	yes	characteristic, complemented
116	#407	C₂×C₂₃	1	1	yes	characteristic, complemented
117	#405	C₂×C₂₃	1	1	yes	characteristic, complemented
118	C₂×C₂₃	—	2	2	no	—
119	C₂×C₂₃	—	2	4	no	—
120	C₂×C₂₃	—	4	2	no	—
121	C₂×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
122	C₂×C₂₃	C₂×#131	1	2	yes	—
123	C₂×#41	C₂²×C₂₃	1	1	yes	characteristic
124	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
125	#131	—	2	4	no	—
126	C₂×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
127	C₂×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
128	C₂×M₁₆	C₂²×C₂₃	1	2	yes	complemented
129	C₂²×C₈	—	2	4	no	—
130	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic
131	C₂²×C₈	C₂²×C₂₃	1	2	yes	—
132	C₂²×C₈	C₂²×C₂₃	1	2	yes	—
133	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic
134	C₂²×C₈	C₂²×C₂₃	1	2	yes	complemented
135	#123	C₄×C₂₃	1	2	yes	—
136	C₂²⋊Q₈	C₄×C₂₃	1	2	yes	—
137	#121	C₄×C₂₃	1	2	yes	—
138	#120	C₄×C₂₃	1	2	yes	—
139	C₄×D₄	C₂²×C₂₃	1	4	yes	—
140	#117	C₂²×C₂₃	1	1	yes	characteristic
141	C₄⋊C₈	C₂²×C₂₃	1	4	yes	complemented
142	C₂²⋊C₈	C₂²×C₂₃	1	4	yes	—
143	C₂²⋊C₈	C₂²×C₂₃	1	4	yes	complemented
144	#97	C₂²×C₂₃	1	2	yes	complemented
145	C₄×C₈	C₂²×C₂₃	1	2	yes	complemented
146	C₂₃	#2929	1	1	yes	characteristic, direct product factor, Sylow
147	C₂⁴	C₂×C₄×C₂₃	1	1	yes	characteristic
148	#41	—	2	2	no	—
149	#41	—	2	2	no	—
150	C₂×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
151	C₂×D₄	C₂×C₄×C₂₃	1	4	yes	—
152	C₂×D₄	C₂×C₄×C₂₃	1	2	yes	—
153	C₂×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	C₂×C₄×C₂₃	1	2	yes	—
157	C₂²×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
158	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
159	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
160	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
161	M₁₆	—	2	2	no	—
162	M₁₆	—	2	2	no	—
163	M₁₆	—	2	4	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	4	no	—
166	C₂×C₈	—	2	4	no	—
167	C₂×C₈	—	2	4	no	—
168	C₂×C₈	D₄×C₂₃	1	4	yes	complemented
169	C₂×C₈	C₂³×C₂₃	1	2	yes	—
170	C₂×C₈	C₂³×C₂₃	1	2	yes	—
171	C₂×C₈	C₂³×C₂₃	1	2	yes	—
172	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
173	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
174	C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
175	C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
176	C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
177	C₂²⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
178	C₂²⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
179	C₄²	C₂³×C₂₃	1	2	yes	—
180	C₂³	—	2	2	no	—
181	C₂³	—	2	4	no	—
182	C₂³	C₂²×C₄×C₂₃	1	2	yes	—
183	C₂³	C₂²×C₄×C₂₃	1	1	yes	characteristic
184	C₂³	C₂²×C₄×C₂₃	1	2	yes	—
185	Q₈	—	2	2	no	—
186	D₄	—	2	4	no	—
187	D₄	—	2	8	no	—
188	D₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	C₂²×C₄×C₂₃	1	4	yes	—
196	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
197	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
198	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
199	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
200	C₂×C₄	#41×C₂₃	1	2	yes	—
201	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
202	C₈	—	2	2	no	—
203	C₈	—	2	2	no	—
204	C₈	—	2	2	no	—
205	C₈	C₂×D₄×C₂₃	1	2	yes	—
206	C₈	C₂×D₄×C₂₃	1	2	yes	—
207	C₂²	—	2	8	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	4	no	—
212	C₂²	—	2	2	no	—
213	C₂²	C₄×D₄×C₂₃	1	2	yes	—
214	C₂²	C₂₃×#131	1	4	yes	complemented
215	C₂²	C₂³×C₄×C₂₃	1	1	yes	characteristic
216	C₄	—	2	4	no	—
217	C₄	—	2	2	no	—
218	C₄	—	2	2	no	—
219	C₄	C₄×D₄×C₂₃	1	2	yes	—
220	C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
221	C₄	C₂×#41×C₂₃	1	1	yes	characteristic
222	C₂	—	2	2	no	—
223	C₂	—	2	4	no	—
224	C₂	—	4	2	no	—
225	C₂	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
226	C₂	C₂×C₂₃×#131	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₂₃	C₂³
commutator	C₂²	C₂³×C₄×C₂₃
Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂³×C₄
prime radical	C₂₃×#525	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#131	C₂	semidirect
2. C₂³×C₈×C₂₃	C₂	semidirect
4. C₂₃×#435	C₂	semidirect
5. C₈×D₄×C₂₃	C₂	semidirect
6. C₂₃×#424	C₂	semidirect
7. C₂₃×#408	C₂	semidirect
8. C₂₃×#407	C₂	semidirect
9. C₂₃×#405	C₂	semidirect
15. C₂×M₁₆×C₂₃	C₂²	semidirect
21. C₂²×C₈×C₂₃	C₂²	semidirect
28. C₂₃×C₄⋊C₈	C₂²	semidirect
30. C₂₃×C₂²⋊C₈	C₂²	semidirect
31. C₂₃×#97	C₂²	semidirect
32. C₄×C₈×C₂₃	C₂²	semidirect
54. C₂×C₈×C₂₃	D₄	semidirect
93. #2929	C₂₃	direct (N × H)
101. C₂²×C₂₃	#131	semidirect
109. C₂×#131	C₂×C₂₃	semidirect
110. C₂³×C₈	C₂×C₂₃	semidirect
112. #435	C₂×C₂₃	semidirect
113. C₈×D₄	C₂×C₂₃	semidirect
114. #424	C₂×C₂₃	semidirect
115. #408	C₂×C₂₃	semidirect
116. #407	C₂×C₂₃	semidirect
117. #405	C₂×C₂₃	semidirect
128. C₂×M₁₆	C₂²×C₂₃	semidirect
134. C₂²×C₈	C₂²×C₂₃	semidirect
141. C₄⋊C₈	C₂²×C₂₃	semidirect
143. C₂²⋊C₈	C₂²×C₂₃	semidirect
144. #97	C₂²×C₂₃	semidirect
145. C₄×C₈	C₂²×C₂₃	semidirect
146. C₂₃	#2929	direct (N × H)
168. C₂×C₈	D₄×C₂₃	semidirect
214. C₂²	C₂₃×#131	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#131	2
C₂³×C₈×C₂₃	2
C₂₃×#525	2
C₂₃×#435	2
C₈×D₄×C₂₃	2
C₂₃×#424	2
C₂₃×#408	2
C₂₃×#407	2
C₂₃×#405	2
#2929	23

Subgroups — C23×#2929 #15670548

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2929 #15670548