Subgroups — C₂₃×#2969 #15670588

Order 2944.

370 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#529	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#436	C₂	1	2	yes	maximal, complemented
6	C₂₃×#435	C₂	1	2	yes	maximal, complemented
7	C₈×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#433	C₂	1	2	yes	maximal, complemented
9	C₂₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#405	C₂	1	2	yes	maximal, complemented
11	C₄×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
15	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
16	C₂₃×#126	C₄	1	2	yes	—
17	C₂₃×#125	C₄	1	1	yes	characteristic
18	C₂₃×#124	C₄	1	1	yes	characteristic
19	C₂₃×#123	C₄	1	2	yes	—
20	C₂₃×C₂²⋊Q₈	C₄	1	1	yes	characteristic
21	C₂₃×#121	C₄	1	1	yes	characteristic
22	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
24	C₄×D₄×C₂₃	C₂²	1	2	yes	—
25	C₂₃×#117	C₂²	1	2	yes	—
26	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
27	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
29	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
30	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
31	C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
32	C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
33	C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
34	C₂₃×#97	C₂²	1	1	yes	characteristic
35	C₂₃×#97	C₂²	1	1	yes	characteristic
36	C₂₃×#97	C₂²	1	2	yes	—
37	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
38	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
39	C₄×C₈×C₂₃	C₂²	1	2	yes	—
40	C₄×C₈×C₂₃	C₂²	1	2	yes	—
41	C₂×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
42	C₂×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
43	C₂×D₄×C₂₃	C₂×C₄	1	2	yes	—
44	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
48	M₁₆×C₂₃	—	2	2	no	—
49	M₁₆×C₂₃	—	2	2	no	—
50	M₁₆×C₂₃	—	2	4	no	—
51	C₂×C₈×C₂₃	—	2	4	no	—
52	C₂×C₈×C₂₃	C₂³	1	2	yes	—
53	C₂×C₈×C₂₃	C₂³	1	2	yes	—
54	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
55	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
56	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
57	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
58	C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
59	C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
60	C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
61	C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
62	C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
63	C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
64	C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
65	C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
66	C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
67	C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
68	C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
69	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
70	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
71	C₄²×C₂₃	C₂³	1	1	yes	characteristic
72	C₄²×C₂₃	C₂³	1	2	yes	—
73	C₄²×C₂₃	C₂³	1	1	yes	characteristic
74	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
75	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
76	Q₈×C₂₃	—	2	2	no	—
77	D₄×C₂₃	—	2	2	no	—
78	D₄×C₂₃	—	4	2	no	—
79	C₂×C₄×C₂₃	—	2	2	no	—
80	C₂×C₄×C₂₃	—	2	2	no	—
81	C₂×C₄×C₂₃	—	2	1	no	—
82	C₂×C₄×C₂₃	—	2	1	no	—
83	C₂×C₄×C₂₃	—	2	2	no	—
84	C₂×C₄×C₂₃	—	4	1	no	—
85	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
94	C₈×C₂₃	—	2	2	no	—
95	C₈×C₂₃	—	2	1	no	—
96	C₈×C₂₃	—	2	2	no	—
97	C₈×C₂₃	—	2	1	no	—
98	C₈×C₂₃	#41	1	2	yes	—
99	C₈×C₂₃	#41	1	2	yes	—
100	#2969	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂²×C₂₃	—	2	2	no	—
102	C₂²×C₂₃	—	2	1	no	—
103	C₂²×C₂₃	—	2	1	no	—
104	C₂²×C₂₃	—	2	2	no	—
105	C₂²×C₂₃	—	2	1	no	—
106	C₂²×C₂₃	—	2	2	no	—
107	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
108	C₄×C₂₃	—	2	2	no	—
109	C₄×C₂₃	—	2	1	no	—
110	C₄×C₂₃	—	2	1	no	—
111	C₄×C₂₃	—	2	2	no	—
112	C₄×C₂₃	—	2	1	no	—
113	C₄×C₂₃	—	2	1	no	—
114	C₄×C₂₃	—	2	2	no	—
115	C₄×C₂₃	#131	1	2	yes	—
116	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
117	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
118	#529	C₂×C₂₃	1	1	yes	characteristic
119	#446	C₂×C₂₃	1	1	yes	characteristic, complemented
120	C₈×Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
121	#436	C₂×C₂₃	1	1	yes	characteristic, complemented
122	#436	C₂×C₂₃	1	2	yes	complemented
123	#435	C₂×C₂₃	1	2	yes	complemented
124	C₈×D₄	C₂×C₂₃	1	1	yes	characteristic, complemented
125	#433	C₂×C₂₃	1	2	yes	complemented
126	#431	C₂×C₂₃	1	1	yes	characteristic, complemented
127	#405	C₂×C₂₃	1	2	yes	complemented
128	C₄×M₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
129	C₂×C₂₃	—	4	1	no	—
130	C₂×C₂₃	—	4	2	no	—
131	C₂×C₂₃	C₄×#41	1	1	yes	characteristic
132	C₂×C₂₃	C₂×#131	1	1	yes	characteristic
133	C₂×C₂₃	#568	1	1	yes	characteristic
134	C₂×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
135	C₂×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
136	C₂×M₁₆	C₂²×C₂₃	1	2	yes	—
137	C₂²×C₈	C₂²×C₂₃	1	2	yes	—
138	#126	C₄×C₂₃	1	2	yes	—
139	#125	C₄×C₂₃	1	1	yes	characteristic
140	#124	C₄×C₂₃	1	1	yes	characteristic
141	#123	C₄×C₂₃	1	2	yes	—
142	C₂²⋊Q₈	C₄×C₂₃	1	1	yes	characteristic
143	#121	C₄×C₂₃	1	1	yes	characteristic
144	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
145	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
146	C₄×D₄	C₂²×C₂₃	1	2	yes	—
147	#117	C₂²×C₂₃	1	2	yes	—
148	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
149	C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
150	C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
151	C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
152	C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
153	C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
154	C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
155	C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
156	#97	C₂²×C₂₃	1	1	yes	characteristic
157	#97	C₂²×C₂₃	1	1	yes	characteristic
158	#97	C₂²×C₂₃	1	2	yes	—
159	C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
160	C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
161	C₄×C₈	C₂²×C₂₃	1	2	yes	—
162	C₄×C₈	C₂²×C₂₃	1	2	yes	—
163	C₂₃	#2969	1	1	yes	characteristic, direct product factor, Sylow
164	C₂×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
165	C₂×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
166	C₂×D₄	C₂×C₄×C₂₃	1	2	yes	—
167	C₂²×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
168	C₂²×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
169	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
170	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
171	M₁₆	—	2	2	no	—
172	M₁₆	—	2	2	no	—
173	M₁₆	—	2	4	no	—
174	C₂×C₈	—	2	4	no	—
175	C₂×C₈	C₂³×C₂₃	1	2	yes	—
176	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
177	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
178	C₂×C₈	C₂³×C₂₃	1	2	yes	—
179	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
180	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
181	C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
182	C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
183	C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
184	C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
185	C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
186	C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
187	C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
188	C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
189	C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
190	C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
191	C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
192	C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
193	C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
194	C₄²	C₂³×C₂₃	1	1	yes	characteristic
195	C₄²	C₂³×C₂₃	1	1	yes	characteristic
196	C₄²	C₂³×C₂₃	1	2	yes	—
197	C₂³	C₂²×C₄×C₂₃	1	2	yes	—
198	C₂³	C₂²×C₄×C₂₃	1	1	yes	characteristic
199	Q₈	—	2	2	no	—
200	D₄	—	2	2	no	—
201	D₄	—	4	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	4	1	no	—
208	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
209	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
210	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
211	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
212	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
213	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
214	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
215	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
216	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
217	C₈	—	2	2	no	—
218	C₈	—	2	1	no	—
219	C₈	—	2	1	no	—
220	C₈	—	2	2	no	—
221	C₈	#41×C₂₃	1	2	yes	—
222	C₈	#41×C₂₃	1	2	yes	—
223	C₂²	—	2	1	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	2	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	2	1	no	—
229	C₂²	C₂³×C₄×C₂₃	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	2	2	no	—
235	C₄	—	2	2	no	—
236	C₄	—	2	1	no	—
237	C₄	C₂₃×#131	1	2	yes	—
238	C₄	C₂×#41×C₂₃	1	1	yes	characteristic
239	C₄	C₂×#41×C₂₃	1	1	yes	characteristic
240	C₂	—	4	1	no	—
241	C₂	—	4	2	no	—
242	C₂	C₄×#41×C₂₃	1	1	yes	characteristic
243	C₂	C₂×C₂₃×#131	1	1	yes	characteristic
244	C₂	C₂₃×#568	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂⁴
commutator	C₂²	C₂³×C₄×C₂₃
Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂³×C₄
prime radical	C₂₃×#121	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#446	C₂	semidirect
3. C₈×Q₈×C₂₃	C₂	semidirect
4. C₂₃×#436	C₂	semidirect
5. C₂₃×#436	C₂	semidirect
6. C₂₃×#435	C₂	semidirect
7. C₈×D₄×C₂₃	C₂	semidirect
8. C₂₃×#433	C₂	semidirect
9. C₂₃×#431	C₂	semidirect
10. C₂₃×#405	C₂	semidirect
11. C₄×M₁₆×C₂₃	C₂	semidirect
100. #2969	C₂₃	direct (N × H)
119. #446	C₂×C₂₃	semidirect
120. C₈×Q₈	C₂×C₂₃	semidirect
121. #436	C₂×C₂₃	semidirect
122. #436	C₂×C₂₃	semidirect
123. #435	C₂×C₂₃	semidirect
124. C₈×D₄	C₂×C₂₃	semidirect
125. #433	C₂×C₂₃	semidirect
126. #431	C₂×C₂₃	semidirect
127. #405	C₂×C₂₃	semidirect
128. C₄×M₁₆	C₂×C₂₃	semidirect
163. C₂₃	#2969	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#529	2
C₂₃×#446	2
C₈×Q₈×C₂₃	2
C₂₃×#436	2
C₂₃×#435	2
C₈×D₄×C₂₃	2
C₂₃×#433	2
C₂₃×#431	2
C₂₃×#405	2
C₄×M₁₆×C₂₃	2
#2969	23

Subgroups — C23×#2969 #15670588

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2969 #15670588