Subgroups — C₂₃×#3232 #15670851

Order 2944.

334 conjugacy classes of subgroups.

The subgroup lattice has 252 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	C₂×C₂₃×#125	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#499	C₂	1	2	yes	maximal, complemented
4	C₂₃×#488	C₂	1	2	yes	maximal, complemented
5	C₂₃×#484	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#483	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#481	C₂	1	1	yes	maximal, characteristic
8	C₂₃×#480	C₂	1	1	yes	maximal, characteristic
9	C₂₃×#479	C₂	1	2	yes	maximal, complemented
10	C₂₃×#478	C₂	1	2	yes	maximal, complemented
11	C₂₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#126	—	2	1	no	—
13	C₂₃×#125	—	2	1	no	—
14	C₂₃×#125	—	2	1	no	—
15	C₂₃×#125	C₂²	1	2	yes	—
16	C₂₃×#125	C₂²	1	2	yes	—
17	C₂₃×#125	C₂²	1	1	yes	characteristic
18	C₂₃×#124	C₂²	1	1	yes	characteristic
19	C₂₃×#123	—	2	1	no	—
20	C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
21	C₂₃×#121	C₂²	1	1	yes	characteristic
22	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₂₃	—	2	1	no	—
24	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
25	C₂₃×#117	—	2	1	no	—
26	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
27	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
28	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
30	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
31	C₂₃×#107	C₂²	1	2	yes	—
32	C₂₃×#107	C₂²	1	2	yes	—
33	C₂₃×#106	C₂²	1	2	yes	—
34	C₂₃×#106	C₂²	1	2	yes	—
35	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
36	C₂₃×#103	C₂²	1	2	yes	—
37	C₂₃×#103	C₂²	1	2	yes	—
38	C₂₃×#102	C₂²	1	2	yes	—
39	C₂₃×#102	C₂²	1	2	yes	—
40	C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
41	C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
42	C₄×C₈×C₂₃	C₂²	1	2	yes	—
43	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
44	C₂×D₄×C₂₃	—	2	1	no	—
45	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₂₃	—	2	1	no	—
47	C₂²×C₄×C₂₃	—	2	1	no	—
48	C₂²×C₄×C₂₃	—	2	1	no	—
49	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
50	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
51	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₂₃	C₂³	1	2	yes	—
53	C₂×C₈×C₂₃	C₂³	1	2	yes	—
54	C₄⋊C₄×C₂₃	—	2	2	no	—
55	C₄⋊C₄×C₂₃	—	2	2	no	—
56	C₄⋊C₄×C₂₃	—	2	2	no	—
57	C₄⋊C₄×C₂₃	—	2	2	no	—
58	C₄⋊C₄×C₂₃	—	2	1	no	—
59	C₄⋊C₄×C₂₃	—	2	1	no	—
60	C₄⋊C₄×C₂₃	—	2	1	no	—
61	C₄⋊C₄×C₂₃	—	2	1	no	—
62	C₄⋊C₄×C₂₃	—	2	1	no	—
63	C₄⋊C₄×C₂₃	—	2	1	no	—
64	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
65	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
66	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
67	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
68	C₂²⋊C₄×C₂₃	—	2	1	no	—
69	C₂²⋊C₄×C₂₃	—	2	1	no	—
70	C₂²⋊C₄×C₂₃	—	2	1	no	—
71	C₂²⋊C₄×C₂₃	—	2	1	no	—
72	C₂²⋊C₄×C₂₃	—	2	1	no	—
73	C₄²×C₂₃	—	2	1	no	—
74	C₄²×C₂₃	D₄	1	1	yes	characteristic
75	C₄²×C₂₃	D₄	1	1	yes	characteristic
76	C₄²×C₂₃	C₂³	1	1	yes	characteristic
77	C₄²×C₂₃	C₂³	1	1	yes	characteristic
78	C₂³×C₂₃	—	2	1	no	—
79	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
80	Q₈×C₂₃	—	2	2	no	—
81	D₄×C₂₃	—	2	2	no	—
82	D₄×C₂₃	—	4	2	no	—
83	C₂×C₄×C₂₃	—	2	2	no	—
84	C₂×C₄×C₂₃	—	2	2	no	—
85	C₂×C₄×C₂₃	—	2	1	no	—
86	C₂×C₄×C₂₃	—	2	1	no	—
87	C₂×C₄×C₂₃	—	2	1	no	—
88	C₂×C₄×C₂₃	—	2	2	no	—
89	C₂×C₄×C₂₃	—	2	2	no	—
90	C₂×C₄×C₂₃	—	4	1	no	—
91	C₂×C₄×C₂₃	—	4	2	no	—
92	C₂×C₄×C₂₃	—	4	2	no	—
93	C₂×C₄×C₂₃	—	4	2	no	—
94	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
100	C₈×C₂₃	—	2	2	no	—
101	C₈×C₂₃	—	2	2	no	—
102	#3232	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂²×C₂₃	—	2	1	no	—
104	C₂²×C₂₃	—	2	1	no	—
105	C₂²×C₂₃	—	4	1	no	—
106	C₂²×C₂₃	—	4	1	no	—
107	C₂²×C₂₃	—	4	1	no	—
108	C₂²×C₂₃	#135	1	2	yes	—
109	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
110	C₄×C₂₃	—	2	1	no	—
111	C₄×C₂₃	—	2	1	no	—
112	C₄×C₂₃	—	2	1	no	—
113	C₄×C₂₃	—	2	1	no	—
114	C₄×C₂₃	—	2	1	no	—
115	C₄×C₂₃	—	4	1	no	—
116	C₄×C₂₃	—	4	2	no	—
117	C₄×C₂₃	—	4	1	no	—
118	C₄×C₂₃	—	4	1	no	—
119	C₄×C₂₃	—	4	2	no	—
120	C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
121	#529	C₂×C₂₃	1	1	yes	characteristic
122	C₂×#125	C₂×C₂₃	1	1	yes	characteristic, complemented
123	#499	C₂×C₂₃	1	2	yes	complemented
124	#488	C₂×C₂₃	1	2	yes	complemented
125	#484	C₂×C₂₃	1	1	yes	characteristic, complemented
126	#483	C₂×C₂₃	1	1	yes	characteristic, complemented
127	#481	C₂×C₂₃	1	1	yes	characteristic
128	#480	C₂×C₂₃	1	1	yes	characteristic
129	#479	C₂×C₂₃	1	2	yes	complemented
130	#478	C₂×C₂₃	1	2	yes	complemented
131	#431	C₂×C₂₃	1	1	yes	characteristic, complemented
132	C₂×C₂₃	—	2	2	no	—
133	C₂×C₂₃	—	8	1	no	—
134	C₂×C₂₃	#536	1	1	yes	characteristic
135	C₂×C₂₃	C₂×#135	1	1	yes	characteristic
136	C₂×C₂₃	#575	1	1	yes	characteristic
137	#126	—	2	1	no	—
138	#125	—	2	1	no	—
139	#125	—	2	1	no	—
140	#125	C₂²×C₂₃	1	2	yes	—
141	#125	C₂²×C₂₃	1	2	yes	—
142	#125	C₂²×C₂₃	1	1	yes	characteristic
143	#124	C₂²×C₂₃	1	1	yes	characteristic
144	#123	—	2	1	no	—
145	C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
146	#121	C₂²×C₂₃	1	1	yes	characteristic
147	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
148	C₄×D₄	—	2	1	no	—
149	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
150	#117	—	2	1	no	—
151	C₂×C₄⋊C₄	—	2	1	no	—
152	C₂×C₄⋊C₄	—	2	1	no	—
153	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
154	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
155	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
156	#107	C₂²×C₂₃	1	2	yes	—
157	#107	C₂²×C₂₃	1	2	yes	—
158	#106	C₂²×C₂₃	1	2	yes	—
159	#106	C₂²×C₂₃	1	2	yes	—
160	C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
161	#103	C₂²×C₂₃	1	2	yes	—
162	#103	C₂²×C₂₃	1	2	yes	—
163	#102	C₂²×C₂₃	1	2	yes	—
164	#102	C₂²×C₂₃	1	2	yes	—
165	C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
166	C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
167	C₄×C₈	C₂²×C₂₃	1	2	yes	—
168	C₂₃	#3232	1	1	yes	characteristic, direct product factor, Sylow
169	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
170	C₂×D₄	—	2	1	no	—
171	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	2	1	no	—
175	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
176	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
177	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
178	C₂×C₈	C₂³×C₂₃	1	2	yes	—
179	C₂×C₈	C₂³×C₂₃	1	2	yes	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	2	1	no	—
185	C₄⋊C₄	—	2	1	no	—
186	C₄⋊C₄	—	2	1	no	—
187	C₄⋊C₄	—	2	1	no	—
188	C₄⋊C₄	—	2	1	no	—
189	C₄⋊C₄	—	2	1	no	—
190	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
191	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
192	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
193	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
194	C₂²⋊C₄	—	2	1	no	—
195	C₂²⋊C₄	—	2	1	no	—
196	C₂²⋊C₄	—	2	1	no	—
197	C₂²⋊C₄	—	2	1	no	—
198	C₂²⋊C₄	—	2	1	no	—
199	C₄²	—	2	1	no	—
200	C₄²	D₄×C₂₃	1	1	yes	characteristic
201	C₄²	D₄×C₂₃	1	1	yes	characteristic
202	C₄²	C₂³×C₂₃	1	1	yes	characteristic
203	C₄²	C₂³×C₂₃	1	1	yes	characteristic
204	C₂³	—	2	1	no	—
205	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
206	Q₈	—	2	2	no	—
207	D₄	—	2	2	no	—
208	D₄	—	4	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	4	1	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
221	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
222	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
223	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
224	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
225	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
226	C₈	—	2	2	no	—
227	C₈	—	2	2	no	—
228	C₂²	—	2	1	no	—
229	C₂²	—	2	1	no	—
230	C₂²	—	4	1	no	—
231	C₂²	—	4	1	no	—
232	C₂²	—	4	1	no	—
233	C₂²	C₂₃×#135	1	2	yes	—
234	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
235	C₄	—	2	1	no	—
236	C₄	—	2	1	no	—
237	C₄	—	2	1	no	—
238	C₄	—	2	1	no	—
239	C₄	—	2	1	no	—
240	C₄	—	4	2	no	—
241	C₄	—	4	1	no	—
242	C₄	—	4	1	no	—
243	C₄	—	4	2	no	—
244	C₄	—	4	1	no	—
245	C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
246	C₂	—	2	2	no	—
247	C₂	—	8	1	no	—
248	C₂	C₂₃×#536	1	1	yes	characteristic
249	C₂	C₂×C₂₃×#135	1	1	yes	characteristic
250	C₂	C₂₃×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
prime radical	C₂₃×#121	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#125	C₂	semidirect
3. C₂₃×#499	C₂	semidirect
4. C₂₃×#488	C₂	semidirect
5. C₂₃×#484	C₂	semidirect
6. C₂₃×#483	C₂	semidirect
9. C₂₃×#479	C₂	semidirect
10. C₂₃×#478	C₂	semidirect
11. C₂₃×#431	C₂	semidirect
102. #3232	C₂₃	direct (N × H)
122. C₂×#125	C₂×C₂₃	semidirect
123. #499	C₂×C₂₃	semidirect
124. #488	C₂×C₂₃	semidirect
125. #484	C₂×C₂₃	semidirect
126. #483	C₂×C₂₃	semidirect
129. #479	C₂×C₂₃	semidirect
130. #478	C₂×C₂₃	semidirect
131. #431	C₂×C₂₃	semidirect
168. C₂₃	#3232	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
C₂×C₂₃×#125	2
C₂₃×#499	2
C₂₃×#488	2
C₂₃×#484	2
C₂₃×#483	2
C₂₃×#481	2
C₂₃×#480	2
C₂₃×#479	2
C₂₃×#478	2
C₂₃×#431	2
#3232	23

Subgroups — C23×#3232 #15670851

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3232 #15670851