Subgroups — C₂₃×#33826 #19181452

Order 5888.

488 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3177	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3176	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2229	C₂	1	2	yes	maximal, complemented
4	C₂₃×#2228	C₂	1	2	yes	maximal, complemented
5	C₂₃×#2220	C₂	1	2	yes	maximal, complemented
6	C₂₃×#2217	C₂	1	2	yes	maximal, complemented
7	C₂₃×#2208	C₂	1	2	yes	maximal, complemented
8	C₂₃×#2207	C₂	1	2	yes	maximal, complemented
9	C₂₃×#2151	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#541	—	2	1	no	—
11	C₂₃×#540	—	2	1	no	—
12	C₂×C₂₃×SD₃₂	C₂²	1	2	yes	—
13	C₂×C₂₃×SD₃₂	C₂²	1	2	yes	—
14	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
16	C₂₃×#499	C₂²	1	1	yes	characteristic
17	C₂₃×#499	C₂²	1	1	yes	characteristic
18	C₂₃×#494	C₂²	1	1	yes	characteristic
19	C₂₃×#493	C₂²	1	1	yes	characteristic
20	C₂₃×#491	—	2	1	no	—
21	C₂₃×#490	—	2	1	no	—
22	C₂₃×#470	—	2	1	no	—
23	C₂₃×#469	—	2	1	no	—
24	C₂₃×#467	C₂²	1	2	yes	—
25	C₂₃×#466	C₂²	1	2	yes	—
26	C₂₃×#442	—	2	1	no	—
27	C₂₃×#441	—	2	1	no	—
28	C₄×Q₁₆×C₂₃	C₂²	1	2	yes	—
29	C₄×D₈×C₂₃	C₂²	1	2	yes	—
30	C₂₃×#405	C₂²	1	1	yes	characteristic
31	C₂₃×#367	C₂²	1	2	yes	—
32	C₂₃×#366	C₂²	1	2	yes	—
33	C₂₃×C₄⋊C₁₆	C₂²	1	2	yes	—
34	C₂₃×C₄⋊C₁₆	C₂²	1	2	yes	—
35	C₂₃×#358	C₂²	1	2	yes	—
36	C₂₃×#358	C₂²	1	2	yes	—
37	C₂₃×#357	C₂²	1	2	yes	—
38	C₂₃×#357	C₂²	1	2	yes	—
39	C₂×Q₁₆×C₂₃	—	2	1	no	—
40	C₂×Q₁₆×C₂₃	C₂³	1	2	yes	—
41	C₂×D₈×C₂₃	—	2	1	no	—
42	C₂×D₈×C₂₃	C₂³	1	2	yes	—
43	C₂×M₁₆×C₂₃	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
45	C₂₃×C₄⋊Q₈	—	2	1	no	—
46	C₂₃×#127	—	2	1	no	—
47	C₂₃×#126	—	4	2	no	—
48	C₂₃×#125	—	2	1	no	—
49	C₂₃×#125	—	2	1	no	—
50	C₂₃×#125	—	4	2	no	—
51	C₂₃×#123	—	4	2	no	—
52	C₂₃×C₂²⋊Q₈	—	2	2	no	—
53	C₂₃×#121	—	2	2	no	—
54	C₂₃×#121	—	4	2	no	—
55	C₄×Q₈×C₂₃	—	2	2	no	—
56	C₄×D₄×C₂₃	—	2	2	no	—
57	C₂₃×#117	D₄	1	1	yes	characteristic
58	C₂₃×SD₃₂	—	4	2	no	—
59	C₂₃×SD₃₂	—	4	2	no	—
60	C₂₃×M₃₂	—	4	1	no	—
61	C₂×C₁₆×C₂₃	—	4	1	no	—
62	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
63	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
64	C₂₃×#107	C₂³	1	2	yes	—
65	C₂₃×#107	C₂³	1	2	yes	—
66	C₂₃×#106	—	2	1	no	—
67	C₂₃×#106	—	2	1	no	—
68	C₂₃×#103	—	2	2	no	—
69	C₂₃×#103	—	2	2	no	—
70	C₂₃×#102	—	2	2	no	—
71	C₂₃×#102	—	2	2	no	—
72	C₂₃×#97	D₄	1	1	yes	characteristic
73	C₂₃×#97	D₄	1	1	yes	characteristic
74	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
76	C₂×Q₈×C₂₃	—	2	2	no	—
77	C₂×D₄×C₂₃	—	2	2	no	—
78	C₂×D₄×C₂₃	—	4	2	no	—
79	C₂×D₄×C₂₃	—	4	2	no	—
80	C₂²×C₄×C₂₃	—	4	2	no	—
81	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
82	Q₁₆×C₂₃	—	2	2	no	—
83	Q₁₆×C₂₃	—	2	2	no	—
84	Q₁₆×C₂₃	—	4	2	no	—
85	D₈×C₂₃	—	2	2	no	—
86	D₈×C₂₃	—	2	2	no	—
87	D₈×C₂₃	—	4	2	no	—
88	M₁₆×C₂₃	—	4	1	no	—
89	C₂×C₈×C₂₃	—	2	1	no	—
90	C₂×C₈×C₂₃	—	2	1	no	—
91	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
95	C₄⋊C₄×C₂₃	—	2	2	no	—
96	C₄⋊C₄×C₂₃	—	2	2	no	—
97	C₄⋊C₄×C₂₃	—	4	2	no	—
98	C₄⋊C₄×C₂₃	—	4	2	no	—
99	C₄⋊C₄×C₂₃	—	4	2	no	—
100	C₄⋊C₄×C₂₃	—	4	2	no	—
101	C₄⋊C₄×C₂₃	—	4	2	no	—
102	C₄⋊C₄×C₂₃	D₈	1	2	yes	—
103	C₂²⋊C₄×C₂₃	—	4	2	no	—
104	C₂²⋊C₄×C₂₃	—	4	2	no	—
105	C₂²⋊C₄×C₂₃	—	4	2	no	—
106	C₂²⋊C₄×C₂₃	D₈	1	2	yes	—
107	C₄²×C₂₃	—	4	2	no	—
108	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
110	C₁₆×C₂₃	—	2	2	no	—
111	C₁₆×C₂₃	—	2	2	no	—
112	#33826	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂³×C₂₃	—	4	2	no	—
114	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
115	Q₈×C₂₃	—	4	2	no	—
116	Q₈×C₂₃	—	4	2	no	—
117	D₄×C₂₃	—	4	2	no	—
118	D₄×C₂₃	—	4	2	no	—
119	D₄×C₂₃	—	16	2	no	—
120	D₄×C₂₃	—	16	2	no	—
121	C₂×C₄×C₂₃	—	2	1	no	—
122	C₂×C₄×C₂₃	—	2	1	no	—
123	C₂×C₄×C₂₃	—	4	2	no	—
124	C₂×C₄×C₂₃	—	4	2	no	—
125	C₂×C₄×C₂₃	—	4	2	no	—
126	C₂×C₄×C₂₃	—	16	2	no	—
127	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
128	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
129	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
131	C₈×C₂₃	—	2	1	no	—
132	C₈×C₂₃	—	2	1	no	—
133	C₈×C₂₃	—	2	1	no	—
134	C₈×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
135	C₈×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
136	#3177	C₂×C₂₃	1	1	yes	characteristic, complemented
137	#3176	C₂×C₂₃	1	1	yes	characteristic
138	#2229	C₂×C₂₃	1	2	yes	complemented
139	#2228	C₂×C₂₃	1	2	yes	complemented
140	#2220	C₂×C₂₃	1	2	yes	complemented
141	#2217	C₂×C₂₃	1	2	yes	complemented
142	#2208	C₂×C₂₃	1	2	yes	complemented
143	#2207	C₂×C₂₃	1	2	yes	complemented
144	#2151	C₂×C₂₃	1	1	yes	characteristic, complemented
145	C₂²×C₂₃	—	2	2	no	—
146	C₂²×C₂₃	—	2	1	no	—
147	C₂²×C₂₃	—	8	2	no	—
148	C₂²×C₂₃	—	8	2	no	—
149	C₂²×C₂₃	—	8	2	no	—
150	C₂²×C₂₃	C₂²×D₈	1	1	yes	characteristic
151	C₄×C₂₃	—	2	2	no	—
152	C₄×C₂₃	—	2	2	no	—
153	C₄×C₂₃	—	2	1	no	—
154	C₄×C₂₃	—	8	2	no	—
155	C₄×C₂₃	—	8	2	no	—
156	C₄×C₂₃	—	8	2	no	—
157	C₄×C₂₃	#537	1	1	yes	characteristic
158	C₄×C₂₃	#575	1	1	yes	characteristic
159	#541	—	2	1	no	—
160	#540	—	2	1	no	—
161	C₂×SD₃₂	C₂²×C₂₃	1	2	yes	—
162	C₂×SD₃₂	C₂²×C₂₃	1	2	yes	—
163	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
164	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
165	#499	C₂²×C₂₃	1	1	yes	characteristic
166	#499	C₂²×C₂₃	1	1	yes	characteristic
167	#494	C₂²×C₂₃	1	1	yes	characteristic
168	#493	C₂²×C₂₃	1	1	yes	characteristic
169	#491	—	2	1	no	—
170	#490	—	2	1	no	—
171	#470	—	2	1	no	—
172	#469	—	2	1	no	—
173	#467	C₂²×C₂₃	1	2	yes	—
174	#466	C₂²×C₂₃	1	2	yes	—
175	#442	—	2	1	no	—
176	#441	—	2	1	no	—
177	C₄×Q₁₆	C₂²×C₂₃	1	2	yes	—
178	C₄×D₈	C₂²×C₂₃	1	2	yes	—
179	#405	C₂²×C₂₃	1	1	yes	characteristic
180	#367	C₂²×C₂₃	1	2	yes	—
181	#366	C₂²×C₂₃	1	2	yes	—
182	C₄⋊C₁₆	C₂²×C₂₃	1	2	yes	—
183	C₄⋊C₁₆	C₂²×C₂₃	1	2	yes	—
184	#358	C₂²×C₂₃	1	2	yes	—
185	#358	C₂²×C₂₃	1	2	yes	—
186	#357	C₂²×C₂₃	1	2	yes	—
187	#357	C₂²×C₂₃	1	2	yes	—
188	C₂×C₂₃	—	4	1	no	—
189	C₂×C₂₃	—	16	2	no	—
190	C₂×C₂₃	#3245	1	1	yes	characteristic
191	C₂×C₂₃	#3416	1	2	yes	—
192	C₂×Q₁₆	—	2	1	no	—
193	C₂×Q₁₆	C₂³×C₂₃	1	2	yes	—
194	C₂×D₈	—	2	1	no	—
195	C₂×D₈	C₂³×C₂₃	1	2	yes	—
196	C₂×M₁₆	D₄×C₂₃	1	1	yes	characteristic
197	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
198	C₄⋊Q₈	—	2	1	no	—
199	#127	—	2	1	no	—
200	#126	—	4	2	no	—
201	#125	—	2	1	no	—
202	#125	—	2	1	no	—
203	#125	—	4	2	no	—
204	#123	—	4	2	no	—
205	C₂²⋊Q₈	—	2	2	no	—
206	#121	—	2	2	no	—
207	#121	—	4	2	no	—
208	C₄×Q₈	—	2	2	no	—
209	C₄×D₄	—	2	2	no	—
210	#117	D₄×C₂₃	1	1	yes	characteristic
211	SD₃₂	—	4	2	no	—
212	SD₃₂	—	4	2	no	—
213	M₃₂	—	4	1	no	—
214	C₂×C₁₆	—	4	1	no	—
215	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
216	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
217	#107	C₂³×C₂₃	1	2	yes	—
218	#107	C₂³×C₂₃	1	2	yes	—
219	#106	—	2	1	no	—
220	#106	—	2	1	no	—
221	#103	—	2	2	no	—
222	#103	—	2	2	no	—
223	#102	—	2	2	no	—
224	#102	—	2	2	no	—
225	#97	D₄×C₂₃	1	1	yes	characteristic
226	#97	D₄×C₂₃	1	1	yes	characteristic
227	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
228	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
229	C₂₃	#33826	1	1	yes	characteristic, direct product factor, Sylow
230	C₂×Q₈	—	2	2	no	—
231	C₂×D₄	—	2	2	no	—
232	C₂×D₄	—	4	2	no	—
233	C₂×D₄	—	4	2	no	—
234	C₂²×C₄	—	4	2	no	—
235	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
236	Q₁₆	—	2	2	no	—
237	Q₁₆	—	2	2	no	—
238	Q₁₆	—	4	2	no	—
239	D₈	—	2	2	no	—
240	D₈	—	2	2	no	—
241	D₈	—	4	2	no	—
242	M₁₆	—	4	1	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
246	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
247	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
248	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	D₈×C₂₃	1	2	yes	—
257	C₂²⋊C₄	—	4	2	no	—
258	C₂²⋊C₄	—	4	2	no	—
259	C₂²⋊C₄	—	4	2	no	—
260	C₂²⋊C₄	D₈×C₂₃	1	2	yes	—
261	C₄²	—	4	2	no	—
262	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
263	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
264	C₁₆	—	2	2	no	—
265	C₁₆	—	2	2	no	—
266	C₂³	—	4	2	no	—
267	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
268	Q₈	—	4	2	no	—
269	Q₈	—	4	2	no	—
270	D₄	—	4	2	no	—
271	D₄	—	4	2	no	—
272	D₄	—	16	2	no	—
273	D₄	—	16	2	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	16	2	no	—
280	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
281	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
282	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
283	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	—	2	1	no	—
287	C₈	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
288	C₈	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
289	C₂²	—	2	1	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	8	2	no	—
292	C₂²	—	8	2	no	—
293	C₂²	—	8	2	no	—
294	C₂²	C₂²×D₈×C₂₃	1	1	yes	characteristic
295	C₄	—	2	1	no	—
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	8	2	no	—
299	C₄	—	8	2	no	—
300	C₄	—	8	2	no	—
301	C₄	C₂₃×#537	1	1	yes	characteristic
302	C₄	C₂₃×#575	1	1	yes	characteristic
303	C₂	—	4	1	no	—
304	C₂	—	16	2	no	—
305	C₂	C₂₃×#3245	1	1	yes	characteristic
306	C₂	C₂₃×#3416	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
prime radical	C₂₃×#3176	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂₃×#117 ◁ C₂₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3177	C₂	semidirect
3. C₂₃×#2229	C₂	semidirect
4. C₂₃×#2228	C₂	semidirect
5. C₂₃×#2220	C₂	semidirect
6. C₂₃×#2217	C₂	semidirect
7. C₂₃×#2208	C₂	semidirect
8. C₂₃×#2207	C₂	semidirect
9. C₂₃×#2151	C₂	semidirect
112. #33826	C₂₃	direct (N × H)
136. #3177	C₂×C₂₃	semidirect
138. #2229	C₂×C₂₃	semidirect
139. #2228	C₂×C₂₃	semidirect
140. #2220	C₂×C₂₃	semidirect
141. #2217	C₂×C₂₃	semidirect
142. #2208	C₂×C₂₃	semidirect
143. #2207	C₂×C₂₃	semidirect
144. #2151	C₂×C₂₃	semidirect
229. C₂₃	#33826	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×#3177	2
C₂₃×#3176	2
C₂₃×#2229	2
C₂₃×#2228	2
C₂₃×#2220	2
C₂₃×#2217	2
C₂₃×#2208	2
C₂₃×#2207	2
C₂₃×#2151	2
#33826	23

Subgroups — C23×#33826 #19181452

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33826 #19181452