Subgroups — C₂₃×#33902 #19181528

Order 5888.

516 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3338	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3334	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2254	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2235	C₂	1	2	yes	maximal, complemented
5	C₂₃×#2233	C₂	1	2	yes	maximal, complemented
6	C₂₃×#2226	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#2225	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#2217	C₂	1	2	yes	maximal, complemented
9	C₄×C₂₃×SD₃₂	C₂	1	1	yes	maximal, characteristic, complemented
10	D₄×C₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₂₃×#367	C₂	1	2	yes	maximal, complemented
12	C₂₃×#554	—	2	1	no	—
13	C₂₃×#547	—	2	1	no	—
14	C₂×C₂₃×SD₃₂	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₂₃	C₂²	1	2	yes	—
16	C₂₃×#500	C₂²	1	1	yes	characteristic
17	C₂₃×#500	C₂²	1	1	yes	characteristic
18	C₂₃×#484	—	2	2	no	—
19	C₂₃×#480	—	2	2	no	—
20	C₂₃×#477	—	2	1	no	—
21	C₂₃×#474	—	2	1	no	—
22	C₂₃×#467	C₂²	1	2	yes	—
23	C₂₃×#466	C₂²	1	2	yes	—
24	C₄×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
25	C₄×D₈×C₂₃	C₂²	1	1	yes	characteristic
26	C₈×D₄×C₂₃	C₂²	1	1	yes	characteristic
27	C₂×C₂₃×#107	C₂²	1	2	yes	—
28	C₂×C₂₃×#107	C₂²	1	2	yes	—
29	C₂₃×#367	C₂²	1	1	yes	characteristic
30	C₂₃×#367	C₂²	1	2	yes	—
31	C₂₃×#367	C₂²	1	2	yes	—
32	C₂₃×#367	C₂²	1	2	yes	—
33	C₂₃×#367	C₂²	1	2	yes	—
34	C₂₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
35	C₂₃×#358	C₂²	1	1	yes	characteristic
36	C₂₃×#358	C₂²	1	2	yes	—
37	C₂₃×#357	C₂²	1	1	yes	characteristic
38	C₂₃×#357	C₂²	1	2	yes	—
39	C₂₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
40	C₄×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
41	C₂×#41×C₂₃	—	4	1	no	—
42	C₂×Q₁₆×C₂₃	C₂³	1	1	yes	characteristic
43	C₂×D₈×C₂₃	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
45	C₂₃×C₄⋊Q₈	—	2	1	no	—
46	C₂₃×C₄⋊Q₈	—	2	1	no	—
47	C₂₃×#125	—	4	1	no	—
48	C₂₃×#123	—	4	2	no	—
49	C₂₃×C₂²⋊Q₈	—	2	2	no	—
50	C₂₃×C₂²⋊Q₈	—	4	1	no	—
51	C₂₃×C₂²⋊Q₈	—	4	2	no	—
52	C₂₃×#121	—	2	2	no	—
53	C₄×Q₈×C₂₃	—	2	1	no	—
54	C₄×D₄×C₂₃	—	2	1	no	—
55	C₄×D₄×C₂₃	—	4	1	no	—
56	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
58	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
59	C₂₃×SD₃₂	—	2	2	no	—
60	C₂×C₁₆×C₂₃	—	2	4	no	—
61	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
62	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
63	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
64	C₂₃×#107	—	2	2	no	—
65	C₂₃×#107	—	2	2	no	—
66	C₂₃×#107	—	2	1	no	—
67	C₂₃×#107	—	2	1	no	—
68	C₂₃×#107	C₂³	1	1	yes	characteristic
69	C₂₃×#107	C₂³	1	2	yes	—
70	C₂₃×#107	C₂³	1	1	yes	characteristic
71	C₂₃×#107	C₂³	1	2	yes	—
72	C₂₃×C₄⋊C₈	D₄	1	1	yes	characteristic
73	C₂₃×#103	—	2	1	no	—
74	C₂₃×#103	—	2	2	no	—
75	C₂₃×#102	—	2	1	no	—
76	C₂₃×#102	—	2	2	no	—
77	C₂₃×C₂²⋊C₈	D₄	1	2	yes	—
78	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
79	#41×C₂₃	—	8	4	no	—
80	C₂×Q₈×C₂₃	—	2	1	no	—
81	C₂×Q₈×C₂₃	—	4	1	no	—
82	C₂×Q₈×C₂₃	—	4	1	no	—
83	C₂×D₄×C₂₃	—	2	1	no	—
84	C₂×D₄×C₂₃	—	4	2	no	—
85	C₂×D₄×C₂₃	D₈	1	1	yes	characteristic
86	C₂²×C₄×C₂₃	—	4	2	no	—
87	C₂²×C₄×C₂₃	—	4	2	no	—
88	C₂²×C₄×C₂₃	—	4	1	no	—
89	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
90	Q₁₆×C₂₃	—	2	1	no	—
91	Q₁₆×C₂₃	—	2	1	no	—
92	D₈×C₂₃	—	2	1	no	—
93	D₈×C₂₃	—	2	1	no	—
94	C₂×C₈×C₂₃	—	2	2	no	—
95	C₂×C₈×C₂₃	—	2	2	no	—
96	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
99	C₄⋊C₄×C₂₃	—	2	1	no	—
100	C₄⋊C₄×C₂₃	—	2	2	no	—
101	C₄⋊C₄×C₂₃	—	2	1	no	—
102	C₄⋊C₄×C₂₃	—	2	2	no	—
103	C₄⋊C₄×C₂₃	—	4	2	no	—
104	C₄⋊C₄×C₂₃	—	4	2	no	—
105	C₄⋊C₄×C₂₃	—	4	1	no	—
106	C₄⋊C₄×C₂₃	—	4	1	no	—
107	C₄⋊C₄×C₂₃	—	4	2	no	—
108	C₄⋊C₄×C₂₃	—	4	1	no	—
109	C₄⋊C₄×C₂₃	D₈	1	1	yes	characteristic
110	C₂²⋊C₄×C₂₃	—	4	2	no	—
111	C₂²⋊C₄×C₂₃	—	4	1	no	—
112	C₂²⋊C₄×C₂₃	—	4	2	no	—
113	C₂²⋊C₄×C₂₃	D₈	1	2	yes	—
114	C₄²×C₂₃	—	4	1	no	—
115	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
116	C₁₆×C₂₃	—	2	2	no	—
117	C₁₆×C₂₃	—	2	1	no	—
118	C₁₆×C₂₃	#41	1	2	yes	complemented
119	#33902	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂³×C₂₃	—	4	1	no	—
121	C₂³×C₂₃	C₂×D₈	1	2	yes	—
122	Q₈×C₂₃	—	4	1	no	—
123	Q₈×C₂₃	—	4	1	no	—
124	Q₈×C₂₃	—	8	2	no	—
125	Q₈×C₂₃	—	16	1	no	—
126	D₄×C₂₃	—	4	1	no	—
127	D₄×C₂₃	—	4	1	no	—
128	D₄×C₂₃	—	8	4	no	—
129	D₄×C₂₃	SD₃₂	1	4	yes	complemented
130	C₂×C₄×C₂₃	—	2	2	no	—
131	C₂×C₄×C₂₃	—	2	2	no	—
132	C₂×C₄×C₂₃	—	4	1	no	—
133	C₂×C₄×C₂₃	—	4	1	no	—
134	C₂×C₄×C₂₃	—	4	2	no	—
135	C₂×C₄×C₂₃	—	4	1	no	—
136	C₂×C₄×C₂₃	—	4	2	no	—
137	C₂×C₄×C₂₃	—	8	4	no	—
138	C₂×C₄×C₂₃	—	8	2	no	—
139	C₂×C₄×C₂₃	—	16	2	no	—
140	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
142	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
144	C₈×C₂₃	—	2	2	no	—
145	C₈×C₂₃	—	2	1	no	—
146	C₈×C₂₃	C₂×#41	1	1	yes	characteristic
147	C₈×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
148	#3338	C₂×C₂₃	1	1	yes	characteristic, complemented
149	#3334	C₂×C₂₃	1	1	yes	characteristic
150	#2254	C₂×C₂₃	1	1	yes	characteristic, complemented
151	#2235	C₂×C₂₃	1	2	yes	complemented
152	#2233	C₂×C₂₃	1	2	yes	complemented
153	#2226	C₂×C₂₃	1	1	yes	characteristic, complemented
154	#2225	C₂×C₂₃	1	1	yes	characteristic, complemented
155	#2217	C₂×C₂₃	1	2	yes	complemented
156	C₄×SD₃₂	C₂×C₂₃	1	1	yes	characteristic, complemented
157	D₄×C₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
158	C₂×#367	C₂×C₂₃	1	2	yes	complemented
159	C₂²×C₂₃	—	2	2	no	—
160	C₂²×C₂₃	—	2	2	no	—
161	C₂²×C₂₃	—	8	1	no	—
162	C₂²×C₂₃	—	8	1	no	—
163	C₂²×C₂₃	—	8	1	no	—
164	C₂²×C₂₃	C₂×SD₃₂	1	4	yes	—
165	C₂²×C₂₃	C₂²×D₈	1	1	yes	characteristic
166	C₄×C₂₃	—	2	2	no	—
167	C₄×C₂₃	—	2	1	no	—
168	C₄×C₂₃	—	8	1	no	—
169	C₄×C₂₃	—	8	1	no	—
170	C₄×C₂₃	—	8	2	no	—
171	C₄×C₂₃	—	8	1	no	—
172	C₄×C₂₃	—	8	2	no	—
173	C₄×C₂₃	C₂×SD₃₂	1	2	yes	—
174	C₄×C₂₃	#547	1	1	yes	characteristic
175	C₄×C₂₃	#578	1	1	yes	characteristic
176	#554	—	2	1	no	—
177	#547	—	2	1	no	—
178	C₂×SD₃₂	C₂²×C₂₃	1	1	yes	characteristic
179	C₂²×C₁₆	C₂²×C₂₃	1	2	yes	—
180	#500	C₂²×C₂₃	1	1	yes	characteristic
181	#500	C₂²×C₂₃	1	1	yes	characteristic
182	#484	—	2	2	no	—
183	#480	—	2	2	no	—
184	#477	—	2	1	no	—
185	#474	—	2	1	no	—
186	#467	C₂²×C₂₃	1	2	yes	—
187	#466	C₂²×C₂₃	1	2	yes	—
188	C₄×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
189	C₄×D₈	C₂²×C₂₃	1	1	yes	characteristic
190	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic
191	C₂×#107	C₂²×C₂₃	1	2	yes	—
192	C₂×#107	C₂²×C₂₃	1	2	yes	—
193	#367	C₂²×C₂₃	1	1	yes	characteristic
194	#367	C₂²×C₂₃	1	2	yes	—
195	#367	C₂²×C₂₃	1	2	yes	—
196	#367	C₂²×C₂₃	1	2	yes	—
197	#367	C₂²×C₂₃	1	2	yes	—
198	C₄⋊C₁₆	C₂²×C₂₃	1	1	yes	characteristic
199	#358	C₂²×C₂₃	1	1	yes	characteristic
200	#358	C₂²×C₂₃	1	2	yes	—
201	#357	C₂²×C₂₃	1	1	yes	characteristic
202	#357	C₂²×C₂₃	1	2	yes	—
203	C₂²⋊C₁₆	C₂²×C₂₃	1	2	yes	—
204	C₄×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
205	C₂×C₂₃	—	2	4	no	—
206	C₂×C₂₃	—	16	1	no	—
207	C₂×C₂₃	#3334	1	1	yes	characteristic
208	C₂×C₂₃	C₂²×SD₃₂	1	1	yes	characteristic
209	C₂×C₂₃	#3416	1	1	yes	characteristic
210	C₂×#41	—	4	1	no	—
211	C₂×Q₁₆	C₂³×C₂₃	1	1	yes	characteristic
212	C₂×D₈	C₂³×C₂₃	1	1	yes	characteristic
213	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
214	C₄⋊Q₈	—	2	1	no	—
215	C₄⋊Q₈	—	2	1	no	—
216	#125	—	4	1	no	—
217	#123	—	4	2	no	—
218	C₂²⋊Q₈	—	2	2	no	—
219	C₂²⋊Q₈	—	4	1	no	—
220	C₂²⋊Q₈	—	4	2	no	—
221	#121	—	2	2	no	—
222	C₄×Q₈	—	2	1	no	—
223	C₄×D₄	—	2	1	no	—
224	C₄×D₄	—	4	1	no	—
225	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
226	C₂×C₄⋊C₄	—	2	2	no	—
227	C₂×C₄⋊C₄	—	2	2	no	—
228	SD₃₂	—	2	2	no	—
229	C₂×C₁₆	—	2	4	no	—
230	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
231	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
232	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
233	#107	—	2	2	no	—
234	#107	—	2	2	no	—
235	#107	—	2	1	no	—
236	#107	—	2	1	no	—
237	#107	C₂³×C₂₃	1	1	yes	characteristic
238	#107	C₂³×C₂₃	1	2	yes	—
239	#107	C₂³×C₂₃	1	1	yes	characteristic
240	#107	C₂³×C₂₃	1	2	yes	—
241	C₄⋊C₈	D₄×C₂₃	1	1	yes	characteristic
242	#103	—	2	1	no	—
243	#103	—	2	2	no	—
244	#102	—	2	1	no	—
245	#102	—	2	2	no	—
246	C₂²⋊C₈	D₄×C₂₃	1	2	yes	—
247	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
248	C₂₃	#33902	1	1	yes	characteristic, direct product factor, Sylow
249	#41	—	8	4	no	—
250	C₂×Q₈	—	2	1	no	—
251	C₂×Q₈	—	4	1	no	—
252	C₂×Q₈	—	4	1	no	—
253	C₂×D₄	—	2	1	no	—
254	C₂×D₄	—	4	2	no	—
255	C₂×D₄	D₈×C₂₃	1	1	yes	characteristic
256	C₂²×C₄	—	4	2	no	—
257	C₂²×C₄	—	4	2	no	—
258	C₂²×C₄	—	4	1	no	—
259	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
260	Q₁₆	—	2	1	no	—
261	Q₁₆	—	2	1	no	—
262	D₈	—	2	1	no	—
263	D₈	—	2	1	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
267	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
268	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
269	C₄⋊C₄	—	2	1	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₄⋊C₄	—	4	2	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₄⋊C₄	D₈×C₂₃	1	1	yes	characteristic
280	C₂²⋊C₄	—	4	2	no	—
281	C₂²⋊C₄	—	4	1	no	—
282	C₂²⋊C₄	—	4	2	no	—
283	C₂²⋊C₄	D₈×C₂₃	1	2	yes	—
284	C₄²	—	4	1	no	—
285	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
286	C₁₆	—	2	2	no	—
287	C₁₆	—	2	1	no	—
288	C₁₆	#41×C₂₃	1	2	yes	complemented
289	C₂³	—	4	1	no	—
290	C₂³	C₂×D₈×C₂₃	1	2	yes	—
291	Q₈	—	4	1	no	—
292	Q₈	—	4	1	no	—
293	Q₈	—	8	2	no	—
294	Q₈	—	16	1	no	—
295	D₄	—	4	1	no	—
296	D₄	—	4	1	no	—
297	D₄	—	8	4	no	—
298	D₄	C₂₃×SD₃₂	1	4	yes	complemented
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	8	4	no	—
307	C₂×C₄	—	8	2	no	—
308	C₂×C₄	—	16	2	no	—
309	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
310	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
311	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
312	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
313	C₈	—	2	2	no	—
314	C₈	—	2	1	no	—
315	C₈	C₂×#41×C₂₃	1	1	yes	characteristic
316	C₈	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
317	C₂²	—	2	2	no	—
318	C₂²	—	2	2	no	—
319	C₂²	—	8	1	no	—
320	C₂²	—	8	1	no	—
321	C₂²	—	8	1	no	—
322	C₂²	C₂×C₂₃×SD₃₂	1	4	yes	—
323	C₂²	C₂²×D₈×C₂₃	1	1	yes	characteristic
324	C₄	—	2	2	no	—
325	C₄	—	2	1	no	—
326	C₄	—	8	1	no	—
327	C₄	—	8	1	no	—
328	C₄	—	8	2	no	—
329	C₄	—	8	1	no	—
330	C₄	—	8	2	no	—
331	C₄	C₂×C₂₃×SD₃₂	1	2	yes	—
332	C₄	C₂₃×#547	1	1	yes	characteristic
333	C₄	C₂₃×#578	1	1	yes	characteristic
334	C₂	—	2	4	no	—
335	C₂	—	16	1	no	—
336	C₂	C₂₃×#3334	1	1	yes	characteristic
337	C₂	C₂²×C₂₃×SD₃₂	1	1	yes	characteristic
338	C₂	C₂₃×#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
prime radical	C₂₃×#3334	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄×D₄×C₂₃ ◁ C₈×D₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3338	C₂	semidirect
3. C₂₃×#2254	C₂	semidirect
4. C₂₃×#2235	C₂	semidirect
5. C₂₃×#2233	C₂	semidirect
6. C₂₃×#2226	C₂	semidirect
7. C₂₃×#2225	C₂	semidirect
8. C₂₃×#2217	C₂	semidirect
9. C₄×C₂₃×SD₃₂	C₂	semidirect
10. D₄×C₁₆×C₂₃	C₂	semidirect
11. C₂×C₂₃×#367	C₂	semidirect
118. C₁₆×C₂₃	#41	semidirect
119. #33902	C₂₃	direct (N × H)
129. D₄×C₂₃	SD₃₂	semidirect
148. #3338	C₂×C₂₃	semidirect
150. #2254	C₂×C₂₃	semidirect
151. #2235	C₂×C₂₃	semidirect
152. #2233	C₂×C₂₃	semidirect
153. #2226	C₂×C₂₃	semidirect
154. #2225	C₂×C₂₃	semidirect
155. #2217	C₂×C₂₃	semidirect
156. C₄×SD₃₂	C₂×C₂₃	semidirect
157. D₄×C₁₆	C₂×C₂₃	semidirect
158. C₂×#367	C₂×C₂₃	semidirect
248. C₂₃	#33902	direct (N × H)
288. C₁₆	#41×C₂₃	semidirect
298. D₄	C₂₃×SD₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3338	2
C₂₃×#3334	2
C₂₃×#2254	2
C₂₃×#2235	2
C₂₃×#2233	2
C₂₃×#2226	2
C₂₃×#2225	2
C₂₃×#2217	2
C₄×C₂₃×SD₃₂	2
D₄×C₁₆×C₂₃	2
C₂×C₂₃×#367	2
#33902	23

Subgroups — C23×#33902 #19181528

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33902 #19181528