Subgroups — C₂₃×#33967 #19181593

Order 5888.

510 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3396	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#3395	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2251	C₂	1	2	yes	maximal, complemented
4	C₂₃×#2248	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#2211	C₂	1	2	yes	maximal, complemented
6	C₂₃×#2209	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#2208	C₂	1	2	yes	maximal, complemented
8	C₂₃×#2207	C₂	1	1	yes	maximal, characteristic
9	C₂₃×#2183	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#2175	C₂	1	2	yes	maximal, complemented
11	C₄×C₂₃×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#559	—	2	1	no	—
13	C₂₃×#557	—	2	1	no	—
14	C₂×C₂₃×Q₃₂	C₂²	1	1	yes	characteristic
15	C₂×C₂₃×Q₃₂	C₂²	1	2	yes	—
16	C₂×C₂₃×SD₃₂	C₂²	1	2	yes	—
17	C₂×C₂₃×SD₃₂	C₂²	1	2	yes	—
18	C₂×C₂₃×SD₃₂	C₂²	1	2	yes	—
19	C₂₃×#495	C₂²	1	2	yes	—
20	C₂₃×#494	C₂²	1	2	yes	—
21	C₂₃×#494	C₂²	1	1	yes	characteristic
22	C₂₃×#493	C₂²	1	1	yes	characteristic
23	C₂₃×#463	—	2	2	no	—
24	C₂₃×#462	—	2	1	no	—
25	C₂₃×#462	—	2	2	no	—
26	C₂₃×#460	—	2	1	no	—
27	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic
28	C₄×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
29	C₄×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
30	C₄×Q₁₆×C₂₃	C₂²	1	2	yes	—
31	C₄×D₈×C₂₃	C₂²	1	2	yes	—
32	C₂₃×#366	C₂²	1	1	yes	characteristic
33	C₂₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
34	C₂₃×C₄⋊C₁₆	C₂²	1	2	yes	—
35	C₂₃×#358	C₂²	1	1	yes	characteristic
36	C₂₃×#358	C₂²	1	1	yes	characteristic
37	C₂₃×#358	C₂²	1	2	yes	—
38	C₂₃×#357	C₂²	1	2	yes	—
39	C₂₃×#346	C₂²	1	2	yes	—
40	C₄×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
41	C₂×Q₁₆×C₂₃	—	2	2	no	—
42	C₂×Q₁₆×C₂₃	—	2	1	no	—
43	C₂×Q₁₆×C₂₃	C₂³	1	2	yes	—
44	C₂×Q₁₆×C₂₃	C₂³	1	1	yes	characteristic
45	C₂×Q₁₆×C₂₃	C₂³	1	1	yes	characteristic
46	C₂×SD₁₆×C₂₃	—	2	2	no	—
47	C₂×D₈×C₂₃	—	2	1	no	—
48	C₂×D₈×C₂₃	C₂³	1	2	yes	—
49	C₂₃×C₄⋊Q₈	—	2	2	no	—
50	C₂₃×C₄⋊Q₈	—	2	1	no	—
51	C₂₃×#127	—	2	1	no	—
52	C₂₃×#125	—	4	1	no	—
53	C₂₃×#125	—	4	2	no	—
54	C₂₃×#124	—	2	2	no	—
55	C₂₃×#124	—	4	1	no	—
56	C₂₃×#123	—	4	2	no	—
57	C₄×Q₈×C₂₃	—	2	2	no	—
58	C₄×Q₈×C₂₃	—	2	1	no	—
59	C₄×Q₈×C₂₃	—	2	1	no	—
60	C₄×Q₈×C₂₃	—	4	1	no	—
61	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
62	C₄×D₄×C₂₃	—	2	2	no	—
63	C₄×D₄×C₂₃	—	4	1	no	—
64	C₂₃×Q₃₂	—	2	2	no	—
65	C₂₃×Q₃₂	—	4	2	no	—
66	C₂₃×SD₃₂	—	2	4	no	—
67	C₂₃×SD₃₂	—	4	2	no	—
68	C₂₃×SD₃₂	—	4	2	no	—
69	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
70	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
71	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
72	C₂₃×#107	C₂³	1	1	yes	characteristic
73	C₂₃×#107	C₂³	1	1	yes	characteristic
74	C₂₃×C₄⋊C₈	D₄	1	1	yes	characteristic
75	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
76	C₂₃×#103	—	2	2	no	—
77	C₂₃×#103	—	2	1	no	—
78	C₂₃×#103	—	2	1	no	—
79	C₂₃×#102	—	2	2	no	—
80	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₂₃	C₂³	1	2	yes	—
82	C₂×Q₈×C₂₃	—	2	1	no	—
83	C₂×Q₈×C₂₃	—	2	2	no	—
84	C₂×Q₈×C₂₃	—	2	1	no	—
85	C₂×Q₈×C₂₃	D₈	1	1	yes	characteristic
86	C₂×D₄×C₂₃	—	2	2	no	—
87	C₂×D₄×C₂₃	—	4	1	no	—
88	C₂×D₄×C₂₃	—	4	1	no	—
89	C₂²×C₄×C₂₃	—	4	2	no	—
90	Q₁₆×C₂₃	—	2	2	no	—
91	Q₁₆×C₂₃	—	2	1	no	—
92	Q₁₆×C₂₃	—	2	1	no	—
93	Q₁₆×C₂₃	—	2	1	no	—
94	Q₁₆×C₂₃	—	2	2	no	—
95	Q₁₆×C₂₃	—	2	1	no	—
96	Q₁₆×C₂₃	—	8	1	no	—
97	Q₁₆×C₂₃	—	8	2	no	—
98	SD₁₆×C₂₃	—	8	2	no	—
99	D₈×C₂₃	—	2	2	no	—
100	D₈×C₂₃	—	2	2	no	—
101	D₈×C₂₃	—	8	1	no	—
102	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
103	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
105	C₄⋊C₄×C₂₃	—	2	1	no	—
106	C₄⋊C₄×C₂₃	—	2	1	no	—
107	C₄⋊C₄×C₂₃	—	4	1	no	—
108	C₄⋊C₄×C₂₃	—	4	2	no	—
109	C₄⋊C₄×C₂₃	—	4	1	no	—
110	C₄⋊C₄×C₂₃	—	4	2	no	—
111	C₄⋊C₄×C₂₃	—	4	1	no	—
112	C₄⋊C₄×C₂₃	—	4	2	no	—
113	C₄⋊C₄×C₂₃	—	4	1	no	—
114	C₄⋊C₄×C₂₃	D₈	1	1	yes	characteristic
115	C₄⋊C₄×C₂₃	D₈	1	2	yes	—
116	C₂²⋊C₄×C₂₃	—	4	2	no	—
117	C₂²⋊C₄×C₂₃	—	4	2	no	—
118	C₂²⋊C₄×C₂₃	—	4	2	no	—
119	C₄²×C₂₃	—	4	1	no	—
120	C₄²×C₂₃	—	4	2	no	—
121	C₄²×C₂₃	—	4	1	no	—
122	C₄²×C₂₃	C₂×D₄	1	2	yes	—
123	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
124	C₁₆×C₂₃	—	2	1	no	—
125	C₁₆×C₂₃	—	2	2	no	—
126	C₁₆×C₂₃	#41	1	2	yes	—
127	#33967	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₂³×C₂₃	—	4	2	no	—
129	Q₈×C₂₃	—	2	2	no	—
130	Q₈×C₂₃	—	4	2	no	—
131	Q₈×C₂₃	—	4	1	no	—
132	Q₈×C₂₃	—	4	1	no	—
133	Q₈×C₂₃	—	4	1	no	—
134	Q₈×C₂₃	—	4	2	no	—
135	Q₈×C₂₃	—	4	1	no	—
136	D₄×C₂₃	—	4	2	no	—
137	D₄×C₂₃	—	4	2	no	—
138	D₄×C₂₃	—	8	2	no	—
139	D₄×C₂₃	—	16	1	no	—
140	C₂×C₄×C₂₃	—	4	1	no	—
141	C₂×C₄×C₂₃	—	4	1	no	—
142	C₂×C₄×C₂₃	—	4	1	no	—
143	C₂×C₄×C₂₃	—	4	2	no	—
144	C₂×C₄×C₂₃	—	4	1	no	—
145	C₂×C₄×C₂₃	—	16	2	no	—
146	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
147	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
148	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
151	C₈×C₂₃	—	2	1	no	—
152	C₈×C₂₃	—	2	2	no	—
153	C₈×C₂₃	C₂×#41	1	1	yes	characteristic
154	C₈×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
155	#3396	C₂×C₂₃	1	1	yes	characteristic
156	#3395	C₂×C₂₃	1	1	yes	characteristic, complemented
157	#2251	C₂×C₂₃	1	2	yes	complemented
158	#2248	C₂×C₂₃	1	1	yes	characteristic
159	#2211	C₂×C₂₃	1	2	yes	complemented
160	#2209	C₂×C₂₃	1	1	yes	characteristic, complemented
161	#2208	C₂×C₂₃	1	2	yes	complemented
162	#2207	C₂×C₂₃	1	1	yes	characteristic
163	#2183	C₂×C₂₃	1	1	yes	characteristic, complemented
164	#2175	C₂×C₂₃	1	2	yes	complemented
165	C₄×Q₃₂	C₂×C₂₃	1	1	yes	characteristic, complemented
166	C₂²×C₂₃	—	8	2	no	—
167	C₂²×C₂₃	—	8	2	no	—
168	C₂²×C₂₃	—	8	2	no	—
169	C₂²×C₂₃	C₂²×D₈	1	1	yes	characteristic
170	C₄×C₂₃	—	2	2	no	—
171	C₄×C₂₃	—	2	2	no	—
172	C₄×C₂₃	—	2	1	no	—
173	C₄×C₂₃	—	8	1	no	—
174	C₄×C₂₃	—	8	2	no	—
175	C₄×C₂₃	—	8	1	no	—
176	C₄×C₂₃	—	8	1	no	—
177	C₄×C₂₃	—	8	1	no	—
178	C₄×C₂₃	#510	1	2	yes	—
179	C₄×C₂₃	#550	1	1	yes	characteristic
180	C₄×C₂₃	#576	1	1	yes	characteristic
181	#559	—	2	1	no	—
182	#557	—	2	1	no	—
183	C₂×Q₃₂	C₂²×C₂₃	1	1	yes	characteristic
184	C₂×Q₃₂	C₂²×C₂₃	1	2	yes	—
185	C₂×SD₃₂	C₂²×C₂₃	1	2	yes	—
186	C₂×SD₃₂	C₂²×C₂₃	1	2	yes	—
187	C₂×SD₃₂	C₂²×C₂₃	1	2	yes	—
188	#495	C₂²×C₂₃	1	2	yes	—
189	#494	C₂²×C₂₃	1	2	yes	—
190	#494	C₂²×C₂₃	1	1	yes	characteristic
191	#493	C₂²×C₂₃	1	1	yes	characteristic
192	#463	—	2	2	no	—
193	#462	—	2	1	no	—
194	#462	—	2	2	no	—
195	#460	—	2	1	no	—
196	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
197	C₄×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
198	C₄×Q₁₆	C₂²×C₂₃	1	2	yes	—
199	C₄×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
200	C₄×D₈	C₂²×C₂₃	1	2	yes	—
201	#366	C₂²×C₂₃	1	1	yes	characteristic
202	C₄⋊C₁₆	C₂²×C₂₃	1	1	yes	characteristic
203	C₄⋊C₁₆	C₂²×C₂₃	1	2	yes	—
204	#358	C₂²×C₂₃	1	1	yes	characteristic
205	#358	C₂²×C₂₃	1	1	yes	characteristic
206	#358	C₂²×C₂₃	1	2	yes	—
207	#357	C₂²×C₂₃	1	2	yes	—
208	#346	C₂²×C₂₃	1	2	yes	—
209	C₄×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
210	C₂×C₂₃	—	16	2	no	—
211	C₂×C₂₃	#3391	1	1	yes	characteristic
212	C₂×C₂₃	C₂×#510	1	1	yes	characteristic
213	C₂×C₂₃	#3416	1	1	yes	characteristic
214	C₂×Q₁₆	—	2	2	no	—
215	C₂×Q₁₆	—	2	1	no	—
216	C₂×Q₁₆	C₂³×C₂₃	1	2	yes	—
217	C₂×Q₁₆	C₂³×C₂₃	1	1	yes	characteristic
218	C₂×Q₁₆	C₂³×C₂₃	1	1	yes	characteristic
219	C₂×SD₁₆	—	2	2	no	—
220	C₂×D₈	—	2	1	no	—
221	C₂×D₈	C₂³×C₂₃	1	2	yes	—
222	C₄⋊Q₈	—	2	2	no	—
223	C₄⋊Q₈	—	2	1	no	—
224	#127	—	2	1	no	—
225	#125	—	4	1	no	—
226	#125	—	4	2	no	—
227	#124	—	2	2	no	—
228	#124	—	4	1	no	—
229	#123	—	4	2	no	—
230	C₄×Q₈	—	2	2	no	—
231	C₄×Q₈	—	2	1	no	—
232	C₄×Q₈	—	2	1	no	—
233	C₄×Q₈	—	4	1	no	—
234	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
235	C₄×D₄	—	2	2	no	—
236	C₄×D₄	—	4	1	no	—
237	Q₃₂	—	2	2	no	—
238	Q₃₂	—	4	2	no	—
239	SD₃₂	—	2	4	no	—
240	SD₃₂	—	4	2	no	—
241	SD₃₂	—	4	2	no	—
242	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
243	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
244	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
245	#107	C₂³×C₂₃	1	1	yes	characteristic
246	#107	C₂³×C₂₃	1	1	yes	characteristic
247	C₄⋊C₈	D₄×C₂₃	1	1	yes	characteristic
248	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
249	#103	—	2	2	no	—
250	#103	—	2	1	no	—
251	#103	—	2	1	no	—
252	#102	—	2	2	no	—
253	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
254	C₄×C₈	C₂³×C₂₃	1	2	yes	—
255	C₂₃	#33967	1	1	yes	characteristic, direct product factor, Sylow
256	C₂×Q₈	—	2	1	no	—
257	C₂×Q₈	—	2	2	no	—
258	C₂×Q₈	—	2	1	no	—
259	C₂×Q₈	D₈×C₂₃	1	1	yes	characteristic
260	C₂×D₄	—	2	2	no	—
261	C₂×D₄	—	4	1	no	—
262	C₂×D₄	—	4	1	no	—
263	C₂²×C₄	—	4	2	no	—
264	Q₁₆	—	2	2	no	—
265	Q₁₆	—	2	1	no	—
266	Q₁₆	—	2	1	no	—
267	Q₁₆	—	2	1	no	—
268	Q₁₆	—	2	2	no	—
269	Q₁₆	—	2	1	no	—
270	Q₁₆	—	8	1	no	—
271	Q₁₆	—	8	2	no	—
272	SD₁₆	—	8	2	no	—
273	D₈	—	2	2	no	—
274	D₈	—	2	2	no	—
275	D₈	—	8	1	no	—
276	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
277	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
278	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
279	C₄⋊C₄	—	2	1	no	—
280	C₄⋊C₄	—	2	1	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	4	2	no	—
283	C₄⋊C₄	—	4	1	no	—
284	C₄⋊C₄	—	4	2	no	—
285	C₄⋊C₄	—	4	1	no	—
286	C₄⋊C₄	—	4	2	no	—
287	C₄⋊C₄	—	4	1	no	—
288	C₄⋊C₄	D₈×C₂₃	1	1	yes	characteristic
289	C₄⋊C₄	D₈×C₂₃	1	2	yes	—
290	C₂²⋊C₄	—	4	2	no	—
291	C₂²⋊C₄	—	4	2	no	—
292	C₂²⋊C₄	—	4	2	no	—
293	C₄²	—	4	1	no	—
294	C₄²	—	4	2	no	—
295	C₄²	—	4	1	no	—
296	C₄²	C₂×D₄×C₂₃	1	2	yes	—
297	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
298	C₁₆	—	2	1	no	—
299	C₁₆	—	2	2	no	—
300	C₁₆	#41×C₂₃	1	2	yes	—
301	C₂³	—	4	2	no	—
302	Q₈	—	2	2	no	—
303	Q₈	—	4	2	no	—
304	Q₈	—	4	1	no	—
305	Q₈	—	4	1	no	—
306	Q₈	—	4	1	no	—
307	Q₈	—	4	2	no	—
308	Q₈	—	4	1	no	—
309	D₄	—	4	2	no	—
310	D₄	—	4	2	no	—
311	D₄	—	8	2	no	—
312	D₄	—	16	1	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	16	2	no	—
319	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
320	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
321	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
322	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
323	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
324	C₈	—	2	1	no	—
325	C₈	—	2	2	no	—
326	C₈	C₂×#41×C₂₃	1	1	yes	characteristic
327	C₈	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
328	C₂²	—	8	2	no	—
329	C₂²	—	8	2	no	—
330	C₂²	—	8	2	no	—
331	C₂²	C₂²×D₈×C₂₃	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	2	1	no	—
335	C₄	—	8	1	no	—
336	C₄	—	8	2	no	—
337	C₄	—	8	1	no	—
338	C₄	—	8	1	no	—
339	C₄	—	8	1	no	—
340	C₄	C₂₃×#510	1	2	yes	—
341	C₄	C₂₃×#550	1	1	yes	characteristic
342	C₄	C₂₃×#576	1	1	yes	characteristic
343	C₂	—	16	2	no	—
344	C₂	C₂₃×#3391	1	1	yes	characteristic
345	C₂	C₂×C₂₃×#510	1	1	yes	characteristic
346	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×#493	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄×Q₈×C₂₃ ◁ C₈×Q₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#3395	C₂	semidirect
3. C₂₃×#2251	C₂	semidirect
5. C₂₃×#2211	C₂	semidirect
6. C₂₃×#2209	C₂	semidirect
7. C₂₃×#2208	C₂	semidirect
9. C₂₃×#2183	C₂	semidirect
10. C₂₃×#2175	C₂	semidirect
11. C₄×C₂₃×Q₃₂	C₂	semidirect
127. #33967	C₂₃	direct (N × H)
156. #3395	C₂×C₂₃	semidirect
157. #2251	C₂×C₂₃	semidirect
159. #2211	C₂×C₂₃	semidirect
160. #2209	C₂×C₂₃	semidirect
161. #2208	C₂×C₂₃	semidirect
163. #2183	C₂×C₂₃	semidirect
164. #2175	C₂×C₂₃	semidirect
165. C₄×Q₃₂	C₂×C₂₃	semidirect
255. C₂₃	#33967	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×#3396	2
C₂₃×#3395	2
C₂₃×#2251	2
C₂₃×#2248	2
C₂₃×#2211	2
C₂₃×#2209	2
C₂₃×#2208	2
C₂₃×#2207	2
C₂₃×#2183	2
C₂₃×#2175	2
C₄×C₂₃×Q₃₂	2
#33967	23

Subgroups — C23×#33967 #19181593

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33967 #19181593