Subgroups — C₂₃×#33900 #19181526

Order 5888.

558 conjugacy classes of subgroups.

The subgroup lattice has 362 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×#3287	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2240	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2232	C₂	1	2	yes	maximal, complemented
5	C₂₃×#2227	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#2220	C₂	1	2	yes	maximal, complemented
7	C₂₃×#2207	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#2189	C₂	1	2	yes	maximal, complemented
9	C₄×C₂₃×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#2169	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#2139	C₂	1	2	yes	maximal, complemented
12	C₂×C₂₃×#135	C₂²	1	2	yes	—
13	C₂₃×#554	—	2	1	no	—
14	C₂₃×#550	—	2	1	no	—
15	C₂×C₂₃×Q₃₂	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×SD₃₂	C₂²	1	2	yes	—
17	C₂×C₂₃×M₃₂	C₂²	1	2	yes	—
18	C₂₃×#500	C₂²	1	1	yes	characteristic
19	C₂₃×#493	C₂²	1	1	yes	characteristic
20	C₂₃×#484	—	2	2	no	—
21	C₂₃×#477	—	2	1	no	—
22	C₂₃×#467	C₂²	1	2	yes	—
23	C₂₃×#466	C₂²	1	2	yes	—
24	C₂₃×#460	—	2	1	no	—
25	C₂₃×#449	—	2	2	no	—
26	C₄×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
27	C₄×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
28	C₈×D₄×C₂₃	C₂²	1	1	yes	characteristic
29	C₂×C₂₃×#107	C₂²	1	2	yes	—
30	C₂₃×#366	C₂²	1	1	yes	characteristic
31	C₂₃×#366	C₂²	1	2	yes	—
32	C₂₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
33	C₂₃×#358	C₂²	1	1	yes	characteristic
34	C₂₃×#358	C₂²	1	1	yes	characteristic
35	C₂₃×#358	C₂²	1	2	yes	—
36	C₂₃×#357	C₂²	1	2	yes	—
37	C₂₃×#357	C₂²	1	2	yes	—
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	—	2	2	no	—
42	C₂₃×#135	—	2	4	no	—
43	C₂₃×#135	—	4	2	no	—
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	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
50	C₂₃×C₄⋊Q₈	—	2	1	no	—
51	C₂₃×#127	—	2	1	no	—
52	C₂₃×#127	—	4	1	no	—
53	C₂₃×#125	—	4	1	no	—
54	C₂₃×C₂²⋊Q₈	—	2	2	no	—
55	C₂₃×C₂²⋊Q₈	—	4	2	no	—
56	C₂₃×#121	—	2	2	no	—
57	C₂₃×#121	—	4	2	no	—
58	C₄×Q₈×C₂₃	—	2	1	no	—
59	C₄×Q₈×C₂₃	—	2	1	no	—
60	C₄×D₄×C₂₃	—	4	1	no	—
61	C₄×D₄×C₂₃	—	4	1	no	—
62	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
64	C₂₃×Q₃₂	—	2	2	no	—
65	C₂₃×SD₃₂	—	4	2	no	—
66	C₂₃×M₃₂	—	4	2	no	—
67	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
68	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
69	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
70	C₂₃×#107	—	2	2	no	—
71	C₂₃×#107	—	2	1	no	—
72	C₂₃×#107	C₂³	1	1	yes	characteristic
73	C₂₃×#107	C₂³	1	2	yes	—
74	C₂₃×#107	C₂³	1	1	yes	characteristic
75	C₂₃×C₄⋊C₈	D₄	1	1	yes	characteristic
76	C₂₃×#103	—	2	1	no	—
77	C₂₃×#103	—	2	1	no	—
78	C₂₃×#103	—	2	2	no	—
79	C₂₃×#102	—	2	2	no	—
80	C₂₃×C₂²⋊C₈	D₄	1	2	yes	—
81	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
82	#41×C₂₃	—	4	4	no	—
83	#41×C₂₃	—	8	2	no	—
84	C₂×Q₈×C₂₃	—	2	1	no	—
85	C₂×Q₈×C₂₃	—	2	1	no	—
86	C₂×Q₈×C₂₃	—	4	1	no	—
87	C₂×D₄×C₂₃	—	2	2	no	—
88	C₂×D₄×C₂₃	—	4	2	no	—
89	C₂×D₄×C₂₃	—	4	2	no	—
90	C₂×D₄×C₂₃	—	4	1	no	—
91	C₂×D₄×C₂₃	—	4	1	no	—
92	C₂×D₄×C₂₃	D₈	1	1	yes	characteristic
93	C₂²×C₄×C₂₃	—	4	2	no	—
94	C₂²×C₄×C₂₃	—	4	2	no	—
95	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
96	Q₁₆×C₂₃	—	2	1	no	—
97	Q₁₆×C₂₃	—	2	1	no	—
98	Q₁₆×C₂₃	—	2	1	no	—
99	Q₁₆×C₂₃	#41	1	2	yes	—
100	SD₁₆×C₂₃	—	4	4	no	—
101	D₈×C₂₃	—	2	2	no	—
102	D₈×C₂₃	—	2	2	no	—
103	D₈×C₂₃	—	8	1	no	—
104	C₂×C₈×C₂₃	—	2	2	no	—
105	C₂×C₈×C₂₃	—	2	2	no	—
106	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
107	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
109	C₄⋊C₄×C₂₃	—	2	1	no	—
110	C₄⋊C₄×C₂₃	—	2	1	no	—
111	C₄⋊C₄×C₂₃	—	2	2	no	—
112	C₄⋊C₄×C₂₃	—	4	2	no	—
113	C₄⋊C₄×C₂₃	—	4	1	no	—
114	C₄⋊C₄×C₂₃	—	4	1	no	—
115	C₄⋊C₄×C₂₃	—	4	1	no	—
116	C₄⋊C₄×C₂₃	—	4	2	no	—
117	C₄⋊C₄×C₂₃	D₈	1	1	yes	characteristic
118	C₂²⋊C₄×C₂₃	—	4	2	no	—
119	C₂²⋊C₄×C₂₃	—	4	2	no	—
120	C₂²⋊C₄×C₂₃	D₈	1	2	yes	—
121	C₄²×C₂₃	—	4	1	no	—
122	C₄²×C₂₃	—	4	1	no	—
123	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
124	C₁₆×C₂₃	—	2	1	no	—
125	C₁₆×C₂₃	—	2	1	no	—
126	C₁₆×C₂₃	—	2	2	no	—
127	#33900	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₂³×C₂₃	—	4	2	no	—
129	C₂³×C₂₃	C₂×D₈	1	2	yes	—
130	Q₈×C₂₃	—	2	2	no	—
131	Q₈×C₂₃	—	4	1	no	—
132	Q₈×C₂₃	—	4	1	no	—
133	Q₈×C₂₃	—	4	1	no	—
134	Q₈×C₂₃	—	16	1	no	—
135	D₄×C₂₃	—	2	2	no	—
136	D₄×C₂₃	—	4	2	no	—
137	D₄×C₂₃	—	4	2	no	—
138	D₄×C₂₃	—	8	4	no	—
139	D₄×C₂₃	—	8	2	no	—
140	D₄×C₂₃	—	16	1	no	—
141	D₄×C₂₃	—	16	2	no	—
142	C₂×C₄×C₂₃	—	2	2	no	—
143	C₂×C₄×C₂₃	—	2	2	no	—
144	C₂×C₄×C₂₃	—	4	1	no	—
145	C₂×C₄×C₂₃	—	4	1	no	—
146	C₂×C₄×C₂₃	—	4	1	no	—
147	C₂×C₄×C₂₃	—	4	1	no	—
148	C₂×C₄×C₂₃	—	4	2	no	—
149	C₂×C₄×C₂₃	—	8	4	no	—
150	C₂×C₄×C₂₃	—	16	2	no	—
151	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
152	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
153	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
154	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
155	C₈×C₂₃	—	2	2	no	—
156	C₈×C₂₃	—	2	1	no	—
157	C₈×C₂₃	C₂×#41	1	1	yes	characteristic
158	C₈×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
159	#3338	C₂×C₂₃	1	1	yes	characteristic, complemented
160	#3287	C₂×C₂₃	1	1	yes	characteristic
161	#2240	C₂×C₂₃	1	1	yes	characteristic, complemented
162	#2232	C₂×C₂₃	1	2	yes	complemented
163	#2227	C₂×C₂₃	1	1	yes	characteristic, complemented
164	#2220	C₂×C₂₃	1	2	yes	complemented
165	#2207	C₂×C₂₃	1	1	yes	characteristic, complemented
166	#2189	C₂×C₂₃	1	2	yes	complemented
167	C₄×Q₃₂	C₂×C₂₃	1	1	yes	characteristic, complemented
168	#2169	C₂×C₂₃	1	1	yes	characteristic, complemented
169	#2139	C₂×C₂₃	1	2	yes	complemented
170	C₂²×C₂₃	—	2	2	no	—
171	C₂²×C₂₃	—	2	2	no	—
172	C₂²×C₂₃	—	2	2	no	—
173	C₂²×C₂₃	—	8	2	no	—
174	C₂²×C₂₃	—	8	2	no	—
175	C₂²×C₂₃	—	8	2	no	—
176	C₂²×C₂₃	C₂²×D₈	1	1	yes	characteristic
177	C₄×C₂₃	—	2	1	no	—
178	C₄×C₂₃	—	2	2	no	—
179	C₄×C₂₃	—	4	2	no	—
180	C₄×C₂₃	—	8	1	no	—
181	C₄×C₂₃	—	8	2	no	—
182	C₄×C₂₃	—	8	1	no	—
183	C₄×C₂₃	—	8	1	no	—
184	C₄×C₂₃	#510	1	2	yes	—
185	C₄×C₂₃	#546	1	1	yes	characteristic
186	C₄×C₂₃	#577	1	1	yes	characteristic
187	C₂×#135	C₂²×C₂₃	1	2	yes	—
188	#554	—	2	1	no	—
189	#550	—	2	1	no	—
190	C₂×Q₃₂	C₂²×C₂₃	1	1	yes	characteristic
191	C₂×SD₃₂	C₂²×C₂₃	1	2	yes	—
192	C₂×M₃₂	C₂²×C₂₃	1	2	yes	—
193	#500	C₂²×C₂₃	1	1	yes	characteristic
194	#493	C₂²×C₂₃	1	1	yes	characteristic
195	#484	—	2	2	no	—
196	#477	—	2	1	no	—
197	#467	C₂²×C₂₃	1	2	yes	—
198	#466	C₂²×C₂₃	1	2	yes	—
199	#460	—	2	1	no	—
200	#449	—	2	2	no	—
201	C₄×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
202	C₄×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
203	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic
204	C₂×#107	C₂²×C₂₃	1	2	yes	—
205	#366	C₂²×C₂₃	1	1	yes	characteristic
206	#366	C₂²×C₂₃	1	2	yes	—
207	C₄⋊C₁₆	C₂²×C₂₃	1	1	yes	characteristic
208	#358	C₂²×C₂₃	1	1	yes	characteristic
209	#358	C₂²×C₂₃	1	2	yes	—
210	#358	C₂²×C₂₃	1	1	yes	characteristic
211	#357	C₂²×C₂₃	1	2	yes	—
212	#357	C₂²×C₂₃	1	2	yes	—
213	#357	C₂²×C₂₃	1	2	yes	—
214	C₂²⋊C₁₆	C₂²×C₂₃	1	2	yes	—
215	C₄×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
216	C₂×C₂₃	—	4	2	no	—
217	C₂×C₂₃	—	16	2	no	—
218	C₂×C₂₃	#3294	1	1	yes	characteristic
219	C₂×C₂₃	C₂×#510	1	1	yes	characteristic
220	C₂×C₂₃	#3415	1	1	yes	characteristic
221	C₂×#41	—	2	2	no	—
222	#135	—	2	4	no	—
223	#135	—	4	2	no	—
224	C₂×Q₁₆	C₂³×C₂₃	1	1	yes	characteristic
225	C₂×Q₁₆	C₂³×C₂₃	1	1	yes	characteristic
226	C₂×SD₁₆	—	2	2	no	—
227	C₂×D₈	—	2	1	no	—
228	C₂×D₈	C₂³×C₂₃	1	2	yes	—
229	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
230	C₄⋊Q₈	—	2	1	no	—
231	#127	—	2	1	no	—
232	#127	—	4	1	no	—
233	#125	—	4	1	no	—
234	C₂²⋊Q₈	—	2	2	no	—
235	C₂²⋊Q₈	—	4	2	no	—
236	#121	—	2	2	no	—
237	#121	—	4	2	no	—
238	C₄×Q₈	—	2	1	no	—
239	C₄×Q₈	—	2	1	no	—
240	C₄×D₄	—	4	1	no	—
241	C₄×D₄	—	4	1	no	—
242	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
243	C₂×C₄⋊C₄	—	2	2	no	—
244	Q₃₂	—	2	2	no	—
245	SD₃₂	—	4	2	no	—
246	M₃₂	—	4	2	no	—
247	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
248	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
249	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
250	#107	—	2	2	no	—
251	#107	—	2	1	no	—
252	#107	C₂³×C₂₃	1	1	yes	characteristic
253	#107	C₂³×C₂₃	1	2	yes	—
254	#107	C₂³×C₂₃	1	1	yes	characteristic
255	C₄⋊C₈	D₄×C₂₃	1	1	yes	characteristic
256	#103	—	2	1	no	—
257	#103	—	2	1	no	—
258	#103	—	2	2	no	—
259	#102	—	2	2	no	—
260	C₂²⋊C₈	D₄×C₂₃	1	2	yes	—
261	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
262	C₂₃	#33900	1	1	yes	characteristic, direct product factor, Sylow
263	#41	—	4	4	no	—
264	#41	—	8	2	no	—
265	C₂×Q₈	—	2	1	no	—
266	C₂×Q₈	—	2	1	no	—
267	C₂×Q₈	—	4	1	no	—
268	C₂×D₄	—	2	2	no	—
269	C₂×D₄	—	4	2	no	—
270	C₂×D₄	—	4	2	no	—
271	C₂×D₄	—	4	1	no	—
272	C₂×D₄	—	4	1	no	—
273	C₂×D₄	D₈×C₂₃	1	1	yes	characteristic
274	C₂²×C₄	—	4	2	no	—
275	C₂²×C₄	—	4	2	no	—
276	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
277	Q₁₆	—	2	1	no	—
278	Q₁₆	—	2	1	no	—
279	Q₁₆	—	2	1	no	—
280	Q₁₆	#41×C₂₃	1	2	yes	—
281	SD₁₆	—	4	4	no	—
282	D₈	—	2	2	no	—
283	D₈	—	2	2	no	—
284	D₈	—	8	1	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
288	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
289	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
290	C₄⋊C₄	—	2	1	no	—
291	C₄⋊C₄	—	2	1	no	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	4	2	no	—
294	C₄⋊C₄	—	4	1	no	—
295	C₄⋊C₄	—	4	1	no	—
296	C₄⋊C₄	—	4	1	no	—
297	C₄⋊C₄	—	4	2	no	—
298	C₄⋊C₄	D₈×C₂₃	1	1	yes	characteristic
299	C₂²⋊C₄	—	4	2	no	—
300	C₂²⋊C₄	—	4	2	no	—
301	C₂²⋊C₄	D₈×C₂₃	1	2	yes	—
302	C₄²	—	4	1	no	—
303	C₄²	—	4	1	no	—
304	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
305	C₁₆	—	2	1	no	—
306	C₁₆	—	2	1	no	—
307	C₁₆	—	2	2	no	—
308	C₂³	—	4	2	no	—
309	C₂³	C₂×D₈×C₂₃	1	2	yes	—
310	Q₈	—	2	2	no	—
311	Q₈	—	4	1	no	—
312	Q₈	—	4	1	no	—
313	Q₈	—	4	1	no	—
314	Q₈	—	16	1	no	—
315	D₄	—	2	2	no	—
316	D₄	—	4	2	no	—
317	D₄	—	4	2	no	—
318	D₄	—	8	4	no	—
319	D₄	—	8	2	no	—
320	D₄	—	16	1	no	—
321	D₄	—	16	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	4	1	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	4	1	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	8	4	no	—
330	C₂×C₄	—	16	2	no	—
331	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
332	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
333	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
334	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
335	C₈	—	2	1	no	—
336	C₈	—	2	2	no	—
337	C₈	C₂×#41×C₂₃	1	1	yes	characteristic
338	C₈	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
339	C₂²	—	2	2	no	—
340	C₂²	—	2	2	no	—
341	C₂²	—	2	2	no	—
342	C₂²	—	8	2	no	—
343	C₂²	—	8	2	no	—
344	C₂²	—	8	2	no	—
345	C₂²	C₂²×D₈×C₂₃	1	1	yes	characteristic
346	C₄	—	2	2	no	—
347	C₄	—	2	1	no	—
348	C₄	—	4	2	no	—
349	C₄	—	8	1	no	—
350	C₄	—	8	1	no	—
351	C₄	—	8	2	no	—
352	C₄	—	8	1	no	—
353	C₄	C₂₃×#510	1	2	yes	—
354	C₄	C₂₃×#546	1	1	yes	characteristic
355	C₄	C₂₃×#577	1	1	yes	characteristic
356	C₂	—	4	2	no	—
357	C₂	—	16	2	no	—
358	C₂	C₂₃×#3294	1	1	yes	characteristic
359	C₂	C₂×C₂₃×#510	1	1	yes	characteristic
360	C₂	C₂₃×#3415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
prime radical	C₂₃×#3287	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×#2240	C₂	semidirect
4. C₂₃×#2232	C₂	semidirect
5. C₂₃×#2227	C₂	semidirect
6. C₂₃×#2220	C₂	semidirect
7. C₂₃×#2207	C₂	semidirect
8. C₂₃×#2189	C₂	semidirect
9. C₄×C₂₃×Q₃₂	C₂	semidirect
10. C₂₃×#2169	C₂	semidirect
11. C₂₃×#2139	C₂	semidirect
127. #33900	C₂₃	direct (N × H)
159. #3338	C₂×C₂₃	semidirect
161. #2240	C₂×C₂₃	semidirect
162. #2232	C₂×C₂₃	semidirect
163. #2227	C₂×C₂₃	semidirect
164. #2220	C₂×C₂₃	semidirect
165. #2207	C₂×C₂₃	semidirect
166. #2189	C₂×C₂₃	semidirect
167. C₄×Q₃₂	C₂×C₂₃	semidirect
168. #2169	C₂×C₂₃	semidirect
169. #2139	C₂×C₂₃	semidirect
262. C₂₃	#33900	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×#3338	2
C₂₃×#3287	2
C₂₃×#2240	2
C₂₃×#2232	2
C₂₃×#2227	2
C₂₃×#2220	2
C₂₃×#2207	2
C₂₃×#2189	2
C₄×C₂₃×Q₃₂	2
C₂₃×#2169	2
C₂₃×#2139	2
#33900	23

Subgroups — C23×#33900 #19181526

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33900 #19181526