Subgroups — C₂₃×#34637 #19182263

Order 5888.

2390 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#532	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2642	C₂	1	2	yes	maximal, complemented
4	C₂₃×#2641	C₂	1	2	yes	maximal, complemented
5	C₂₃×#2640	C₂	1	4	yes	maximal, complemented
6	C₂₃×#2639	C₂	1	8	yes	maximal, complemented
7	C₂₃×#2398	C₂	1	8	yes	maximal, complemented
8	C₂₃×#2297	C₂	1	4	yes	maximal, complemented
9	C₂×C₂₃×#376	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₂₃	C₂²	1	2	yes	—
11	C₂₃×#533	—	2	8	no	—
12	C₂₃×#532	—	2	8	no	—
13	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
15	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×#125	C₂²	1	2	yes	—
17	C₂×C₂₃×#124	C₂²	1	2	yes	—
18	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	4	yes	—
19	C₂×C₂₃×#121	C₂²	1	4	yes	—
20	C₂×C₄×Q₈×C₂₃	C₂²	1	4	yes	—
21	C₂×C₄×D₄×C₂₃	C₂²	1	4	yes	—
22	C₂×C₂₃×#117	C₂²	1	2	yes	—
23	C₂²×C₄²×C₂₃	C₂²	1	2	yes	—
24	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
25	C₂₃×#400	C₂²	1	16	yes	—
26	C₂₃×#397	C₂²	1	16	yes	—
27	C₂₃×#394	C₂²	1	16	yes	—
28	C₂₃×#393	C₂²	1	16	yes	—
29	C₂₃×#391	C₂²	1	8	yes	—
30	C₂₃×#390	C₂²	1	8	yes	—
31	C₂₃×#386	C₂²	1	16	yes	—
32	C₄×C₂²⋊C₄×C₂₃	C₂²	1	16	yes	—
33	C₂₃×#376	C₂²	1	4	yes	—
34	C₂₃×#376	C₂²	1	2	yes	—
35	C₂₃×#376	C₂²	1	2	yes	—
36	C₂×C₂₃×#95	C₂²	1	4	yes	—
37	C₂×#41×C₂₃	—	2	8	no	—
38	C₂×#41×C₂₃	—	2	16	no	—
39	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
40	C₂²×Q₈×C₂₃	C₂³	1	4	yes	—
41	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
42	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
43	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
44	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
45	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
46	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
47	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
48	C₂₃×C₄⋊Q₈	—	2	4	no	—
49	C₂₃×C₄⋊Q₈	—	2	8	no	—
50	C₂₃×#127	—	2	4	no	—
51	C₂₃×#125	—	2	8	no	—
52	C₂₃×#124	—	2	8	no	—
53	C₂₃×C₂²⋊Q₈	—	2	16	no	—
54	C₂₃×#121	—	2	16	no	—
55	C₄×Q₈×C₂₃	—	2	16	no	—
56	C₄×D₄×C₂₃	—	2	16	no	—
57	C₂₃×#117	—	2	8	no	—
58	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
59	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
60	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
61	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
62	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
63	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
64	C₂×C₂²⋊C₄×C₂₃	C₂³	1	32	yes	—
65	C₂×C₄²×C₂₃	—	2	8	no	—
66	C₂×C₄²×C₂₃	D₄	1	8	yes	—
67	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
68	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
69	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
70	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
72	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
73	C₂₃×#95	C₂³	1	8	yes	—
74	C₂₃×#95	C₂³	1	8	yes	—
75	C₂₃×#95	C₂³	1	32	yes	—
76	C₂⁴×C₂₃	C₂⁴	1	4	yes	—
77	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
78	#41×C₂₃	—	4	32	no	—
79	C₂×Q₈×C₂₃	—	2	8	no	—
80	C₂×Q₈×C₂₃	—	2	4	no	—
81	C₂×Q₈×C₂₃	—	2	8	no	—
82	C₂×Q₈×C₂₃	—	4	8	no	—
83	C₂×D₄×C₂₃	—	2	8	no	—
84	C₂×D₄×C₂₃	—	2	16	no	—
85	C₂×D₄×C₂₃	—	2	8	no	—
86	C₂×D₄×C₂₃	—	2	4	no	—
87	C₂×D₄×C₂₃	—	2	8	no	—
88	C₂×D₄×C₂₃	—	4	8	no	—
89	C₂²×C₄×C₂₃	—	2	8	no	—
90	C₂²×C₄×C₂₃	—	2	8	no	—
91	C₂²×C₄×C₂₃	—	2	16	no	—
92	C₂²×C₄×C₂₃	—	4	4	no	—
93	C₂²×C₄×C₂₃	—	4	8	no	—
94	C₂²×C₄×C₂₃	C₂×D₄	1	8	yes	—
95	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
96	C₂²×C₄×C₂₃	#41	1	8	yes	—
97	C₂²×C₄×C₂₃	C₂⁴	1	8	yes	—
98	C₂²×C₄×C₂₃	C₂⁴	1	8	yes	—
99	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
100	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
101	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
102	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
103	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
104	C₄⋊C₄×C₂₃	—	2	8	no	—
105	C₄⋊C₄×C₂₃	—	2	8	no	—
106	C₄⋊C₄×C₂₃	—	2	16	no	—
107	C₄⋊C₄×C₂₃	—	2	8	no	—
108	C₂²⋊C₄×C₂₃	—	2	8	no	—
109	C₂²⋊C₄×C₂₃	—	2	16	no	—
110	C₂²⋊C₄×C₂₃	—	4	32	no	—
111	C₄²×C₂₃	—	2	8	no	—
112	C₄²×C₂₃	—	2	8	no	—
113	C₄²×C₂₃	—	2	8	no	—
114	C₄²×C₂₃	C₂×D₄	1	8	yes	—
115	C₄²×C₂₃	C₂×D₄	1	4	yes	—
116	C₄²×C₂₃	C₂×D₄	1	4	yes	—
117	#34637	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₂³×C₂₃	—	2	4	no	—
119	C₂³×C₂₃	—	2	4	no	—
120	C₂³×C₂₃	—	2	8	no	—
121	C₂³×C₂₃	—	2	16	no	—
122	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
123	C₂³×C₂₃	C₂×#41	1	4	yes	—
124	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
125	Q₈×C₂₃	—	4	16	no	—
126	Q₈×C₂₃	—	4	8	no	—
127	D₄×C₂₃	—	4	16	no	—
128	D₄×C₂₃	—	4	16	no	—
129	D₄×C₂₃	—	4	8	no	—
130	C₂×C₄×C₂₃	—	2	4	no	—
131	C₂×C₄×C₂₃	—	2	4	no	—
132	C₂×C₄×C₂₃	—	2	8	no	—
133	C₂×C₄×C₂₃	—	2	4	no	—
134	C₂×C₄×C₂₃	—	2	8	no	—
135	C₂×C₄×C₂₃	—	2	8	no	—
136	C₂×C₄×C₂₃	—	2	16	no	—
137	C₂×C₄×C₂₃	—	2	16	no	—
138	C₂×C₄×C₂₃	—	4	8	no	—
139	C₂×C₄×C₂₃	—	4	16	no	—
140	C₂×C₄×C₂₃	—	4	16	no	—
141	C₂×C₄×C₂₃	#124	1	16	yes	—
142	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
143	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
144	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
145	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
146	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
147	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
148	C₂×#533	C₂×C₂₃	1	1	yes	characteristic, complemented
149	C₂×#532	C₂×C₂₃	1	1	yes	characteristic
150	#2642	C₂×C₂₃	1	2	yes	complemented
151	#2641	C₂×C₂₃	1	2	yes	complemented
152	#2640	C₂×C₂₃	1	4	yes	complemented
153	#2639	C₂×C₂₃	1	8	yes	complemented
154	#2398	C₂×C₂₃	1	8	yes	complemented
155	#2297	C₂×C₂₃	1	4	yes	complemented
156	C₂×#376	C₂×C₂₃	1	1	yes	characteristic, complemented
157	C₂²×C₂₃	—	2	4	no	—
158	C₂²×C₂₃	—	2	8	no	—
159	C₂²×C₂₃	—	4	4	no	—
160	C₂²×C₂₃	—	4	8	no	—
161	C₂²×C₂₃	—	4	16	no	—
162	C₂²×C₂₃	C₂×#124	1	4	yes	—
163	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
164	C₂²×C₂₃	C₂²×#41	1	2	yes	—
165	C₂²×C₂₃	#585	1	4	yes	—
166	C₄×C₂₃	—	2	4	no	—
167	C₄×C₂₃	—	2	4	no	—
168	C₄×C₂₃	—	4	4	no	—
169	C₄×C₂₃	—	4	8	no	—
170	C₄×C₂₃	—	4	8	no	—
171	C₄×C₂₃	C₂×#124	1	4	yes	—
172	C₄×C₂₃	C₂×#124	1	4	yes	—
173	C₂²×#41	C₂²×C₂₃	1	2	yes	—
174	#533	—	2	8	no	—
175	#532	—	2	8	no	—
176	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
177	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
178	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
179	C₂×#125	C₂²×C₂₃	1	2	yes	—
180	C₂×#124	C₂²×C₂₃	1	2	yes	—
181	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	4	yes	—
182	C₂×#121	C₂²×C₂₃	1	4	yes	—
183	C₂×C₄×Q₈	C₂²×C₂₃	1	4	yes	—
184	C₂×C₄×D₄	C₂²×C₂₃	1	4	yes	—
185	C₂×#117	C₂²×C₂₃	1	2	yes	—
186	C₂²×C₄²	C₂²×C₂₃	1	2	yes	—
187	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
188	#400	C₂²×C₂₃	1	16	yes	—
189	#397	C₂²×C₂₃	1	16	yes	—
190	#394	C₂²×C₂₃	1	16	yes	—
191	#393	C₂²×C₂₃	1	16	yes	—
192	#391	C₂²×C₂₃	1	8	yes	—
193	#390	C₂²×C₂₃	1	8	yes	—
194	#386	C₂²×C₂₃	1	16	yes	—
195	C₄×C₂²⋊C₄	C₂²×C₂₃	1	16	yes	—
196	#376	C₂²×C₂₃	1	2	yes	—
197	#376	C₂²×C₂₃	1	2	yes	—
198	#376	C₂²×C₂₃	1	4	yes	—
199	C₂×#95	C₂²×C₂₃	1	4	yes	—
200	C₂×C₂₃	—	2	4	no	—
201	C₂×C₂₃	—	8	4	no	—
202	C₂×C₂₃	C₂²×#124	1	1	yes	characteristic
203	C₂×C₂₃	#3449	1	2	yes	—
204	C₂×C₂₃	#3455	1	4	yes	—
205	C₂×#41	—	2	8	no	—
206	C₂×#41	—	2	16	no	—
207	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
208	C₂²×Q₈	C₂³×C₂₃	1	4	yes	—
209	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
210	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
211	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
212	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
213	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
214	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
215	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
216	C₄⋊Q₈	—	2	4	no	—
217	C₄⋊Q₈	—	2	8	no	—
218	#127	—	2	4	no	—
219	#125	—	2	8	no	—
220	#124	—	2	8	no	—
221	C₂²⋊Q₈	—	2	16	no	—
222	#121	—	2	16	no	—
223	C₄×Q₈	—	2	16	no	—
224	C₄×D₄	—	2	16	no	—
225	#117	—	2	8	no	—
226	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
227	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
228	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
229	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
230	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
231	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
232	C₂×C₂²⋊C₄	C₂³×C₂₃	1	32	yes	—
233	C₂×C₄²	—	2	8	no	—
234	C₂×C₄²	D₄×C₂₃	1	8	yes	—
235	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
236	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
237	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
238	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
239	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
240	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
241	#95	C₂³×C₂₃	1	8	yes	—
242	#95	C₂³×C₂₃	1	8	yes	—
243	#95	C₂³×C₂₃	1	32	yes	—
244	C₂₃	#34637	1	1	yes	characteristic, direct product factor, Sylow
245	C₂⁴	C₂⁴×C₂₃	1	4	yes	—
246	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
247	#41	—	4	32	no	—
248	C₂×Q₈	—	2	8	no	—
249	C₂×Q₈	—	2	4	no	—
250	C₂×Q₈	—	2	8	no	—
251	C₂×Q₈	—	4	8	no	—
252	C₂×D₄	—	2	8	no	—
253	C₂×D₄	—	2	16	no	—
254	C₂×D₄	—	2	8	no	—
255	C₂×D₄	—	2	4	no	—
256	C₂×D₄	—	2	8	no	—
257	C₂×D₄	—	4	8	no	—
258	C₂²×C₄	—	2	8	no	—
259	C₂²×C₄	—	2	8	no	—
260	C₂²×C₄	—	2	16	no	—
261	C₂²×C₄	—	4	4	no	—
262	C₂²×C₄	—	4	8	no	—
263	C₂²×C₄	C₂×D₄×C₂₃	1	8	yes	—
264	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
265	C₂²×C₄	#41×C₂₃	1	8	yes	—
266	C₂²×C₄	C₂⁴×C₂₃	1	8	yes	—
267	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
268	C₂²×C₄	C₂⁴×C₂₃	1	8	yes	—
269	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
270	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
271	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
272	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
273	C₄⋊C₄	—	2	8	no	—
274	C₄⋊C₄	—	2	8	no	—
275	C₄⋊C₄	—	2	16	no	—
276	C₄⋊C₄	—	2	8	no	—
277	C₂²⋊C₄	—	2	8	no	—
278	C₂²⋊C₄	—	2	16	no	—
279	C₂²⋊C₄	—	4	32	no	—
280	C₄²	—	2	8	no	—
281	C₄²	—	2	8	no	—
282	C₄²	—	2	8	no	—
283	C₄²	C₂×D₄×C₂₃	1	8	yes	—
284	C₄²	C₂×D₄×C₂₃	1	4	yes	—
285	C₄²	C₂×D₄×C₂₃	1	4	yes	—
286	C₂³	—	2	4	no	—
287	C₂³	—	2	4	no	—
288	C₂³	—	2	8	no	—
289	C₂³	—	2	16	no	—
290	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
291	C₂³	C₂×#41×C₂₃	1	4	yes	—
292	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
293	Q₈	—	4	16	no	—
294	Q₈	—	4	8	no	—
295	D₄	—	4	16	no	—
296	D₄	—	4	16	no	—
297	D₄	—	4	8	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	2	8	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	2	8	no	—
303	C₂×C₄	—	2	8	no	—
304	C₂×C₄	—	2	16	no	—
305	C₂×C₄	—	2	16	no	—
306	C₂×C₄	—	4	8	no	—
307	C₂×C₄	—	4	16	no	—
308	C₂×C₄	—	4	16	no	—
309	C₂×C₄	C₂₃×#124	1	16	yes	—
310	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
311	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
312	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
313	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
314	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
315	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
316	C₂²	—	2	4	no	—
317	C₂²	—	2	8	no	—
318	C₂²	—	4	4	no	—
319	C₂²	—	4	8	no	—
320	C₂²	—	4	16	no	—
321	C₂²	C₂×C₂₃×#124	1	4	yes	—
322	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
323	C₂²	C₂²×#41×C₂₃	1	2	yes	—
324	C₂²	C₂₃×#585	1	4	yes	—
325	C₄	—	2	4	no	—
326	C₄	—	2	4	no	—
327	C₄	—	4	4	no	—
328	C₄	—	4	8	no	—
329	C₄	—	4	8	no	—
330	C₄	C₂×C₂₃×#124	1	4	yes	—
331	C₄	C₂×C₂₃×#124	1	4	yes	—
332	C₂	—	2	4	no	—
333	C₂	—	8	4	no	—
334	C₂	C₂²×C₂₃×#124	1	1	yes	characteristic
335	C₂	C₂₃×#3449	1	2	yes	—
336	C₂	C₂₃×#3455	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₂₃
socle	C₂³×C₂₃	C₂⁵
prime radical	C₂×C₂₃×#532	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#533	C₂	semidirect
3. C₂₃×#2642	C₂	semidirect
4. C₂₃×#2641	C₂	semidirect
5. C₂₃×#2640	C₂	semidirect
6. C₂₃×#2639	C₂	semidirect
7. C₂₃×#2398	C₂	semidirect
8. C₂₃×#2297	C₂	semidirect
9. C₂×C₂₃×#376	C₂	semidirect
117. #34637	C₂₃	direct (N × H)
148. C₂×#533	C₂×C₂₃	semidirect
150. #2642	C₂×C₂₃	semidirect
151. #2641	C₂×C₂₃	semidirect
152. #2640	C₂×C₂₃	semidirect
153. #2639	C₂×C₂₃	semidirect
154. #2398	C₂×C₂₃	semidirect
155. #2297	C₂×C₂₃	semidirect
156. C₂×#376	C₂×C₂₃	semidirect
244. C₂₃	#34637	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#533	2
C₂×C₂₃×#532	2
C₂₃×#2642	2
C₂₃×#2641	2
C₂₃×#2640	2
C₂₃×#2639	2
C₂₃×#2398	2
C₂₃×#2297	2
C₂×C₂₃×#376	2
#34637	23

Subgroups — C23×#34637 #19182263

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#34637 #19182263