Subgroups — C₂₃×#23822 #19171448

Order 5888.

934 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₂₃×#127	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2100	C₂	1	2	yes	maximal, complemented
4	C₂₃×#2071	C₂	1	2	yes	maximal, complemented
5	C₂₃×#2041	C₂	1	2	yes	maximal
6	C₂₃×#2023	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1762	C₂	1	4	yes	maximal, complemented
8	C₂₃×#1745	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
12	C₂×C₄×D₄×C₂₃	—	2	1	no	—
13	C₂×C₄×D₄×C₂₃	—	2	1	no	—
14	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
15	C₂×C₂₃×#102	C₂²	1	4	yes	—
16	C₂×C₂₃×#102	C₂²	1	4	yes	—
17	C₂×C₂₃×#97	C₂²	1	2	yes	—
18	C₂₃×#391	—	2	1	no	—
19	C₂₃×#390	—	2	1	no	—
20	C₂₃×#390	—	2	1	no	—
21	C₂₃×#390	C₂²	1	2	yes	—
22	C₂₃×#389	—	2	1	no	—
23	C₂₃×#389	C₂²	1	2	yes	—
24	C₂₃×#384	C₂²	1	1	yes	characteristic
25	C₂₃×#384	C₂²	1	1	yes	characteristic
26	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
27	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
28	C₄³×C₂₃	C₂²	1	1	yes	characteristic
29	C₂₃×#340	C₂²	1	4	yes	—
30	C₂₃×#340	C₂²	1	4	yes	—
31	C₂₃×#336	C₂²	1	2	yes	—
32	C₂₃×#336	C₂²	1	2	yes	—
33	C₂²×Q₈×C₂₃	—	2	1	no	—
34	C₂²×D₄×C₂₃	—	2	1	no	—
35	C₂²×D₄×C₂₃	—	2	1	no	—
36	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
37	C₂³×C₄×C₂₃	—	2	2	no	—
38	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
39	C₂₃×C₄⋊Q₈	—	2	4	no	—
40	C₂₃×#127	—	2	4	no	—
41	C₄×Q₈×C₂₃	—	4	4	no	—
42	C₄×D₄×C₂₃	—	2	8	no	—
43	C₄×D₄×C₂₃	—	4	4	no	—
44	C₄×D₄×C₂₃	—	4	4	no	—
45	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
46	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
47	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
48	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
49	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
50	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
51	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
52	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
53	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
54	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
55	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
56	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
57	C₂×C₄²×C₂₃	—	2	2	no	—
58	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
59	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
60	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
61	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
62	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
63	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
65	C₂₃×#102	—	2	8	no	—
66	C₂₃×#102	—	2	8	no	—
67	C₂₃×#97	—	2	4	no	—
68	C₂₃×#95	—	2	2	no	—
69	C₂₃×#95	—	2	2	no	—
70	C₂⁴×C₂₃	—	2	2	no	—
71	C₂×Q₈×C₂₃	—	2	4	no	—
72	C₂×Q₈×C₂₃	—	4	2	no	—
73	C₂×Q₈×C₂₃	—	4	2	no	—
74	C₂×D₄×C₂₃	—	2	4	no	—
75	C₂×D₄×C₂₃	—	2	4	no	—
76	C₂×D₄×C₂₃	—	2	4	no	—
77	C₂×D₄×C₂₃	—	2	4	no	—
78	C₂×D₄×C₂₃	—	2	2	no	—
79	C₂×D₄×C₂₃	—	4	2	no	—
80	C₂×D₄×C₂₃	—	4	2	no	—
81	C₂×D₄×C₂₃	—	4	2	no	—
82	C₂×D₄×C₂₃	—	4	2	no	—
83	C₂×D₄×C₂₃	#41	1	4	yes	—
84	C₂²×C₄×C₂₃	—	2	2	no	—
85	C₂²×C₄×C₂₃	—	2	2	no	—
86	C₂²×C₄×C₂₃	—	2	2	no	—
87	C₂²×C₄×C₂₃	—	4	4	no	—
88	C₂²×C₄×C₂₃	—	4	4	no	—
89	C₂²×C₄×C₂₃	—	4	4	no	—
90	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
97	C₂×C₈×C₂₃	—	2	4	no	—
98	C₂×C₈×C₂₃	—	2	4	no	—
99	C₂×C₈×C₂₃	—	2	4	no	—
100	C₄⋊C₄×C₂₃	—	2	4	no	—
101	C₄⋊C₄×C₂₃	—	2	2	no	—
102	C₄⋊C₄×C₂₃	—	2	4	no	—
103	C₄⋊C₄×C₂₃	—	2	4	no	—
104	C₄⋊C₄×C₂₃	—	4	2	no	—
105	C₄⋊C₄×C₂₃	—	4	2	no	—
106	C₄⋊C₄×C₂₃	—	4	2	no	—
107	C₄⋊C₄×C₂₃	—	4	2	no	—
108	C₄⋊C₄×C₂₃	—	4	4	no	—
109	C₄⋊C₄×C₂₃	#41	1	4	yes	—
110	C₂²⋊C₄×C₂₃	—	4	4	no	—
111	C₂²⋊C₄×C₂₃	—	4	4	no	—
112	C₂²⋊C₄×C₂₃	—	4	4	no	—
113	C₄²×C₂₃	—	2	2	no	—
114	C₄²×C₂₃	—	2	2	no	—
115	C₄²×C₂₃	—	2	2	no	—
116	C₄²×C₂₃	—	2	2	no	—
117	C₄²×C₂₃	—	2	2	no	—
118	C₄²×C₂₃	—	2	2	no	—
119	C₄²×C₂₃	—	4	4	no	—
120	C₄²×C₂₃	#41	1	4	yes	—
121	#23822	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₂³×C₂₃	—	2	4	no	—
123	C₂³×C₂₃	—	4	2	no	—
124	C₂³×C₂₃	—	4	2	no	—
125	C₂³×C₂₃	—	4	2	no	—
126	C₂³×C₂₃	—	4	2	no	—
127	C₂³×C₂₃	—	4	2	no	—
128	C₂³×C₂₃	—	4	2	no	—
129	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
130	Q₈×C₂₃	—	4	8	no	—
131	D₄×C₂₃	—	2	8	no	—
132	D₄×C₂₃	—	4	8	no	—
133	D₄×C₂₃	—	4	8	no	—
134	D₄×C₂₃	—	4	4	no	—
135	C₂×C₄×C₂₃	—	2	1	no	—
136	C₂×C₄×C₂₃	—	2	1	no	—
137	C₂×C₄×C₂₃	—	2	1	no	—
138	C₂×C₄×C₂₃	—	2	1	no	—
139	C₂×C₄×C₂₃	—	2	1	no	—
140	C₂×C₄×C₂₃	—	2	1	no	—
141	C₂×C₄×C₂₃	—	2	1	no	—
142	C₂×C₄×C₂₃	—	2	1	no	—
143	C₂×C₄×C₂₃	—	2	1	no	—
144	C₂×C₄×C₂₃	—	2	1	no	—
145	C₂×C₄×C₂₃	—	2	1	no	—
146	C₂×C₄×C₂₃	—	2	1	no	—
147	C₂×C₄×C₂₃	—	2	4	no	—
148	C₂×C₄×C₂₃	—	4	2	no	—
149	C₂×C₄×C₂₃	—	4	2	no	—
150	C₂×C₄×C₂₃	—	4	2	no	—
151	C₂×C₄×C₂₃	—	4	2	no	—
152	C₂×C₄×C₂₃	—	4	2	no	—
153	C₂×C₄×C₂₃	—	4	2	no	—
154	C₂×C₄×C₂₃	—	4	2	no	—
155	C₂×C₄×C₂₃	—	4	2	no	—
156	C₂×C₄×C₂₃	—	8	8	no	—
157	C₂×C₄×C₂₃	#124	1	2	yes	—
158	C₂×C₄×C₂₃	#124	1	2	yes	—
159	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
160	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
161	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
162	C₂×C₄×C₂₃	#137	1	2	yes	—
163	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
164	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
165	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
166	C₈×C₂₃	—	4	4	no	—
167	C₄×C₄⋊Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
168	C₄×#127	C₂×C₂₃	1	1	yes	characteristic
169	#2100	C₂×C₂₃	1	2	yes	complemented
170	#2071	C₂×C₂₃	1	2	yes	complemented
171	#2041	C₂×C₂₃	1	2	yes	—
172	#2023	C₂×C₂₃	1	2	yes	complemented
173	#1762	C₂×C₂₃	1	4	yes	complemented
174	#1745	C₂×C₂₃	1	1	yes	characteristic, complemented
175	C₂²×C₂₃	—	4	4	no	—
176	C₂²×C₂₃	—	4	4	no	—
177	C₂²×C₂₃	—	4	4	no	—
178	C₂²×C₂₃	—	8	2	no	—
179	C₂²×C₂₃	—	8	2	no	—
180	C₂²×C₂₃	—	8	2	no	—
181	C₂²×C₂₃	—	8	2	no	—
182	C₂²×C₂₃	#525	1	1	yes	characteristic
183	C₂²×C₂₃	#525	1	1	yes	characteristic
184	C₂²×C₂₃	C₂×#124	1	1	yes	characteristic
185	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
186	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
187	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
188	C₂²×C₂₃	C₂×#137	1	1	yes	characteristic
189	C₄×C₂₃	—	2	2	no	—
190	C₄×C₂₃	—	2	2	no	—
191	C₄×C₂₃	—	2	2	no	—
192	C₄×C₂₃	—	2	2	no	—
193	C₄×C₂₃	—	2	2	no	—
194	C₄×C₂₃	—	2	2	no	—
195	C₄×C₂₃	—	4	4	no	—
196	C₄×C₂₃	—	8	2	no	—
197	C₄×C₂₃	—	8	2	no	—
198	C₄×C₂₃	#529	1	4	yes	—
199	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
200	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
201	C₂×C₄×Q₈	—	2	1	no	—
202	C₂×C₄×D₄	—	2	1	no	—
203	C₂×C₄×D₄	—	2	1	no	—
204	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
205	C₂×#102	C₂²×C₂₃	1	4	yes	—
206	C₂×#102	C₂²×C₂₃	1	4	yes	—
207	C₂×#97	C₂²×C₂₃	1	2	yes	—
208	#391	—	2	1	no	—
209	#390	—	2	1	no	—
210	#390	—	2	1	no	—
211	#390	C₂²×C₂₃	1	2	yes	—
212	#389	—	2	1	no	—
213	#389	C₂²×C₂₃	1	2	yes	—
214	#384	C₂²×C₂₃	1	1	yes	characteristic
215	#384	C₂²×C₂₃	1	1	yes	characteristic
216	C₄×C₄⋊C₄	—	2	1	no	—
217	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
218	C₄³	C₂²×C₂₃	1	1	yes	characteristic
219	#340	C₂²×C₂₃	1	4	yes	—
220	#340	C₂²×C₂₃	1	4	yes	—
221	#336	C₂²×C₂₃	1	2	yes	—
222	#336	C₂²×C₂₃	1	2	yes	—
223	C₂×C₂₃	—	8	4	no	—
224	C₂×C₂₃	#2398	1	1	yes	characteristic
225	C₂×C₂₃	#3109	1	1	yes	characteristic
226	C₂×C₂₃	#3109	1	1	yes	characteristic
227	C₂×C₂₃	#3110	1	1	yes	characteristic
228	C₂×C₂₃	#3112	1	1	yes	characteristic
229	C₂×C₂₃	#3138	1	1	yes	characteristic
230	C₂×C₂₃	#3141	1	1	yes	characteristic
231	C₂²×Q₈	—	2	1	no	—
232	C₂²×D₄	—	2	1	no	—
233	C₂²×D₄	—	2	1	no	—
234	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
235	C₂³×C₄	—	2	2	no	—
236	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
237	C₄⋊Q₈	—	2	4	no	—
238	#127	—	2	4	no	—
239	C₄×Q₈	—	4	4	no	—
240	C₄×D₄	—	2	8	no	—
241	C₄×D₄	—	4	4	no	—
242	C₄×D₄	—	4	4	no	—
243	C₂×C₄⋊C₄	—	2	1	no	—
244	C₂×C₄⋊C₄	—	2	1	no	—
245	C₂×C₄⋊C₄	—	2	1	no	—
246	C₂×C₄⋊C₄	—	2	1	no	—
247	C₂×C₄⋊C₄	—	2	2	no	—
248	C₂×C₄⋊C₄	—	2	1	no	—
249	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
250	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
251	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
252	C₂×C₂²⋊C₄	—	2	2	no	—
253	C₂×C₂²⋊C₄	—	2	2	no	—
254	C₂×C₂²⋊C₄	—	2	2	no	—
255	C₂×C₄²	—	2	2	no	—
256	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
257	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
258	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
259	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
260	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
261	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
262	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
263	#102	—	2	8	no	—
264	#102	—	2	8	no	—
265	#97	—	2	4	no	—
266	#95	—	2	2	no	—
267	#95	—	2	2	no	—
268	C₂₃	#23822	1	1	yes	characteristic, direct product factor, Sylow
269	C₂⁴	—	2	2	no	—
270	C₂×Q₈	—	2	4	no	—
271	C₂×Q₈	—	4	2	no	—
272	C₂×Q₈	—	4	2	no	—
273	C₂×D₄	—	2	4	no	—
274	C₂×D₄	—	2	4	no	—
275	C₂×D₄	—	2	4	no	—
276	C₂×D₄	—	2	4	no	—
277	C₂×D₄	—	2	2	no	—
278	C₂×D₄	—	4	2	no	—
279	C₂×D₄	—	4	2	no	—
280	C₂×D₄	—	4	2	no	—
281	C₂×D₄	—	4	2	no	—
282	C₂×D₄	#41×C₂₃	1	4	yes	—
283	C₂²×C₄	—	2	2	no	—
284	C₂²×C₄	—	2	2	no	—
285	C₂²×C₄	—	2	2	no	—
286	C₂²×C₄	—	4	4	no	—
287	C₂²×C₄	—	4	4	no	—
288	C₂²×C₄	—	4	4	no	—
289	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
290	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
291	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
292	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
293	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
294	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
295	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
296	C₂×C₈	—	2	4	no	—
297	C₂×C₈	—	2	4	no	—
298	C₂×C₈	—	2	4	no	—
299	C₄⋊C₄	—	2	4	no	—
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	2	4	no	—
302	C₄⋊C₄	—	2	4	no	—
303	C₄⋊C₄	—	4	2	no	—
304	C₄⋊C₄	—	4	2	no	—
305	C₄⋊C₄	—	4	2	no	—
306	C₄⋊C₄	—	4	4	no	—
307	C₄⋊C₄	—	4	2	no	—
308	C₄⋊C₄	#41×C₂₃	1	4	yes	—
309	C₂²⋊C₄	—	4	4	no	—
310	C₂²⋊C₄	—	4	4	no	—
311	C₂²⋊C₄	—	4	4	no	—
312	C₄²	—	2	2	no	—
313	C₄²	—	2	2	no	—
314	C₄²	—	2	2	no	—
315	C₄²	—	2	2	no	—
316	C₄²	—	2	2	no	—
317	C₄²	—	2	2	no	—
318	C₄²	—	4	4	no	—
319	C₄²	#41×C₂₃	1	4	yes	—
320	C₂³	—	2	4	no	—
321	C₂³	—	4	2	no	—
322	C₂³	—	4	2	no	—
323	C₂³	—	4	2	no	—
324	C₂³	—	4	2	no	—
325	C₂³	—	4	2	no	—
326	C₂³	—	4	2	no	—
327	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
328	Q₈	—	4	8	no	—
329	D₄	—	2	8	no	—
330	D₄	—	4	8	no	—
331	D₄	—	4	8	no	—
332	D₄	—	4	4	no	—
333	C₂×C₄	—	2	1	no	—
334	C₂×C₄	—	2	1	no	—
335	C₂×C₄	—	2	1	no	—
336	C₂×C₄	—	2	1	no	—
337	C₂×C₄	—	2	1	no	—
338	C₂×C₄	—	2	1	no	—
339	C₂×C₄	—	2	1	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	4	no	—
346	C₂×C₄	—	4	2	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	4	2	no	—
349	C₂×C₄	—	4	2	no	—
350	C₂×C₄	—	4	2	no	—
351	C₂×C₄	—	4	2	no	—
352	C₂×C₄	—	4	2	no	—
353	C₂×C₄	—	4	2	no	—
354	C₂×C₄	—	8	8	no	—
355	C₂×C₄	C₂₃×#124	1	2	yes	—
356	C₂×C₄	C₂₃×#124	1	2	yes	—
357	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
358	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
359	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
360	C₂×C₄	C₂₃×#137	1	2	yes	—
361	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
362	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
363	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
364	C₈	—	4	4	no	—
365	C₂²	—	4	4	no	—
366	C₂²	—	4	4	no	—
367	C₂²	—	4	4	no	—
368	C₂²	—	8	2	no	—
369	C₂²	—	8	2	no	—
370	C₂²	—	8	2	no	—
371	C₂²	—	8	2	no	—
372	C₂²	C₂₃×#525	1	1	yes	characteristic
373	C₂²	C₂₃×#525	1	1	yes	characteristic
374	C₂²	C₂×C₂₃×#124	1	1	yes	characteristic
375	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
376	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
377	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
378	C₂²	C₂×C₂₃×#137	1	1	yes	characteristic
379	C₄	—	2	2	no	—
380	C₄	—	2	2	no	—
381	C₄	—	2	2	no	—
382	C₄	—	2	2	no	—
383	C₄	—	2	2	no	—
384	C₄	—	2	2	no	—
385	C₄	—	4	4	no	—
386	C₄	—	8	2	no	—
387	C₄	—	8	2	no	—
388	C₄	C₂₃×#529	1	4	yes	—
389	C₂	—	8	4	no	—
390	C₂	C₂₃×#2398	1	1	yes	characteristic
391	C₂	C₂₃×#3109	1	1	yes	characteristic
392	C₂	C₂₃×#3109	1	1	yes	characteristic
393	C₂	C₂₃×#3110	1	1	yes	characteristic
394	C₂	C₂₃×#3112	1	1	yes	characteristic
395	C₂	C₂₃×#3138	1	1	yes	characteristic
396	C₂	C₂₃×#3141	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂×C₂₃×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₄³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄×C₂₃×C₄⋊Q₈	C₂	semidirect
3. C₂₃×#2100	C₂	semidirect
4. C₂₃×#2071	C₂	semidirect
6. C₂₃×#2023	C₂	semidirect
7. C₂₃×#1762	C₂	semidirect
8. C₂₃×#1745	C₂	semidirect
121. #23822	C₂₃	direct (N × H)
167. C₄×C₄⋊Q₈	C₂×C₂₃	semidirect
169. #2100	C₂×C₂₃	semidirect
170. #2071	C₂×C₂₃	semidirect
172. #2023	C₂×C₂₃	semidirect
173. #1762	C₂×C₂₃	semidirect
174. #1745	C₂×C₂₃	semidirect
268. C₂₃	#23822	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₂₃×C₄⋊Q₈	2
C₄×C₂₃×#127	2
C₂₃×#2100	2
C₂₃×#2071	2
C₂₃×#2041	2
C₂₃×#2023	2
C₂₃×#1762	2
C₂₃×#1745	2
#23822	23

Subgroups — C23×#23822 #19171448

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#23822 #19171448