Subgroups — C₂₃×#19388 #19167014

Order 5888.

662 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3210	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3110	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3100	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#2972	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1681	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1679	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1669	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1665	C₂	1	2	yes	maximal, complemented
9	C₂₃×#1572	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1553	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1544	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#533	C₂²	1	1	yes	characteristic
13	C₂₃×#532	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×#494	C₂²	1	1	yes	characteristic, complemented
16	C₂₃×#492	C₂²	1	1	yes	characteristic, complemented
17	C₂₃×#487	C₂²	1	1	yes	characteristic, complemented
18	C₂₃×#486	C₂²	1	1	yes	characteristic, complemented
19	C₂₃×#484	—	2	1	no	—
20	C₂₃×#480	—	2	1	no	—
21	C₂₃×#474	C₂²	1	2	yes	—
22	C₂₃×#474	C₂²	1	2	yes	—
23	C₂₃×#464	—	2	2	no	—
24	C₂₃×#461	C₂²	1	2	yes	—
25	C₂₃×#459	C₂²	1	2	yes	—
26	C₂₃×#452	—	2	2	no	—
27	C₂₃×#450	—	2	1	no	—
28	C₂₃×#447	—	2	1	no	—
29	C₂₃×#440	—	2	2	no	—
30	C₄×D₈×C₂₃	—	2	2	no	—
31	C₈×D₄×C₂₃	—	2	2	no	—
32	C₈×D₄×C₂₃	C₂²	1	2	yes	—
33	C₈×D₄×C₂₃	C₂²	1	2	yes	—
34	C₂₃×#432	C₂²	1	1	yes	characteristic
35	C₂₃×#431	C₂²	1	1	yes	characteristic
36	C₂₃×#431	C₂²	1	1	yes	characteristic
37	C₂₃×#431	C₂²	1	1	yes	characteristic
38	C₂×C₂₃×C₂²⋊C₈	—	2	2	no	—
39	C₂₃×#335	C₂²	1	2	yes	complemented
40	C₂₃×#334	C₂²	1	2	yes	complemented
41	C₂₃×#332	C₂²	1	2	yes	complemented
42	C₂₃×#332	C₂²	1	2	yes	complemented
43	C₂₃×#332	C₂²	1	2	yes	complemented
44	C₂₃×#331	C₂²	1	2	yes	complemented
45	C₂×#41×C₂₃	—	2	1	no	—
46	C₂²×D₄×C₂₃	D₄	1	1	yes	characteristic
47	C₂³×C₄×C₂₃	—	2	1	no	—
48	C₂×Q₁₆×C₂₃	—	2	2	no	—
49	C₂×SD₁₆×C₂₃	—	2	2	no	—
50	C₂×SD₁₆×C₂₃	—	2	2	no	—
51	C₂×D₈×C₂₃	—	2	2	no	—
52	C₂²×C₈×C₂₃	—	2	2	no	—
53	C₂²×C₈×C₂₃	—	2	2	no	—
54	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
55	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
56	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
57	C₂₃×#127	C₂³	1	1	yes	characteristic
58	C₂₃×#125	—	2	1	no	—
59	C₂₃×#124	—	2	1	no	—
60	C₂₃×C₂²⋊Q₈	—	2	1	no	—
61	C₂₃×C₂²⋊Q₈	—	2	1	no	—
62	C₂₃×#121	—	2	1	no	—
63	C₂₃×#121	—	2	1	no	—
64	C₄×Q₈×C₂₃	—	2	1	no	—
65	C₄×Q₈×C₂₃	—	2	1	no	—
66	C₄×D₄×C₂₃	—	2	1	no	—
67	C₄×D₄×C₂₃	—	2	1	no	—
68	C₄×D₄×C₂₃	—	2	1	no	—
69	C₄×D₄×C₂₃	—	2	1	no	—
70	C₄×D₄×C₂₃	D₄	1	2	yes	—
71	C₄×D₄×C₂₃	C₂³	1	2	yes	—
72	C₂₃×#117	—	2	1	no	—
73	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
74	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
75	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
76	C₂₃×#107	—	2	2	no	—
77	C₂₃×#107	—	2	2	no	—
78	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
79	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
80	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
81	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
82	C₂₃×#103	—	2	2	no	—
83	C₂₃×#103	—	2	2	no	—
84	C₂₃×#102	—	2	2	no	—
85	C₂₃×#102	—	2	2	no	—
86	C₂₃×#102	—	2	2	no	—
87	C₂₃×#102	—	2	2	no	—
88	C₂₃×C₂²⋊C₈	—	2	1	no	—
89	C₂₃×C₂²⋊C₈	—	2	1	no	—
90	C₂₃×C₂²⋊C₈	—	2	1	no	—
91	C₂₃×C₂²⋊C₈	—	2	2	no	—
92	C₂₃×C₂²⋊C₈	—	2	2	no	—
93	C₂₃×C₂²⋊C₈	D₄	1	2	yes	—
94	C₂₃×#97	—	2	1	no	—
95	C₄×C₈×C₂₃	—	2	1	no	—
96	C₄×C₈×C₂₃	—	2	1	no	—
97	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
98	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
99	C₂⁴×C₂₃	—	2	1	no	—
100	#41×C₂₃	—	8	2	no	—
101	C₂×Q₈×C₂₃	—	2	1	no	—
102	C₂×Q₈×C₂₃	—	2	1	no	—
103	C₂×Q₈×C₂₃	—	2	1	no	—
104	C₂×D₄×C₂₃	—	2	4	no	—
105	C₂×D₄×C₂₃	—	2	1	no	—
106	C₂×D₄×C₂₃	—	2	1	no	—
107	C₂×D₄×C₂₃	—	2	1	no	—
108	C₂×D₄×C₂₃	—	2	1	no	—
109	C₂×D₄×C₂₃	—	4	1	no	—
110	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
111	C₂²×C₄×C₂₃	—	2	1	no	—
112	C₂²×C₄×C₂₃	—	2	2	no	—
113	C₂²×C₄×C₂₃	—	2	2	no	—
114	C₂²×C₄×C₂₃	—	4	1	no	—
115	C₂²×C₄×C₂₃	—	8	2	no	—
116	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
119	Q₁₆×C₂₃	—	8	2	no	—
120	SD₁₆×C₂₃	—	4	4	no	—
121	SD₁₆×C₂₃	—	8	2	no	—
122	D₈×C₂₃	—	4	4	no	—
123	C₂×C₈×C₂₃	—	2	2	no	—
124	C₂×C₈×C₂₃	—	2	2	no	—
125	C₂×C₈×C₂₃	—	2	2	no	—
126	C₂×C₈×C₂₃	—	8	2	no	—
127	C₂×C₈×C₂₃	—	8	2	no	—
128	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
129	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
130	C₄⋊C₄×C₂₃	—	2	1	no	—
131	C₄⋊C₄×C₂₃	—	2	1	no	—
132	C₄⋊C₄×C₂₃	—	2	1	no	—
133	C₄⋊C₄×C₂₃	—	2	1	no	—
134	C₄⋊C₄×C₂₃	—	2	1	no	—
135	C₄⋊C₄×C₂₃	—	2	1	no	—
136	C₄⋊C₄×C₂₃	—	2	1	no	—
137	C₄⋊C₄×C₂₃	—	4	1	no	—
138	C₄⋊C₄×C₂₃	—	4	1	no	—
139	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
140	C₂²⋊C₄×C₂₃	—	2	2	no	—
141	C₂²⋊C₄×C₂₃	—	2	2	no	—
142	C₂²⋊C₄×C₂₃	—	4	1	no	—
143	C₂²⋊C₄×C₂₃	—	4	1	no	—
144	C₂²⋊C₄×C₂₃	—	4	1	no	—
145	C₄²×C₂₃	—	4	1	no	—
146	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
147	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
148	C₄²×C₂₃	#41	1	1	yes	characteristic
149	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
150	#19388	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
151	C₂³×C₂₃	—	2	4	no	—
152	C₂³×C₂₃	—	2	2	no	—
153	C₂³×C₂₃	—	4	1	no	—
154	C₂³×C₂₃	—	8	1	no	—
155	C₂³×C₂₃	—	8	1	no	—
156	C₂³×C₂₃	#121	1	1	yes	characteristic
157	Q₈×C₂₃	—	4	2	no	—
158	Q₈×C₂₃	—	4	2	no	—
159	Q₈×C₂₃	—	4	2	no	—
160	D₄×C₂₃	—	2	4	no	—
161	D₄×C₂₃	—	2	2	no	—
162	D₄×C₂₃	—	4	2	no	—
163	D₄×C₂₃	—	4	2	no	—
164	D₄×C₂₃	—	8	1	no	—
165	D₄×C₂₃	—	8	2	no	—
166	D₄×C₂₃	#135	1	4	yes	—
167	C₂×C₄×C₂₃	—	2	2	no	—
168	C₂×C₄×C₂₃	—	2	2	no	—
169	C₂×C₄×C₂₃	—	2	2	no	—
170	C₂×C₄×C₂₃	—	4	1	no	—
171	C₂×C₄×C₂₃	—	4	2	no	—
172	C₂×C₄×C₂₃	—	4	2	no	—
173	C₂×C₄×C₂₃	—	4	1	no	—
174	C₂×C₄×C₂₃	—	4	1	no	—
175	C₂×C₄×C₂₃	—	4	1	no	—
176	C₂×C₄×C₂₃	—	8	2	no	—
177	C₂×C₄×C₂₃	—	8	2	no	—
178	C₂×C₄×C₂₃	—	8	1	no	—
179	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
180	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
181	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
182	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
183	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
184	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
185	C₈×C₂₃	—	2	2	no	—
186	C₈×C₂₃	—	2	2	no	—
187	C₈×C₂₃	—	4	2	no	—
188	C₈×C₂₃	—	4	2	no	—
189	C₈×C₂₃	—	4	2	no	—
190	#3210	C₂×C₂₃	1	1	yes	characteristic, complemented
191	#3110	C₂×C₂₃	1	1	yes	characteristic, complemented
192	#3100	C₂×C₂₃	1	1	yes	characteristic
193	#2972	C₂×C₂₃	1	1	yes	characteristic, complemented
194	#1681	C₂×C₂₃	1	2	yes	complemented
195	#1679	C₂×C₂₃	1	2	yes	complemented
196	#1669	C₂×C₂₃	1	2	yes	complemented
197	#1665	C₂×C₂₃	1	2	yes	complemented
198	#1572	C₂×C₂₃	1	1	yes	characteristic, complemented
199	#1553	C₂×C₂₃	1	1	yes	characteristic, complemented
200	#1544	C₂×C₂₃	1	1	yes	characteristic, complemented
201	C₂²×C₂₃	—	2	1	no	—
202	C₂²×C₂₃	—	2	1	no	—
203	C₂²×C₂₃	—	2	4	no	—
204	C₂²×C₂₃	—	2	1	no	—
205	C₂²×C₂₃	—	4	2	no	—
206	C₂²×C₂₃	—	4	2	no	—
207	C₂²×C₂₃	—	8	4	no	—
208	C₂²×C₂₃	—	8	1	no	—
209	C₂²×C₂₃	—	8	1	no	—
210	C₂²×C₂₃	—	8	1	no	—
211	C₂²×C₂₃	C₂×#121	1	1	yes	characteristic
212	C₄×C₂₃	—	2	1	no	—
213	C₄×C₂₃	—	2	1	no	—
214	C₄×C₂₃	—	2	1	no	—
215	C₄×C₂₃	—	2	1	no	—
216	C₄×C₂₃	—	4	2	no	—
217	C₄×C₂₃	—	8	1	no	—
218	C₄×C₂₃	—	8	1	no	—
219	C₄×C₂₃	—	8	1	no	—
220	C₄×C₂₃	C₂×#135	1	2	yes	—
221	C₄×C₂₃	#576	1	2	yes	—
222	#533	C₂²×C₂₃	1	1	yes	characteristic
223	#532	C₂²×C₂₃	1	1	yes	characteristic
224	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
225	#494	C₂²×C₂₃	1	1	yes	characteristic, complemented
226	#492	C₂²×C₂₃	1	1	yes	characteristic, complemented
227	#487	C₂²×C₂₃	1	1	yes	characteristic, complemented
228	#486	C₂²×C₂₃	1	1	yes	characteristic, complemented
229	#484	—	2	1	no	—
230	#480	—	2	1	no	—
231	#474	C₂²×C₂₃	1	2	yes	—
232	#474	C₂²×C₂₃	1	2	yes	—
233	#464	—	2	2	no	—
234	#461	C₂²×C₂₃	1	2	yes	—
235	#459	C₂²×C₂₃	1	2	yes	—
236	#452	—	2	2	no	—
237	#450	—	2	1	no	—
238	#447	—	2	1	no	—
239	#440	—	2	2	no	—
240	C₄×D₈	—	2	2	no	—
241	C₈×D₄	—	2	2	no	—
242	C₈×D₄	C₂²×C₂₃	1	2	yes	—
243	C₈×D₄	C₂²×C₂₃	1	2	yes	—
244	#432	C₂²×C₂₃	1	1	yes	characteristic
245	#431	C₂²×C₂₃	1	1	yes	characteristic
246	#431	C₂²×C₂₃	1	1	yes	characteristic
247	#431	C₂²×C₂₃	1	1	yes	characteristic
248	C₂×C₂²⋊C₈	—	2	2	no	—
249	#335	C₂²×C₂₃	1	2	yes	complemented
250	#334	C₂²×C₂₃	1	2	yes	complemented
251	#332	C₂²×C₂₃	1	2	yes	complemented
252	#332	C₂²×C₂₃	1	2	yes	complemented
253	#332	C₂²×C₂₃	1	2	yes	complemented
254	#331	C₂²×C₂₃	1	2	yes	complemented
255	C₂×C₂₃	—	4	1	no	—
256	C₂×C₂₃	—	4	4	no	—
257	C₂×C₂₃	—	16	1	no	—
258	C₂×C₂₃	#3041	1	1	yes	characteristic
259	C₂×C₂₃	#3050	1	1	yes	characteristic
260	C₂×C₂₃	#3069	1	1	yes	characteristic
261	C₂×#41	—	2	1	no	—
262	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
263	C₂³×C₄	—	2	1	no	—
264	C₂×Q₁₆	—	2	2	no	—
265	C₂×SD₁₆	—	2	2	no	—
266	C₂×SD₁₆	—	2	2	no	—
267	C₂×D₈	—	2	2	no	—
268	C₂²×C₈	—	2	2	no	—
269	C₂²×C₈	—	2	2	no	—
270	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
271	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
272	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
273	#127	C₂³×C₂₃	1	1	yes	characteristic
274	#125	—	2	1	no	—
275	#124	—	2	1	no	—
276	C₂²⋊Q₈	—	2	1	no	—
277	C₂²⋊Q₈	—	2	1	no	—
278	#121	—	2	1	no	—
279	#121	—	2	1	no	—
280	C₄×Q₈	—	2	1	no	—
281	C₄×Q₈	—	2	1	no	—
282	C₄×D₄	—	2	1	no	—
283	C₄×D₄	—	2	1	no	—
284	C₄×D₄	—	2	1	no	—
285	C₄×D₄	—	2	1	no	—
286	C₄×D₄	D₄×C₂₃	1	2	yes	—
287	C₄×D₄	C₂³×C₂₃	1	2	yes	—
288	#117	—	2	1	no	—
289	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
290	C₂×C₂²⋊C₄	—	2	1	no	—
291	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
292	#107	—	2	2	no	—
293	#107	—	2	2	no	—
294	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
295	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
296	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
297	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
298	#103	—	2	2	no	—
299	#103	—	2	2	no	—
300	#102	—	2	2	no	—
301	#102	—	2	2	no	—
302	#102	—	2	2	no	—
303	#102	—	2	2	no	—
304	C₂²⋊C₈	—	2	1	no	—
305	C₂²⋊C₈	—	2	1	no	—
306	C₂²⋊C₈	—	2	1	no	—
307	C₂²⋊C₈	—	2	2	no	—
308	C₂²⋊C₈	—	2	2	no	—
309	C₂²⋊C₈	D₄×C₂₃	1	2	yes	—
310	#97	—	2	1	no	—
311	C₄×C₈	—	2	1	no	—
312	C₄×C₈	—	2	1	no	—
313	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
314	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
315	C₂₃	#19388	1	1	yes	characteristic, direct product factor, Sylow
316	C₂⁴	—	2	1	no	—
317	#41	—	8	2	no	—
318	C₂×Q₈	—	2	1	no	—
319	C₂×Q₈	—	2	1	no	—
320	C₂×Q₈	—	2	1	no	—
321	C₂×D₄	—	2	4	no	—
322	C₂×D₄	—	2	1	no	—
323	C₂×D₄	—	2	1	no	—
324	C₂×D₄	—	2	1	no	—
325	C₂×D₄	—	2	1	no	—
326	C₂×D₄	—	4	1	no	—
327	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
328	C₂²×C₄	—	2	1	no	—
329	C₂²×C₄	—	2	2	no	—
330	C₂²×C₄	—	2	2	no	—
331	C₂²×C₄	—	4	1	no	—
332	C₂²×C₄	—	8	2	no	—
333	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
334	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
335	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
336	Q₁₆	—	8	2	no	—
337	SD₁₆	—	4	4	no	—
338	SD₁₆	—	8	2	no	—
339	D₈	—	4	4	no	—
340	C₂×C₈	—	2	2	no	—
341	C₂×C₈	—	2	2	no	—
342	C₂×C₈	—	2	2	no	—
343	C₂×C₈	—	8	2	no	—
344	C₂×C₈	—	8	2	no	—
345	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
346	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
347	C₄⋊C₄	—	2	1	no	—
348	C₄⋊C₄	—	2	1	no	—
349	C₄⋊C₄	—	2	1	no	—
350	C₄⋊C₄	—	2	1	no	—
351	C₄⋊C₄	—	2	1	no	—
352	C₄⋊C₄	—	2	1	no	—
353	C₄⋊C₄	—	2	1	no	—
354	C₄⋊C₄	—	4	1	no	—
355	C₄⋊C₄	—	4	1	no	—
356	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
357	C₂²⋊C₄	—	2	2	no	—
358	C₂²⋊C₄	—	2	2	no	—
359	C₂²⋊C₄	—	4	1	no	—
360	C₂²⋊C₄	—	4	1	no	—
361	C₂²⋊C₄	—	4	1	no	—
362	C₄²	—	4	1	no	—
363	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
364	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
365	C₄²	#41×C₂₃	1	1	yes	characteristic
366	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
367	C₂³	—	2	4	no	—
368	C₂³	—	2	2	no	—
369	C₂³	—	4	1	no	—
370	C₂³	—	8	1	no	—
371	C₂³	—	8	1	no	—
372	C₂³	C₂₃×#121	1	1	yes	characteristic
373	Q₈	—	4	2	no	—
374	Q₈	—	4	2	no	—
375	Q₈	—	4	2	no	—
376	D₄	—	2	4	no	—
377	D₄	—	2	2	no	—
378	D₄	—	4	2	no	—
379	D₄	—	4	2	no	—
380	D₄	—	8	1	no	—
381	D₄	—	8	2	no	—
382	D₄	C₂₃×#135	1	4	yes	—
383	C₂×C₄	—	2	2	no	—
384	C₂×C₄	—	2	2	no	—
385	C₂×C₄	—	2	2	no	—
386	C₂×C₄	—	4	1	no	—
387	C₂×C₄	—	4	2	no	—
388	C₂×C₄	—	4	2	no	—
389	C₂×C₄	—	4	1	no	—
390	C₂×C₄	—	4	1	no	—
391	C₂×C₄	—	4	1	no	—
392	C₂×C₄	—	8	2	no	—
393	C₂×C₄	—	8	2	no	—
394	C₂×C₄	—	8	1	no	—
395	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
396	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
397	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
398	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
399	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
400	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
401	C₈	—	2	2	no	—
402	C₈	—	2	2	no	—
403	C₈	—	4	2	no	—
404	C₈	—	4	2	no	—
405	C₈	—	4	2	no	—
406	C₂²	—	2	1	no	—
407	C₂²	—	2	1	no	—
408	C₂²	—	2	4	no	—
409	C₂²	—	2	1	no	—
410	C₂²	—	4	2	no	—
411	C₂²	—	4	2	no	—
412	C₂²	—	8	4	no	—
413	C₂²	—	8	1	no	—
414	C₂²	—	8	1	no	—
415	C₂²	—	8	1	no	—
416	C₂²	C₂×C₂₃×#121	1	1	yes	characteristic
417	C₄	—	2	1	no	—
418	C₄	—	2	1	no	—
419	C₄	—	2	1	no	—
420	C₄	—	2	1	no	—
421	C₄	—	4	2	no	—
422	C₄	—	8	1	no	—
423	C₄	—	8	1	no	—
424	C₄	—	8	1	no	—
425	C₄	C₂×C₂₃×#135	1	2	yes	—
426	C₄	C₂₃×#576	1	2	yes	—
427	C₂	—	4	1	no	—
428	C₂	—	4	4	no	—
429	C₂	—	16	1	no	—
430	C₂	C₂₃×#3041	1	1	yes	characteristic
431	C₂	C₂₃×#3050	1	1	yes	characteristic
432	C₂	C₂₃×#3069	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#3100	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3210	C₂	semidirect
2. C₂₃×#3110	C₂	semidirect
4. C₂₃×#2972	C₂	semidirect
5. C₂₃×#1681	C₂	semidirect
6. C₂₃×#1679	C₂	semidirect
7. C₂₃×#1669	C₂	semidirect
8. C₂₃×#1665	C₂	semidirect
9. C₂₃×#1572	C₂	semidirect
10. C₂₃×#1553	C₂	semidirect
11. C₂₃×#1544	C₂	semidirect
15. C₂₃×#494	C₂²	semidirect
16. C₂₃×#492	C₂²	semidirect
17. C₂₃×#487	C₂²	semidirect
18. C₂₃×#486	C₂²	semidirect
39. C₂₃×#335	C₂²	semidirect
40. C₂₃×#334	C₂²	semidirect
41. C₂₃×#332	C₂²	semidirect
42. C₂₃×#332	C₂²	semidirect
43. C₂₃×#332	C₂²	semidirect
44. C₂₃×#331	C₂²	semidirect
150. #19388	C₂₃	direct (N × H)
190. #3210	C₂×C₂₃	semidirect
191. #3110	C₂×C₂₃	semidirect
193. #2972	C₂×C₂₃	semidirect
194. #1681	C₂×C₂₃	semidirect
195. #1679	C₂×C₂₃	semidirect
196. #1669	C₂×C₂₃	semidirect
197. #1665	C₂×C₂₃	semidirect
198. #1572	C₂×C₂₃	semidirect
199. #1553	C₂×C₂₃	semidirect
200. #1544	C₂×C₂₃	semidirect
225. #494	C₂²×C₂₃	semidirect
226. #492	C₂²×C₂₃	semidirect
227. #487	C₂²×C₂₃	semidirect
228. #486	C₂²×C₂₃	semidirect
249. #335	C₂²×C₂₃	semidirect
250. #334	C₂²×C₂₃	semidirect
251. #332	C₂²×C₂₃	semidirect
252. #332	C₂²×C₂₃	semidirect
253. #332	C₂²×C₂₃	semidirect
254. #331	C₂²×C₂₃	semidirect
315. C₂₃	#19388	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×#3210	2
C₂₃×#3110	2
C₂₃×#3100	2
C₂₃×#2972	2
C₂₃×#1681	2
C₂₃×#1679	2
C₂₃×#1669	2
C₂₃×#1665	2
C₂₃×#1572	2
C₂₃×#1553	2
C₂₃×#1544	2
#19388	23

Subgroups — C23×#19388 #19167014

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19388 #19167014