Subgroups — C₂₃×#33358 #19180984

Order 5888.

1252 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3449	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#410	C₂	1	2	yes	maximal, complemented
3	C₂₃×#2129	C₂	1	2	yes	maximal, complemented
4	C₂₃×#2128	C₂	1	2	yes	maximal, complemented
5	C₂₃×#2127	C₂	1	4	yes	maximal, complemented
6	C₂₃×#2126	C₂	1	4	yes	maximal, complemented
7	C₂²×#41×C₂₃	C₄	1	1	yes	characteristic, complemented
8	C₂²×#41×C₂₃	C₂²	1	1	yes	characteristic
9	C₂₃×#537	—	2	2	no	—
10	C₂₃×#536	C₄	1	2	yes	complemented
11	C₂₃×#536	C₄	1	1	yes	characteristic, complemented
12	C₂₃×#536	C₂²	1	1	yes	characteristic
13	C₂₃×#535	C₄	1	1	yes	characteristic, complemented
14	C₂₃×#535	C₂²	1	2	yes	—
15	C₂₃×#535	C₂²	1	1	yes	characteristic
16	C₂₃×#534	—	2	2	no	—
17	C₂₃×#532	C₄	1	2	yes	complemented
18	C₂₃×#532	C₂²	1	2	yes	—
19	C₂₃×#525	—	2	2	no	—
20	C₂₃×#525	—	2	2	no	—
21	C₂×C₂₃×#117	—	2	2	no	—
22	C₂×C₂₃×#117	C₄	1	1	yes	characteristic, complemented
23	C₂₃×#410	—	2	2	no	—
24	C₂₃×#410	—	2	2	no	—
25	C₂₃×#410	C₂²	1	4	yes	—
26	C₂₃×#410	C₂²	1	4	yes	complemented
27	C₂×C₂₃×#99	—	2	2	no	—
28	C₂×C₂₃×#99	C₂²	1	4	yes	complemented
29	C₂₃×#354	C₂²	1	8	yes	complemented
30	C₂₃×#353	C₂²	1	8	yes	complemented
31	C₂×#41×C₂₃	—	2	1	no	—
32	C₂×#41×C₂₃	—	2	1	no	—
33	C₂×#41×C₂₃	—	4	2	no	—
34	C₂×#41×C₂₃	—	4	2	no	—
35	C₂×#41×C₂₃	—	4	2	no	—
36	C₂×#41×C₂₃	—	4	2	no	—
37	C₂×#41×C₂₃	C₂×C₄	1	2	yes	—
38	C₂×#41×C₂₃	C₂×C₄	1	1	yes	characteristic
39	C₂×#41×C₂₃	C₂×C₄	1	1	yes	characteristic
40	C₂×#41×C₂₃	C₂×C₄	1	2	yes	complemented
41	C₂×#41×C₂₃	D₄	1	2	yes	—
42	C₂×#41×C₂₃	D₄	1	2	yes	—
43	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
44	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
45	C₂²×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
46	C₂²×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic, complemented
47	C₂²×D₄×C₂₃	—	2	1	no	—
48	C₂²×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic, complemented
49	C₂²×D₄×C₂₃	D₄	1	2	yes	—
50	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
51	C₂³×C₄×C₂₃	—	2	1	no	—
52	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
53	C₂³×C₄×C₂₃	D₄	1	2	yes	—
54	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	complemented
55	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	complemented
56	C₂₃×#127	C₂×C₄	1	2	yes	—
57	C₂₃×#127	C₂³	1	2	yes	—
58	C₂₃×#124	C₂×C₄	1	2	yes	—
59	C₂₃×#124	C₂×C₄	1	2	yes	complemented
60	C₂₃×#124	C₂×C₄	1	2	yes	complemented
61	C₂₃×#124	C₂³	1	2	yes	—
62	C₂₃×#123	—	2	2	no	—
63	C₂₃×#123	—	2	2	no	—
64	C₂₃×#123	—	2	4	no	—
65	C₂₃×C₂²⋊Q₈	—	2	4	no	—
66	C₂₃×C₂²⋊Q₈	—	2	2	no	—
67	C₂₃×C₂²⋊Q₈	—	2	2	no	—
68	C₂₃×#121	—	2	4	no	—
69	C₂₃×#121	—	2	4	no	—
70	C₂₃×#121	—	2	4	no	—
71	C₂₃×#121	—	2	2	no	—
72	C₂₃×#121	—	2	2	no	—
73	C₂₃×#120	—	2	4	no	—
74	C₂₃×#120	—	2	2	no	—
75	C₂₃×#120	—	2	2	no	—
76	C₄×D₄×C₂₃	—	2	4	no	—
77	C₄×D₄×C₂₃	—	2	4	no	—
78	C₂₃×#117	—	2	1	no	—
79	C₂₃×#117	—	2	1	no	—
80	C₂₃×#117	—	2	4	no	—
81	C₂₃×#117	—	2	4	no	—
82	C₂₃×#117	—	4	2	no	—
83	C₂₃×#117	—	4	2	no	—
84	C₂₃×#117	C₂×C₄	1	2	yes	—
85	C₂₃×#117	C₂×C₄	1	1	yes	characteristic, complemented
86	C₂₃×#117	C₂×C₄	1	1	yes	characteristic, complemented
87	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
88	C₂×C₄⋊C₄×C₂₃	—	4	2	no	—
89	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
90	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
91	C₂×C₄²×C₂₃	—	4	2	no	—
92	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	complemented
93	C₂₃×#99	—	2	4	no	—
94	C₂₃×#99	—	2	4	no	—
95	C₂₃×#99	—	2	4	no	—
96	C₂₃×#99	C₂³	1	8	yes	—
97	C₂⁴×C₂₃	—	2	1	no	—
98	C₂⁴×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
99	C₂⁴×C₂₃	C₂×D₄	1	2	yes	—
100	#41×C₂₃	—	4	2	no	—
101	#41×C₂₃	—	4	1	no	—
102	#41×C₂₃	—	4	1	no	—
103	#41×C₂₃	—	4	2	no	—
104	#41×C₂₃	—	4	2	no	—
105	#41×C₂₃	—	4	2	no	—
106	#41×C₂₃	—	4	2	no	—
107	#41×C₂₃	—	4	2	no	—
108	#41×C₂₃	—	4	1	no	—
109	#41×C₂₃	—	4	1	no	—
110	#41×C₂₃	—	4	4	no	—
111	#41×C₂₃	—	4	4	no	—
112	#41×C₂₃	—	4	2	no	—
113	#41×C₂₃	—	4	2	no	—
114	#41×C₂₃	—	4	2	no	—
115	#41×C₂₃	—	4	2	no	—
116	C₂×Q₈×C₂₃	—	2	1	no	—
117	C₂×Q₈×C₂₃	—	4	2	no	—
118	C₂×Q₈×C₂₃	—	4	2	no	—
119	C₂×Q₈×C₂₃	C₂²⋊C₄	1	2	yes	—
120	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
121	C₂×Q₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
122	C₂×Q₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
123	C₂×D₄×C₂₃	—	2	2	no	—
124	C₂×D₄×C₂₃	—	2	2	no	—
125	C₂×D₄×C₂₃	—	2	4	no	—
126	C₂×D₄×C₂₃	—	2	4	no	—
127	C₂×D₄×C₂₃	—	2	1	no	—
128	C₂×D₄×C₂₃	—	4	2	no	—
129	C₂×D₄×C₂₃	—	4	2	no	—
130	C₂×D₄×C₂₃	—	4	2	no	—
131	C₂×D₄×C₂₃	—	4	2	no	—
132	C₂×D₄×C₂₃	—	4	4	no	—
133	C₂×D₄×C₂₃	C₂²⋊C₄	1	2	yes	—
134	C₂×D₄×C₂₃	C₂²⋊C₄	1	2	yes	—
135	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
136	C₂×D₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
137	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
138	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
139	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
140	C₂×D₄×C₂₃	C₂⁴	1	1	yes	characteristic
141	C₂²×C₄×C₂₃	—	2	2	no	—
142	C₂²×C₄×C₂₃	—	2	2	no	—
143	C₂²×C₄×C₂₃	—	2	2	no	—
144	C₂²×C₄×C₂₃	—	2	2	no	—
145	C₂²×C₄×C₂₃	—	2	2	no	—
146	C₂²×C₄×C₂₃	—	2	2	no	—
147	C₂²×C₄×C₂₃	—	2	1	no	—
148	C₂²×C₄×C₂₃	—	2	2	no	—
149	C₂²×C₄×C₂₃	—	2	2	no	—
150	C₂²×C₄×C₂₃	—	2	2	no	—
151	C₂²×C₄×C₂₃	—	4	2	no	—
152	C₂²×C₄×C₂₃	—	4	2	no	—
153	C₂²×C₄×C₂₃	—	4	4	no	—
154	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
155	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
156	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
157	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
158	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
159	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
160	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
161	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
162	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
163	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
164	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
165	C₄⋊C₄×C₂₃	—	2	2	no	—
166	C₄⋊C₄×C₂₃	—	2	1	no	—
167	C₄⋊C₄×C₂₃	—	2	1	no	—
168	C₄⋊C₄×C₂₃	—	2	4	no	—
169	C₄⋊C₄×C₂₃	—	2	4	no	—
170	C₄⋊C₄×C₂₃	—	4	4	no	—
171	C₄⋊C₄×C₂₃	—	4	4	no	—
172	C₂²⋊C₄×C₂₃	—	2	2	no	—
173	C₂²⋊C₄×C₂₃	—	2	1	no	—
174	C₂²⋊C₄×C₂₃	—	2	1	no	—
175	C₂²⋊C₄×C₂₃	—	2	4	no	—
176	C₂²⋊C₄×C₂₃	—	2	4	no	—
177	C₂²⋊C₄×C₂₃	—	2	4	no	—
178	C₂²⋊C₄×C₂₃	—	2	4	no	—
179	C₂²⋊C₄×C₂₃	—	2	8	no	—
180	C₂²⋊C₄×C₂₃	—	4	4	no	—
181	C₄²×C₂₃	—	2	2	no	—
182	C₄²×C₂₃	—	4	4	no	—
183	C₄²×C₂₃	—	4	4	no	—
184	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
185	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
186	#33358	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
187	C₂³×C₂₃	—	2	2	no	—
188	C₂³×C₂₃	—	2	2	no	—
189	C₂³×C₂₃	—	2	2	no	—
190	C₂³×C₂₃	—	2	2	no	—
191	C₂³×C₂₃	—	4	2	no	—
192	C₂³×C₂₃	—	8	2	no	—
193	C₂³×C₂₃	—	8	1	no	—
194	C₂³×C₂₃	—	8	2	no	—
195	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
196	C₂³×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
197	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
198	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
199	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
200	Q₈×C₂₃	—	4	2	no	—
201	Q₈×C₂₃	—	4	1	no	—
202	Q₈×C₂₃	—	4	1	no	—
203	Q₈×C₂₃	—	4	2	no	—
204	Q₈×C₂₃	—	4	2	no	—
205	D₄×C₂₃	—	4	2	no	—
206	D₄×C₂₃	—	4	2	no	—
207	D₄×C₂₃	—	4	1	no	—
208	D₄×C₂₃	—	4	2	no	—
209	D₄×C₂₃	—	4	2	no	—
210	D₄×C₂₃	—	4	2	no	—
211	D₄×C₂₃	—	4	1	no	—
212	D₄×C₂₃	—	4	2	no	—
213	D₄×C₂₃	—	4	4	no	—
214	D₄×C₂₃	—	4	4	no	—
215	D₄×C₂₃	—	4	2	no	—
216	D₄×C₂₃	—	8	4	no	—
217	D₄×C₂₃	—	8	4	no	—
218	C₂×C₄×C₂₃	—	2	2	no	—
219	C₂×C₄×C₂₃	—	2	2	no	—
220	C₂×C₄×C₂₃	—	2	2	no	—
221	C₂×C₄×C₂₃	—	2	2	no	—
222	C₂×C₄×C₂₃	—	2	2	no	—
223	C₂×C₄×C₂₃	—	2	2	no	—
224	C₂×C₄×C₂₃	—	2	1	no	—
225	C₂×C₄×C₂₃	—	2	2	no	—
226	C₂×C₄×C₂₃	—	2	2	no	—
227	C₂×C₄×C₂₃	—	2	2	no	—
228	C₂×C₄×C₂₃	—	2	2	no	—
229	C₂×C₄×C₂₃	—	2	2	no	—
230	C₂×C₄×C₂₃	—	2	1	no	—
231	C₂×C₄×C₂₃	—	2	2	no	—
232	C₂×C₄×C₂₃	—	2	1	no	—
233	C₂×C₄×C₂₃	—	2	2	no	—
234	C₂×C₄×C₂₃	—	2	1	no	—
235	C₂×C₄×C₂₃	—	2	2	no	—
236	C₂×C₄×C₂₃	—	2	2	no	—
237	C₂×C₄×C₂₃	—	2	2	no	—
238	C₂×C₄×C₂₃	—	4	4	no	—
239	C₂×C₄×C₂₃	—	4	2	no	—
240	C₂×C₄×C₂₃	—	4	2	no	—
241	C₂×C₄×C₂₃	—	4	2	no	—
242	C₂×C₄×C₂₃	—	4	8	no	—
243	C₂×C₄×C₂₃	—	8	2	no	—
244	C₂×C₄×C₂₃	—	8	8	no	—
245	C₂×C₄×C₂₃	#99	1	2	yes	—
246	C₂×C₄×C₂₃	#99	1	2	yes	—
247	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
248	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
249	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
250	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
251	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
252	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
253	#3449	C₂×C₂₃	1	1	yes	characteristic
254	C₂×#410	C₂×C₂₃	1	2	yes	complemented
255	#2129	C₂×C₂₃	1	2	yes	complemented
256	#2128	C₂×C₂₃	1	2	yes	complemented
257	#2127	C₂×C₂₃	1	4	yes	complemented
258	#2126	C₂×C₂₃	1	4	yes	complemented
259	C₂²×C₂₃	—	2	2	no	—
260	C₂²×C₂₃	—	2	1	no	—
261	C₂²×C₂₃	—	2	1	no	—
262	C₂²×C₂₃	—	2	2	no	—
263	C₂²×C₂₃	—	4	1	no	—
264	C₂²×C₂₃	—	4	1	no	—
265	C₂²×C₂₃	—	4	2	no	—
266	C₂²×C₂₃	—	4	2	no	—
267	C₂²×C₂₃	—	4	1	no	—
268	C₂²×C₂₃	—	4	4	no	—
269	C₂²×C₂₃	—	4	4	no	—
270	C₂²×C₂₃	—	4	2	no	—
271	C₂²×C₂₃	—	8	2	no	—
272	C₂²×C₂₃	—	8	2	no	—
273	C₂²×C₂₃	—	8	4	no	—
274	C₂²×C₂₃	C₂×#99	1	2	yes	—
275	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
276	C₄×C₂₃	—	2	1	no	—
277	C₄×C₂₃	—	2	2	no	—
278	C₄×C₂₃	—	2	2	no	—
279	C₄×C₂₃	—	2	1	no	—
280	C₄×C₂₃	—	4	2	no	—
281	C₄×C₂₃	—	4	2	no	—
282	C₄×C₂₃	—	4	2	no	—
283	C₄×C₂₃	—	8	8	no	—
284	C₄×C₂₃	C₂×#99	1	1	yes	characteristic
285	C₄×C₂₃	C₂×#99	1	2	yes	—
286	C₄×C₂₃	C₂×#99	1	1	yes	characteristic
287	C₂²×#41	C₄×C₂₃	1	1	yes	characteristic, complemented
288	C₂²×#41	C₂²×C₂₃	1	1	yes	characteristic
289	#537	—	2	2	no	—
290	#536	C₄×C₂₃	1	2	yes	complemented
291	#536	C₄×C₂₃	1	1	yes	characteristic, complemented
292	#536	C₂²×C₂₃	1	1	yes	characteristic
293	#535	C₄×C₂₃	1	1	yes	characteristic, complemented
294	#535	C₂²×C₂₃	1	2	yes	—
295	#535	C₂²×C₂₃	1	1	yes	characteristic
296	#534	—	2	2	no	—
297	#532	C₄×C₂₃	1	2	yes	complemented
298	#532	C₂²×C₂₃	1	2	yes	—
299	#525	—	2	2	no	—
300	#525	—	2	2	no	—
301	C₂×#117	—	2	2	no	—
302	C₂×#117	C₄×C₂₃	1	1	yes	characteristic, complemented
303	#410	—	2	2	no	—
304	#410	—	2	2	no	—
305	#410	C₂²×C₂₃	1	4	yes	—
306	#410	C₂²×C₂₃	1	4	yes	complemented
307	C₂×#99	—	2	2	no	—
308	C₂×#99	C₂²×C₂₃	1	4	yes	complemented
309	#354	C₂²×C₂₃	1	8	yes	complemented
310	#353	C₂²×C₂₃	1	8	yes	complemented
311	C₂×C₂₃	—	2	1	no	—
312	C₂×C₂₃	—	2	2	no	—
313	C₂×C₂₃	—	4	2	no	—
314	C₂×C₂₃	—	4	2	no	—
315	C₂×C₂₃	—	4	1	no	—
316	C₂×C₂₃	—	4	1	no	—
317	C₂×C₂₃	—	8	2	no	—
318	C₂×C₂₃	C₂²×#99	1	1	yes	characteristic
319	C₂×#41	—	2	1	no	—
320	C₂×#41	—	2	1	no	—
321	C₂×#41	—	4	2	no	—
322	C₂×#41	—	4	2	no	—
323	C₂×#41	—	4	2	no	—
324	C₂×#41	—	4	2	no	—
325	C₂×#41	C₂×C₄×C₂₃	1	2	yes	—
326	C₂×#41	C₂×C₄×C₂₃	1	1	yes	characteristic
327	C₂×#41	C₂×C₄×C₂₃	1	1	yes	characteristic
328	C₂×#41	C₂×C₄×C₂₃	1	2	yes	complemented
329	C₂×#41	D₄×C₂₃	1	2	yes	—
330	C₂×#41	D₄×C₂₃	1	2	yes	—
331	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
332	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
333	C₂²×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
334	C₂²×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic, complemented
335	C₂²×D₄	—	2	1	no	—
336	C₂²×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic, complemented
337	C₂²×D₄	D₄×C₂₃	1	2	yes	—
338	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
339	C₂³×C₄	—	2	1	no	—
340	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
341	C₂³×C₄	D₄×C₂₃	1	2	yes	—
342	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	complemented
343	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	complemented
344	#127	C₂×C₄×C₂₃	1	2	yes	—
345	#127	C₂³×C₂₃	1	2	yes	—
346	#124	C₂×C₄×C₂₃	1	2	yes	—
347	#124	C₂×C₄×C₂₃	1	2	yes	complemented
348	#124	C₂×C₄×C₂₃	1	2	yes	complemented
349	#124	C₂³×C₂₃	1	2	yes	—
350	#123	—	2	2	no	—
351	#123	—	2	2	no	—
352	#123	—	2	4	no	—
353	C₂²⋊Q₈	—	2	4	no	—
354	C₂²⋊Q₈	—	2	2	no	—
355	C₂²⋊Q₈	—	2	2	no	—
356	#121	—	2	4	no	—
357	#121	—	2	4	no	—
358	#121	—	2	4	no	—
359	#121	—	2	2	no	—
360	#121	—	2	2	no	—
361	#120	—	2	4	no	—
362	#120	—	2	2	no	—
363	#120	—	2	2	no	—
364	C₄×D₄	—	2	4	no	—
365	C₄×D₄	—	2	4	no	—
366	#117	—	2	1	no	—
367	#117	—	2	1	no	—
368	#117	—	2	4	no	—
369	#117	—	2	4	no	—
370	#117	—	4	2	no	—
371	#117	—	4	2	no	—
372	#117	C₂×C₄×C₂₃	1	2	yes	—
373	#117	C₂×C₄×C₂₃	1	1	yes	characteristic, complemented
374	#117	C₂×C₄×C₂₃	1	1	yes	characteristic, complemented
375	C₂×C₄⋊C₄	—	2	1	no	—
376	C₂×C₄⋊C₄	—	4	2	no	—
377	C₂×C₂²⋊C₄	—	2	1	no	—
378	C₂×C₂²⋊C₄	—	2	4	no	—
379	C₂×C₄²	—	4	2	no	—
380	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	complemented
381	#99	—	2	4	no	—
382	#99	—	2	4	no	—
383	#99	—	2	4	no	—
384	#99	C₂³×C₂₃	1	8	yes	—
385	C₂₃	#33358	1	1	yes	characteristic, direct product factor, Sylow
386	C₂⁴	—	2	1	no	—
387	C₂⁴	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
388	C₂⁴	C₂×D₄×C₂₃	1	2	yes	—
389	#41	—	4	2	no	—
390	#41	—	4	1	no	—
391	#41	—	4	1	no	—
392	#41	—	4	2	no	—
393	#41	—	4	2	no	—
394	#41	—	4	2	no	—
395	#41	—	4	2	no	—
396	#41	—	4	1	no	—
397	#41	—	4	2	no	—
398	#41	—	4	1	no	—
399	#41	—	4	4	no	—
400	#41	—	4	4	no	—
401	#41	—	4	2	no	—
402	#41	—	4	2	no	—
403	#41	—	4	2	no	—
404	#41	—	4	2	no	—
405	C₂×Q₈	—	2	1	no	—
406	C₂×Q₈	—	4	2	no	—
407	C₂×Q₈	—	4	2	no	—
408	C₂×Q₈	C₂²⋊C₄×C₂₃	1	2	yes	—
409	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
410	C₂×Q₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
411	C₂×Q₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
412	C₂×D₄	—	2	2	no	—
413	C₂×D₄	—	2	2	no	—
414	C₂×D₄	—	2	4	no	—
415	C₂×D₄	—	2	4	no	—
416	C₂×D₄	—	2	1	no	—
417	C₂×D₄	—	4	2	no	—
418	C₂×D₄	—	4	2	no	—
419	C₂×D₄	—	4	2	no	—
420	C₂×D₄	—	4	2	no	—
421	C₂×D₄	—	4	4	no	—
422	C₂×D₄	C₂²⋊C₄×C₂₃	1	2	yes	—
423	C₂×D₄	C₂²⋊C₄×C₂₃	1	2	yes	—
424	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
425	C₂×D₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
426	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
427	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
428	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
429	C₂×D₄	C₂⁴×C₂₃	1	1	yes	characteristic
430	C₂²×C₄	—	2	2	no	—
431	C₂²×C₄	—	2	2	no	—
432	C₂²×C₄	—	2	2	no	—
433	C₂²×C₄	—	2	2	no	—
434	C₂²×C₄	—	2	2	no	—
435	C₂²×C₄	—	2	2	no	—
436	C₂²×C₄	—	2	1	no	—
437	C₂²×C₄	—	2	2	no	—
438	C₂²×C₄	—	2	2	no	—
439	C₂²×C₄	—	2	2	no	—
440	C₂²×C₄	—	4	2	no	—
441	C₂²×C₄	—	4	2	no	—
442	C₂²×C₄	—	4	4	no	—
443	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
444	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
445	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
446	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
447	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
448	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
449	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
450	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
451	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
452	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
453	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
454	C₄⋊C₄	—	2	2	no	—
455	C₄⋊C₄	—	2	1	no	—
456	C₄⋊C₄	—	2	1	no	—
457	C₄⋊C₄	—	2	4	no	—
458	C₄⋊C₄	—	2	4	no	—
459	C₄⋊C₄	—	4	4	no	—
460	C₄⋊C₄	—	4	4	no	—
461	C₂²⋊C₄	—	2	2	no	—
462	C₂²⋊C₄	—	2	1	no	—
463	C₂²⋊C₄	—	2	1	no	—
464	C₂²⋊C₄	—	2	4	no	—
465	C₂²⋊C₄	—	2	4	no	—
466	C₂²⋊C₄	—	2	4	no	—
467	C₂²⋊C₄	—	2	4	no	—
468	C₂²⋊C₄	—	2	8	no	—
469	C₂²⋊C₄	—	4	4	no	—
470	C₄²	—	2	2	no	—
471	C₄²	—	4	4	no	—
472	C₄²	—	4	4	no	—
473	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
474	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
475	C₂³	—	2	2	no	—
476	C₂³	—	2	2	no	—
477	C₂³	—	2	2	no	—
478	C₂³	—	2	2	no	—
479	C₂³	—	4	2	no	—
480	C₂³	—	8	2	no	—
481	C₂³	—	8	1	no	—
482	C₂³	—	8	2	no	—
483	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
484	C₂³	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
485	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
486	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
487	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
488	Q₈	—	4	2	no	—
489	Q₈	—	4	1	no	—
490	Q₈	—	4	1	no	—
491	Q₈	—	4	2	no	—
492	Q₈	—	4	2	no	—
493	D₄	—	4	2	no	—
494	D₄	—	4	2	no	—
495	D₄	—	4	1	no	—
496	D₄	—	4	2	no	—
497	D₄	—	4	2	no	—
498	D₄	—	4	2	no	—
499	D₄	—	4	1	no	—
500	D₄	—	4	2	no	—
501	D₄	—	4	4	no	—
502	D₄	—	4	4	no	—
503	D₄	—	4	2	no	—
504	D₄	—	8	4	no	—
505	D₄	—	8	4	no	—
506	C₂×C₄	—	2	2	no	—
507	C₂×C₄	—	2	2	no	—
508	C₂×C₄	—	2	2	no	—
509	C₂×C₄	—	2	2	no	—
510	C₂×C₄	—	2	2	no	—
511	C₂×C₄	—	2	2	no	—
512	C₂×C₄	—	2	2	no	—
513	C₂×C₄	—	2	1	no	—
514	C₂×C₄	—	2	1	no	—
515	C₂×C₄	—	2	2	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	2	2	no	—
518	C₂×C₄	—	2	2	no	—
519	C₂×C₄	—	2	2	no	—
520	C₂×C₄	—	2	2	no	—
521	C₂×C₄	—	2	1	no	—
522	C₂×C₄	—	2	1	no	—
523	C₂×C₄	—	2	2	no	—
524	C₂×C₄	—	2	2	no	—
525	C₂×C₄	—	2	2	no	—
526	C₂×C₄	—	4	4	no	—
527	C₂×C₄	—	4	2	no	—
528	C₂×C₄	—	4	2	no	—
529	C₂×C₄	—	4	2	no	—
530	C₂×C₄	—	4	8	no	—
531	C₂×C₄	—	8	2	no	—
532	C₂×C₄	—	8	8	no	—
533	C₂×C₄	C₂₃×#99	1	2	yes	—
534	C₂×C₄	C₂₃×#99	1	2	yes	—
535	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
536	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
537	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
538	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
539	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
540	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
541	C₂²	—	2	2	no	—
542	C₂²	—	2	1	no	—
543	C₂²	—	2	1	no	—
544	C₂²	—	2	2	no	—
545	C₂²	—	4	1	no	—
546	C₂²	—	4	1	no	—
547	C₂²	—	4	2	no	—
548	C₂²	—	4	2	no	—
549	C₂²	—	4	1	no	—
550	C₂²	—	4	4	no	—
551	C₂²	—	4	4	no	—
552	C₂²	—	4	2	no	—
553	C₂²	—	8	2	no	—
554	C₂²	—	8	2	no	—
555	C₂²	—	8	4	no	—
556	C₂²	C₂×C₂₃×#99	1	2	yes	—
557	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
558	C₄	—	2	1	no	—
559	C₄	—	2	2	no	—
560	C₄	—	2	2	no	—
561	C₄	—	2	1	no	—
562	C₄	—	4	2	no	—
563	C₄	—	4	2	no	—
564	C₄	—	4	2	no	—
565	C₄	—	8	8	no	—
566	C₄	C₂×C₂₃×#99	1	1	yes	characteristic
567	C₄	C₂×C₂₃×#99	1	1	yes	characteristic
568	C₄	C₂×C₂₃×#99	1	2	yes	—
569	C₂	—	2	1	no	—
570	C₂	—	2	2	no	—
571	C₂	—	4	2	no	—
572	C₂	—	4	2	no	—
573	C₂	—	4	1	no	—
574	C₂	—	4	1	no	—
575	C₂	—	8	2	no	—
576	C₂	C₂²×C₂₃×#99	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₂₃	C₂×#99
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂×D₄	C₂⁴×C₂₃
socle	C₂×C₂₃	C₂²×#99
prime radical	C₂₃×#3449	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ C₂²×#41×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#410	C₂	semidirect
3. C₂₃×#2129	C₂	semidirect
4. C₂₃×#2128	C₂	semidirect
5. C₂₃×#2127	C₂	semidirect
6. C₂₃×#2126	C₂	semidirect
7. C₂²×#41×C₂₃	C₄	semidirect
10. C₂₃×#536	C₄	semidirect
11. C₂₃×#536	C₄	semidirect
13. C₂₃×#535	C₄	semidirect
17. C₂₃×#532	C₄	semidirect
22. C₂×C₂₃×#117	C₄	semidirect
26. C₂₃×#410	C₂²	semidirect
28. C₂×C₂₃×#99	C₂²	semidirect
29. C₂₃×#354	C₂²	semidirect
30. C₂₃×#353	C₂²	semidirect
40. C₂×#41×C₂₃	C₂×C₄	semidirect
46. C₂²×Q₈×C₂₃	C₂×C₄	semidirect
48. C₂²×D₄×C₂₃	C₂×C₄	semidirect
54. C₂₃×C₄⋊Q₈	C₂×C₄	semidirect
55. C₂₃×C₄⋊Q₈	C₂×C₄	semidirect
59. C₂₃×#124	C₂×C₄	semidirect
60. C₂₃×#124	C₂×C₄	semidirect
85. C₂₃×#117	C₂×C₄	semidirect
86. C₂₃×#117	C₂×C₄	semidirect
92. C₂×C₄²×C₂₃	C₂×C₄	semidirect
186. #33358	C₂₃	direct (N × H)
254. C₂×#410	C₂×C₂₃	semidirect
255. #2129	C₂×C₂₃	semidirect
256. #2128	C₂×C₂₃	semidirect
257. #2127	C₂×C₂₃	semidirect
258. #2126	C₂×C₂₃	semidirect
287. C₂²×#41	C₄×C₂₃	semidirect
290. #536	C₄×C₂₃	semidirect
291. #536	C₄×C₂₃	semidirect
293. #535	C₄×C₂₃	semidirect
297. #532	C₄×C₂₃	semidirect
302. C₂×#117	C₄×C₂₃	semidirect
306. #410	C₂²×C₂₃	semidirect
308. C₂×#99	C₂²×C₂₃	semidirect
309. #354	C₂²×C₂₃	semidirect
310. #353	C₂²×C₂₃	semidirect
328. C₂×#41	C₂×C₄×C₂₃	semidirect
334. C₂²×Q₈	C₂×C₄×C₂₃	semidirect
336. C₂²×D₄	C₂×C₄×C₂₃	semidirect
342. C₄⋊Q₈	C₂×C₄×C₂₃	semidirect
343. C₄⋊Q₈	C₂×C₄×C₂₃	semidirect
347. #124	C₂×C₄×C₂₃	semidirect
348. #124	C₂×C₄×C₂₃	semidirect
373. #117	C₂×C₄×C₂₃	semidirect
374. #117	C₂×C₄×C₂₃	semidirect
380. C₂×C₄²	C₂×C₄×C₂₃	semidirect
385. C₂₃	#33358	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×#3449	2
C₂×C₂₃×#410	2
C₂₃×#2129	2
C₂₃×#2128	2
C₂₃×#2127	2
C₂₃×#2126	2
#33358	23

Subgroups — C23×#33358 #19180984

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33358 #19180984