Subgroups — C₂₃×#60705 #19208331

Order 5888.

1846 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3455	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#541	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3178	C₂	1	8	yes	maximal, complemented
4	C₂₃×#3177	C₂	1	4	yes	maximal, complemented
5	C₂₃×#3174	C₂	1	4	yes	maximal, complemented
6	C₂₃×#3166	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#3152	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#3142	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#3139	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#3056	C₂	1	4	yes	maximal, complemented
11	C₄×C₂₃×#137	C₂	1	4	yes	maximal, complemented
12	C₂₃×#2873	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×C₂₃×#137	C₂²	1	4	yes	—
14	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
15	C₂₃×#542	C₂²	1	4	yes	—
16	C₂₃×#541	—	2	4	no	—
17	C₂₃×#541	C₂²	1	4	yes	—
18	C₂₃×#541	C₂²	1	2	yes	—
19	C₂₃×#541	C₂²	1	4	yes	—
20	C₂₃×#540	C₂²	1	2	yes	—
21	C₂₃×#539	—	2	2	no	—
22	C₂₃×#538	—	2	2	no	—
23	C₂₃×#533	C₂²	1	1	yes	characteristic
24	C₂₃×#532	C₂²	1	1	yes	characteristic
25	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
26	C₂₃×#529	—	2	4	no	—
27	C₂×C₂₃×#126	—	2	2	no	—
28	C₂×C₂₃×#125	—	2	2	no	—
29	C₂×C₂₃×#125	C₂²	1	1	yes	characteristic
30	C₂₃×#525	C₂²	1	2	yes	—
31	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
32	C₄×#41×C₂₃	C₂²	1	2	yes	—
33	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	—
34	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
35	C₂₃×#501	C₂²	1	4	yes	complemented
36	C₂₃×#499	C₂²	1	4	yes	complemented
37	C₂₃×#497	C₂²	1	4	yes	complemented
38	C₂₃×#494	C₂²	1	2	yes	complemented
39	C₂₃×#492	C₂²	1	2	yes	complemented
40	C₂₃×#491	C₂²	1	4	yes	complemented
41	C₂₃×#491	C₂²	1	2	yes	complemented
42	C₂₃×#490	C₂²	1	2	yes	complemented
43	C₂₃×#488	C₂²	1	4	yes	complemented
44	C₂₃×#487	C₂²	1	4	yes	complemented
45	C₂₃×#470	C₂²	1	8	yes	—
46	C₂₃×#468	C₂²	1	8	yes	—
47	C₂₃×#467	C₂²	1	8	yes	—
48	C₂₃×#465	C₂²	1	8	yes	—
49	C₂₃×#441	C₂²	1	8	yes	complemented
50	C₂₃×#440	C₂²	1	8	yes	complemented
51	C₄×Q₁₆×C₂₃	C₂²	1	8	yes	complemented
52	C₄×SD₁₆×C₂₃	C₂²	1	8	yes	complemented
53	C₂₃×#428	C₂²	1	4	yes	—
54	C₂₃×#419	C₂²	1	4	yes	—
55	C₂₃×#417	C₂²	1	4	yes	—
56	C₂₃×#405	C₂²	1	4	yes	—
57	C₂₃×#405	C₂²	1	4	yes	—
58	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
59	C₄×M₁₆×C₂₃	C₂²	1	2	yes	—
60	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
61	C₂×#41×C₂₃	C₂³	1	2	yes	—
62	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
63	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
64	C₂₃×#137	—	2	16	no	—
65	C₂×Q₁₆×C₂₃	C₂³	1	8	yes	—
66	C₂×SD₁₆×C₂₃	C₂³	1	8	yes	—
67	C₂×M₁₆×C₂₃	—	2	8	no	—
68	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
69	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
70	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
71	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
72	C₂₃×C₄⋊Q₈	—	2	2	no	—
73	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
74	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
75	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
76	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
77	C₂₃×#127	C₂³	1	1	yes	characteristic
78	C₂₃×#126	—	2	4	no	—
79	C₂₃×#126	—	2	8	no	—
80	C₂₃×#126	—	2	8	no	—
81	C₂₃×#126	—	2	4	no	—
82	C₂₃×#125	—	2	8	no	—
83	C₂₃×#125	—	2	8	no	—
84	C₂₃×#125	—	2	4	no	—
85	C₂₃×#125	—	2	2	no	—
86	C₂₃×#125	C₂³	1	2	yes	—
87	C₂₃×#125	C₂³	1	2	yes	—
88	C₂₃×#125	C₂³	1	2	yes	—
89	C₂₃×#124	—	2	4	no	—
90	C₂₃×#124	C₂³	1	2	yes	—
91	C₂₃×#123	—	2	2	no	—
92	C₂₃×#123	—	2	4	no	—
93	C₂₃×#123	—	2	4	no	—
94	C₂₃×C₂²⋊Q₈	—	2	4	no	—
95	C₂₃×C₂²⋊Q₈	—	2	2	no	—
96	C₂₃×C₂²⋊Q₈	—	2	4	no	—
97	C₂₃×C₂²⋊Q₈	C₂³	1	8	yes	—
98	C₂₃×C₂²⋊Q₈	C₂³	1	4	yes	—
99	C₂₃×#121	—	2	2	no	—
100	C₂₃×#121	—	2	4	no	—
101	C₂₃×#121	C₂³	1	4	yes	—
102	C₂₃×#120	—	2	2	no	—
103	C₄×Q₈×C₂₃	—	2	4	no	—
104	C₄×Q₈×C₂₃	—	2	2	no	—
105	C₄×Q₈×C₂₃	C₂³	1	8	yes	—
106	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
107	C₄×D₄×C₂₃	—	2	4	no	—
108	C₄×D₄×C₂₃	—	2	4	no	—
109	C₄×D₄×C₂₃	—	2	2	no	—
110	C₄×D₄×C₂₃	C₂³	1	4	yes	—
111	C₂₃×#117	—	2	2	no	—
112	C₂₃×#117	D₄	1	4	yes	—
113	C₂₃×#117	C₂³	1	2	yes	—
114	C₂₃×#117	C₂³	1	2	yes	—
115	C₂₃×#117	C₂³	1	2	yes	—
116	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
117	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
118	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
119	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
120	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
121	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
122	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
123	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
124	C₂×C₄²×C₂₃	—	2	2	no	—
125	C₂×C₄²×C₂₃	—	2	2	no	—
126	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
127	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
128	C₂₃×#107	C₂³	1	8	yes	—
129	C₂₃×#106	C₂³	1	8	yes	—
130	C₂₃×#103	C₂³	1	8	yes	—
131	C₂₃×#103	C₂³	1	8	yes	—
132	C₂₃×#103	C₂³	1	8	yes	—
133	C₂₃×#102	C₂³	1	8	yes	—
134	C₂₃×#97	C₂³	1	4	yes	—
135	C₂₃×#97	C₂³	1	2	yes	—
136	C₂₃×#97	C₂³	1	2	yes	—
137	C₄×C₈×C₂₃	C₂³	1	4	yes	—
138	C₄×C₈×C₂₃	C₂³	1	2	yes	—
139	C₄×C₈×C₂₃	C₂³	1	2	yes	—
140	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
141	#41×C₂₃	—	2	8	no	—
142	C₂×Q₈×C₂₃	—	2	2	no	—
143	C₂×Q₈×C₂₃	—	2	2	no	—
144	C₂×Q₈×C₂₃	—	2	8	no	—
145	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
146	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
147	C₂×D₄×C₂₃	—	2	2	no	—
148	C₂×D₄×C₂₃	—	2	4	no	—
149	C₂×D₄×C₂₃	—	2	2	no	—
150	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
151	C₂²×C₄×C₂₃	—	2	4	no	—
152	C₂²×C₄×C₂₃	—	2	2	no	—
153	C₂²×C₄×C₂₃	—	2	2	no	—
154	C₂²×C₄×C₂₃	—	2	2	no	—
155	C₂²×C₄×C₂₃	—	2	4	no	—
156	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
158	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
159	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
160	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
161	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
162	Q₁₆×C₂₃	—	2	16	no	—
163	SD₁₆×C₂₃	—	2	16	no	—
164	M₁₆×C₂₃	—	2	4	no	—
165	M₁₆×C₂₃	—	2	8	no	—
166	M₁₆×C₂₃	#41	1	8	yes	—
167	C₂×C₈×C₂₃	—	2	8	no	—
168	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
169	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
170	C₄⋊C₄×C₂₃	—	2	8	no	—
171	C₄⋊C₄×C₂₃	—	2	8	no	—
172	C₄⋊C₄×C₂₃	—	2	4	no	—
173	C₄⋊C₄×C₂₃	—	2	4	no	—
174	C₄⋊C₄×C₂₃	—	2	4	no	—
175	C₄⋊C₄×C₂₃	—	2	4	no	—
176	C₄⋊C₄×C₂₃	—	2	2	no	—
177	C₄⋊C₄×C₂₃	—	2	4	no	—
178	C₄⋊C₄×C₂₃	—	2	2	no	—
179	C₄⋊C₄×C₂₃	—	2	4	no	—
180	C₄⋊C₄×C₂₃	—	2	4	no	—
181	C₄⋊C₄×C₂₃	—	2	2	no	—
182	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
183	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
184	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
185	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
186	C₂²⋊C₄×C₂₃	—	2	8	no	—
187	C₂²⋊C₄×C₂₃	—	2	2	no	—
188	C₂²⋊C₄×C₂₃	—	2	4	no	—
189	C₂²⋊C₄×C₂₃	—	2	4	no	—
190	C₂²⋊C₄×C₂₃	—	2	4	no	—
191	C₂²⋊C₄×C₂₃	—	2	4	no	—
192	C₂²⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
193	C₂²⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
194	C₄²×C₂₃	—	2	2	no	—
195	C₄²×C₂₃	—	2	4	no	—
196	C₄²×C₂₃	—	2	8	no	—
197	C₄²×C₂₃	C₂×D₄	1	2	yes	—
198	C₄²×C₂₃	C₂×D₄	1	2	yes	—
199	C₄²×C₂₃	C₂⁴	1	2	yes	—
200	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
201	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
202	#60705	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
203	C₂³×C₂₃	—	2	2	no	—
204	C₂³×C₂₃	—	2	2	no	—
205	C₂³×C₂₃	—	4	2	no	—
206	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
207	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
208	Q₈×C₂₃	—	2	8	no	—
209	Q₈×C₂₃	—	2	4	no	—
210	Q₈×C₂₃	—	4	4	no	—
211	Q₈×C₂₃	—	8	2	no	—
212	D₄×C₂₃	—	2	4	no	—
213	D₄×C₂₃	—	4	4	no	—
214	D₄×C₂₃	—	8	2	no	—
215	D₄×C₂₃	—	8	4	no	—
216	C₂×C₄×C₂₃	—	2	2	no	—
217	C₂×C₄×C₂₃	—	2	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	4	no	—
222	C₂×C₄×C₂₃	—	2	4	no	—
223	C₂×C₄×C₂₃	—	2	4	no	—
224	C₂×C₄×C₂₃	—	4	2	no	—
225	C₂×C₄×C₂₃	—	4	2	no	—
226	C₂×C₄×C₂₃	—	4	4	no	—
227	C₂×C₄×C₂₃	—	4	4	no	—
228	C₂×C₄×C₂₃	—	4	4	no	—
229	C₂×C₄×C₂₃	—	8	4	no	—
230	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
231	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
232	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
233	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
234	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
235	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
236	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
237	C₈×C₂₃	—	2	4	no	—
238	C₈×C₂₃	C₂×#41	1	8	yes	—
239	#3455	C₂×C₂₃	1	1	yes	characteristic
240	C₂×#541	C₂×C₂₃	1	1	yes	characteristic, complemented
241	#3178	C₂×C₂₃	1	8	yes	complemented
242	#3177	C₂×C₂₃	1	4	yes	complemented
243	#3174	C₂×C₂₃	1	4	yes	complemented
244	#3166	C₂×C₂₃	1	1	yes	characteristic, complemented
245	#3152	C₂×C₂₃	1	1	yes	characteristic, complemented
246	#3142	C₂×C₂₃	1	1	yes	characteristic, complemented
247	#3139	C₂×C₂₃	1	1	yes	characteristic, complemented
248	#3056	C₂×C₂₃	1	4	yes	complemented
249	C₄×#137	C₂×C₂₃	1	4	yes	complemented
250	#2873	C₂×C₂₃	1	1	yes	characteristic, complemented
251	C₂²×C₂₃	—	2	2	no	—
252	C₂²×C₂₃	—	2	4	no	—
253	C₂²×C₂₃	—	2	2	no	—
254	C₂²×C₂₃	—	4	4	no	—
255	C₂²×C₂₃	—	4	2	no	—
256	C₂²×C₂₃	—	8	2	no	—
257	C₂²×C₂₃	#532	1	2	yes	—
258	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
259	C₄×C₂₃	—	2	4	no	—
260	C₄×C₂₃	—	2	2	no	—
261	C₄×C₂₃	—	2	2	no	—
262	C₄×C₂₃	—	2	2	no	—
263	C₄×C₂₃	—	4	4	no	—
264	C₄×C₂₃	—	4	4	no	—
265	C₄×C₂₃	—	4	2	no	—
266	C₄×C₂₃	—	4	4	no	—
267	C₄×C₂₃	#532	1	2	yes	—
268	C₄×C₂₃	C₂²×#41	1	2	yes	—
269	C₂×#137	C₂²×C₂₃	1	4	yes	—
270	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
271	#542	C₂²×C₂₃	1	4	yes	—
272	#541	—	2	4	no	—
273	#541	C₂²×C₂₃	1	4	yes	—
274	#541	C₂²×C₂₃	1	2	yes	—
275	#541	C₂²×C₂₃	1	4	yes	—
276	#540	C₂²×C₂₃	1	2	yes	—
277	#539	—	2	2	no	—
278	#538	—	2	2	no	—
279	#533	C₂²×C₂₃	1	1	yes	characteristic
280	#532	C₂²×C₂₃	1	1	yes	characteristic
281	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
282	#529	—	2	4	no	—
283	C₂×#126	—	2	2	no	—
284	C₂×#125	—	2	2	no	—
285	C₂×#125	C₂²×C₂₃	1	1	yes	characteristic
286	#525	C₂²×C₂₃	1	2	yes	—
287	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
288	C₄×#41	C₂²×C₂₃	1	2	yes	—
289	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	—
290	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
291	#501	C₂²×C₂₃	1	4	yes	complemented
292	#499	C₂²×C₂₃	1	4	yes	complemented
293	#497	C₂²×C₂₃	1	4	yes	complemented
294	#494	C₂²×C₂₃	1	2	yes	complemented
295	#492	C₂²×C₂₃	1	2	yes	complemented
296	#491	C₂²×C₂₃	1	4	yes	complemented
297	#491	C₂²×C₂₃	1	2	yes	complemented
298	#490	C₂²×C₂₃	1	2	yes	complemented
299	#488	C₂²×C₂₃	1	4	yes	complemented
300	#487	C₂²×C₂₃	1	4	yes	complemented
301	#470	C₂²×C₂₃	1	8	yes	—
302	#468	C₂²×C₂₃	1	8	yes	—
303	#467	C₂²×C₂₃	1	8	yes	—
304	#465	C₂²×C₂₃	1	8	yes	—
305	#441	C₂²×C₂₃	1	8	yes	complemented
306	#440	C₂²×C₂₃	1	8	yes	complemented
307	C₄×Q₁₆	C₂²×C₂₃	1	8	yes	complemented
308	C₄×SD₁₆	C₂²×C₂₃	1	8	yes	complemented
309	#428	C₂²×C₂₃	1	4	yes	—
310	#419	C₂²×C₂₃	1	4	yes	—
311	#417	C₂²×C₂₃	1	4	yes	—
312	#405	C₂²×C₂₃	1	4	yes	—
313	#405	C₂²×C₂₃	1	4	yes	—
314	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
315	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
316	C₄×M₁₆	C₂²×C₂₃	1	2	yes	—
317	C₂×C₂₃	—	2	2	no	—
318	C₂×C₂₃	—	4	2	no	—
319	C₂×C₂₃	—	8	2	no	—
320	C₂×C₂₃	C₂×#532	1	1	yes	characteristic
321	C₂×C₂₃	#3586	1	2	yes	—
322	C₂×#41	C₂³×C₂₃	1	2	yes	—
323	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
324	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
325	#137	—	2	16	no	—
326	C₂×Q₁₆	C₂³×C₂₃	1	8	yes	—
327	C₂×SD₁₆	C₂³×C₂₃	1	8	yes	—
328	C₂×M₁₆	—	2	8	no	—
329	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
330	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
331	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
332	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
333	C₄⋊Q₈	—	2	2	no	—
334	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
335	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
336	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
337	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
338	#127	C₂³×C₂₃	1	1	yes	characteristic
339	#126	—	2	4	no	—
340	#126	—	2	8	no	—
341	#126	—	2	8	no	—
342	#126	—	2	4	no	—
343	#125	—	2	8	no	—
344	#125	—	2	8	no	—
345	#125	—	2	4	no	—
346	#125	—	2	2	no	—
347	#125	C₂³×C₂₃	1	2	yes	—
348	#125	C₂³×C₂₃	1	2	yes	—
349	#125	C₂³×C₂₃	1	2	yes	—
350	#124	—	2	4	no	—
351	#124	C₂³×C₂₃	1	2	yes	—
352	#123	—	2	2	no	—
353	#123	—	2	4	no	—
354	#123	—	2	4	no	—
355	C₂²⋊Q₈	—	2	4	no	—
356	C₂²⋊Q₈	—	2	2	no	—
357	C₂²⋊Q₈	—	2	4	no	—
358	C₂²⋊Q₈	C₂³×C₂₃	1	8	yes	—
359	C₂²⋊Q₈	C₂³×C₂₃	1	4	yes	—
360	#121	—	2	2	no	—
361	#121	—	2	4	no	—
362	#121	C₂³×C₂₃	1	4	yes	—
363	#120	—	2	2	no	—
364	C₄×Q₈	—	2	4	no	—
365	C₄×Q₈	—	2	2	no	—
366	C₄×Q₈	C₂³×C₂₃	1	8	yes	—
367	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
368	C₄×D₄	—	2	4	no	—
369	C₄×D₄	—	2	4	no	—
370	C₄×D₄	—	2	2	no	—
371	C₄×D₄	C₂³×C₂₃	1	4	yes	—
372	#117	—	2	2	no	—
373	#117	D₄×C₂₃	1	4	yes	—
374	#117	C₂³×C₂₃	1	2	yes	—
375	#117	C₂³×C₂₃	1	2	yes	—
376	#117	C₂³×C₂₃	1	2	yes	—
377	C₂×C₄⋊C₄	—	2	2	no	—
378	C₂×C₄⋊C₄	—	2	2	no	—
379	C₂×C₄⋊C₄	—	2	2	no	—
380	C₂×C₄⋊C₄	—	2	2	no	—
381	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
382	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
383	C₂×C₂²⋊C₄	—	2	2	no	—
384	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
385	C₂×C₄²	—	2	2	no	—
386	C₂×C₄²	—	2	2	no	—
387	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
388	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
389	#107	C₂³×C₂₃	1	8	yes	—
390	#106	C₂³×C₂₃	1	8	yes	—
391	#103	C₂³×C₂₃	1	8	yes	—
392	#103	C₂³×C₂₃	1	8	yes	—
393	#103	C₂³×C₂₃	1	8	yes	—
394	#102	C₂³×C₂₃	1	8	yes	—
395	#97	C₂³×C₂₃	1	4	yes	—
396	#97	C₂³×C₂₃	1	2	yes	—
397	#97	C₂³×C₂₃	1	2	yes	—
398	C₄×C₈	C₂³×C₂₃	1	4	yes	—
399	C₄×C₈	C₂³×C₂₃	1	2	yes	—
400	C₄×C₈	C₂³×C₂₃	1	2	yes	—
401	C₂₃	#60705	1	1	yes	characteristic, direct product factor, Sylow
402	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
403	#41	—	2	8	no	—
404	C₂×Q₈	—	2	2	no	—
405	C₂×Q₈	—	2	2	no	—
406	C₂×Q₈	—	2	8	no	—
407	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
408	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
409	C₂×D₄	—	2	2	no	—
410	C₂×D₄	—	2	4	no	—
411	C₂×D₄	—	2	2	no	—
412	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
413	C₂²×C₄	—	2	4	no	—
414	C₂²×C₄	—	2	2	no	—
415	C₂²×C₄	—	2	2	no	—
416	C₂²×C₄	—	2	2	no	—
417	C₂²×C₄	—	2	4	no	—
418	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
419	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
420	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
421	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
422	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
423	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
424	Q₁₆	—	2	16	no	—
425	SD₁₆	—	2	16	no	—
426	M₁₆	—	2	4	no	—
427	M₁₆	—	2	8	no	—
428	M₁₆	#41×C₂₃	1	8	yes	—
429	C₂×C₈	—	2	8	no	—
430	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
431	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
432	C₄⋊C₄	—	2	8	no	—
433	C₄⋊C₄	—	2	8	no	—
434	C₄⋊C₄	—	2	4	no	—
435	C₄⋊C₄	—	2	4	no	—
436	C₄⋊C₄	—	2	4	no	—
437	C₄⋊C₄	—	2	4	no	—
438	C₄⋊C₄	—	2	2	no	—
439	C₄⋊C₄	—	2	4	no	—
440	C₄⋊C₄	—	2	2	no	—
441	C₄⋊C₄	—	2	4	no	—
442	C₄⋊C₄	—	2	4	no	—
443	C₄⋊C₄	—	2	2	no	—
444	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
445	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
446	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
447	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
448	C₂²⋊C₄	—	2	8	no	—
449	C₂²⋊C₄	—	2	2	no	—
450	C₂²⋊C₄	—	2	4	no	—
451	C₂²⋊C₄	—	2	4	no	—
452	C₂²⋊C₄	—	2	4	no	—
453	C₂²⋊C₄	—	2	4	no	—
454	C₂²⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
455	C₂²⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
456	C₄²	—	2	2	no	—
457	C₄²	—	2	4	no	—
458	C₄²	—	2	8	no	—
459	C₄²	C₂×D₄×C₂₃	1	2	yes	—
460	C₄²	C₂×D₄×C₂₃	1	2	yes	—
461	C₄²	C₂⁴×C₂₃	1	2	yes	—
462	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
463	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
464	C₂³	—	2	2	no	—
465	C₂³	—	2	2	no	—
466	C₂³	—	4	2	no	—
467	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
468	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
469	Q₈	—	2	8	no	—
470	Q₈	—	2	4	no	—
471	Q₈	—	4	4	no	—
472	Q₈	—	8	2	no	—
473	D₄	—	2	4	no	—
474	D₄	—	4	4	no	—
475	D₄	—	8	2	no	—
476	D₄	—	8	4	no	—
477	C₂×C₄	—	2	2	no	—
478	C₂×C₄	—	2	2	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	4	no	—
481	C₂×C₄	—	2	2	no	—
482	C₂×C₄	—	2	4	no	—
483	C₂×C₄	—	2	4	no	—
484	C₂×C₄	—	2	4	no	—
485	C₂×C₄	—	4	2	no	—
486	C₂×C₄	—	4	2	no	—
487	C₂×C₄	—	4	4	no	—
488	C₂×C₄	—	4	4	no	—
489	C₂×C₄	—	4	4	no	—
490	C₂×C₄	—	8	4	no	—
491	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
492	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
493	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
494	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
495	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
496	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
497	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
498	C₈	—	2	4	no	—
499	C₈	C₂×#41×C₂₃	1	8	yes	—
500	C₂²	—	2	2	no	—
501	C₂²	—	2	2	no	—
502	C₂²	—	2	4	no	—
503	C₂²	—	4	2	no	—
504	C₂²	—	4	4	no	—
505	C₂²	—	8	2	no	—
506	C₂²	C₂₃×#532	1	2	yes	—
507	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
508	C₄	—	2	2	no	—
509	C₄	—	2	4	no	—
510	C₄	—	2	2	no	—
511	C₄	—	2	2	no	—
512	C₄	—	4	2	no	—
513	C₄	—	4	4	no	—
514	C₄	—	4	4	no	—
515	C₄	—	4	4	no	—
516	C₄	C₂₃×#532	1	2	yes	—
517	C₄	C₂²×#41×C₂₃	1	2	yes	—
518	C₂	—	2	2	no	—
519	C₂	—	4	2	no	—
520	C₂	—	8	2	no	—
521	C₂	C₂×C₂₃×#532	1	1	yes	characteristic
522	C₂	C₂₃×#3586	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂₃×#3455	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₂₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#541	C₂	semidirect
3. C₂₃×#3178	C₂	semidirect
4. C₂₃×#3177	C₂	semidirect
5. C₂₃×#3174	C₂	semidirect
6. C₂₃×#3166	C₂	semidirect
7. C₂₃×#3152	C₂	semidirect
8. C₂₃×#3142	C₂	semidirect
9. C₂₃×#3139	C₂	semidirect
10. C₂₃×#3056	C₂	semidirect
11. C₄×C₂₃×#137	C₂	semidirect
12. C₂₃×#2873	C₂	semidirect
35. C₂₃×#501	C₂²	semidirect
36. C₂₃×#499	C₂²	semidirect
37. C₂₃×#497	C₂²	semidirect
38. C₂₃×#494	C₂²	semidirect
39. C₂₃×#492	C₂²	semidirect
40. C₂₃×#491	C₂²	semidirect
41. C₂₃×#491	C₂²	semidirect
42. C₂₃×#490	C₂²	semidirect
43. C₂₃×#488	C₂²	semidirect
44. C₂₃×#487	C₂²	semidirect
49. C₂₃×#441	C₂²	semidirect
50. C₂₃×#440	C₂²	semidirect
51. C₄×Q₁₆×C₂₃	C₂²	semidirect
52. C₄×SD₁₆×C₂₃	C₂²	semidirect
202. #60705	C₂₃	direct (N × H)
240. C₂×#541	C₂×C₂₃	semidirect
241. #3178	C₂×C₂₃	semidirect
242. #3177	C₂×C₂₃	semidirect
243. #3174	C₂×C₂₃	semidirect
244. #3166	C₂×C₂₃	semidirect
245. #3152	C₂×C₂₃	semidirect
246. #3142	C₂×C₂₃	semidirect
247. #3139	C₂×C₂₃	semidirect
248. #3056	C₂×C₂₃	semidirect
249. C₄×#137	C₂×C₂₃	semidirect
250. #2873	C₂×C₂₃	semidirect
291. #501	C₂²×C₂₃	semidirect
292. #499	C₂²×C₂₃	semidirect
293. #497	C₂²×C₂₃	semidirect
294. #494	C₂²×C₂₃	semidirect
295. #492	C₂²×C₂₃	semidirect
296. #491	C₂²×C₂₃	semidirect
297. #491	C₂²×C₂₃	semidirect
298. #490	C₂²×C₂₃	semidirect
299. #488	C₂²×C₂₃	semidirect
300. #487	C₂²×C₂₃	semidirect
305. #441	C₂²×C₂₃	semidirect
306. #440	C₂²×C₂₃	semidirect
307. C₄×Q₁₆	C₂²×C₂₃	semidirect
308. C₄×SD₁₆	C₂²×C₂₃	semidirect
401. C₂₃	#60705	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×#3455	2
C₂×C₂₃×#541	2
C₂₃×#3178	2
C₂₃×#3177	2
C₂₃×#3174	2
C₂₃×#3166	2
C₂₃×#3152	2
C₂₃×#3142	2
C₂₃×#3139	2
C₂₃×#3056	2
C₄×C₂₃×#137	2
C₂₃×#2873	2
#60705	23

Subgroups — C23×#60705 #19208331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#60705 #19208331