Subgroups — C₂₃×#25469 #19173095

Order 5888.

838 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#474	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2546	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2538	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#2086	C₂	1	2	yes	maximal, complemented
6	C₂₃×#2067	C₂	1	2	yes	maximal
7	C₂₃×#1990	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1947	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1942	C₂	1	2	yes	maximal, complemented
10	C₂₃×#1934	C₂	1	2	yes	maximal
11	C₄×C₂₃×#102	C₂	1	1	yes	maximal, characteristic
12	C₂×C₂₃×C₄⋊Q₈	—	2	1	no	—
13	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
15	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×#474	—	2	4	no	—
18	C₈×Q₈×C₂₃	—	2	4	no	—
19	C₂×C₂₃×#107	C₂²	1	2	yes	—
20	C₂×C₂₃×#107	C₂²	1	2	yes	—
21	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
22	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₂₃×#102	C₂²	1	2	yes	—
24	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
25	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
28	C₂₃×#398	—	2	2	no	—
29	C₂₃×#396	—	2	2	no	—
30	C₂₃×#391	—	2	1	no	—
31	C₂₃×#390	C₂²	1	2	yes	—
32	C₂₃×#389	C₂²	1	2	yes	—
33	C₂₃×#389	C₂²	1	2	yes	—
34	C₂₃×#389	C₂²	1	2	yes	—
35	C₂₃×#384	C₂²	1	1	yes	characteristic
36	C₂₃×#380	—	2	1	no	—
37	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
38	C₂₃×#340	C₂²	1	2	yes	—
39	C₂₃×#340	C₂²	1	2	yes	—
40	C₂₃×#340	C₂²	1	1	yes	characteristic
41	C₂₃×#340	C₂²	1	2	yes	—
42	C₂₃×#340	C₂²	1	1	yes	characteristic
43	C₂²×Q₈×C₂₃	—	2	1	no	—
44	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
45	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₂₃	—	2	1	no	—
47	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
49	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
50	C₂₃×C₄⋊Q₈	—	2	2	no	—
51	C₂₃×C₄⋊Q₈	—	2	2	no	—
52	C₂₃×C₄⋊Q₈	—	4	4	no	—
53	C₂₃×C₂²⋊Q₈	—	4	4	no	—
54	C₄×Q₈×C₂₃	—	2	2	no	—
55	C₄×Q₈×C₂₃	—	2	2	no	—
56	C₄×D₄×C₂₃	—	2	2	no	—
57	C₄×D₄×C₂₃	—	2	2	no	—
58	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
59	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
60	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
61	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
62	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
63	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
64	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
66	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
67	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
68	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
71	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
72	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
73	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
74	C₂×C₄²×C₂₃	—	2	1	no	—
75	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
76	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
77	C₂₃×#107	—	2	4	no	—
78	C₂₃×#107	—	2	4	no	—
79	C₂₃×C₄⋊C₈	—	2	4	no	—
80	C₂₃×C₄⋊C₈	D₄	1	4	yes	—
81	C₂₃×#102	—	2	4	no	—
82	C₂₃×#102	—	2	2	no	—
83	C₂₃×#102	Q₈	1	4	yes	complemented
84	C₄×C₈×C₂₃	—	2	4	no	—
85	C₄×C₈×C₂₃	—	2	2	no	—
86	C₂₃×#95	—	2	1	no	—
87	C₂₃×#95	—	2	2	no	—
88	C₂₃×#95	—	2	2	no	—
89	C₂₃×#95	—	2	2	no	—
90	C₂⁴×C₂₃	—	2	1	no	—
91	C₂×Q₈×C₂₃	—	2	2	no	—
92	C₂×Q₈×C₂₃	—	2	2	no	—
93	C₂×Q₈×C₂₃	—	4	2	no	—
94	C₂×Q₈×C₂₃	—	4	2	no	—
95	C₂×Q₈×C₂₃	—	4	2	no	—
96	C₂×Q₈×C₂₃	D₈	1	4	yes	—
97	C₂×D₄×C₂₃	—	2	1	no	—
98	C₂×D₄×C₂₃	—	2	1	no	—
99	C₂×D₄×C₂₃	—	2	1	no	—
100	C₂×D₄×C₂₃	—	2	1	no	—
101	C₂×D₄×C₂₃	—	2	1	no	—
102	C₂×D₄×C₂₃	C₂×Q₈	1	2	yes	—
103	C₂²×C₄×C₂₃	—	2	1	no	—
104	C₂²×C₄×C₂₃	—	2	1	no	—
105	C₂²×C₄×C₂₃	—	2	2	no	—
106	C₂²×C₄×C₂₃	—	2	1	no	—
107	C₂²×C₄×C₂₃	—	2	2	no	—
108	C₂²×C₄×C₂₃	—	4	2	no	—
109	C₂²×C₄×C₂₃	—	4	2	no	—
110	C₂²×C₄×C₂₃	—	4	2	no	—
111	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
112	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
114	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
116	C₂×C₈×C₂₃	—	2	1	no	—
117	C₂×C₈×C₂₃	—	2	2	no	—
118	C₂×C₈×C₂₃	—	2	2	no	—
119	C₂×C₈×C₂₃	—	2	1	no	—
120	C₂×C₈×C₂₃	—	2	1	no	—
121	C₂×C₈×C₂₃	—	2	1	no	—
122	C₂×C₈×C₂₃	—	2	1	no	—
123	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
124	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
125	C₄⋊C₄×C₂₃	—	2	4	no	—
126	C₄⋊C₄×C₂₃	—	2	2	no	—
127	C₄⋊C₄×C₂₃	—	2	1	no	—
128	C₄⋊C₄×C₂₃	—	2	1	no	—
129	C₄⋊C₄×C₂₃	—	2	2	no	—
130	C₄⋊C₄×C₂₃	—	2	1	no	—
131	C₄⋊C₄×C₂₃	—	2	2	no	—
132	C₄⋊C₄×C₂₃	—	2	1	no	—
133	C₄⋊C₄×C₂₃	—	2	2	no	—
134	C₄⋊C₄×C₂₃	—	2	2	no	—
135	C₄⋊C₄×C₂₃	—	4	2	no	—
136	C₄⋊C₄×C₂₃	—	4	2	no	—
137	C₄⋊C₄×C₂₃	—	4	4	no	—
138	C₄⋊C₄×C₂₃	—	4	4	no	—
139	C₄⋊C₄×C₂₃	—	4	4	no	—
140	C₄⋊C₄×C₂₃	—	8	1	no	—
141	C₄⋊C₄×C₂₃	C₂×Q₈	1	2	yes	—
142	C₄⋊C₄×C₂₃	#41	1	2	yes	—
143	C₂²⋊C₄×C₂₃	—	4	2	no	—
144	C₂²⋊C₄×C₂₃	—	4	4	no	—
145	C₂²⋊C₄×C₂₃	—	8	2	no	—
146	C₄²×C₂₃	—	2	2	no	—
147	C₄²×C₂₃	—	2	2	no	—
148	C₄²×C₂₃	—	4	2	no	—
149	C₄²×C₂₃	C₂×D₄	1	2	yes	—
150	C₄²×C₂₃	#41	1	2	yes	—
151	#25469	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
152	C₂³×C₂₃	—	2	2	no	—
153	C₂³×C₂₃	—	4	1	no	—
154	C₂³×C₂₃	—	4	1	no	—
155	C₂³×C₂₃	—	4	1	no	—
156	C₂³×C₂₃	—	4	1	no	—
157	C₂³×C₂₃	—	4	1	no	—
158	C₂³×C₂₃	—	4	1	no	—
159	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
160	Q₈×C₂₃	—	2	8	no	—
161	Q₈×C₂₃	—	8	4	no	—
162	D₄×C₂₃	—	2	2	no	—
163	D₄×C₂₃	—	2	2	no	—
164	D₄×C₂₃	—	4	1	no	—
165	D₄×C₂₃	—	4	1	no	—
166	C₂×C₄×C₂₃	—	2	2	no	—
167	C₂×C₄×C₂₃	—	2	2	no	—
168	C₂×C₄×C₂₃	—	2	1	no	—
169	C₂×C₄×C₂₃	—	2	1	no	—
170	C₂×C₄×C₂₃	—	2	2	no	—
171	C₂×C₄×C₂₃	—	2	2	no	—
172	C₂×C₄×C₂₃	—	2	2	no	—
173	C₂×C₄×C₂₃	—	2	2	no	—
174	C₂×C₄×C₂₃	—	2	2	no	—
175	C₂×C₄×C₂₃	—	4	1	no	—
176	C₂×C₄×C₂₃	—	4	1	no	—
177	C₂×C₄×C₂₃	—	4	1	no	—
178	C₂×C₄×C₂₃	—	4	1	no	—
179	C₂×C₄×C₂₃	—	4	1	no	—
180	C₂×C₄×C₂₃	—	4	2	no	—
181	C₂×C₄×C₂₃	—	4	2	no	—
182	C₂×C₄×C₂₃	—	4	2	no	—
183	C₂×C₄×C₂₃	—	4	1	no	—
184	C₂×C₄×C₂₃	—	4	2	no	—
185	C₂×C₄×C₂₃	—	4	1	no	—
186	C₂×C₄×C₂₃	—	4	2	no	—
187	C₂×C₄×C₂₃	—	4	2	no	—
188	C₂×C₄×C₂₃	—	8	4	no	—
189	C₂×C₄×C₂₃	#121	1	2	yes	—
190	C₂×C₄×C₂₃	#121	1	2	yes	—
191	C₂×C₄×C₂₃	C₂×D₈	1	4	yes	—
192	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
193	C₂×C₄×C₂₃	#135	1	2	yes	—
194	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
195	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
196	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
197	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
198	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
199	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
200	C₈×C₂₃	—	2	4	no	—
201	C₈×C₂₃	—	2	2	no	—
202	C₈×C₂₃	—	4	1	no	—
203	C₂×#474	C₂×C₂₃	1	1	yes	characteristic
204	C₂×C₈×Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
205	#2546	C₂×C₂₃	1	1	yes	characteristic, complemented
206	#2538	C₂×C₂₃	1	1	yes	characteristic
207	#2086	C₂×C₂₃	1	2	yes	complemented
208	#2067	C₂×C₂₃	1	2	yes	—
209	#1990	C₂×C₂₃	1	1	yes	characteristic, complemented
210	#1947	C₂×C₂₃	1	1	yes	characteristic, complemented
211	#1942	C₂×C₂₃	1	2	yes	complemented
212	#1934	C₂×C₂₃	1	2	yes	—
213	C₄×#102	C₂×C₂₃	1	1	yes	characteristic
214	C₂²×C₂₃	—	4	2	no	—
215	C₂²×C₂₃	—	4	2	no	—
216	C₂²×C₂₃	—	4	2	no	—
217	C₂²×C₂₃	—	8	1	no	—
218	C₂²×C₂₃	—	8	1	no	—
219	C₂²×C₂₃	—	8	1	no	—
220	C₂²×C₂₃	—	8	1	no	—
221	C₂²×C₂₃	C₂×#121	1	1	yes	characteristic
222	C₂²×C₂₃	#548	1	1	yes	characteristic
223	C₂²×C₂₃	D₄×Q₈	1	1	yes	characteristic
224	C₂²×C₂₃	C₂²×D₈	1	1	yes	characteristic
225	C₂²×C₂₃	C₂×#135	1	1	yes	characteristic
226	C₂²×C₂₃	#577	1	1	yes	characteristic
227	C₂²×C₂₃	#578	1	1	yes	characteristic
228	C₄×C₂₃	—	2	4	no	—
229	C₄×C₂₃	—	2	2	no	—
230	C₄×C₂₃	—	4	2	no	—
231	C₄×C₂₃	—	4	2	no	—
232	C₄×C₂₃	—	4	2	no	—
233	C₄×C₂₃	—	8	2	no	—
234	C₄×C₂₃	—	8	1	no	—
235	C₄×C₂₃	—	8	2	no	—
236	C₄×C₂₃	#466	1	4	yes	—
237	C₄×C₂₃	#533	1	1	yes	characteristic
238	C₄×C₂₃	#541	1	1	yes	characteristic
239	C₄×C₂₃	#547	1	1	yes	characteristic
240	C₄×C₂₃	D₄×Q₈	1	1	yes	characteristic
241	C₂×C₄⋊Q₈	—	2	1	no	—
242	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
243	C₂×C₂²⋊Q₈	—	2	1	no	—
244	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
245	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
246	#474	—	2	4	no	—
247	C₈×Q₈	—	2	4	no	—
248	C₂×#107	C₂²×C₂₃	1	2	yes	—
249	C₂×#107	C₂²×C₂₃	1	2	yes	—
250	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
251	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
252	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
253	C₂×#102	C₂²×C₂₃	1	2	yes	—
254	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
255	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
256	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
257	#398	—	2	2	no	—
258	#396	—	2	2	no	—
259	#391	—	2	1	no	—
260	#390	C₂²×C₂₃	1	2	yes	—
261	#389	C₂²×C₂₃	1	2	yes	—
262	#389	C₂²×C₂₃	1	2	yes	—
263	#389	C₂²×C₂₃	1	2	yes	—
264	#384	C₂²×C₂₃	1	1	yes	characteristic
265	#380	—	2	1	no	—
266	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
267	#340	C₂²×C₂₃	1	2	yes	—
268	#340	C₂²×C₂₃	1	1	yes	characteristic
269	#340	C₂²×C₂₃	1	2	yes	—
270	#340	C₂²×C₂₃	1	2	yes	—
271	#340	C₂²×C₂₃	1	1	yes	characteristic
272	C₂×C₂₃	—	8	2	no	—
273	C₂×C₂₃	#2429	1	1	yes	characteristic
274	C₂×C₂₃	C₂×#466	1	1	yes	characteristic
275	C₂×C₂₃	#3063	1	1	yes	characteristic
276	C₂×C₂₃	#3358	1	1	yes	characteristic
277	C₂×C₂₃	#3364	1	1	yes	characteristic
278	C₂×C₂₃	Q₈×D₈	1	1	yes	characteristic
279	C₂×C₂₃	#3381	1	1	yes	characteristic
280	C₂²×Q₈	—	2	1	no	—
281	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
282	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
283	C₂³×C₄	—	2	1	no	—
284	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
285	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
286	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
287	C₄⋊Q₈	—	2	2	no	—
288	C₄⋊Q₈	—	2	2	no	—
289	C₄⋊Q₈	—	4	4	no	—
290	C₂²⋊Q₈	—	4	4	no	—
291	C₄×Q₈	—	2	2	no	—
292	C₄×Q₈	—	2	2	no	—
293	C₄×D₄	—	2	2	no	—
294	C₄×D₄	—	2	2	no	—
295	C₂×C₄⋊C₄	—	2	1	no	—
296	C₂×C₄⋊C₄	—	2	2	no	—
297	C₂×C₄⋊C₄	—	2	2	no	—
298	C₂×C₄⋊C₄	—	2	2	no	—
299	C₂×C₄⋊C₄	—	2	1	no	—
300	C₂×C₄⋊C₄	—	2	1	no	—
301	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
302	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
303	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
304	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
305	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
306	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
307	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
308	C₂×C₂²⋊C₄	—	2	1	no	—
309	C₂×C₂²⋊C₄	—	2	2	no	—
310	C₂×C₂²⋊C₄	—	2	2	no	—
311	C₂×C₄²	—	2	1	no	—
312	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
313	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
314	#107	—	2	4	no	—
315	#107	—	2	4	no	—
316	C₄⋊C₈	—	2	4	no	—
317	C₄⋊C₈	D₄×C₂₃	1	4	yes	—
318	#102	—	2	4	no	—
319	#102	—	2	2	no	—
320	#102	Q₈×C₂₃	1	4	yes	complemented
321	C₄×C₈	—	2	4	no	—
322	C₄×C₈	—	2	2	no	—
323	#95	—	2	1	no	—
324	#95	—	2	2	no	—
325	#95	—	2	2	no	—
326	#95	—	2	2	no	—
327	C₂₃	#25469	1	1	yes	characteristic, direct product factor, Sylow
328	C₂⁴	—	2	1	no	—
329	C₂×Q₈	—	2	2	no	—
330	C₂×Q₈	—	2	2	no	—
331	C₂×Q₈	—	4	2	no	—
332	C₂×Q₈	—	4	2	no	—
333	C₂×Q₈	—	4	2	no	—
334	C₂×Q₈	D₈×C₂₃	1	4	yes	—
335	C₂×D₄	—	2	1	no	—
336	C₂×D₄	—	2	1	no	—
337	C₂×D₄	—	2	1	no	—
338	C₂×D₄	—	2	1	no	—
339	C₂×D₄	—	2	1	no	—
340	C₂×D₄	C₂×Q₈×C₂₃	1	2	yes	—
341	C₂²×C₄	—	2	1	no	—
342	C₂²×C₄	—	2	1	no	—
343	C₂²×C₄	—	2	1	no	—
344	C₂²×C₄	—	2	2	no	—
345	C₂²×C₄	—	2	2	no	—
346	C₂²×C₄	—	4	2	no	—
347	C₂²×C₄	—	4	2	no	—
348	C₂²×C₄	—	4	2	no	—
349	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
350	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
351	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
352	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
353	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
354	C₂×C₈	—	2	1	no	—
355	C₂×C₈	—	2	2	no	—
356	C₂×C₈	—	2	2	no	—
357	C₂×C₈	—	2	1	no	—
358	C₂×C₈	—	2	1	no	—
359	C₂×C₈	—	2	1	no	—
360	C₂×C₈	—	2	1	no	—
361	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
362	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
363	C₄⋊C₄	—	2	4	no	—
364	C₄⋊C₄	—	2	2	no	—
365	C₄⋊C₄	—	2	1	no	—
366	C₄⋊C₄	—	2	1	no	—
367	C₄⋊C₄	—	2	2	no	—
368	C₄⋊C₄	—	2	1	no	—
369	C₄⋊C₄	—	2	2	no	—
370	C₄⋊C₄	—	2	1	no	—
371	C₄⋊C₄	—	2	2	no	—
372	C₄⋊C₄	—	2	2	no	—
373	C₄⋊C₄	—	4	2	no	—
374	C₄⋊C₄	—	4	2	no	—
375	C₄⋊C₄	—	4	4	no	—
376	C₄⋊C₄	—	4	4	no	—
377	C₄⋊C₄	—	4	4	no	—
378	C₄⋊C₄	—	8	1	no	—
379	C₄⋊C₄	C₂×Q₈×C₂₃	1	2	yes	—
380	C₄⋊C₄	#41×C₂₃	1	2	yes	—
381	C₂²⋊C₄	—	4	2	no	—
382	C₂²⋊C₄	—	4	4	no	—
383	C₂²⋊C₄	—	8	2	no	—
384	C₄²	—	2	2	no	—
385	C₄²	—	2	2	no	—
386	C₄²	—	4	2	no	—
387	C₄²	C₂×D₄×C₂₃	1	2	yes	—
388	C₄²	#41×C₂₃	1	2	yes	—
389	C₂³	—	2	2	no	—
390	C₂³	—	4	1	no	—
391	C₂³	—	4	1	no	—
392	C₂³	—	4	1	no	—
393	C₂³	—	4	1	no	—
394	C₂³	—	4	1	no	—
395	C₂³	—	4	1	no	—
396	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
397	Q₈	—	2	8	no	—
398	Q₈	—	8	4	no	—
399	D₄	—	2	2	no	—
400	D₄	—	2	2	no	—
401	D₄	—	4	1	no	—
402	D₄	—	4	1	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	1	no	—
405	C₂×C₄	—	2	1	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	2	no	—
412	C₂×C₄	—	4	1	no	—
413	C₂×C₄	—	4	1	no	—
414	C₂×C₄	—	4	1	no	—
415	C₂×C₄	—	4	1	no	—
416	C₂×C₄	—	4	2	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	4	1	no	—
419	C₂×C₄	—	4	2	no	—
420	C₂×C₄	—	4	1	no	—
421	C₂×C₄	—	4	2	no	—
422	C₂×C₄	—	4	2	no	—
423	C₂×C₄	—	4	1	no	—
424	C₂×C₄	—	4	2	no	—
425	C₂×C₄	—	8	4	no	—
426	C₂×C₄	C₂₃×#121	1	2	yes	—
427	C₂×C₄	C₂₃×#121	1	2	yes	—
428	C₂×C₄	C₂×D₈×C₂₃	1	4	yes	—
429	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
430	C₂×C₄	C₂₃×#135	1	2	yes	—
431	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
432	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
433	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
434	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
435	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
436	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
437	C₈	—	2	4	no	—
438	C₈	—	2	2	no	—
439	C₈	—	4	1	no	—
440	C₂²	—	4	2	no	—
441	C₂²	—	4	2	no	—
442	C₂²	—	4	2	no	—
443	C₂²	—	8	1	no	—
444	C₂²	—	8	1	no	—
445	C₂²	—	8	1	no	—
446	C₂²	—	8	1	no	—
447	C₂²	C₂×C₂₃×#121	1	1	yes	characteristic
448	C₂²	C₂₃×#548	1	1	yes	characteristic
449	C₂²	D₄×Q₈×C₂₃	1	1	yes	characteristic
450	C₂²	C₂²×D₈×C₂₃	1	1	yes	characteristic
451	C₂²	C₂×C₂₃×#135	1	1	yes	characteristic
452	C₂²	C₂₃×#577	1	1	yes	characteristic
453	C₂²	C₂₃×#578	1	1	yes	characteristic
454	C₄	—	2	4	no	—
455	C₄	—	2	2	no	—
456	C₄	—	4	2	no	—
457	C₄	—	4	2	no	—
458	C₄	—	4	2	no	—
459	C₄	—	8	1	no	—
460	C₄	—	8	2	no	—
461	C₄	—	8	2	no	—
462	C₄	C₂₃×#466	1	4	yes	—
463	C₄	C₂₃×#533	1	1	yes	characteristic
464	C₄	C₂₃×#541	1	1	yes	characteristic
465	C₄	C₂₃×#547	1	1	yes	characteristic
466	C₄	D₄×Q₈×C₂₃	1	1	yes	characteristic
467	C₂	—	8	2	no	—
468	C₂	C₂₃×#2429	1	1	yes	characteristic
469	C₂	C₂×C₂₃×#466	1	1	yes	characteristic
470	C₂	C₂₃×#3063	1	1	yes	characteristic
471	C₂	C₂₃×#3358	1	1	yes	characteristic
472	C₂	C₂₃×#3364	1	1	yes	characteristic
473	C₂	Q₈×D₈×C₂₃	1	1	yes	characteristic
474	C₂	C₂₃×#3381	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂²×D₄×C₂₃	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄×Q₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₈×Q₈×C₂₃	C₂	semidirect
3. C₂₃×#2546	C₂	semidirect
5. C₂₃×#2086	C₂	semidirect
7. C₂₃×#1990	C₂	semidirect
8. C₂₃×#1947	C₂	semidirect
9. C₂₃×#1942	C₂	semidirect
83. C₂₃×#102	Q₈	semidirect
151. #25469	C₂₃	direct (N × H)
204. C₂×C₈×Q₈	C₂×C₂₃	semidirect
205. #2546	C₂×C₂₃	semidirect
207. #2086	C₂×C₂₃	semidirect
209. #1990	C₂×C₂₃	semidirect
210. #1947	C₂×C₂₃	semidirect
211. #1942	C₂×C₂₃	semidirect
320. #102	Q₈×C₂₃	semidirect
327. C₂₃	#25469	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×#474	2
C₂×C₈×Q₈×C₂₃	2
C₂₃×#2546	2
C₂₃×#2538	2
C₂₃×#2086	2
C₂₃×#2067	2
C₂₃×#1990	2
C₂₃×#1947	2
C₂₃×#1942	2
C₂₃×#1934	2
C₄×C₂₃×#102	2
#25469	23

Subgroups — C23×#25469 #19173095

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#25469 #19173095