Subgroups — C₂₃×#10870 #19158496

Order 5888.

774 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×#108	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂₃×#107	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#405	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×#343	C₂	1	2	yes	maximal
5	C₂₃×#1736	C₂	1	2	yes	maximal
6	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
7	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
11	C₂×C₂₃×#108	—	2	2	no	—
12	C₂×C₂₃×#108	—	2	2	no	—
13	C₂×C₂₃×#108	C₄	1	4	yes	complemented
14	C₂×C₂₃×#107	—	2	1	no	—
15	C₂×C₂₃×#107	—	2	1	no	—
16	C₂×C₂₃×#107	—	2	2	no	—
17	C₂×C₂₃×#107	C₄	1	4	yes	complemented
18	C₂₃×#405	—	2	2	no	—
19	C₂₃×#405	C₄	1	4	yes	complemented
20	C₂×C₂₃×#97	—	2	1	no	—
21	C₂×C₄×C₈×C₂₃	—	2	1	no	—
22	C₂₃×#343	—	2	4	no	—
23	C₂₃×#340	—	2	4	no	—
24	C₂³×C₄×C₂₃	—	2	2	no	—
25	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
26	C₂×M₁₆×C₂₃	—	2	1	no	—
27	C₂×M₁₆×C₂₃	—	2	1	no	—
28	C₂×M₁₆×C₂₃	—	2	1	no	—
29	C₂×M₁₆×C₂₃	—	2	2	no	—
30	C₂×M₁₆×C₂₃	—	2	2	no	—
31	C₂×M₁₆×C₂₃	—	2	2	no	—
32	C₂×M₁₆×C₂₃	—	2	2	no	—
33	C₂×M₁₆×C₂₃	—	2	2	no	—
34	C₂×M₁₆×C₂₃	C₂×C₄	1	4	yes	—
35	C₂×M₁₆×C₂₃	C₂×C₄	1	2	yes	—
36	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
37	C₂×M₁₆×C₂₃	Q₈	1	2	yes	—
38	C₂²×C₈×C₂₃	—	2	1	no	—
39	C₂²×C₈×C₂₃	—	2	2	no	—
40	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
41	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₂₃	D₄	1	2	yes	—
46	C₂²×C₈×C₂₃	D₄	1	2	yes	—
47	C₂²×C₈×C₂₃	Q₈	1	2	yes	—
48	C₂₃×#117	—	2	1	no	—
49	C₂₃×#117	—	2	1	no	—
50	C₂₃×#117	C₂×C₄	1	2	yes	—
51	C₂₃×#117	Q₈	1	2	yes	—
52	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
53	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
54	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
55	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
56	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
57	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
58	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
59	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
60	C₂×C₄²×C₂₃	—	2	1	no	—
61	C₂₃×#108	—	2	8	no	—
62	C₂₃×#107	—	2	4	no	—
63	C₂₃×#107	—	2	4	no	—
64	C₂₃×#97	—	2	4	no	—
65	C₄×C₈×C₂₃	—	2	4	no	—
66	C₂⁴×C₂₃	#41	1	1	yes	characteristic
67	C₂²×C₄×C₂₃	—	2	2	no	—
68	C₂²×C₄×C₂₃	—	2	4	no	—
69	C₂²×C₄×C₂₃	—	4	2	no	—
70	C₂²×C₄×C₂₃	—	4	2	no	—
71	C₂²×C₄×C₂₃	—	8	4	no	—
72	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
81	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
82	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
83	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
84	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
85	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
86	M₁₆×C₂₃	—	2	2	no	—
87	M₁₆×C₂₃	—	2	2	no	—
88	M₁₆×C₂₃	—	2	2	no	—
89	M₁₆×C₂₃	—	2	4	no	—
90	M₁₆×C₂₃	—	2	4	no	—
91	M₁₆×C₂₃	—	4	2	no	—
92	M₁₆×C₂₃	—	4	2	no	—
93	M₁₆×C₂₃	C₄⋊C₄	1	4	yes	—
94	C₂×C₈×C₂₃	—	2	2	no	—
95	C₂×C₈×C₂₃	—	2	2	no	—
96	C₂×C₈×C₂₃	—	2	2	no	—
97	C₂×C₈×C₂₃	—	2	4	no	—
98	C₂×C₈×C₂₃	—	4	2	no	—
99	C₂×C₈×C₂₃	—	4	2	no	—
100	C₂×C₈×C₂₃	C₄²	1	4	yes	—
101	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
102	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
103	C₂×C₈×C₂₃	C₂²⋊C₄	1	4	yes	—
104	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
105	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
106	C₂×C₈×C₂₃	C₄⋊C₄	1	4	yes	—
107	C₂×C₈×C₂₃	C₄⋊C₄	1	4	yes	—
108	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
109	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
110	C₄⋊C₄×C₂₃	—	2	2	no	—
111	C₄⋊C₄×C₂₃	—	2	4	no	—
112	C₄⋊C₄×C₂₃	—	2	4	no	—
113	C₄⋊C₄×C₂₃	—	4	2	no	—
114	C₄⋊C₄×C₂₃	—	4	2	no	—
115	C₄⋊C₄×C₂₃	C₄⋊C₄	1	4	yes	—
116	C₂²⋊C₄×C₂₃	—	2	2	no	—
117	C₂²⋊C₄×C₂₃	—	2	2	no	—
118	C₄²×C₂₃	—	2	2	no	—
119	C₄²×C₂₃	—	2	2	no	—
120	#10870	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂³×C₂₃	—	2	2	no	—
122	C₂³×C₂₃	—	2	2	no	—
123	C₂³×C₂₃	#117	1	1	yes	characteristic
124	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
125	C₂³×C₂₃	C₄×Q₈	1	1	yes	characteristic
126	C₂³×C₂₃	#121	1	1	yes	characteristic
127	C₂³×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
128	C₂³×C₂₃	#124	1	1	yes	characteristic
129	C₂³×C₂₃	#125	1	1	yes	characteristic
130	C₂×C₄×C₂₃	—	2	4	no	—
131	C₂×C₄×C₂₃	—	4	2	no	—
132	C₂×C₄×C₂₃	—	4	2	no	—
133	C₂×C₄×C₂₃	—	4	4	no	—
134	C₂×C₄×C₂₃	—	8	4	no	—
135	C₂×C₄×C₂₃	—	8	4	no	—
136	C₂×C₄×C₂₃	—	8	4	no	—
137	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
138	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
147	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
148	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
149	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
150	C₂×C₄×C₂₃	C₄×Q₈	1	1	yes	characteristic
151	C₂×C₄×C₂₃	C₄×Q₈	1	1	yes	characteristic
152	C₂×C₄×C₂₃	C₄×Q₈	1	1	yes	characteristic
153	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
154	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
155	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
156	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
157	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
158	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
159	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
160	C₂×C₄×C₂₃	#125	1	1	yes	characteristic
161	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
162	C₂×C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
163	C₂×C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
164	C₂×C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
165	C₈×C₂₃	—	2	4	no	—
166	C₈×C₂₃	—	4	4	no	—
167	C₈×C₂₃	#95	1	4	yes	—
168	C₈×C₂₃	#95	1	4	yes	—
169	C₂²×#108	C₂×C₂₃	1	1	yes	characteristic
170	C₂²×#107	C₂×C₂₃	1	1	yes	characteristic
171	C₂×#405	C₂×C₂₃	1	1	yes	characteristic
172	C₂×#343	C₂×C₂₃	1	2	yes	—
173	#1736	C₂×C₂₃	1	2	yes	—
174	C₂²×C₂₃	—	2	2	no	—
175	C₂²×C₂₃	—	2	2	no	—
176	C₂²×C₂₃	—	2	2	no	—
177	C₂²×C₂₃	—	2	2	no	—
178	C₂²×C₂₃	—	2	1	no	—
179	C₂²×C₂₃	—	2	1	no	—
180	C₂²×C₂₃	—	2	1	no	—
181	C₂²×C₂₃	—	2	1	no	—
182	C₂²×C₂₃	C₄×C₄⋊C₄	1	1	yes	characteristic
183	C₂²×C₂₃	#380	1	1	yes	characteristic
184	C₂²×C₂₃	#382	1	1	yes	characteristic
185	C₂²×C₂₃	#389	1	1	yes	characteristic
186	C₂²×C₂₃	#389	1	1	yes	characteristic
187	C₂²×C₂₃	#390	1	1	yes	characteristic
188	C₂²×C₂₃	#391	1	1	yes	characteristic
189	C₂²×C₂₃	#472	1	2	yes	—
190	C₂²×C₂₃	#473	1	2	yes	—
191	C₄×C₂₃	—	4	4	no	—
192	C₄×C₂₃	—	8	4	no	—
193	C₄×C₂₃	C₂×#95	1	1	yes	characteristic
194	C₄×C₂₃	C₄×C₄⋊C₄	1	1	yes	characteristic
195	C₄×C₂₃	#380	1	1	yes	characteristic
196	C₄×C₂₃	#384	1	1	yes	characteristic
197	C₄×C₂₃	#389	1	1	yes	characteristic
198	C₄×C₂₃	#389	1	1	yes	characteristic
199	C₄×C₂₃	#390	1	1	yes	characteristic
200	C₄×C₂₃	#391	1	1	yes	characteristic
201	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
202	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
203	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
204	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
205	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
206	C₂×#108	—	2	2	no	—
207	C₂×#108	—	2	2	no	—
208	C₂×#108	C₄×C₂₃	1	4	yes	complemented
209	C₂×#107	—	2	1	no	—
210	C₂×#107	—	2	1	no	—
211	C₂×#107	—	2	2	no	—
212	C₂×#107	C₄×C₂₃	1	4	yes	complemented
213	#405	—	2	2	no	—
214	#405	C₄×C₂₃	1	4	yes	complemented
215	C₂×#97	—	2	1	no	—
216	C₂×C₄×C₈	—	2	1	no	—
217	#343	—	2	4	no	—
218	#340	—	2	4	no	—
219	C₂×C₂₃	—	2	1	no	—
220	C₂×C₂₃	—	2	1	no	—
221	C₂×C₂₃	—	2	1	no	—
222	C₂×C₂₃	—	2	1	no	—
223	C₂×C₂₃	C₄⋊#95	1	1	yes	characteristic
224	C₂×C₂₃	#1816	1	2	yes	—
225	C₂×C₂₃	#1952	1	2	yes	—
226	C₂×C₂₃	#1978	1	1	yes	characteristic
227	C₂×C₂₃	#1979	1	1	yes	characteristic
228	C₂³×C₄	—	2	2	no	—
229	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
230	C₂×M₁₆	—	2	1	no	—
231	C₂×M₁₆	—	2	1	no	—
232	C₂×M₁₆	—	2	1	no	—
233	C₂×M₁₆	—	2	2	no	—
234	C₂×M₁₆	—	2	2	no	—
235	C₂×M₁₆	—	2	2	no	—
236	C₂×M₁₆	—	2	2	no	—
237	C₂×M₁₆	—	2	2	no	—
238	C₂×M₁₆	C₂×C₄×C₂₃	1	4	yes	—
239	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
240	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
241	C₂×M₁₆	Q₈×C₂₃	1	2	yes	—
242	C₂²×C₈	—	2	1	no	—
243	C₂²×C₈	—	2	2	no	—
244	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
245	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
246	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
247	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
248	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
249	C₂²×C₈	D₄×C₂₃	1	2	yes	—
250	C₂²×C₈	D₄×C₂₃	1	2	yes	—
251	C₂²×C₈	Q₈×C₂₃	1	2	yes	—
252	#117	—	2	1	no	—
253	#117	—	2	1	no	—
254	#117	C₂×C₄×C₂₃	1	2	yes	—
255	#117	Q₈×C₂₃	1	2	yes	—
256	C₂×C₄⋊C₄	—	2	2	no	—
257	C₂×C₄⋊C₄	—	2	2	no	—
258	C₂×C₄⋊C₄	—	2	2	no	—
259	C₂×C₄⋊C₄	—	2	2	no	—
260	C₂×C₄⋊C₄	—	2	2	no	—
261	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
262	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
263	C₂×C₂²⋊C₄	—	2	1	no	—
264	C₂×C₄²	—	2	1	no	—
265	#108	—	2	8	no	—
266	#107	—	2	4	no	—
267	#107	—	2	4	no	—
268	#97	—	2	4	no	—
269	C₄×C₈	—	2	4	no	—
270	C₂₃	#10870	1	1	yes	characteristic, direct product factor, Sylow
271	C₂⁴	#41×C₂₃	1	1	yes	characteristic
272	C₂²×C₄	—	2	2	no	—
273	C₂²×C₄	—	2	4	no	—
274	C₂²×C₄	—	4	2	no	—
275	C₂²×C₄	—	4	2	no	—
276	C₂²×C₄	—	8	4	no	—
277	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
278	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
279	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
280	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
281	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
282	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
283	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
284	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
285	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
286	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
287	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
288	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
289	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
290	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
291	M₁₆	—	2	2	no	—
292	M₁₆	—	2	2	no	—
293	M₁₆	—	2	2	no	—
294	M₁₆	—	2	4	no	—
295	M₁₆	—	2	4	no	—
296	M₁₆	—	4	2	no	—
297	M₁₆	—	4	2	no	—
298	M₁₆	C₄⋊C₄×C₂₃	1	4	yes	—
299	C₂×C₈	—	2	2	no	—
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	2	4	no	—
303	C₂×C₈	—	4	2	no	—
304	C₂×C₈	—	4	2	no	—
305	C₂×C₈	C₄²×C₂₃	1	4	yes	—
306	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
307	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
308	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
309	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
310	C₂×C₈	C₂²⋊C₄×C₂₃	1	4	yes	—
311	C₂×C₈	C₄⋊C₄×C₂₃	1	4	yes	—
312	C₂×C₈	C₄⋊C₄×C₂₃	1	4	yes	—
313	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
314	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
315	C₄⋊C₄	—	2	2	no	—
316	C₄⋊C₄	—	2	4	no	—
317	C₄⋊C₄	—	2	4	no	—
318	C₄⋊C₄	—	4	2	no	—
319	C₄⋊C₄	—	4	2	no	—
320	C₄⋊C₄	C₄⋊C₄×C₂₃	1	4	yes	—
321	C₂²⋊C₄	—	2	2	no	—
322	C₂²⋊C₄	—	2	2	no	—
323	C₄²	—	2	2	no	—
324	C₄²	—	2	2	no	—
325	C₂³	—	2	2	no	—
326	C₂³	—	2	2	no	—
327	C₂³	C₂₃×#117	1	1	yes	characteristic
328	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
329	C₂³	C₄×Q₈×C₂₃	1	1	yes	characteristic
330	C₂³	C₂₃×#121	1	1	yes	characteristic
331	C₂³	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
332	C₂³	C₂₃×#124	1	1	yes	characteristic
333	C₂³	C₂₃×#125	1	1	yes	characteristic
334	C₂×C₄	—	2	4	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	4	4	no	—
338	C₂×C₄	—	8	4	no	—
339	C₂×C₄	—	8	4	no	—
340	C₂×C₄	—	8	4	no	—
341	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
342	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
343	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
344	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
345	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
346	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
347	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
348	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
349	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
350	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
351	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
352	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
353	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
354	C₂×C₄	C₄×Q₈×C₂₃	1	1	yes	characteristic
355	C₂×C₄	C₄×Q₈×C₂₃	1	1	yes	characteristic
356	C₂×C₄	C₄×Q₈×C₂₃	1	1	yes	characteristic
357	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
358	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
359	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
360	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
361	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
362	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
363	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
364	C₂×C₄	C₂₃×#125	1	1	yes	characteristic
365	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
366	C₂×C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
367	C₂×C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
368	C₂×C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
369	C₈	—	2	4	no	—
370	C₈	—	4	4	no	—
371	C₈	C₂₃×#95	1	4	yes	—
372	C₈	C₂₃×#95	1	4	yes	—
373	C₂²	—	2	2	no	—
374	C₂²	—	2	2	no	—
375	C₂²	—	2	1	no	—
376	C₂²	—	2	1	no	—
377	C₂²	—	2	1	no	—
378	C₂²	—	2	1	no	—
379	C₂²	—	2	2	no	—
380	C₂²	—	2	2	no	—
381	C₂²	C₄×C₄⋊C₄×C₂₃	1	1	yes	characteristic
382	C₂²	C₂₃×#380	1	1	yes	characteristic
383	C₂²	C₂₃×#382	1	1	yes	characteristic
384	C₂²	C₂₃×#389	1	1	yes	characteristic
385	C₂²	C₂₃×#389	1	1	yes	characteristic
386	C₂²	C₂₃×#390	1	1	yes	characteristic
387	C₂²	C₂₃×#391	1	1	yes	characteristic
388	C₂²	C₂₃×#472	1	2	yes	—
389	C₂²	C₂₃×#473	1	2	yes	—
390	C₄	—	4	4	no	—
391	C₄	—	8	4	no	—
392	C₄	C₂×C₂₃×#95	1	1	yes	characteristic
393	C₄	C₄×C₄⋊C₄×C₂₃	1	1	yes	characteristic
394	C₄	C₂₃×#380	1	1	yes	characteristic
395	C₄	C₂₃×#384	1	1	yes	characteristic
396	C₄	C₂₃×#389	1	1	yes	characteristic
397	C₄	C₂₃×#389	1	1	yes	characteristic
398	C₄	C₂₃×#390	1	1	yes	characteristic
399	C₄	C₂₃×#391	1	1	yes	characteristic
400	C₂	—	2	1	no	—
401	C₂	—	2	1	no	—
402	C₂	—	2	1	no	—
403	C₂	—	2	1	no	—
404	C₂	C₂₃×C₄⋊#95	1	1	yes	characteristic
405	C₂	C₂₃×#1816	1	2	yes	—
406	C₂	C₂₃×#1952	1	2	yes	—
407	C₂	C₂₃×#1978	1	1	yes	characteristic
408	C₂	C₂₃×#1979	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	#121
commutator	C₂×C₄	C₂×C₄²×C₂₃
Frattini, cosocle	C₂³×C₄	C₂³×C₂₃
prime radical	C₂⁴×C₂₃	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂³×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₂×C₂₃×#108	C₄	semidirect
17. C₂×C₂₃×#107	C₄	semidirect
19. C₂₃×#405	C₄	semidirect
120. #10870	C₂₃	direct (N × H)
208. C₂×#108	C₄×C₂₃	semidirect
212. C₂×#107	C₄×C₂₃	semidirect
214. #405	C₄×C₂₃	semidirect
270. C₂₃	#10870	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₂₃×#108	2
C₂²×C₂₃×#107	2
C₂×C₂₃×#405	2
C₂×C₂₃×#343	2
C₂₃×#1736	2
#10870	23

Subgroups — C23×#10870 #19158496

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#10870 #19158496