Subgroups — C₂₃×#26108 #19173734

Order 5888.

702 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2552	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2099	C₂	1	2	yes	maximal
4	C₂₃×#2088	C₂	1	2	yes	maximal
5	C₂₃×#1948	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#1939	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#1865	C₂	1	2	yes	maximal
8	C₂₃×#1834	C₂	1	1	yes	maximal, characteristic
9	C₂₃×#1803	C₂	1	2	yes	maximal
10	C₂₃×#1803	C₂	1	1	yes	maximal, characteristic
11	C₄×C₂₃×#103	C₂	1	1	yes	maximal, characteristic
12	C₂×C₂₃×#125	—	2	1	no	—
13	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	—
14	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₂₃×#106	C₂²	1	2	yes	—
16	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
18	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
19	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×#103	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
22	C₂₃×#398	—	2	2	no	—
23	C₂₃×#389	—	2	2	no	—
24	C₂₃×#389	—	2	1	no	—
25	C₂₃×#389	C₂²	1	2	yes	—
26	C₂₃×#389	C₂²	1	2	yes	—
27	C₂₃×#384	—	2	1	no	—
28	C₂₃×#384	C₂²	1	1	yes	characteristic
29	C₂₃×#384	C₂²	1	1	yes	characteristic
30	C₂₃×#384	C₂²	1	2	yes	—
31	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
32	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
33	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
34	C₂₃×#340	C₂²	1	2	yes	—
35	C₂₃×#340	C₂²	1	1	yes	characteristic
36	C₂₃×#340	C₂²	1	1	yes	characteristic
37	C₂₃×#340	C₂²	1	2	yes	—
38	C₂₃×#340	C₂²	1	2	yes	—
39	C₂₃×#340	C₂²	1	2	yes	—
40	C₂₃×#336	C₂²	1	2	yes	—
41	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
44	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
45	C₂₃×#125	—	4	4	no	—
46	C₄×Q₈×C₂₃	—	2	2	no	—
47	C₄×Q₈×C₂₃	—	2	4	no	—
48	C₄×Q₈×C₂₃	—	4	2	no	—
49	C₄×Q₈×C₂₃	Q₈	1	4	yes	—
50	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
51	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
52	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
53	C₂×C₄⋊C₄×C₂₃	—	2	1	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	—	2	1	no	—
58	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
59	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
60	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
64	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₂₃	—	2	1	no	—
68	C₂×C₄²×C₂₃	—	2	2	no	—
69	C₂×C₄²×C₂₃	—	2	1	no	—
70	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
71	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
72	C₂₃×#106	—	4	2	no	—
73	C₂₃×C₄⋊C₈	—	2	2	no	—
74	C₂₃×C₄⋊C₈	—	4	2	no	—
75	C₂₃×#103	—	2	2	no	—
76	C₂₃×#103	—	2	4	no	—
77	C₂₃×#103	—	2	2	no	—
78	C₄×C₈×C₂₃	—	2	2	no	—
79	C₂₃×#95	—	2	2	no	—
80	C₂₃×#95	—	2	1	no	—
81	C₂₃×#95	—	2	2	no	—
82	C₂₃×#95	D₄	1	2	yes	—
83	C₂×Q₈×C₂₃	—	2	1	no	—
84	C₂×Q₈×C₂₃	—	2	1	no	—
85	C₂×Q₈×C₂₃	—	2	1	no	—
86	C₂×Q₈×C₂₃	C₂×Q₈	1	2	yes	—
87	C₂×Q₈×C₂₃	#41	1	2	yes	—
88	C₂×Q₈×C₂₃	#41	1	2	yes	—
89	C₂²×C₄×C₂₃	—	2	1	no	—
90	C₂²×C₄×C₂₃	—	2	1	no	—
91	C₂²×C₄×C₂₃	—	2	2	no	—
92	C₂²×C₄×C₂₃	—	2	1	no	—
93	C₂²×C₄×C₂₃	—	2	1	no	—
94	C₂²×C₄×C₂₃	—	2	2	no	—
95	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
96	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
98	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
100	C₂×C₈×C₂₃	—	2	1	no	—
101	C₂×C₈×C₂₃	—	2	2	no	—
102	C₂×C₈×C₂₃	—	2	1	no	—
103	C₂×C₈×C₂₃	—	2	2	no	—
104	C₂×C₈×C₂₃	—	2	1	no	—
105	C₂×C₈×C₂₃	—	2	1	no	—
106	C₂×C₈×C₂₃	—	2	1	no	—
107	C₂×C₈×C₂₃	—	2	2	no	—
108	C₂×C₈×C₂₃	—	2	1	no	—
109	C₄⋊C₄×C₂₃	—	2	4	no	—
110	C₄⋊C₄×C₂₃	—	2	2	no	—
111	C₄⋊C₄×C₂₃	—	2	1	no	—
112	C₄⋊C₄×C₂₃	—	2	1	no	—
113	C₄⋊C₄×C₂₃	—	2	1	no	—
114	C₄⋊C₄×C₂₃	—	2	2	no	—
115	C₄⋊C₄×C₂₃	—	2	2	no	—
116	C₄⋊C₄×C₂₃	—	2	1	no	—
117	C₄⋊C₄×C₂₃	—	2	2	no	—
118	C₄⋊C₄×C₂₃	—	2	1	no	—
119	C₄⋊C₄×C₂₃	—	2	2	no	—
120	C₄⋊C₄×C₂₃	—	4	4	no	—
121	C₄⋊C₄×C₂₃	—	4	4	no	—
122	C₄⋊C₄×C₂₃	—	4	4	no	—
123	C₄⋊C₄×C₂₃	—	4	2	no	—
124	C₄⋊C₄×C₂₃	—	4	2	no	—
125	C₄⋊C₄×C₂₃	—	8	1	no	—
126	C₄⋊C₄×C₂₃	—	8	1	no	—
127	C₄⋊C₄×C₂₃	—	8	2	no	—
128	C₄⋊C₄×C₂₃	—	8	2	no	—
129	C₄⋊C₄×C₂₃	SD₁₆	1	4	yes	—
130	C₄⋊C₄×C₂₃	C₂×Q₈	1	2	yes	—
131	C₄²×C₂₃	—	2	4	no	—
132	C₄²×C₂₃	—	2	2	no	—
133	C₄²×C₂₃	—	2	2	no	—
134	C₄²×C₂₃	—	4	4	no	—
135	C₄²×C₂₃	—	4	2	no	—
136	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
137	C₄²×C₂₃	#41	1	2	yes	—
138	#26108	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
139	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
140	Q₈×C₂₃	—	2	2	no	—
141	Q₈×C₂₃	—	2	2	no	—
142	Q₈×C₂₃	—	2	2	no	—
143	Q₈×C₂₃	#125	1	4	yes	complemented
144	C₂×C₄×C₂₃	—	2	2	no	—
145	C₂×C₄×C₂₃	—	2	2	no	—
146	C₂×C₄×C₂₃	—	2	2	no	—
147	C₂×C₄×C₂₃	—	2	2	no	—
148	C₂×C₄×C₂₃	—	2	1	no	—
149	C₂×C₄×C₂₃	—	2	1	no	—
150	C₂×C₄×C₂₃	—	2	2	no	—
151	C₂×C₄×C₂₃	—	2	2	no	—
152	C₂×C₄×C₂₃	—	2	2	no	—
153	C₂×C₄×C₂₃	—	2	2	no	—
154	C₂×C₄×C₂₃	—	2	2	no	—
155	C₂×C₄×C₂₃	—	4	1	no	—
156	C₂×C₄×C₂₃	—	4	1	no	—
157	C₂×C₄×C₂₃	—	4	2	no	—
158	C₂×C₄×C₂₃	—	4	1	no	—
159	C₂×C₄×C₂₃	—	4	1	no	—
160	C₂×C₄×C₂₃	—	4	2	no	—
161	C₂×C₄×C₂₃	—	4	1	no	—
162	C₂×C₄×C₂₃	—	4	1	no	—
163	C₂×C₄×C₂₃	—	4	2	no	—
164	C₂×C₄×C₂₃	—	4	2	no	—
165	C₂×C₄×C₂₃	—	4	2	no	—
166	C₂×C₄×C₂₃	—	4	1	no	—
167	C₂×C₄×C₂₃	—	4	2	no	—
168	C₂×C₄×C₂₃	—	4	1	no	—
169	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
170	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
171	C₂×C₄×C₂₃	C₂×SD₁₆	1	4	yes	—
172	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
173	C₂×C₄×C₂₃	#137	1	2	yes	—
174	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
175	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
176	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
177	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
178	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
179	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
180	C₈×C₂₃	—	4	1	no	—
181	C₈×C₂₃	—	4	2	no	—
182	C₈×C₂₃	—	4	1	no	—
183	#2552	C₂×C₂₃	1	1	yes	characteristic
184	Q₈×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic
185	#2099	C₂×C₂₃	1	2	yes	—
186	#2088	C₂×C₂₃	1	2	yes	—
187	#1948	C₂×C₂₃	1	1	yes	characteristic
188	#1939	C₂×C₂₃	1	1	yes	characteristic
189	#1865	C₂×C₂₃	1	2	yes	—
190	#1834	C₂×C₂₃	1	1	yes	characteristic
191	#1803	C₂×C₂₃	1	2	yes	—
192	#1803	C₂×C₂₃	1	1	yes	characteristic
193	C₄×#103	C₂×C₂₃	1	1	yes	characteristic
194	C₂²×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
195	C₂²×C₂₃	#546	1	1	yes	characteristic
196	C₂²×C₂₃	#548	1	1	yes	characteristic
197	C₂²×C₂₃	C₂²×SD₁₆	1	1	yes	characteristic
198	C₂²×C₂₃	C₂×#137	1	1	yes	characteristic
199	C₂²×C₂₃	#576	1	1	yes	characteristic
200	C₂²×C₂₃	#578	1	1	yes	characteristic
201	C₄×C₂₃	—	2	2	no	—
202	C₄×C₂₃	—	2	4	no	—
203	C₄×C₂₃	—	2	4	no	—
204	C₄×C₂₃	—	4	2	no	—
205	C₄×C₂₃	—	4	2	no	—
206	C₄×C₂₃	—	8	2	no	—
207	C₄×C₂₃	—	8	1	no	—
208	C₄×C₂₃	—	8	2	no	—
209	C₄×C₂₃	—	8	1	no	—
210	C₄×C₂₃	#475	1	4	yes	—
211	C₄×C₂₃	C₂×#125	1	1	yes	characteristic
212	C₄×C₂₃	#544	1	1	yes	characteristic
213	C₄×C₂₃	#552	1	1	yes	characteristic
214	C₄×C₂₃	#553	1	1	yes	characteristic
215	C₂×#125	—	2	1	no	—
216	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	—
217	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
218	C₂×#106	C₂²×C₂₃	1	2	yes	—
219	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
220	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
221	C₂×#103	C₂²×C₂₃	1	2	yes	—
222	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
223	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
224	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
225	#398	—	2	2	no	—
226	#389	—	2	1	no	—
227	#389	—	2	2	no	—
228	#389	C₂²×C₂₃	1	2	yes	—
229	#389	C₂²×C₂₃	1	2	yes	—
230	#384	—	2	1	no	—
231	#384	C₂²×C₂₃	1	1	yes	characteristic
232	#384	C₂²×C₂₃	1	2	yes	—
233	#384	C₂²×C₂₃	1	1	yes	characteristic
234	C₄×C₄⋊C₄	—	2	1	no	—
235	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
236	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
237	#340	C₂²×C₂₃	1	2	yes	—
238	#340	C₂²×C₂₃	1	2	yes	—
239	#340	C₂²×C₂₃	1	2	yes	—
240	#340	C₂²×C₂₃	1	1	yes	characteristic
241	#340	C₂²×C₂₃	1	2	yes	—
242	#340	C₂²×C₂₃	1	1	yes	characteristic
243	#336	C₂²×C₂₃	1	2	yes	—
244	C₂×C₂₃	#2442	1	1	yes	characteristic
245	C₂×C₂₃	C₂×#475	1	1	yes	characteristic
246	C₂×C₂₃	#3083	1	1	yes	characteristic
247	C₂×C₂₃	#3298	1	1	yes	characteristic
248	C₂×C₂₃	#3333	1	1	yes	characteristic
249	C₂×C₂₃	#3362	1	1	yes	characteristic
250	C₂×C₂₃	#3374	1	1	yes	characteristic
251	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
252	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
253	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
254	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
255	#125	—	4	4	no	—
256	C₄×Q₈	—	2	2	no	—
257	C₄×Q₈	—	2	4	no	—
258	C₄×Q₈	—	4	2	no	—
259	C₄×Q₈	Q₈×C₂₃	1	4	yes	—
260	C₂×C₄⋊C₄	—	2	1	no	—
261	C₂×C₄⋊C₄	—	2	2	no	—
262	C₂×C₄⋊C₄	—	2	1	no	—
263	C₂×C₄⋊C₄	—	2	2	no	—
264	C₂×C₄⋊C₄	—	2	2	no	—
265	C₂×C₄⋊C₄	—	2	1	no	—
266	C₂×C₄⋊C₄	—	2	2	no	—
267	C₂×C₄⋊C₄	—	2	1	no	—
268	C₂×C₄⋊C₄	—	2	2	no	—
269	C₂×C₄⋊C₄	—	2	1	no	—
270	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
271	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
272	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
273	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
274	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
275	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
276	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
277	C₂×C₄²	—	2	1	no	—
278	C₂×C₄²	—	2	2	no	—
279	C₂×C₄²	—	2	1	no	—
280	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
281	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
282	#106	—	4	2	no	—
283	C₄⋊C₈	—	2	2	no	—
284	C₄⋊C₈	—	4	2	no	—
285	#103	—	2	4	no	—
286	#103	—	2	2	no	—
287	#103	—	2	2	no	—
288	C₄×C₈	—	2	2	no	—
289	#95	—	2	2	no	—
290	#95	—	2	1	no	—
291	#95	—	2	2	no	—
292	#95	D₄×C₂₃	1	2	yes	—
293	C₂₃	#26108	1	1	yes	characteristic, direct product factor, Sylow
294	C₂×Q₈	—	2	1	no	—
295	C₂×Q₈	—	2	1	no	—
296	C₂×Q₈	—	2	1	no	—
297	C₂×Q₈	C₂×Q₈×C₂₃	1	2	yes	—
298	C₂×Q₈	#41×C₂₃	1	2	yes	—
299	C₂×Q₈	#41×C₂₃	1	2	yes	—
300	C₂²×C₄	—	2	1	no	—
301	C₂²×C₄	—	2	1	no	—
302	C₂²×C₄	—	2	1	no	—
303	C₂²×C₄	—	2	1	no	—
304	C₂²×C₄	—	2	2	no	—
305	C₂²×C₄	—	2	2	no	—
306	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
307	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
308	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
309	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
310	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
311	C₂×C₈	—	2	1	no	—
312	C₂×C₈	—	2	1	no	—
313	C₂×C₈	—	2	1	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	—	2	1	no	—
316	C₂×C₈	—	2	2	no	—
317	C₂×C₈	—	2	1	no	—
318	C₂×C₈	—	2	2	no	—
319	C₂×C₈	—	2	1	no	—
320	C₄⋊C₄	—	2	4	no	—
321	C₄⋊C₄	—	2	2	no	—
322	C₄⋊C₄	—	2	1	no	—
323	C₄⋊C₄	—	2	1	no	—
324	C₄⋊C₄	—	2	2	no	—
325	C₄⋊C₄	—	2	1	no	—
326	C₄⋊C₄	—	2	2	no	—
327	C₄⋊C₄	—	2	1	no	—
328	C₄⋊C₄	—	2	2	no	—
329	C₄⋊C₄	—	2	1	no	—
330	C₄⋊C₄	—	2	2	no	—
331	C₄⋊C₄	—	4	4	no	—
332	C₄⋊C₄	—	4	4	no	—
333	C₄⋊C₄	—	4	4	no	—
334	C₄⋊C₄	—	4	2	no	—
335	C₄⋊C₄	—	4	2	no	—
336	C₄⋊C₄	—	8	1	no	—
337	C₄⋊C₄	—	8	1	no	—
338	C₄⋊C₄	—	8	2	no	—
339	C₄⋊C₄	—	8	2	no	—
340	C₄⋊C₄	SD₁₆×C₂₃	1	4	yes	—
341	C₄⋊C₄	C₂×Q₈×C₂₃	1	2	yes	—
342	C₄²	—	2	4	no	—
343	C₄²	—	2	2	no	—
344	C₄²	—	2	2	no	—
345	C₄²	—	4	4	no	—
346	C₄²	—	4	2	no	—
347	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
348	C₄²	#41×C₂₃	1	2	yes	—
349	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
350	Q₈	—	2	2	no	—
351	Q₈	—	2	2	no	—
352	Q₈	—	2	2	no	—
353	Q₈	C₂₃×#125	1	4	yes	complemented
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	4	1	no	—
366	C₂×C₄	—	4	1	no	—
367	C₂×C₄	—	4	1	no	—
368	C₂×C₄	—	4	2	no	—
369	C₂×C₄	—	4	1	no	—
370	C₂×C₄	—	4	1	no	—
371	C₂×C₄	—	4	2	no	—
372	C₂×C₄	—	4	2	no	—
373	C₂×C₄	—	4	2	no	—
374	C₂×C₄	—	4	1	no	—
375	C₂×C₄	—	4	2	no	—
376	C₂×C₄	—	4	1	no	—
377	C₂×C₄	—	4	2	no	—
378	C₂×C₄	—	4	1	no	—
379	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
380	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
381	C₂×C₄	C₂×SD₁₆×C₂₃	1	4	yes	—
382	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
383	C₂×C₄	C₂₃×#137	1	2	yes	—
384	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
385	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
386	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
387	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
388	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
389	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
390	C₈	—	4	1	no	—
391	C₈	—	4	1	no	—
392	C₈	—	4	2	no	—
393	C₂²	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
394	C₂²	C₂₃×#546	1	1	yes	characteristic
395	C₂²	C₂₃×#548	1	1	yes	characteristic
396	C₂²	C₂²×SD₁₆×C₂₃	1	1	yes	characteristic
397	C₂²	C₂×C₂₃×#137	1	1	yes	characteristic
398	C₂²	C₂₃×#576	1	1	yes	characteristic
399	C₂²	C₂₃×#578	1	1	yes	characteristic
400	C₄	—	2	4	no	—
401	C₄	—	2	4	no	—
402	C₄	—	2	2	no	—
403	C₄	—	4	2	no	—
404	C₄	—	4	2	no	—
405	C₄	—	8	1	no	—
406	C₄	—	8	2	no	—
407	C₄	—	8	1	no	—
408	C₄	—	8	2	no	—
409	C₄	C₂₃×#475	1	4	yes	—
410	C₄	C₂×C₂₃×#125	1	1	yes	characteristic
411	C₄	C₂₃×#544	1	1	yes	characteristic
412	C₄	C₂₃×#552	1	1	yes	characteristic
413	C₄	C₂₃×#553	1	1	yes	characteristic
414	C₂	C₂₃×#2442	1	1	yes	characteristic
415	C₂	C₂×C₂₃×#475	1	1	yes	characteristic
416	C₂	C₂₃×#3083	1	1	yes	characteristic
417	C₂	C₂₃×#3298	1	1	yes	characteristic
418	C₂	C₂₃×#3333	1	1	yes	characteristic
419	C₂	C₂₃×#3362	1	1	yes	characteristic
420	C₂	C₂₃×#3374	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
138. #26108	C₂₃	direct (N × H)
143. Q₈×C₂₃	#125	semidirect
293. C₂₃	#26108	direct (N × H)
353. Q₈	C₂₃×#125	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#2552	2
Q₈×C₄⋊C₄×C₂₃	2
C₂₃×#2099	2
C₂₃×#2088	2
C₂₃×#1948	2
C₂₃×#1939	2
C₂₃×#1865	2
C₂₃×#1834	2
C₂₃×#1803	2
C₄×C₂₃×#103	2
#26108	23

Subgroups — C23×#26108 #19173734

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#26108 #19173734