Subgroups — C₂₃×#8014 #19155640

Order 5888.

850 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#431	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2285	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1294	C₂	1	4	yes	maximal
5	C₂×C₂₃×#117	—	2	1	no	—
6	C₂²×C₄⋊C₄×C₂₃	—	2	1	no	—
7	C₂²×C₄²×C₂₃	—	2	1	no	—
8	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
9	C₂₃×#431	—	2	2	no	—
10	C₂₃×#431	C₄	1	4	yes	complemented
11	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
12	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₂₃	—	2	1	no	—
14	C₂₃×#384	C₄	1	2	yes	complemented
15	C₂₃×#382	—	2	1	no	—
16	C₂₃×#380	—	2	1	no	—
17	C₂₃×#380	C₂²	1	2	yes	—
18	C₄×C₄⋊C₄×C₂₃	—	2	2	no	—
19	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
20	C₄×C₄⋊C₄×C₂₃	C₄	1	4	yes	complemented
21	C₄×C₄⋊C₄×C₂₃	C₄	1	2	yes	complemented
22	C₂×C₂₃×#95	C₂²	1	2	yes	—
23	C₂³×C₄×C₂₃	—	2	2	no	—
24	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₂₃	—	2	2	no	—
28	C₂₃×#117	—	4	2	no	—
29	C₂₃×#117	—	4	2	no	—
30	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
31	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
32	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
34	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
35	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
36	C₂×C₄²×C₂₃	—	2	4	no	—
37	C₂×C₄²×C₂₃	—	2	2	no	—
38	C₂×C₄²×C₂₃	—	4	2	no	—
39	C₂×C₄²×C₂₃	—	4	2	no	—
40	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₂₃	D₄	1	2	yes	—
45	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₂₃	Q₈	1	2	yes	—
48	C₂₃×C₄⋊C₈	—	2	4	no	—
49	C₂₃×C₂²⋊C₈	—	2	2	no	—
50	C₂₃×C₂²⋊C₈	C₂×C₄	1	4	yes	—
51	C₄×C₈×C₂₃	—	2	4	no	—
52	C₂₃×#95	—	2	2	no	—
53	C₂₃×#95	—	2	2	no	—
54	C₂₃×#95	—	2	2	no	—
55	C₂₃×#95	C₂×C₄	1	4	yes	—
56	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₂₃	—	2	4	no	—
58	C₂²×C₄×C₂₃	—	2	8	no	—
59	C₂²×C₄×C₂₃	—	2	4	no	—
60	C₂²×C₄×C₂₃	—	2	4	no	—
61	C₂²×C₄×C₂₃	—	4	2	no	—
62	C₂²×C₄×C₂₃	—	4	4	no	—
63	C₂²×C₄×C₂₃	—	4	2	no	—
64	C₂²×C₄×C₂₃	—	4	2	no	—
65	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
72	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
73	C₂²×C₄×C₂₃	D₈	1	2	yes	—
74	C₂²×C₄×C₂₃	SD₁₆	1	2	yes	—
75	C₂²×C₄×C₂₃	SD₁₆	1	2	yes	—
76	C₂²×C₄×C₂₃	Q₁₆	1	2	yes	—
77	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
83	C₂×C₈×C₂₃	—	2	4	no	—
84	C₂×C₈×C₂₃	—	4	4	no	—
85	C₄⋊C₄×C₂₃	—	2	16	no	—
86	C₄⋊C₄×C₂₃	—	4	4	no	—
87	C₄⋊C₄×C₂₃	—	4	4	no	—
88	C₄⋊C₄×C₂₃	—	4	4	no	—
89	C₂²⋊C₄×C₂₃	—	4	2	no	—
90	C₂²⋊C₄×C₂₃	—	4	2	no	—
91	C₄²×C₂₃	—	4	8	no	—
92	C₄²×C₂₃	—	4	4	no	—
93	C₄²×C₂₃	C₄²	1	2	yes	—
94	C₄²×C₂₃	C₄²	1	2	yes	—
95	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
96	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
97	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
98	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
99	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
100	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
101	#8014	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	C₂³×C₂₃	—	2	2	no	—
103	C₂³×C₂₃	—	2	2	no	—
104	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
108	C₂³×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
109	C₂³×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
110	C₂³×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
111	C₂×C₄×C₂₃	—	2	2	no	—
112	C₂×C₄×C₂₃	—	2	2	no	—
113	C₂×C₄×C₂₃	—	2	2	no	—
114	C₂×C₄×C₂₃	—	2	4	no	—
115	C₂×C₄×C₂₃	—	2	4	no	—
116	C₂×C₄×C₂₃	—	2	8	no	—
117	C₂×C₄×C₂₃	—	4	1	no	—
118	C₂×C₄×C₂₃	—	4	1	no	—
119	C₂×C₄×C₂₃	—	4	1	no	—
120	C₂×C₄×C₂₃	—	4	1	no	—
121	C₂×C₄×C₂₃	—	4	4	no	—
122	C₂×C₄×C₂₃	—	4	8	no	—
123	C₂×C₄×C₂₃	—	4	8	no	—
124	C₂×C₄×C₂₃	—	4	4	no	—
125	C₂×C₄×C₂₃	—	4	4	no	—
126	C₂×C₄×C₂₃	—	4	4	no	—
127	C₂×C₄×C₂₃	—	4	4	no	—
128	C₂×C₄×C₂₃	#95	1	2	yes	—
129	C₂×C₄×C₂₃	#95	1	2	yes	—
130	C₂×C₄×C₂₃	#95	1	2	yes	—
131	C₂×C₄×C₂₃	#95	1	2	yes	—
132	C₂×C₄×C₂₃	#102	1	4	yes	—
133	C₂×C₄×C₂₃	#102	1	2	yes	—
134	C₂×C₄×C₂₃	#102	1	2	yes	—
135	C₂×C₄×C₂₃	#103	1	4	yes	—
136	C₂×C₄×C₂₃	#103	1	2	yes	—
137	C₂×C₄×C₂₃	#103	1	2	yes	—
138	C₂×C₄×C₂₃	#104	1	4	yes	—
139	C₂×C₄×C₂₃	#106	1	4	yes	—
140	C₂×C₄×C₂₃	#107	1	4	yes	—
141	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
142	C₂×C₄×C₂₃	C₂×C₂²⋊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	C₈×C₂₃	—	4	4	no	—
146	C₂×#431	C₂×C₂₃	1	1	yes	characteristic
147	#2285	C₂×C₂₃	1	1	yes	characteristic
148	C₂×C₄×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic
149	#1294	C₂×C₂₃	1	4	yes	—
150	C₂²×C₂₃	—	2	1	no	—
151	C₂²×C₂₃	—	2	1	no	—
152	C₂²×C₂₃	—	2	1	no	—
153	C₂²×C₂₃	—	2	1	no	—
154	C₂²×C₂₃	—	2	2	no	—
155	C₂²×C₂₃	—	2	2	no	—
156	C₂²×C₂₃	—	2	2	no	—
157	C₂²×C₂₃	—	2	2	no	—
158	C₂²×C₂₃	C₂×#95	1	1	yes	characteristic
159	C₂²×C₂₃	#410	1	2	yes	—
160	C₂²×C₂₃	C₂×#102	1	1	yes	characteristic
161	C₂²×C₂₃	C₂×#102	1	1	yes	characteristic
162	C₂²×C₂₃	C₂×#103	1	1	yes	characteristic
163	C₂²×C₂₃	C₂×#103	1	1	yes	characteristic
164	C₂²×C₂₃	C₂×#104	1	2	yes	—
165	C₂²×C₂₃	C₂×#106	1	1	yes	characteristic
166	C₂²×C₂₃	C₂×#107	1	1	yes	characteristic
167	C₄×C₂₃	—	2	2	no	—
168	C₄×C₂₃	—	2	1	no	—
169	C₄×C₂₃	—	2	1	no	—
170	C₄×C₂₃	—	2	1	no	—
171	C₄×C₂₃	—	2	1	no	—
172	C₄×C₂₃	—	4	8	no	—
173	C₄×C₂₃	—	8	4	no	—
174	C₄×C₂₃	#339	1	4	yes	—
175	C₄×C₂₃	#340	1	4	yes	—
176	C₄×C₂₃	#340	1	4	yes	—
177	C₂×#117	—	2	1	no	—
178	C₂²×C₄⋊C₄	—	2	1	no	—
179	C₂²×C₄²	—	2	1	no	—
180	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
181	#431	—	2	2	no	—
182	#431	C₄×C₂₃	1	4	yes	complemented
183	C₂×C₄⋊C₈	—	2	1	no	—
184	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
185	C₂×C₄×C₈	—	2	1	no	—
186	#384	C₄×C₂₃	1	2	yes	complemented
187	#382	—	2	1	no	—
188	#380	—	2	1	no	—
189	#380	C₂²×C₂₃	1	2	yes	—
190	C₄×C₄⋊C₄	—	2	2	no	—
191	C₄×C₄⋊C₄	—	2	1	no	—
192	C₄×C₄⋊C₄	C₄×C₂₃	1	2	yes	complemented
193	C₄×C₄⋊C₄	C₄×C₂₃	1	4	yes	complemented
194	C₂×#95	C₂²×C₂₃	1	2	yes	—
195	C₂×C₂₃	—	2	1	no	—
196	C₂×C₂₃	—	2	1	no	—
197	C₂×C₂₃	—	2	1	no	—
198	C₂×C₂₃	—	2	1	no	—
199	C₂×C₂₃	#1502	1	2	yes	—
200	C₂×C₂₃	#1503	1	2	yes	—
201	C₂×C₂₃	C₂×#339	1	1	yes	characteristic
202	C₂×C₂₃	C₂×#340	1	1	yes	characteristic
203	C₂×C₂₃	#1740	1	1	yes	characteristic
204	C₂³×C₄	—	2	2	no	—
205	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
206	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
207	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
208	C₂²×C₈	—	2	2	no	—
209	#117	—	4	2	no	—
210	#117	—	4	2	no	—
211	C₂×C₄⋊C₄	—	2	8	no	—
212	C₂×C₄⋊C₄	—	2	4	no	—
213	C₂×C₄⋊C₄	—	2	2	no	—
214	C₂×C₄⋊C₄	—	2	2	no	—
215	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
216	C₂×C₂²⋊C₄	—	2	2	no	—
217	C₂×C₄²	—	2	2	no	—
218	C₂×C₄²	—	2	4	no	—
219	C₂×C₄²	—	4	2	no	—
220	C₂×C₄²	—	4	2	no	—
221	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
222	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
223	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
224	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
225	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
226	C₂×C₄²	D₄×C₂₃	1	2	yes	—
227	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
228	C₂×C₄²	Q₈×C₂₃	1	2	yes	—
229	C₄⋊C₈	—	2	4	no	—
230	C₂²⋊C₈	—	2	2	no	—
231	C₂²⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
232	C₄×C₈	—	2	4	no	—
233	#95	—	2	2	no	—
234	#95	—	2	2	no	—
235	#95	—	2	2	no	—
236	#95	C₂×C₄×C₂₃	1	4	yes	—
237	C₂₃	#8014	1	1	yes	characteristic, direct product factor, Sylow
238	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
239	C₂²×C₄	—	2	4	no	—
240	C₂²×C₄	—	2	8	no	—
241	C₂²×C₄	—	2	4	no	—
242	C₂²×C₄	—	2	4	no	—
243	C₂²×C₄	—	4	2	no	—
244	C₂²×C₄	—	4	4	no	—
245	C₂²×C₄	—	4	2	no	—
246	C₂²×C₄	—	4	2	no	—
247	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
248	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
249	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
250	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
251	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
252	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
253	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
254	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
255	C₂²×C₄	D₈×C₂₃	1	2	yes	—
256	C₂²×C₄	SD₁₆×C₂₃	1	2	yes	—
257	C₂²×C₄	SD₁₆×C₂₃	1	2	yes	—
258	C₂²×C₄	Q₁₆×C₂₃	1	2	yes	—
259	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
260	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
261	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
262	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
263	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
264	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
265	C₂×C₈	—	2	4	no	—
266	C₂×C₈	—	4	4	no	—
267	C₄⋊C₄	—	2	16	no	—
268	C₄⋊C₄	—	4	4	no	—
269	C₄⋊C₄	—	4	4	no	—
270	C₄⋊C₄	—	4	4	no	—
271	C₂²⋊C₄	—	4	2	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₄²	—	4	8	no	—
274	C₄²	—	4	4	no	—
275	C₄²	C₄²×C₂₃	1	2	yes	—
276	C₄²	C₄²×C₂₃	1	2	yes	—
277	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
278	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
279	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
280	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
281	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
282	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
283	C₂³	—	2	2	no	—
284	C₂³	—	2	2	no	—
285	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
286	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
287	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
288	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
289	C₂³	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
290	C₂³	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
291	C₂³	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	2	8	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	—	4	8	no	—
304	C₂×C₄	—	4	8	no	—
305	C₂×C₄	—	4	4	no	—
306	C₂×C₄	—	4	4	no	—
307	C₂×C₄	—	4	4	no	—
308	C₂×C₄	—	4	4	no	—
309	C₂×C₄	C₂₃×#95	1	2	yes	—
310	C₂×C₄	C₂₃×#95	1	2	yes	—
311	C₂×C₄	C₂₃×#95	1	2	yes	—
312	C₂×C₄	C₂₃×#95	1	2	yes	—
313	C₂×C₄	C₂₃×#102	1	4	yes	—
314	C₂×C₄	C₂₃×#102	1	2	yes	—
315	C₂×C₄	C₂₃×#102	1	2	yes	—
316	C₂×C₄	C₂₃×#103	1	4	yes	—
317	C₂×C₄	C₂₃×#103	1	2	yes	—
318	C₂×C₄	C₂₃×#103	1	2	yes	—
319	C₂×C₄	C₂₃×#104	1	4	yes	—
320	C₂×C₄	C₂₃×#106	1	4	yes	—
321	C₂×C₄	C₂₃×#107	1	4	yes	—
322	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
323	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
324	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
325	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
326	C₈	—	4	4	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	2	no	—
332	C₂²	—	2	2	no	—
333	C₂²	—	2	2	no	—
334	C₂²	—	2	2	no	—
335	C₂²	C₂×C₂₃×#95	1	1	yes	characteristic
336	C₂²	C₂₃×#410	1	2	yes	—
337	C₂²	C₂×C₂₃×#102	1	1	yes	characteristic
338	C₂²	C₂×C₂₃×#102	1	1	yes	characteristic
339	C₂²	C₂×C₂₃×#103	1	1	yes	characteristic
340	C₂²	C₂×C₂₃×#103	1	1	yes	characteristic
341	C₂²	C₂×C₂₃×#104	1	2	yes	—
342	C₂²	C₂×C₂₃×#106	1	1	yes	characteristic
343	C₂²	C₂×C₂₃×#107	1	1	yes	characteristic
344	C₄	—	2	2	no	—
345	C₄	—	2	1	no	—
346	C₄	—	2	1	no	—
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	4	8	no	—
350	C₄	—	8	4	no	—
351	C₄	C₂₃×#339	1	4	yes	—
352	C₄	C₂₃×#340	1	4	yes	—
353	C₄	C₂₃×#340	1	4	yes	—
354	C₂	—	2	1	no	—
355	C₂	—	2	1	no	—
356	C₂	—	2	1	no	—
357	C₂	—	2	1	no	—
358	C₂	C₂₃×#1502	1	2	yes	—
359	C₂	C₂₃×#1503	1	2	yes	—
360	C₂	C₂×C₂₃×#339	1	1	yes	characteristic
361	C₂	C₂×C₂₃×#340	1	1	yes	characteristic
362	C₂	C₂₃×#1740	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₂₃
Frattini, cosocle	C₂³×C₄	C₂³×C₂₃
socle	C₂³×C₂₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₂₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₂₃×#431	C₄	semidirect
14. C₂₃×#384	C₄	semidirect
20. C₄×C₄⋊C₄×C₂₃	C₄	semidirect
21. C₄×C₄⋊C₄×C₂₃	C₄	semidirect
101. #8014	C₂₃	direct (N × H)
182. #431	C₄×C₂₃	semidirect
186. #384	C₄×C₂₃	semidirect
192. C₄×C₄⋊C₄	C₄×C₂₃	semidirect
193. C₄×C₄⋊C₄	C₄×C₂₃	semidirect
237. C₂₃	#8014	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×#431	2
C₂₃×#2285	2
C₂×C₄×C₄⋊C₄×C₂₃	2
C₂₃×#1294	2
#8014	23

Subgroups — C23×#8014 #19155640

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8014 #19155640