Subgroups — C₂₃×#60564 #19208190

Order 5888.

2218 conjugacy classes of subgroups.

The subgroup lattice has 546 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, complemented
2	C₂×C₂₃×#518	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3123	C₂	1	4	yes	maximal, complemented
4	C₂₃×#3122	C₂	1	4	yes	maximal, complemented
5	C₂₃×#3118	C₂	1	8	yes	maximal, complemented
6	C₂₃×#3093	C₂	1	2	yes	maximal, complemented
7	C₂₃×#3077	C₂	1	2	yes	maximal, complemented
8	C₂₃×#3036	C₂	1	2	yes	maximal, complemented
9	C₂₃×#3003	C₂	1	2	yes	maximal, complemented
10	C₄×C₂₃×AGL(1,ℤ/8)	C₂	1	4	yes	maximal, complemented
11	C₂₃×#2921	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×D₄×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×C₂₃×AGL(1,ℤ/8)	C₂²	1	4	yes	—
14	C₂₃×#542	C₂²	1	4	yes	complemented
15	C₂₃×#541	C₂²	1	2	yes	complemented
16	C₂₃×#540	C₂²	1	2	yes	complemented
17	C₂₃×#539	—	2	2	no	—
18	C₂₃×#538	—	2	2	no	—
19	C₂₃×#533	C₂²	1	1	yes	characteristic, complemented
20	C₂₃×#532	C₂²	1	1	yes	characteristic, complemented
21	C₂₃×#529	—	2	4	no	—
22	C₂₃×#525	C₂²	1	2	yes	—
23	C₂₃×#518	—	2	4	no	—
24	C₂₃×#518	C₂²	1	4	yes	—
25	C₂₃×#518	C₂²	1	4	yes	—
26	C₄×#41×C₂₃	C₂²	1	2	yes	complemented
27	C₂×C₄×D₄×C₂₃	—	2	2	no	—
28	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
29	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
30	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
31	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
32	C₂²×C₂²⋊C₄×C₂₃	—	2	2	no	—
33	C₂₃×#485	C₂²	1	8	yes	complemented
34	C₂₃×#482	C₂²	1	8	yes	complemented
35	C₂₃×#478	C₂²	1	8	yes	complemented
36	C₂₃×#476	C₂²	1	4	yes	complemented
37	C₂₃×#474	C₂²	1	4	yes	complemented
38	C₂₃×#463	C₂²	1	8	yes	complemented
39	C₂₃×#461	C₂²	1	4	yes	complemented
40	C₂₃×#459	C₂²	1	4	yes	complemented
41	C₂₃×#450	C₂²	1	8	yes	—
42	C₂₃×#447	C₂²	1	8	yes	—
43	C₂₃×#442	C₂²	1	8	yes	complemented
44	C₂₃×#440	C₂²	1	8	yes	complemented
45	C₄×SD₁₆×C₂₃	C₂²	1	8	yes	complemented
46	C₄×D₈×C₂₃	C₂²	1	8	yes	complemented
47	C₂₃×#433	C₂²	1	4	yes	complemented
48	C₂₃×#433	C₂²	1	4	yes	complemented
49	C₂₃×#428	C₂²	1	4	yes	complemented
50	C₂₃×#423	C₂²	1	2	yes	complemented
51	C₂₃×#418	C₂²	1	4	yes	—
52	C₂₃×#417	C₂²	1	4	yes	complemented
53	C₂₃×#407	C₂²	1	2	yes	—
54	C₄×M₁₆×C₂₃	C₂²	1	2	yes	complemented
55	C₂⁵×C₂₃	—	2	1	no	—
56	C₂×#41×C₂₃	C₂³	1	2	yes	—
57	C₂²×D₄×C₂₃	—	2	8	no	—
58	C₂²×D₄×C₂₃	—	2	1	no	—
59	C₂²×D₄×C₂₃	—	2	2	no	—
60	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
61	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
62	C₂³×C₄×C₂₃	—	2	2	no	—
63	C₂³×C₄×C₂₃	—	2	2	no	—
64	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
65	C₂₃×AGL(1,ℤ/8)	—	2	16	no	—
66	C₂×SD₁₆×C₂₃	C₂³	1	8	yes	—
67	C₂×D₈×C₂₃	C₂³	1	8	yes	—
68	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
69	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic, complemented
70	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	complemented
71	C₂₃×#127	C₂³	1	1	yes	characteristic, complemented
72	C₂₃×#126	—	2	4	no	—
73	C₂₃×#126	—	2	4	no	—
74	C₂₃×#125	—	2	4	no	—
75	C₂₃×#125	C₂³	1	2	yes	complemented
76	C₂₃×#124	—	2	4	no	—
77	C₂₃×#124	C₂³	1	2	yes	complemented
78	C₂₃×#123	—	2	2	no	—
79	C₂₃×#123	—	2	4	no	—
80	C₂₃×#123	—	2	4	no	—
81	C₂₃×C₂²⋊Q₈	—	2	2	no	—
82	C₂₃×C₂²⋊Q₈	—	2	4	no	—
83	C₂₃×C₂²⋊Q₈	C₂³	1	4	yes	—
84	C₂₃×#121	—	2	2	no	—
85	C₂₃×#121	—	2	4	no	—
86	C₂₃×#121	C₂³	1	4	yes	—
87	C₂₃×#120	—	2	2	no	—
88	C₄×Q₈×C₂₃	—	2	2	no	—
89	C₄×Q₈×C₂₃	C₂³	1	4	yes	complemented
90	C₄×D₄×C₂₃	—	2	4	no	—
91	C₄×D₄×C₂₃	—	2	4	no	—
92	C₄×D₄×C₂₃	—	2	8	no	—
93	C₄×D₄×C₂₃	—	2	8	no	—
94	C₄×D₄×C₂₃	—	2	4	no	—
95	C₄×D₄×C₂₃	—	2	2	no	—
96	C₄×D₄×C₂₃	—	2	2	no	—
97	C₄×D₄×C₂₃	—	2	2	no	—
98	C₄×D₄×C₂₃	C₂³	1	8	yes	—
99	C₄×D₄×C₂₃	C₂³	1	4	yes	—
100	C₄×D₄×C₂₃	C₂³	1	4	yes	—
101	C₄×D₄×C₂₃	C₂³	1	4	yes	complemented
102	C₂₃×#117	—	2	2	no	—
103	C₂₃×#117	—	2	2	no	—
104	C₂₃×#117	D₄	1	4	yes	—
105	C₂₃×#117	C₂³	1	2	yes	—
106	C₂₃×#117	C₂³	1	4	yes	—
107	C₂₃×#117	C₂³	1	2	yes	—
108	C₂₃×#117	C₂³	1	2	yes	—
109	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
110	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
111	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
112	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
113	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
114	C₂×C₂²⋊C₄×C₂₃	—	2	8	no	—
115	C₂×C₂²⋊C₄×C₂₃	—	2	8	no	—
116	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
117	C₂×C₄²×C₂₃	—	2	1	no	—
118	C₂×C₄²×C₂₃	—	2	2	no	—
119	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
120	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
121	C₂₃×#107	C₂³	1	8	yes	complemented
122	C₂₃×#106	C₂³	1	8	yes	complemented
123	C₂₃×C₄⋊C₈	C₂³	1	4	yes	complemented
124	C₂₃×C₄⋊C₈	C₂³	1	4	yes	complemented
125	C₂₃×#103	C₂³	1	8	yes	complemented
126	C₂₃×#102	C₂³	1	8	yes	—
127	C₂₃×#102	C₂³	1	8	yes	—
128	C₂₃×#102	C₂³	1	8	yes	complemented
129	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	—
130	C₂₃×#97	C₂³	1	4	yes	complemented
131	C₄×C₈×C₂₃	C₂³	1	4	yes	complemented
132	C₂⁴×C₂₃	—	2	8	no	—
133	C₂⁴×C₂₃	—	2	2	no	—
134	C₂⁴×C₂₃	—	2	4	no	—
135	C₂⁴×C₂₃	—	8	4	no	—
136	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
137	#41×C₂₃	—	2	8	no	—
138	C₂×Q₈×C₂₃	—	2	2	no	—
139	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
140	C₂×D₄×C₂₃	—	2	8	no	—
141	C₂×D₄×C₂₃	—	2	16	no	—
142	C₂×D₄×C₂₃	—	2	16	no	—
143	C₂×D₄×C₂₃	—	2	2	no	—
144	C₂×D₄×C₂₃	—	2	4	no	—
145	C₂×D₄×C₂₃	—	2	4	no	—
146	C₂×D₄×C₂₃	—	2	2	no	—
147	C₂×D₄×C₂₃	—	2	2	no	—
148	C₂×D₄×C₂₃	—	2	4	no	—
149	C₂×D₄×C₂₃	#41	1	8	yes	—
150	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
151	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
152	C₂²×C₄×C₂₃	—	2	4	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	—	2	8	no	—
157	C₂²×C₄×C₂₃	—	2	4	no	—
158	C₂²×C₄×C₂₃	—	2	4	no	—
159	C₂²×C₄×C₂₃	—	8	4	no	—
160	C₂²×C₄×C₂₃	—	8	4	no	—
161	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
163	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
164	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
165	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
166	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
167	SD₁₆×C₂₃	—	2	16	no	—
168	D₈×C₂₃	—	2	16	no	—
169	M₁₆×C₂₃	—	2	8	no	—
170	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
171	C₄⋊C₄×C₂₃	—	2	2	no	—
172	C₄⋊C₄×C₂₃	—	2	4	no	—
173	C₄⋊C₄×C₂₃	—	2	2	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	C₂×D₄	1	4	yes	—
178	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
179	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
180	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
181	C₂²⋊C₄×C₂₃	—	2	4	no	—
182	C₂²⋊C₄×C₂₃	—	2	2	no	—
183	C₂²⋊C₄×C₂₃	—	2	4	no	—
184	C₂²⋊C₄×C₂₃	—	2	8	no	—
185	C₂²⋊C₄×C₂₃	—	2	4	no	—
186	C₂²⋊C₄×C₂₃	—	2	4	no	—
187	C₂²⋊C₄×C₂₃	—	2	4	no	—
188	C₂²⋊C₄×C₂₃	—	2	4	no	—
189	C₂²⋊C₄×C₂₃	—	2	4	no	—
190	C₂²⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
191	C₂²⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
192	C₄²×C₂₃	—	2	4	no	—
193	C₄²×C₂₃	—	2	2	no	—
194	C₄²×C₂₃	—	2	4	no	—
195	C₄²×C₂₃	C₂×D₄	1	2	yes	—
196	C₄²×C₂₃	C₂×D₄	1	2	yes	—
197	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
198	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
199	C₄²×C₂₃	C₂⁴	1	2	yes	—
200	#60564	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
201	C₂³×C₂₃	—	2	16	no	—
202	C₂³×C₂₃	—	2	2	no	—
203	C₂³×C₂₃	—	2	8	no	—
204	C₂³×C₂₃	—	2	4	no	—
205	C₂³×C₂₃	—	2	2	no	—
206	C₂³×C₂₃	—	4	2	no	—
207	C₂³×C₂₃	—	4	4	no	—
208	C₂³×C₂₃	—	4	4	no	—
209	C₂³×C₂₃	—	8	8	no	—
210	C₂³×C₂₃	—	8	16	no	—
211	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
212	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
213	Q₈×C₂₃	—	2	4	no	—
214	Q₈×C₂₃	—	8	2	no	—
215	D₄×C₂₃	—	2	16	no	—
216	D₄×C₂₃	—	2	8	no	—
217	D₄×C₂₃	—	2	4	no	—
218	D₄×C₂₃	—	2	4	no	—
219	D₄×C₂₃	—	4	4	no	—
220	D₄×C₂₃	—	8	2	no	—
221	D₄×C₂₃	—	8	4	no	—
222	D₄×C₂₃	C₂×#41	1	8	yes	—
223	C₂×C₄×C₂₃	—	2	2	no	—
224	C₂×C₄×C₂₃	—	2	4	no	—
225	C₂×C₄×C₂₃	—	2	4	no	—
226	C₂×C₄×C₂₃	—	2	4	no	—
227	C₂×C₄×C₂₃	—	2	2	no	—
228	C₂×C₄×C₂₃	—	2	4	no	—
229	C₂×C₄×C₂₃	—	4	2	no	—
230	C₂×C₄×C₂₃	—	4	2	no	—
231	C₂×C₄×C₂₃	—	4	4	no	—
232	C₂×C₄×C₂₃	—	4	4	no	—
233	C₂×C₄×C₂₃	—	8	4	no	—
234	C₂×C₄×C₂₃	—	8	8	no	—
235	C₂×C₄×C₂₃	—	8	4	no	—
236	C₂×C₄×C₂₃	—	8	4	no	—
237	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
238	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
239	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
240	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
241	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
242	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
243	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
244	C₈×C₂₃	—	2	8	no	—
245	#3455	C₂×C₂₃	1	1	yes	characteristic, complemented
246	C₂×#518	C₂×C₂₃	1	1	yes	characteristic, complemented
247	#3123	C₂×C₂₃	1	4	yes	complemented
248	#3122	C₂×C₂₃	1	4	yes	complemented
249	#3118	C₂×C₂₃	1	8	yes	complemented
250	#3093	C₂×C₂₃	1	2	yes	complemented
251	#3077	C₂×C₂₃	1	2	yes	complemented
252	#3036	C₂×C₂₃	1	2	yes	complemented
253	#3003	C₂×C₂₃	1	2	yes	complemented
254	C₄×AGL(1,ℤ/8)	C₂×C₂₃	1	4	yes	complemented
255	#2921	C₂×C₂₃	1	1	yes	characteristic, complemented
256	C₂²×C₂₃	—	2	2	no	—
257	C₂²×C₂₃	—	2	4	no	—
258	C₂²×C₂₃	—	2	2	no	—
259	C₂²×C₂₃	—	2	8	no	—
260	C₂²×C₂₃	—	4	16	no	—
261	C₂²×C₂₃	—	4	4	no	—
262	C₂²×C₂₃	—	4	4	no	—
263	C₂²×C₂₃	—	4	4	no	—
264	C₂²×C₂₃	—	4	4	no	—
265	C₂²×C₂₃	—	4	2	no	—
266	C₂²×C₂₃	—	8	16	no	—
267	C₂²×C₂₃	#525	1	2	yes	—
268	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
269	C₄×C₂₃	—	2	4	no	—
270	C₄×C₂₃	—	2	2	no	—
271	C₄×C₂₃	—	2	2	no	—
272	C₄×C₂₃	—	2	2	no	—
273	C₄×C₂₃	—	4	4	no	—
274	C₄×C₂₃	—	4	4	no	—
275	C₄×C₂₃	—	4	2	no	—
276	C₄×C₂₃	#525	1	2	yes	—
277	C₄×C₂₃	C₂²×#41	1	2	yes	—
278	C₂³×D₄	C₂²×C₂₃	1	1	yes	characteristic
279	C₂×AGL(1,ℤ/8)	C₂²×C₂₃	1	4	yes	—
280	#542	C₂²×C₂₃	1	4	yes	complemented
281	#541	C₂²×C₂₃	1	2	yes	complemented
282	#540	C₂²×C₂₃	1	2	yes	complemented
283	#539	—	2	2	no	—
284	#538	—	2	2	no	—
285	#533	C₂²×C₂₃	1	1	yes	characteristic, complemented
286	#532	C₂²×C₂₃	1	1	yes	characteristic, complemented
287	#529	—	2	4	no	—
288	#525	C₂²×C₂₃	1	2	yes	—
289	#518	—	2	4	no	—
290	#518	C₂²×C₂₃	1	4	yes	—
291	#518	C₂²×C₂₃	1	4	yes	—
292	C₄×#41	C₂²×C₂₃	1	2	yes	complemented
293	C₂×C₄×D₄	—	2	2	no	—
294	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
295	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
296	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
297	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
298	C₂²×C₂²⋊C₄	—	2	2	no	—
299	#485	C₂²×C₂₃	1	8	yes	complemented
300	#482	C₂²×C₂₃	1	8	yes	complemented
301	#478	C₂²×C₂₃	1	8	yes	complemented
302	#476	C₂²×C₂₃	1	4	yes	complemented
303	#474	C₂²×C₂₃	1	4	yes	complemented
304	#463	C₂²×C₂₃	1	8	yes	complemented
305	#461	C₂²×C₂₃	1	4	yes	complemented
306	#459	C₂²×C₂₃	1	4	yes	complemented
307	#450	C₂²×C₂₃	1	8	yes	—
308	#447	C₂²×C₂₃	1	8	yes	—
309	#442	C₂²×C₂₃	1	8	yes	complemented
310	#440	C₂²×C₂₃	1	8	yes	complemented
311	C₄×SD₁₆	C₂²×C₂₃	1	8	yes	complemented
312	C₄×D₈	C₂²×C₂₃	1	8	yes	complemented
313	#433	C₂²×C₂₃	1	4	yes	complemented
314	#433	C₂²×C₂₃	1	4	yes	complemented
315	#428	C₂²×C₂₃	1	4	yes	complemented
316	#423	C₂²×C₂₃	1	2	yes	complemented
317	#418	C₂²×C₂₃	1	4	yes	—
318	#417	C₂²×C₂₃	1	4	yes	complemented
319	#407	C₂²×C₂₃	1	2	yes	—
320	C₄×M₁₆	C₂²×C₂₃	1	2	yes	complemented
321	C₂×C₂₃	—	2	2	no	—
322	C₂×C₂₃	—	4	2	no	—
323	C₂×C₂₃	—	4	8	no	—
324	C₂×C₂₃	—	8	2	no	—
325	C₂×C₂₃	C₂×#525	1	1	yes	characteristic
326	C₂×C₂₃	#3585	1	2	yes	—
327	C₂⁵	—	2	1	no	—
328	C₂×#41	C₂³×C₂₃	1	2	yes	—
329	C₂²×D₄	—	2	8	no	—
330	C₂²×D₄	—	2	1	no	—
331	C₂²×D₄	—	2	2	no	—
332	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
333	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
334	C₂³×C₄	—	2	2	no	—
335	C₂³×C₄	—	2	2	no	—
336	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
337	AGL(1,ℤ/8)	—	2	16	no	—
338	C₂×SD₁₆	C₂³×C₂₃	1	8	yes	—
339	C₂×D₈	C₂³×C₂₃	1	8	yes	—
340	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
341	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic, complemented
342	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	complemented
343	#127	C₂³×C₂₃	1	1	yes	characteristic, complemented
344	#126	—	2	4	no	—
345	#126	—	2	4	no	—
346	#125	—	2	4	no	—
347	#125	C₂³×C₂₃	1	2	yes	complemented
348	#124	—	2	4	no	—
349	#124	C₂³×C₂₃	1	2	yes	complemented
350	#123	—	2	2	no	—
351	#123	—	2	4	no	—
352	#123	—	2	4	no	—
353	C₂²⋊Q₈	—	2	2	no	—
354	C₂²⋊Q₈	—	2	4	no	—
355	C₂²⋊Q₈	C₂³×C₂₃	1	4	yes	—
356	#121	—	2	2	no	—
357	#121	—	2	4	no	—
358	#121	C₂³×C₂₃	1	4	yes	—
359	#120	—	2	2	no	—
360	C₄×Q₈	—	2	2	no	—
361	C₄×Q₈	C₂³×C₂₃	1	4	yes	complemented
362	C₄×D₄	—	2	4	no	—
363	C₄×D₄	—	2	4	no	—
364	C₄×D₄	—	2	8	no	—
365	C₄×D₄	—	2	8	no	—
366	C₄×D₄	—	2	4	no	—
367	C₄×D₄	—	2	2	no	—
368	C₄×D₄	—	2	2	no	—
369	C₄×D₄	—	2	2	no	—
370	C₄×D₄	C₂³×C₂₃	1	8	yes	—
371	C₄×D₄	C₂³×C₂₃	1	4	yes	—
372	C₄×D₄	C₂³×C₂₃	1	4	yes	—
373	C₄×D₄	C₂³×C₂₃	1	4	yes	complemented
374	#117	—	2	2	no	—
375	#117	—	2	2	no	—
376	#117	D₄×C₂₃	1	4	yes	—
377	#117	C₂³×C₂₃	1	4	yes	—
378	#117	C₂³×C₂₃	1	2	yes	—
379	#117	C₂³×C₂₃	1	2	yes	—
380	#117	C₂³×C₂₃	1	2	yes	—
381	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
382	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
383	C₂×C₂²⋊C₄	—	2	1	no	—
384	C₂×C₂²⋊C₄	—	2	2	no	—
385	C₂×C₂²⋊C₄	—	2	2	no	—
386	C₂×C₂²⋊C₄	—	2	8	no	—
387	C₂×C₂²⋊C₄	—	2	8	no	—
388	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
389	C₂×C₄²	—	2	1	no	—
390	C₂×C₄²	—	2	2	no	—
391	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
392	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
393	#107	C₂³×C₂₃	1	8	yes	complemented
394	#106	C₂³×C₂₃	1	8	yes	complemented
395	C₄⋊C₈	C₂³×C₂₃	1	4	yes	complemented
396	C₄⋊C₈	C₂³×C₂₃	1	4	yes	complemented
397	#103	C₂³×C₂₃	1	8	yes	complemented
398	#102	C₂³×C₂₃	1	8	yes	—
399	#102	C₂³×C₂₃	1	8	yes	—
400	#102	C₂³×C₂₃	1	8	yes	complemented
401	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	—
402	#97	C₂³×C₂₃	1	4	yes	complemented
403	C₄×C₈	C₂³×C₂₃	1	4	yes	complemented
404	C₂₃	#60564	1	1	yes	characteristic, direct product factor, Sylow
405	C₂⁴	—	2	8	no	—
406	C₂⁴	—	2	2	no	—
407	C₂⁴	—	2	4	no	—
408	C₂⁴	—	8	4	no	—
409	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
410	#41	—	2	8	no	—
411	C₂×Q₈	—	2	2	no	—
412	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
413	C₂×D₄	—	2	8	no	—
414	C₂×D₄	—	2	16	no	—
415	C₂×D₄	—	2	16	no	—
416	C₂×D₄	—	2	2	no	—
417	C₂×D₄	—	2	4	no	—
418	C₂×D₄	—	2	4	no	—
419	C₂×D₄	—	2	2	no	—
420	C₂×D₄	—	2	2	no	—
421	C₂×D₄	—	2	4	no	—
422	C₂×D₄	#41×C₂₃	1	8	yes	—
423	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
424	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
425	C₂²×C₄	—	2	4	no	—
426	C₂²×C₄	—	2	2	no	—
427	C₂²×C₄	—	2	2	no	—
428	C₂²×C₄	—	2	4	no	—
429	C₂²×C₄	—	2	8	no	—
430	C₂²×C₄	—	2	4	no	—
431	C₂²×C₄	—	2	4	no	—
432	C₂²×C₄	—	8	4	no	—
433	C₂²×C₄	—	8	4	no	—
434	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
435	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
436	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
437	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
438	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
439	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
440	SD₁₆	—	2	16	no	—
441	D₈	—	2	16	no	—
442	M₁₆	—	2	8	no	—
443	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
444	C₄⋊C₄	—	2	2	no	—
445	C₄⋊C₄	—	2	4	no	—
446	C₄⋊C₄	—	2	2	no	—
447	C₄⋊C₄	—	2	4	no	—
448	C₄⋊C₄	—	2	4	no	—
449	C₄⋊C₄	—	2	2	no	—
450	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
451	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
452	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
453	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
454	C₂²⋊C₄	—	2	4	no	—
455	C₂²⋊C₄	—	2	2	no	—
456	C₂²⋊C₄	—	2	4	no	—
457	C₂²⋊C₄	—	2	8	no	—
458	C₂²⋊C₄	—	2	4	no	—
459	C₂²⋊C₄	—	2	4	no	—
460	C₂²⋊C₄	—	2	4	no	—
461	C₂²⋊C₄	—	2	4	no	—
462	C₂²⋊C₄	—	2	4	no	—
463	C₂²⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
464	C₂²⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
465	C₄²	—	2	4	no	—
466	C₄²	—	2	2	no	—
467	C₄²	—	2	4	no	—
468	C₄²	C₂×D₄×C₂₃	1	2	yes	—
469	C₄²	C₂×D₄×C₂₃	1	2	yes	—
470	C₄²	C₂⁴×C₂₃	1	2	yes	—
471	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
472	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
473	C₂³	—	2	16	no	—
474	C₂³	—	2	4	no	—
475	C₂³	—	2	2	no	—
476	C₂³	—	2	8	no	—
477	C₂³	—	2	2	no	—
478	C₂³	—	4	2	no	—
479	C₂³	—	4	4	no	—
480	C₂³	—	4	4	no	—
481	C₂³	—	8	8	no	—
482	C₂³	—	8	16	no	—
483	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
484	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
485	Q₈	—	2	4	no	—
486	Q₈	—	8	2	no	—
487	D₄	—	2	16	no	—
488	D₄	—	2	4	no	—
489	D₄	—	2	4	no	—
490	D₄	—	2	8	no	—
491	D₄	—	4	4	no	—
492	D₄	—	8	2	no	—
493	D₄	—	8	4	no	—
494	D₄	C₂×#41×C₂₃	1	8	yes	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	4	no	—
497	C₂×C₄	—	2	4	no	—
498	C₂×C₄	—	2	4	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	2	4	no	—
501	C₂×C₄	—	4	2	no	—
502	C₂×C₄	—	4	2	no	—
503	C₂×C₄	—	4	4	no	—
504	C₂×C₄	—	4	4	no	—
505	C₂×C₄	—	8	8	no	—
506	C₂×C₄	—	8	4	no	—
507	C₂×C₄	—	8	4	no	—
508	C₂×C₄	—	8	4	no	—
509	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
510	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
511	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
512	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
513	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
514	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
515	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
516	C₈	—	2	8	no	—
517	C₂²	—	2	2	no	—
518	C₂²	—	2	2	no	—
519	C₂²	—	2	8	no	—
520	C₂²	—	2	4	no	—
521	C₂²	—	4	4	no	—
522	C₂²	—	4	16	no	—
523	C₂²	—	4	4	no	—
524	C₂²	—	4	4	no	—
525	C₂²	—	4	2	no	—
526	C₂²	—	4	4	no	—
527	C₂²	—	8	16	no	—
528	C₂²	C₂₃×#525	1	2	yes	—
529	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
530	C₄	—	2	2	no	—
531	C₄	—	2	4	no	—
532	C₄	—	2	2	no	—
533	C₄	—	2	2	no	—
534	C₄	—	4	2	no	—
535	C₄	—	4	4	no	—
536	C₄	—	4	4	no	—
537	C₄	C₂₃×#525	1	2	yes	—
538	C₄	C₂²×#41×C₂₃	1	2	yes	—
539	C₂	—	2	2	no	—
540	C₂	—	4	2	no	—
541	C₂	—	4	8	no	—
542	C₂	—	8	2	no	—
543	C₂	C₂×C₂₃×#525	1	1	yes	characteristic
544	C₂	C₂₃×#3585	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
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
1. C₂₃×#3455	C₂	semidirect
2. C₂×C₂₃×#518	C₂	semidirect
3. C₂₃×#3123	C₂	semidirect
4. C₂₃×#3122	C₂	semidirect
5. C₂₃×#3118	C₂	semidirect
6. C₂₃×#3093	C₂	semidirect
7. C₂₃×#3077	C₂	semidirect
8. C₂₃×#3036	C₂	semidirect
9. C₂₃×#3003	C₂	semidirect
10. C₄×C₂₃×AGL(1,ℤ/8)	C₂	semidirect
11. C₂₃×#2921	C₂	semidirect
14. C₂₃×#542	C₂²	semidirect
15. C₂₃×#541	C₂²	semidirect
16. C₂₃×#540	C₂²	semidirect
19. C₂₃×#533	C₂²	semidirect
20. C₂₃×#532	C₂²	semidirect
26. C₄×#41×C₂₃	C₂²	semidirect
33. C₂₃×#485	C₂²	semidirect
34. C₂₃×#482	C₂²	semidirect
35. C₂₃×#478	C₂²	semidirect
36. C₂₃×#476	C₂²	semidirect
37. C₂₃×#474	C₂²	semidirect
38. C₂₃×#463	C₂²	semidirect
39. C₂₃×#461	C₂²	semidirect
40. C₂₃×#459	C₂²	semidirect
43. C₂₃×#442	C₂²	semidirect
44. C₂₃×#440	C₂²	semidirect
45. C₄×SD₁₆×C₂₃	C₂²	semidirect
46. C₄×D₈×C₂₃	C₂²	semidirect
47. C₂₃×#433	C₂²	semidirect
48. C₂₃×#433	C₂²	semidirect
49. C₂₃×#428	C₂²	semidirect
50. C₂₃×#423	C₂²	semidirect
52. C₂₃×#417	C₂²	semidirect
54. C₄×M₁₆×C₂₃	C₂²	semidirect
69. C₂₃×C₄⋊Q₈	C₂³	semidirect
70. C₂₃×C₄⋊Q₈	C₂³	semidirect
71. C₂₃×#127	C₂³	semidirect
75. C₂₃×#125	C₂³	semidirect
77. C₂₃×#124	C₂³	semidirect
89. C₄×Q₈×C₂₃	C₂³	semidirect
101. C₄×D₄×C₂₃	C₂³	semidirect
121. C₂₃×#107	C₂³	semidirect
122. C₂₃×#106	C₂³	semidirect
123. C₂₃×C₄⋊C₈	C₂³	semidirect
124. C₂₃×C₄⋊C₈	C₂³	semidirect
125. C₂₃×#103	C₂³	semidirect
128. C₂₃×#102	C₂³	semidirect
130. C₂₃×#97	C₂³	semidirect
131. C₄×C₈×C₂₃	C₂³	semidirect
200. #60564	C₂₃	direct (N × H)
245. #3455	C₂×C₂₃	semidirect
246. C₂×#518	C₂×C₂₃	semidirect
247. #3123	C₂×C₂₃	semidirect
248. #3122	C₂×C₂₃	semidirect
249. #3118	C₂×C₂₃	semidirect
250. #3093	C₂×C₂₃	semidirect
251. #3077	C₂×C₂₃	semidirect
252. #3036	C₂×C₂₃	semidirect
253. #3003	C₂×C₂₃	semidirect
254. C₄×AGL(1,ℤ/8)	C₂×C₂₃	semidirect
255. #2921	C₂×C₂₃	semidirect
280. #542	C₂²×C₂₃	semidirect
281. #541	C₂²×C₂₃	semidirect
282. #540	C₂²×C₂₃	semidirect
285. #533	C₂²×C₂₃	semidirect
286. #532	C₂²×C₂₃	semidirect
292. C₄×#41	C₂²×C₂₃	semidirect
299. #485	C₂²×C₂₃	semidirect
300. #482	C₂²×C₂₃	semidirect
301. #478	C₂²×C₂₃	semidirect
302. #476	C₂²×C₂₃	semidirect
303. #474	C₂²×C₂₃	semidirect
304. #463	C₂²×C₂₃	semidirect
305. #461	C₂²×C₂₃	semidirect
306. #459	C₂²×C₂₃	semidirect
309. #442	C₂²×C₂₃	semidirect
310. #440	C₂²×C₂₃	semidirect
311. C₄×SD₁₆	C₂²×C₂₃	semidirect
312. C₄×D₈	C₂²×C₂₃	semidirect
313. #433	C₂²×C₂₃	semidirect
314. #433	C₂²×C₂₃	semidirect
315. #428	C₂²×C₂₃	semidirect
316. #423	C₂²×C₂₃	semidirect
318. #417	C₂²×C₂₃	semidirect
320. C₄×M₁₆	C₂²×C₂₃	semidirect
341. C₄⋊Q₈	C₂³×C₂₃	semidirect
342. C₄⋊Q₈	C₂³×C₂₃	semidirect
343. #127	C₂³×C₂₃	semidirect
347. #125	C₂³×C₂₃	semidirect
349. #124	C₂³×C₂₃	semidirect
361. C₄×Q₈	C₂³×C₂₃	semidirect
373. C₄×D₄	C₂³×C₂₃	semidirect
393. #107	C₂³×C₂₃	semidirect
394. #106	C₂³×C₂₃	semidirect
395. C₄⋊C₈	C₂³×C₂₃	semidirect
396. C₄⋊C₈	C₂³×C₂₃	semidirect
397. #103	C₂³×C₂₃	semidirect
400. #102	C₂³×C₂₃	semidirect
402. #97	C₂³×C₂₃	semidirect
403. C₄×C₈	C₂³×C₂₃	semidirect
404. C₂₃	#60564	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×#518	2
C₂₃×#3123	2
C₂₃×#3122	2
C₂₃×#3118	2
C₂₃×#3093	2
C₂₃×#3077	2
C₂₃×#3036	2
C₂₃×#3003	2
C₄×C₂₃×AGL(1,ℤ/8)	2
C₂₃×#2921	2
#60564	23

Subgroups — C23×#60564 #19208190

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#60564 #19208190