Subgroups — C₂₃×#60140 #19207766

Order 5888.

2722 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×#131	C₂	1	2	yes	maximal, complemented
2	C₂₃×#3449	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2932	C₂	1	4	yes	maximal, complemented
4	C₂₃×#2931	C₂	1	4	yes	maximal, complemented
5	C₂₃×#2930	C₂	1	8	yes	maximal, complemented
6	C₂₃×#2929	C₂	1	8	yes	maximal, complemented
7	C₂₃×#2905	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#2879	C₂	1	2	yes	maximal, complemented
9	C₂×C₂₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₂₃	C₂²	1	2	yes	—
11	C₂×C₂₃×#131	—	2	16	no	—
12	C₂×C₂₃×#131	C₂²	1	4	yes	—
13	C₂×C₂₃×#131	C₂²	1	4	yes	—
14	C₂×C₂₃×#131	C₂²	1	4	yes	complemented
15	C₂×C₂₃×#131	C₂²	1	4	yes	complemented
16	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
17	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
18	C₂²×M₁₆×C₂₃	C₂²	1	4	yes	complemented
19	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
20	C₂³×C₈×C₂₃	C₂²	1	4	yes	complemented
21	C₂₃×#537	C₄	1	4	yes	—
22	C₂₃×#536	C₄	1	4	yes	—
23	C₂₃×#535	C₄	1	4	yes	—
24	C₂₃×#534	C₄	1	4	yes	—
25	C₂₃×#532	C₂²	1	4	yes	—
26	C₂₃×#525	C₂²	1	8	yes	—
27	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
28	C₂₃×#436	C₂²	1	16	yes	complemented
29	C₂₃×#435	C₂²	1	32	yes	complemented
30	C₈×D₄×C₂₃	C₂²	1	16	yes	complemented
31	C₂₃×#424	C₂²	1	8	yes	complemented
32	C₂₃×#423	C₂²	1	4	yes	complemented
33	C₂₃×#408	C₂²	1	8	yes	—
34	C₂₃×#408	C₂²	1	8	yes	complemented
35	C₂₃×#407	C₂²	1	8	yes	complemented
36	C₂₃×#405	C₂²	1	4	yes	complemented
37	C₂₃×#405	C₂²	1	4	yes	complemented
38	C₂×C₂₃×#97	C₂²	1	2	yes	complemented
39	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	complemented
40	C₂×#41×C₂₃	—	2	8	no	—
41	C₂×#41×C₂₃	—	2	8	no	—
42	C₂×#41×C₂₃	C₂×C₄	1	4	yes	—
43	C₂×#41×C₂₃	C₂×C₄	1	4	yes	—
44	C₂×#41×C₂₃	C₂³	1	4	yes	—
45	C₂×#41×C₂₃	C₂³	1	4	yes	—
46	C₂²×Q₈×C₂₃	C₂×C₄	1	2	yes	—
47	C₂²×D₄×C₂₃	C₂×C₄	1	4	yes	—
48	C₂²×D₄×C₂₃	C₂×C₄	1	2	yes	—
49	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
50	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
51	C₂₃×#131	—	2	16	no	—
52	C₂₃×#131	—	2	16	no	—
53	C₂₃×#131	—	2	16	no	—
54	C₂₃×#131	—	2	16	no	—
55	C₂×M₁₆×C₂₃	—	2	8	no	—
56	C₂×M₁₆×C₂₃	—	2	4	no	—
57	C₂×M₁₆×C₂₃	—	2	16	no	—
58	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
59	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
60	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
61	C₂×M₁₆×C₂₃	C₂³	1	8	yes	—
62	C₂×M₁₆×C₂₃	C₂³	1	8	yes	—
63	C₂²×C₈×C₂₃	—	2	16	no	—
64	C₂²×C₈×C₂₃	D₄	1	8	yes	complemented
65	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
66	C₂²×C₈×C₂₃	C₂³	1	8	yes	—
67	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
68	C₂²×C₈×C₂₃	C₂³	1	8	yes	—
69	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
70	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
72	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
73	C₂₃×C₄⋊Q₈	C₂×C₄	1	4	yes	—
74	C₂₃×#127	C₂×C₄	1	4	yes	—
75	C₂₃×#124	C₂×C₄	1	8	yes	—
76	C₂₃×#123	C₂×C₄	1	16	yes	—
77	C₂₃×C₂²⋊Q₈	C₂×C₄	1	16	yes	—
78	C₂₃×#121	C₂×C₄	1	16	yes	—
79	C₂₃×#121	C₂×C₄	1	16	yes	—
80	C₂₃×#120	C₂×C₄	1	16	yes	—
81	C₄×D₄×C₂₃	C₂³	1	16	yes	—
82	C₂₃×#117	C₂×C₄	1	4	yes	—
83	C₂₃×#117	C₂³	1	4	yes	—
84	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
85	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
86	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
87	C₂₃×C₄⋊C₈	C₂³	1	16	yes	—
88	C₂₃×C₂²⋊C₈	C₂³	1	32	yes	—
89	C₂₃×#97	C₂³	1	8	yes	—
90	C₄×C₈×C₂₃	C₂³	1	4	yes	—
91	C₄×C₈×C₂₃	C₂³	1	4	yes	—
92	C₂⁴×C₂₃	C₂²×C₄	1	4	yes	—
93	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
94	#41×C₂₃	—	2	16	no	—
95	#41×C₂₃	—	2	16	no	—
96	#41×C₂₃	—	2	8	no	—
97	#41×C₂₃	—	2	8	no	—
98	#41×C₂₃	—	2	8	no	—
99	#41×C₂₃	—	2	8	no	—
100	C₂×Q₈×C₂₃	—	2	8	no	—
101	C₂×Q₈×C₂₃	C₂²×C₄	1	4	yes	—
102	C₂×Q₈×C₂₃	C₂²×C₄	1	4	yes	—
103	C₂×D₄×C₂₃	—	2	16	no	—
104	C₂×D₄×C₂₃	—	2	8	no	—
105	C₂×D₄×C₂₃	C₂²×C₄	1	8	yes	—
106	C₂×D₄×C₂₃	C₂²×C₄	1	8	yes	—
107	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
108	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
109	C₂×D₄×C₂₃	C₂²×C₄	1	16	yes	—
110	C₂²×C₄×C₂₃	—	2	8	no	—
111	C₂²×C₄×C₂₃	—	2	8	no	—
112	C₂²×C₄×C₂₃	C₂²×C₄	1	8	yes	—
113	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
114	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
115	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
116	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
117	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
118	C₂²×C₄×C₂₃	#41	1	4	yes	—
119	C₂²×C₄×C₂₃	C₂⁴	1	8	yes	—
120	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
121	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
122	M₁₆×C₂₃	—	2	8	no	—
123	M₁₆×C₂₃	—	2	8	no	—
124	M₁₆×C₂₃	—	2	8	no	—
125	M₁₆×C₂₃	—	2	16	no	—
126	M₁₆×C₂₃	—	2	16	no	—
127	C₂×C₈×C₂₃	—	2	8	no	—
128	C₂×C₈×C₂₃	—	2	4	no	—
129	C₂×C₈×C₂₃	—	2	16	no	—
130	C₂×C₈×C₂₃	—	2	16	no	—
131	C₂×C₈×C₂₃	C₂×D₄	1	8	yes	—
132	C₂×C₈×C₂₃	C₂×D₄	1	8	yes	—
133	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
134	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
135	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
136	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
137	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
138	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
139	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
140	C₄⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
141	C₄⋊C₄×C₂₃	C₂²×C₄	1	16	yes	—
142	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
143	C₂²⋊C₄×C₂₃	C₂²×C₄	1	32	yes	—
144	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
145	C₄²×C₂₃	C₂⁴	1	4	yes	—
146	#60140	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
147	C₂³×C₂₃	—	2	8	no	—
148	C₂³×C₂₃	—	4	2	no	—
149	C₂³×C₂₃	—	4	8	no	—
150	C₂³×C₂₃	C₄×D₄	1	4	yes	—
151	C₂³×C₂₃	C₂³×C₄	1	8	yes	—
152	C₂³×C₂₃	C₂³×C₄	1	2	yes	—
153	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
154	Q₈×C₂₃	—	2	8	no	—
155	Q₈×C₂₃	—	2	8	no	—
156	D₄×C₂₃	—	2	16	no	—
157	D₄×C₂₃	—	2	16	no	—
158	D₄×C₂₃	—	2	8	no	—
159	D₄×C₂₃	—	2	8	no	—
160	D₄×C₂₃	—	4	16	no	—
161	C₂×C₄×C₂₃	—	2	8	no	—
162	C₂×C₄×C₂₃	—	2	16	no	—
163	C₂×C₄×C₂₃	—	2	8	no	—
164	C₂×C₄×C₂₃	—	2	8	no	—
165	C₂×C₄×C₂₃	—	4	4	no	—
166	C₂×C₄×C₂₃	C₄×D₄	1	8	yes	—
167	C₂×C₄×C₂₃	C₄×D₄	1	4	yes	—
168	C₂×C₄×C₂₃	C₂³×C₄	1	8	yes	—
169	C₂×C₄×C₂₃	C₂³×C₄	1	8	yes	—
170	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
171	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
172	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
173	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
174	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
175	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
176	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
177	C₈×C₂₃	—	2	8	no	—
178	C₈×C₂₃	—	2	4	no	—
179	C₈×C₂₃	C₂²×D₄	1	2	yes	—
180	C₈×C₂₃	C₂²×D₄	1	4	yes	—
181	C₈×C₂₃	C₂²×D₄	1	2	yes	—
182	C₂²×#131	C₂×C₂₃	1	2	yes	complemented
183	#3449	C₂×C₂₃	1	1	yes	characteristic
184	#2932	C₂×C₂₃	1	4	yes	complemented
185	#2931	C₂×C₂₃	1	4	yes	complemented
186	#2930	C₂×C₂₃	1	8	yes	complemented
187	#2929	C₂×C₂₃	1	8	yes	complemented
188	#2905	C₂×C₂₃	1	1	yes	characteristic, complemented
189	#2879	C₂×C₂₃	1	2	yes	complemented
190	C₂×#405	C₂×C₂₃	1	1	yes	characteristic, complemented
191	C₂²×C₂₃	—	2	2	no	—
192	C₂²×C₂₃	—	2	4	no	—
193	C₂²×C₂₃	—	2	8	no	—
194	C₂²×C₂₃	—	2	16	no	—
195	C₂²×C₂₃	—	4	4	no	—
196	C₂²×C₂₃	—	4	16	no	—
197	C₂²×C₂₃	C₂×C₄×D₄	1	4	yes	—
198	C₂²×C₂₃	C₂×C₄×D₄	1	2	yes	—
199	C₂²×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
200	C₄×C₂₃	—	2	8	no	—
201	C₄×C₂₃	—	2	8	no	—
202	C₄×C₂₃	C₂×C₄×D₄	1	4	yes	—
203	C₄×C₂₃	C₂×C₄×D₄	1	2	yes	—
204	C₄×C₂₃	C₂³×D₄	1	1	yes	characteristic
205	C₄×C₂₃	C₂²×#41	1	1	yes	characteristic
206	C₂²×#41	C₂²×C₂₃	1	2	yes	—
207	C₂×#131	—	2	16	no	—
208	C₂×#131	C₂²×C₂₃	1	4	yes	—
209	C₂×#131	C₂²×C₂₃	1	4	yes	—
210	C₂×#131	C₂²×C₂₃	1	4	yes	complemented
211	C₂×#131	C₂²×C₂₃	1	4	yes	complemented
212	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
213	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
214	C₂²×M₁₆	C₂²×C₂₃	1	4	yes	complemented
215	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
216	C₂³×C₈	C₂²×C₂₃	1	4	yes	complemented
217	#537	C₄×C₂₃	1	4	yes	—
218	#536	C₄×C₂₃	1	4	yes	—
219	#535	C₄×C₂₃	1	4	yes	—
220	#534	C₄×C₂₃	1	4	yes	—
221	#532	C₂²×C₂₃	1	4	yes	—
222	#525	C₂²×C₂₃	1	8	yes	—
223	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
224	#436	C₂²×C₂₃	1	16	yes	complemented
225	#435	C₂²×C₂₃	1	32	yes	complemented
226	C₈×D₄	C₂²×C₂₃	1	16	yes	complemented
227	#424	C₂²×C₂₃	1	8	yes	complemented
228	#423	C₂²×C₂₃	1	4	yes	complemented
229	#408	C₂²×C₂₃	1	8	yes	—
230	#408	C₂²×C₂₃	1	8	yes	complemented
231	#407	C₂²×C₂₃	1	8	yes	complemented
232	#405	C₂²×C₂₃	1	4	yes	complemented
233	#405	C₂²×C₂₃	1	4	yes	complemented
234	C₂×#97	C₂²×C₂₃	1	2	yes	complemented
235	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	complemented
236	C₂×C₂₃	—	2	4	no	—
237	C₂×C₂₃	—	2	2	no	—
238	C₂×C₂₃	—	4	8	no	—
239	C₂×C₂₃	C₂²×C₄×D₄	1	1	yes	characteristic
240	C₂×C₂₃	#3573	1	2	yes	—
241	C₂×#41	—	2	8	no	—
242	C₂×#41	—	2	8	no	—
243	C₂×#41	C₂×C₄×C₂₃	1	4	yes	—
244	C₂×#41	C₂×C₄×C₂₃	1	4	yes	—
245	C₂×#41	C₂³×C₂₃	1	4	yes	—
246	C₂×#41	C₂³×C₂₃	1	4	yes	—
247	C₂²×Q₈	C₂×C₄×C₂₃	1	2	yes	—
248	C₂²×D₄	C₂×C₄×C₂₃	1	4	yes	—
249	C₂²×D₄	C₂×C₄×C₂₃	1	2	yes	—
250	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
251	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
252	#131	—	2	16	no	—
253	#131	—	2	16	no	—
254	#131	—	2	16	no	—
255	#131	—	2	16	no	—
256	C₂×M₁₆	—	2	8	no	—
257	C₂×M₁₆	—	2	4	no	—
258	C₂×M₁₆	—	2	16	no	—
259	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
260	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
261	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
262	C₂×M₁₆	C₂³×C₂₃	1	8	yes	—
263	C₂×M₁₆	C₂³×C₂₃	1	8	yes	—
264	C₂²×C₈	—	2	16	no	—
265	C₂²×C₈	D₄×C₂₃	1	8	yes	complemented
266	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
267	C₂²×C₈	C₂³×C₂₃	1	8	yes	—
268	C₂²×C₈	C₂³×C₂₃	1	8	yes	—
269	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
270	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
271	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
272	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
273	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
274	C₄⋊Q₈	C₂×C₄×C₂₃	1	4	yes	—
275	#127	C₂×C₄×C₂₃	1	4	yes	—
276	#124	C₂×C₄×C₂₃	1	8	yes	—
277	#123	C₂×C₄×C₂₃	1	16	yes	—
278	C₂²⋊Q₈	C₂×C₄×C₂₃	1	16	yes	—
279	#121	C₂×C₄×C₂₃	1	16	yes	—
280	#121	C₂×C₄×C₂₃	1	16	yes	—
281	#120	C₂×C₄×C₂₃	1	16	yes	—
282	C₄×D₄	C₂³×C₂₃	1	16	yes	—
283	#117	C₂×C₄×C₂₃	1	4	yes	—
284	#117	C₂³×C₂₃	1	4	yes	—
285	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
286	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
287	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
288	C₄⋊C₈	C₂³×C₂₃	1	16	yes	—
289	C₂²⋊C₈	C₂³×C₂₃	1	32	yes	—
290	#97	C₂³×C₂₃	1	8	yes	—
291	C₄×C₈	C₂³×C₂₃	1	4	yes	—
292	C₄×C₈	C₂³×C₂₃	1	4	yes	—
293	C₂₃	#60140	1	1	yes	characteristic, direct product factor, Sylow
294	C₂⁴	C₂²×C₄×C₂₃	1	4	yes	—
295	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
296	#41	—	2	16	no	—
297	#41	—	2	16	no	—
298	#41	—	2	8	no	—
299	#41	—	2	8	no	—
300	#41	—	2	8	no	—
301	#41	—	2	8	no	—
302	C₂×Q₈	—	2	8	no	—
303	C₂×Q₈	C₂²×C₄×C₂₃	1	4	yes	—
304	C₂×Q₈	C₂²×C₄×C₂₃	1	4	yes	—
305	C₂×D₄	—	2	16	no	—
306	C₂×D₄	—	2	8	no	—
307	C₂×D₄	C₂²×C₄×C₂₃	1	8	yes	—
308	C₂×D₄	C₂²×C₄×C₂₃	1	8	yes	—
309	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
310	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
311	C₂×D₄	C₂²×C₄×C₂₃	1	16	yes	—
312	C₂²×C₄	—	2	8	no	—
313	C₂²×C₄	—	2	8	no	—
314	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
315	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
316	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
317	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
318	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
319	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
320	C₂²×C₄	#41×C₂₃	1	4	yes	—
321	C₂²×C₄	C₂⁴×C₂₃	1	8	yes	—
322	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
323	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
324	M₁₆	—	2	8	no	—
325	M₁₆	—	2	8	no	—
326	M₁₆	—	2	8	no	—
327	M₁₆	—	2	16	no	—
328	M₁₆	—	2	16	no	—
329	C₂×C₈	—	2	8	no	—
330	C₂×C₈	—	2	4	no	—
331	C₂×C₈	—	2	16	no	—
332	C₂×C₈	—	2	16	no	—
333	C₂×C₈	C₂×D₄×C₂₃	1	8	yes	—
334	C₂×C₈	C₂×D₄×C₂₃	1	8	yes	—
335	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
336	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
337	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
338	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
339	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
340	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
341	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
342	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
343	C₄⋊C₄	C₂²×C₄×C₂₃	1	16	yes	—
344	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
345	C₂²⋊C₄	C₂²×C₄×C₂₃	1	32	yes	—
346	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
347	C₄²	C₂⁴×C₂₃	1	4	yes	—
348	C₂³	—	2	8	no	—
349	C₂³	—	4	2	no	—
350	C₂³	—	4	8	no	—
351	C₂³	C₄×D₄×C₂₃	1	4	yes	—
352	C₂³	C₂³×C₄×C₂₃	1	8	yes	—
353	C₂³	C₂³×C₄×C₂₃	1	2	yes	—
354	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
355	Q₈	—	2	8	no	—
356	Q₈	—	2	8	no	—
357	D₄	—	2	16	no	—
358	D₄	—	2	16	no	—
359	D₄	—	2	8	no	—
360	D₄	—	2	8	no	—
361	D₄	—	4	16	no	—
362	C₂×C₄	—	2	8	no	—
363	C₂×C₄	—	2	8	no	—
364	C₂×C₄	—	2	8	no	—
365	C₂×C₄	—	2	16	no	—
366	C₂×C₄	—	4	4	no	—
367	C₂×C₄	C₄×D₄×C₂₃	1	8	yes	—
368	C₂×C₄	C₄×D₄×C₂₃	1	4	yes	—
369	C₂×C₄	C₂³×C₄×C₂₃	1	8	yes	—
370	C₂×C₄	C₂³×C₄×C₂₃	1	8	yes	—
371	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
372	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
373	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
374	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
375	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
376	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
377	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
378	C₈	—	2	8	no	—
379	C₈	—	2	4	no	—
380	C₈	C₂²×D₄×C₂₃	1	2	yes	—
381	C₈	C₂²×D₄×C₂₃	1	4	yes	—
382	C₈	C₂²×D₄×C₂₃	1	2	yes	—
383	C₂²	—	2	2	no	—
384	C₂²	—	2	4	no	—
385	C₂²	—	2	8	no	—
386	C₂²	—	2	16	no	—
387	C₂²	—	4	4	no	—
388	C₂²	—	4	16	no	—
389	C₂²	C₂×C₄×D₄×C₂₃	1	4	yes	—
390	C₂²	C₂×C₄×D₄×C₂₃	1	2	yes	—
391	C₂²	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
392	C₄	—	2	8	no	—
393	C₄	—	2	8	no	—
394	C₄	C₂×C₄×D₄×C₂₃	1	4	yes	—
395	C₄	C₂×C₄×D₄×C₂₃	1	2	yes	—
396	C₄	C₂³×D₄×C₂₃	1	1	yes	characteristic
397	C₄	C₂²×#41×C₂₃	1	1	yes	characteristic
398	C₂	—	2	4	no	—
399	C₂	—	2	2	no	—
400	C₂	—	4	8	no	—
401	C₂	C₂²×C₄×D₄×C₂₃	1	1	yes	characteristic
402	C₂	C₂₃×#3573	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₂₃	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₂₃
Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
socle	C₂²×C₂₃	C₂⁴×C₄
prime radical	C₂₃×#3449	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₂₃×#131	C₂	semidirect
3. C₂₃×#2932	C₂	semidirect
4. C₂₃×#2931	C₂	semidirect
5. C₂₃×#2930	C₂	semidirect
6. C₂₃×#2929	C₂	semidirect
7. C₂₃×#2905	C₂	semidirect
8. C₂₃×#2879	C₂	semidirect
9. C₂×C₂₃×#405	C₂	semidirect
14. C₂×C₂₃×#131	C₂²	semidirect
15. C₂×C₂₃×#131	C₂²	semidirect
18. C₂²×M₁₆×C₂₃	C₂²	semidirect
20. C₂³×C₈×C₂₃	C₂²	semidirect
28. C₂₃×#436	C₂²	semidirect
29. C₂₃×#435	C₂²	semidirect
30. C₈×D₄×C₂₃	C₂²	semidirect
31. C₂₃×#424	C₂²	semidirect
32. C₂₃×#423	C₂²	semidirect
34. C₂₃×#408	C₂²	semidirect
35. C₂₃×#407	C₂²	semidirect
36. C₂₃×#405	C₂²	semidirect
37. C₂₃×#405	C₂²	semidirect
38. C₂×C₂₃×#97	C₂²	semidirect
39. C₂×C₄×C₈×C₂₃	C₂²	semidirect
64. C₂²×C₈×C₂₃	D₄	semidirect
146. #60140	C₂₃	direct (N × H)
182. C₂²×#131	C₂×C₂₃	semidirect
184. #2932	C₂×C₂₃	semidirect
185. #2931	C₂×C₂₃	semidirect
186. #2930	C₂×C₂₃	semidirect
187. #2929	C₂×C₂₃	semidirect
188. #2905	C₂×C₂₃	semidirect
189. #2879	C₂×C₂₃	semidirect
190. C₂×#405	C₂×C₂₃	semidirect
210. C₂×#131	C₂²×C₂₃	semidirect
211. C₂×#131	C₂²×C₂₃	semidirect
214. C₂²×M₁₆	C₂²×C₂₃	semidirect
216. C₂³×C₈	C₂²×C₂₃	semidirect
224. #436	C₂²×C₂₃	semidirect
225. #435	C₂²×C₂₃	semidirect
226. C₈×D₄	C₂²×C₂₃	semidirect
227. #424	C₂²×C₂₃	semidirect
228. #423	C₂²×C₂₃	semidirect
230. #408	C₂²×C₂₃	semidirect
231. #407	C₂²×C₂₃	semidirect
232. #405	C₂²×C₂₃	semidirect
233. #405	C₂²×C₂₃	semidirect
234. C₂×#97	C₂²×C₂₃	semidirect
235. C₂×C₄×C₈	C₂²×C₂₃	semidirect
265. C₂²×C₈	D₄×C₂₃	semidirect
293. C₂₃	#60140	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×#131	2
C₂₃×#3449	2
C₂₃×#2932	2
C₂₃×#2931	2
C₂₃×#2930	2
C₂₃×#2929	2
C₂₃×#2905	2
C₂₃×#2879	2
C₂×C₂₃×#405	2
#60140	23

Subgroups — C23×#60140 #19207766

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#60140 #19207766