Subgroups — C₂₃×#22131 #19169757

Order 5888.

636 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3076	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2960	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2922	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1986	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1950	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1946	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1938	C₂	1	2	yes	maximal
8	C₂₃×#1929	C₂	1	2	yes	maximal
9	C₂₃×#1922	C₂	1	2	yes	maximal, complemented
10	C₂₃×#1806	C₂	1	1	yes	maximal, characteristic
11	C₂₃×#1770	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#533	C₂²	1	1	yes	characteristic
13	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×#479	C₄	1	2	yes	complemented
15	C₂₃×#478	C₄	1	2	yes	complemented
16	C₂₃×#477	C₄	1	1	yes	characteristic, complemented
17	C₂₃×#476	C₄	1	1	yes	characteristic, complemented
18	C₂₃×#475	C₄	1	1	yes	characteristic, complemented
19	C₂₃×#474	C₄	1	1	yes	characteristic, complemented
20	C₂₃×#446	—	2	1	no	—
21	C₂₃×#446	—	2	1	no	—
22	C₈×Q₈×C₂₃	—	2	1	no	—
23	C₈×Q₈×C₂₃	—	2	1	no	—
24	C₂₃×#433	—	2	1	no	—
25	C₂₃×#433	—	2	1	no	—
26	C₂₃×#431	—	2	1	no	—
27	C₂₃×#431	—	2	1	no	—
28	C₂×C₂₃×#108	C₂²	1	2	yes	—
29	C₂₃×#427	C₂²	1	2	yes	—
30	C₂₃×#424	C₂²	1	1	yes	characteristic
31	C₂₃×#424	C₂²	1	1	yes	characteristic
32	C₂₃×#424	C₂²	1	2	yes	—
33	C₂₃×#423	C₂²	1	1	yes	characteristic
34	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
35	C₂×C₂₃×#104	C₂²	1	2	yes	—
36	C₂₃×#416	C₂²	1	2	yes	—
37	C₂₃×#408	C₂²	1	1	yes	characteristic
38	C₂₃×#408	C₂²	1	1	yes	characteristic
39	C₂₃×#405	C₂²	1	2	yes	—
40	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
41	C₂×C₂₃×#97	C₂²	1	2	yes	—
42	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
43	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
44	C₂₃×#341	C₂²	1	2	yes	—
45	C₂₃×#339	C₂²	1	2	yes	—
46	C₂₃×#339	C₂²	1	2	yes	—
47	C₂₃×#339	C₂²	1	2	yes	—
48	C₂₃×#336	C₂²	1	2	yes	—
49	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
50	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
51	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
52	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
53	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
54	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
56	C₂₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
57	C₂₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
58	C₂₃×#125	C₂×C₄	1	2	yes	—
59	C₂₃×C₂²⋊Q₈	—	2	1	no	—
60	C₂₃×C₂²⋊Q₈	—	2	1	no	—
61	C₄×Q₈×C₂₃	—	2	1	no	—
62	C₄×Q₈×C₂₃	—	2	1	no	—
63	C₄×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
64	C₄×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
65	C₄×D₄×C₂₃	—	2	1	no	—
66	C₄×D₄×C₂₃	—	2	1	no	—
67	C₄×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
68	C₄×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
69	C₂₃×#117	—	2	1	no	—
70	C₂₃×#117	C₂³	1	2	yes	—
71	C₂₃×#117	C₂³	1	1	yes	characteristic
72	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
73	C₂×C₄²×C₂₃	—	2	1	no	—
74	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
75	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
76	C₂₃×#108	—	4	2	no	—
77	C₂₃×#107	C₂×C₄	1	2	yes	—
78	C₂₃×#107	C₂×C₄	1	2	yes	—
79	C₂₃×#106	C₂×C₄	1	2	yes	—
80	C₂₃×#106	C₂×C₄	1	2	yes	—
81	C₂₃×C₄⋊C₈	—	2	1	no	—
82	C₂₃×C₄⋊C₈	—	2	1	no	—
83	C₂₃×C₄⋊C₈	—	2	1	no	—
84	C₂₃×C₄⋊C₈	—	2	1	no	—
85	C₂₃×C₄⋊C₈	—	2	2	no	—
86	C₂₃×C₄⋊C₈	—	2	2	no	—
87	C₂₃×C₄⋊C₈	—	2	1	no	—
88	C₂₃×C₄⋊C₈	—	2	1	no	—
89	C₂₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
90	C₂₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
91	C₂₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
92	C₂₃×#104	—	4	2	no	—
93	C₂₃×#103	C₂×C₄	1	2	yes	—
94	C₂₃×#103	C₂×C₄	1	2	yes	—
95	C₂₃×#102	C₂×C₄	1	2	yes	—
96	C₂₃×#102	C₂×C₄	1	2	yes	—
97	C₂₃×C₂²⋊C₈	—	2	1	no	—
98	C₂₃×C₂²⋊C₈	—	2	1	no	—
99	C₂₃×C₂²⋊C₈	—	2	1	no	—
100	C₂₃×C₂²⋊C₈	—	2	1	no	—
101	C₂₃×#97	—	2	2	no	—
102	C₂₃×#97	—	2	1	no	—
103	C₂₃×#97	—	4	2	no	—
104	C₄×C₈×C₂₃	—	2	2	no	—
105	C₄×C₈×C₂₃	—	2	1	no	—
106	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
107	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
108	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
109	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
110	C₂₃×#95	—	2	1	no	—
111	#41×C₂₃	—	2	1	no	—
112	#41×C₂₃	—	2	1	no	—
113	#41×C₂₃	—	2	1	no	—
114	#41×C₂₃	—	2	1	no	—
115	C₂×Q₈×C₂₃	—	2	1	no	—
116	C₂×Q₈×C₂₃	—	2	1	no	—
117	C₂×Q₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
118	C₂×D₄×C₂₃	—	2	1	no	—
119	C₂×D₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
120	C₂²×C₄×C₂₃	—	2	1	no	—
121	C₂²×C₄×C₂₃	—	2	1	no	—
122	C₂²×C₄×C₂₃	—	2	1	no	—
123	C₂²×C₄×C₂₃	—	2	2	no	—
124	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
125	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
126	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
127	M₁₆×C₂₃	—	2	2	no	—
128	M₁₆×C₂₃	—	2	1	no	—
129	M₁₆×C₂₃	—	2	1	no	—
130	M₁₆×C₂₃	—	2	1	no	—
131	M₁₆×C₂₃	—	2	1	no	—
132	M₁₆×C₂₃	—	2	2	no	—
133	C₂×C₈×C₂₃	—	2	2	no	—
134	C₂×C₈×C₂₃	—	2	1	no	—
135	C₂×C₈×C₂₃	—	2	2	no	—
136	C₂×C₈×C₂₃	—	2	1	no	—
137	C₂×C₈×C₂₃	—	2	1	no	—
138	C₂×C₈×C₂₃	—	2	1	no	—
139	C₂×C₈×C₂₃	—	2	2	no	—
140	C₂×C₈×C₂₃	—	2	1	no	—
141	C₂×C₈×C₂₃	—	2	1	no	—
142	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
143	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
144	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
145	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
146	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
147	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
148	C₄⋊C₄×C₂₃	—	2	1	no	—
149	C₄⋊C₄×C₂₃	—	2	1	no	—
150	C₄⋊C₄×C₂₃	—	2	2	no	—
151	C₄⋊C₄×C₂₃	—	2	2	no	—
152	C₄⋊C₄×C₂₃	—	2	1	no	—
153	C₄⋊C₄×C₂₃	—	2	1	no	—
154	C₄⋊C₄×C₂₃	—	4	2	no	—
155	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
156	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
157	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
158	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
159	C₂²⋊C₄×C₂₃	—	2	1	no	—
160	C₂²⋊C₄×C₂₃	—	2	2	no	—
161	C₂²⋊C₄×C₂₃	—	2	1	no	—
162	C₂²⋊C₄×C₂₃	—	2	1	no	—
163	C₄²×C₂₃	—	2	1	no	—
164	C₄²×C₂₃	—	2	1	no	—
165	C₄²×C₂₃	—	2	1	no	—
166	C₄²×C₂₃	—	2	2	no	—
167	C₄²×C₂₃	—	2	2	no	—
168	C₄²×C₂₃	—	2	2	no	—
169	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
170	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
171	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
172	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
173	#22131	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
174	C₂³×C₂₃	—	2	1	no	—
175	C₂³×C₂₃	C₂×#41	1	1	yes	characteristic
176	Q₈×C₂₃	—	2	1	no	—
177	Q₈×C₂₃	—	2	1	no	—
178	Q₈×C₂₃	—	4	2	no	—
179	Q₈×C₂₃	—	4	2	no	—
180	D₄×C₂₃	—	2	1	no	—
181	D₄×C₂₃	—	2	1	no	—
182	D₄×C₂₃	—	4	1	no	—
183	D₄×C₂₃	—	4	1	no	—
184	C₂×C₄×C₂₃	—	2	2	no	—
185	C₂×C₄×C₂₃	—	2	1	no	—
186	C₂×C₄×C₂₃	—	2	1	no	—
187	C₂×C₄×C₂₃	—	2	1	no	—
188	C₂×C₄×C₂₃	—	2	2	no	—
189	C₂×C₄×C₂₃	—	2	2	no	—
190	C₂×C₄×C₂₃	—	4	1	no	—
191	C₂×C₄×C₂₃	—	4	2	no	—
192	C₂×C₄×C₂₃	—	4	1	no	—
193	C₂×C₄×C₂₃	—	4	1	no	—
194	C₂×C₄×C₂₃	—	4	2	no	—
195	C₂×C₄×C₂₃	—	4	2	no	—
196	C₂×C₄×C₂₃	—	4	2	no	—
197	C₂×C₄×C₂₃	—	4	2	no	—
198	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
199	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
200	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
201	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
202	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
203	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
204	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
205	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
206	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
207	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
208	C₈×C₂₃	—	2	2	no	—
209	C₈×C₂₃	—	2	2	no	—
210	C₈×C₂₃	—	2	1	no	—
211	C₈×C₂₃	—	2	1	no	—
212	C₈×C₂₃	—	2	1	no	—
213	C₈×C₂₃	—	2	1	no	—
214	C₈×C₂₃	—	4	1	no	—
215	C₈×C₂₃	—	4	2	no	—
216	C₈×C₂₃	—	4	1	no	—
217	#3076	C₂×C₂₃	1	1	yes	characteristic
218	#2960	C₂×C₂₃	1	1	yes	characteristic, complemented
219	#2922	C₂×C₂₃	1	1	yes	characteristic
220	#1986	C₂×C₂₃	1	1	yes	characteristic, complemented
221	#1950	C₂×C₂₃	1	1	yes	characteristic, complemented
222	#1946	C₂×C₂₃	1	2	yes	complemented
223	#1938	C₂×C₂₃	1	2	yes	—
224	#1929	C₂×C₂₃	1	2	yes	—
225	#1922	C₂×C₂₃	1	2	yes	complemented
226	#1806	C₂×C₂₃	1	1	yes	characteristic
227	#1770	C₂×C₂₃	1	1	yes	characteristic, complemented
228	C₂²×C₂₃	—	2	1	no	—
229	C₂²×C₂₃	—	2	1	no	—
230	C₂²×C₂₃	—	4	1	no	—
231	C₂²×C₂₃	—	4	1	no	—
232	C₂²×C₂₃	—	4	1	no	—
233	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
234	C₂²×C₂₃	#541	1	1	yes	characteristic
235	C₂²×C₂₃	#542	1	1	yes	characteristic
236	C₄×C₂₃	—	2	1	no	—
237	C₄×C₂₃	—	2	1	no	—
238	C₄×C₂₃	—	2	1	no	—
239	C₄×C₂₃	—	4	1	no	—
240	C₄×C₂₃	—	4	1	no	—
241	C₄×C₂₃	—	4	2	no	—
242	C₄×C₂₃	—	4	1	no	—
243	C₄×C₂₃	—	4	2	no	—
244	C₄×C₂₃	—	8	2	no	—
245	C₄×C₂₃	#444	1	2	yes	—
246	C₄×C₂₃	#519	1	1	yes	characteristic
247	C₄×C₂₃	#520	1	1	yes	characteristic
248	C₄×C₂₃	#536	1	1	yes	characteristic
249	C₄×C₂₃	#537	1	1	yes	characteristic
250	#533	C₂²×C₂₃	1	1	yes	characteristic
251	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
252	#479	C₄×C₂₃	1	2	yes	complemented
253	#478	C₄×C₂₃	1	2	yes	complemented
254	#477	C₄×C₂₃	1	1	yes	characteristic, complemented
255	#476	C₄×C₂₃	1	1	yes	characteristic, complemented
256	#475	C₄×C₂₃	1	1	yes	characteristic, complemented
257	#474	C₄×C₂₃	1	1	yes	characteristic, complemented
258	#446	—	2	1	no	—
259	#446	—	2	1	no	—
260	C₈×Q₈	—	2	1	no	—
261	C₈×Q₈	—	2	1	no	—
262	#433	—	2	1	no	—
263	#433	—	2	1	no	—
264	#431	—	2	1	no	—
265	#431	—	2	1	no	—
266	C₂×#108	C₂²×C₂₃	1	2	yes	—
267	#427	C₂²×C₂₃	1	2	yes	—
268	#424	C₂²×C₂₃	1	1	yes	characteristic
269	#424	C₂²×C₂₃	1	1	yes	characteristic
270	#424	C₂²×C₂₃	1	2	yes	—
271	#423	C₂²×C₂₃	1	1	yes	characteristic
272	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
273	C₂×#104	C₂²×C₂₃	1	2	yes	—
274	#416	C₂²×C₂₃	1	2	yes	—
275	#408	C₂²×C₂₃	1	1	yes	characteristic
276	#408	C₂²×C₂₃	1	1	yes	characteristic
277	#405	C₂²×C₂₃	1	2	yes	—
278	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
279	C₂×#97	C₂²×C₂₃	1	2	yes	—
280	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
281	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
282	#341	C₂²×C₂₃	1	2	yes	—
283	#339	C₂²×C₂₃	1	2	yes	—
284	#339	C₂²×C₂₃	1	2	yes	—
285	#339	C₂²×C₂₃	1	2	yes	—
286	#336	C₂²×C₂₃	1	2	yes	—
287	C₂×C₂₃	—	2	1	no	—
288	C₂×C₂₃	—	2	1	no	—
289	C₂×C₂₃	—	8	1	no	—
290	C₂×C₂₃	#2323	1	1	yes	characteristic
291	C₂×C₂₃	C₂×#444	1	1	yes	characteristic
292	C₂×C₂₃	#2955	1	1	yes	characteristic
293	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
294	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
295	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
296	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
297	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
298	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
299	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
300	C₄⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
301	C₄⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
302	#125	C₂×C₄×C₂₃	1	2	yes	—
303	C₂²⋊Q₈	—	2	1	no	—
304	C₂²⋊Q₈	—	2	1	no	—
305	C₄×Q₈	—	2	1	no	—
306	C₄×Q₈	—	2	1	no	—
307	C₄×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
308	C₄×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
309	C₄×D₄	—	2	1	no	—
310	C₄×D₄	—	2	1	no	—
311	C₄×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
312	C₄×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
313	#117	—	2	1	no	—
314	#117	C₂³×C₂₃	1	1	yes	characteristic
315	#117	C₂³×C₂₃	1	2	yes	—
316	C₂×C₄⋊C₄	—	2	1	no	—
317	C₂×C₄²	—	2	1	no	—
318	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
319	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
320	#108	—	4	2	no	—
321	#107	C₂×C₄×C₂₃	1	2	yes	—
322	#107	C₂×C₄×C₂₃	1	2	yes	—
323	#106	C₂×C₄×C₂₃	1	2	yes	—
324	#106	C₂×C₄×C₂₃	1	2	yes	—
325	C₄⋊C₈	—	2	1	no	—
326	C₄⋊C₈	—	2	1	no	—
327	C₄⋊C₈	—	2	1	no	—
328	C₄⋊C₈	—	2	1	no	—
329	C₄⋊C₈	—	2	2	no	—
330	C₄⋊C₈	—	2	2	no	—
331	C₄⋊C₈	—	2	1	no	—
332	C₄⋊C₈	—	2	1	no	—
333	C₄⋊C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
334	C₄⋊C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
335	C₄⋊C₈	C₂×C₄×C₂₃	1	2	yes	—
336	#104	—	4	2	no	—
337	#103	C₂×C₄×C₂₃	1	2	yes	—
338	#103	C₂×C₄×C₂₃	1	2	yes	—
339	#102	C₂×C₄×C₂₃	1	2	yes	—
340	#102	C₂×C₄×C₂₃	1	2	yes	—
341	C₂²⋊C₈	—	2	1	no	—
342	C₂²⋊C₈	—	2	1	no	—
343	C₂²⋊C₈	—	2	1	no	—
344	C₂²⋊C₈	—	2	1	no	—
345	#97	—	2	2	no	—
346	#97	—	2	1	no	—
347	#97	—	4	2	no	—
348	C₄×C₈	—	2	2	no	—
349	C₄×C₈	—	2	1	no	—
350	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
351	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
352	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
353	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
354	#95	—	2	1	no	—
355	C₂₃	#22131	1	1	yes	characteristic, direct product factor, Sylow
356	#41	—	2	1	no	—
357	#41	—	2	1	no	—
358	#41	—	2	1	no	—
359	#41	—	2	1	no	—
360	C₂×Q₈	—	2	1	no	—
361	C₂×Q₈	—	2	1	no	—
362	C₂×Q₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
363	C₂×D₄	—	2	1	no	—
364	C₂×D₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
365	C₂²×C₄	—	2	1	no	—
366	C₂²×C₄	—	2	1	no	—
367	C₂²×C₄	—	2	1	no	—
368	C₂²×C₄	—	2	2	no	—
369	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
370	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
371	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
372	M₁₆	—	2	2	no	—
373	M₁₆	—	2	1	no	—
374	M₁₆	—	2	1	no	—
375	M₁₆	—	2	1	no	—
376	M₁₆	—	2	1	no	—
377	M₁₆	—	2	2	no	—
378	C₂×C₈	—	2	1	no	—
379	C₂×C₈	—	2	2	no	—
380	C₂×C₈	—	2	2	no	—
381	C₂×C₈	—	2	1	no	—
382	C₂×C₈	—	2	1	no	—
383	C₂×C₈	—	2	1	no	—
384	C₂×C₈	—	2	1	no	—
385	C₂×C₈	—	2	1	no	—
386	C₂×C₈	—	2	2	no	—
387	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
388	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
389	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
390	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
391	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
392	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
393	C₄⋊C₄	—	2	1	no	—
394	C₄⋊C₄	—	2	1	no	—
395	C₄⋊C₄	—	2	2	no	—
396	C₄⋊C₄	—	2	2	no	—
397	C₄⋊C₄	—	2	1	no	—
398	C₄⋊C₄	—	2	1	no	—
399	C₄⋊C₄	—	4	2	no	—
400	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
401	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
402	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
403	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
404	C₂²⋊C₄	—	2	1	no	—
405	C₂²⋊C₄	—	2	2	no	—
406	C₂²⋊C₄	—	2	1	no	—
407	C₂²⋊C₄	—	2	1	no	—
408	C₄²	—	2	1	no	—
409	C₄²	—	2	1	no	—
410	C₄²	—	2	2	no	—
411	C₄²	—	2	1	no	—
412	C₄²	—	2	2	no	—
413	C₄²	—	2	2	no	—
414	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
415	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
416	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
417	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
418	C₂³	—	2	1	no	—
419	C₂³	C₂×#41×C₂₃	1	1	yes	characteristic
420	Q₈	—	2	1	no	—
421	Q₈	—	2	1	no	—
422	Q₈	—	4	2	no	—
423	Q₈	—	4	2	no	—
424	D₄	—	2	1	no	—
425	D₄	—	2	1	no	—
426	D₄	—	4	1	no	—
427	D₄	—	4	1	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	1	no	—
430	C₂×C₄	—	2	1	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	2	no	—
433	C₂×C₄	—	2	2	no	—
434	C₂×C₄	—	4	1	no	—
435	C₂×C₄	—	4	2	no	—
436	C₂×C₄	—	4	2	no	—
437	C₂×C₄	—	4	1	no	—
438	C₂×C₄	—	4	1	no	—
439	C₂×C₄	—	4	2	no	—
440	C₂×C₄	—	4	2	no	—
441	C₂×C₄	—	4	2	no	—
442	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
443	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
444	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
445	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
446	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
447	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
448	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
449	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
450	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
451	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
452	C₈	—	2	2	no	—
453	C₈	—	2	2	no	—
454	C₈	—	2	1	no	—
455	C₈	—	2	1	no	—
456	C₈	—	2	1	no	—
457	C₈	—	2	1	no	—
458	C₈	—	4	1	no	—
459	C₈	—	4	1	no	—
460	C₈	—	4	2	no	—
461	C₂²	—	2	1	no	—
462	C₂²	—	2	1	no	—
463	C₂²	—	4	1	no	—
464	C₂²	—	4	1	no	—
465	C₂²	—	4	1	no	—
466	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
467	C₂²	C₂₃×#541	1	1	yes	characteristic
468	C₂²	C₂₃×#542	1	1	yes	characteristic
469	C₄	—	2	1	no	—
470	C₄	—	2	1	no	—
471	C₄	—	2	1	no	—
472	C₄	—	4	1	no	—
473	C₄	—	4	2	no	—
474	C₄	—	4	1	no	—
475	C₄	—	4	2	no	—
476	C₄	—	4	1	no	—
477	C₄	—	8	2	no	—
478	C₄	C₂₃×#444	1	2	yes	—
479	C₄	C₂₃×#519	1	1	yes	characteristic
480	C₄	C₂₃×#520	1	1	yes	characteristic
481	C₄	C₂₃×#536	1	1	yes	characteristic
482	C₄	C₂₃×#537	1	1	yes	characteristic
483	C₂	—	2	1	no	—
484	C₂	—	2	1	no	—
485	C₂	—	8	1	no	—
486	C₂	C₂₃×#2323	1	1	yes	characteristic
487	C₂	C₂×C₂₃×#444	1	1	yes	characteristic
488	C₂	C₂₃×#2955	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂×C₄×D₄
prime radical	C₂×#41×C₂₃	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2960	C₂	semidirect
4. C₂₃×#1986	C₂	semidirect
5. C₂₃×#1950	C₂	semidirect
6. C₂₃×#1946	C₂	semidirect
9. C₂₃×#1922	C₂	semidirect
11. C₂₃×#1770	C₂	semidirect
14. C₂₃×#479	C₄	semidirect
15. C₂₃×#478	C₄	semidirect
16. C₂₃×#477	C₄	semidirect
17. C₂₃×#476	C₄	semidirect
18. C₂₃×#475	C₄	semidirect
19. C₂₃×#474	C₄	semidirect
173. #22131	C₂₃	direct (N × H)
218. #2960	C₂×C₂₃	semidirect
220. #1986	C₂×C₂₃	semidirect
221. #1950	C₂×C₂₃	semidirect
222. #1946	C₂×C₂₃	semidirect
225. #1922	C₂×C₂₃	semidirect
227. #1770	C₂×C₂₃	semidirect
252. #479	C₄×C₂₃	semidirect
253. #478	C₄×C₂₃	semidirect
254. #477	C₄×C₂₃	semidirect
255. #476	C₄×C₂₃	semidirect
256. #475	C₄×C₂₃	semidirect
257. #474	C₄×C₂₃	semidirect
355. C₂₃	#22131	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×#3076	2
C₂₃×#2960	2
C₂₃×#2922	2
C₂₃×#1986	2
C₂₃×#1950	2
C₂₃×#1946	2
C₂₃×#1938	2
C₂₃×#1929	2
C₂₃×#1922	2
C₂₃×#1806	2
C₂₃×#1770	2
#22131	23

Subgroups — C23×#22131 #19169757

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#22131 #19169757