Subgroups — C₂₃×#21100 #19168726

Order 5888.

1030 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	C₂×C₂₃×#424	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#421	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2896	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×#408	C₂	1	1	yes	maximal, characteristic
5	C₂×C₂₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1938	C₂	1	4	yes	maximal, complemented
7	C₂₃×#1937	C₂	1	4	yes	maximal, complemented
8	C₂₃×#1738	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1733	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₂₃	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
12	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
17	C₂₃×#424	—	2	2	no	—
18	C₂₃×#424	C₂²	1	2	yes	complemented
19	C₂₃×#424	C₂²	1	2	yes	complemented
20	C₂×C₂₃×C₄⋊C₈	—	2	2	no	—
21	C₂₃×#421	—	2	4	no	—
22	C₂×C₂₃×#104	C₂²	1	4	yes	—
23	C₂₃×#419	C₄	1	2	yes	complemented
24	C₂₃×#418	C₄	1	2	yes	complemented
25	C₂₃×#417	C₄	1	4	yes	complemented
26	C₂₃×#416	C₂²	1	4	yes	—
27	C₂₃×#408	—	2	2	no	—
28	C₂₃×#408	C₂²	1	2	yes	—
29	C₂₃×#408	C₂²	1	2	yes	—
30	C₂×C₂₃×C₂²⋊C₈	—	2	2	no	—
31	C₂₃×#405	—	2	4	no	—
32	C₂×C₂₃×#97	C₂²	1	2	yes	complemented
33	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	complemented
34	C₂₃×#341	C₂²	1	4	yes	complemented
35	C₂₃×#339	C₂²	1	4	yes	complemented
36	C₂×#41×C₂₃	—	2	1	no	—
37	C₂×#41×C₂₃	—	2	1	no	—
38	C₂×#41×C₂₃	—	2	2	no	—
39	C₂×#41×C₂₃	—	2	2	no	—
40	C₂×#41×C₂₃	—	2	2	no	—
41	C₂×#41×C₂₃	—	2	2	no	—
42	C₂×#41×C₂₃	C₂×C₄	1	2	yes	—
43	C₂×#41×C₂₃	C₂³	1	2	yes	—
44	C₂²×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×D₄×C₂₃	—	2	1	no	—
46	C₂²×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
47	C₂³×C₄×C₂₃	—	2	1	no	—
48	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
49	C₂×M₁₆×C₂₃	—	2	2	no	—
50	C₂×M₁₆×C₂₃	—	2	2	no	—
51	C₂×M₁₆×C₂₃	—	2	1	no	—
52	C₂×M₁₆×C₂₃	—	2	1	no	—
53	C₂×M₁₆×C₂₃	—	2	2	no	—
54	C₂×M₁₆×C₂₃	—	2	2	no	—
55	C₂×M₁₆×C₂₃	—	2	2	no	—
56	C₂×M₁₆×C₂₃	—	2	2	no	—
57	C₂×M₁₆×C₂₃	C₂×C₄	1	2	yes	—
58	C₂×M₁₆×C₂₃	C₂×C₄	1	2	yes	—
59	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
60	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
61	C₂²×C₈×C₂₃	—	2	2	no	—
62	C₂²×C₈×C₂₃	—	2	2	no	—
63	C₂²×C₈×C₂₃	—	2	1	no	—
64	C₂²×C₈×C₂₃	—	2	1	no	—
65	C₂²×C₈×C₂₃	D₄	1	2	yes	—
66	C₂²×C₈×C₂₃	D₄	1	2	yes	—
67	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
68	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
70	C₂₃×#117	—	2	1	no	—
71	C₂₃×#117	—	2	1	no	—
72	C₂₃×#117	—	2	2	no	—
73	C₂₃×#117	C₂×C₄	1	2	yes	—
74	C₂₃×#117	D₄	1	4	yes	—
75	C₂₃×#117	C₂³	1	2	yes	—
76	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
77	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
78	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
79	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
80	C₂×C₄²×C₂₃	—	2	1	no	—
81	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
82	C₂₃×C₄⋊C₈	—	2	4	no	—
83	C₂₃×C₄⋊C₈	—	2	4	no	—
84	C₂₃×#104	—	2	8	no	—
85	C₂₃×#103	C₂×C₄	1	8	yes	—
86	C₂₃×#102	C₂×C₄	1	8	yes	—
87	C₂₃×C₂²⋊C₈	—	2	4	no	—
88	C₂₃×C₂²⋊C₈	—	2	4	no	—
89	C₂₃×#97	—	2	4	no	—
90	C₄×C₈×C₂₃	—	2	4	no	—
91	C₂⁴×C₂₃	—	2	1	no	—
92	C₂⁴×C₂₃	#41	1	1	yes	characteristic
93	#41×C₂₃	—	2	4	no	—
94	#41×C₂₃	—	2	4	no	—
95	#41×C₂₃	—	2	2	no	—
96	#41×C₂₃	—	2	2	no	—
97	#41×C₂₃	—	2	2	no	—
98	#41×C₂₃	—	2	2	no	—
99	#41×C₂₃	—	4	2	no	—
100	#41×C₂₃	—	4	2	no	—
101	#41×C₂₃	—	4	2	no	—
102	#41×C₂₃	—	4	2	no	—
103	C₂×Q₈×C₂₃	—	2	1	no	—
104	C₂×Q₈×C₂₃	—	2	2	no	—
105	C₂×Q₈×C₂₃	—	2	2	no	—
106	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
107	C₂×D₄×C₂₃	—	2	2	no	—
108	C₂×D₄×C₂₃	—	2	2	no	—
109	C₂×D₄×C₂₃	—	2	2	no	—
110	C₂×D₄×C₂₃	—	2	2	no	—
111	C₂×D₄×C₂₃	—	2	1	no	—
112	C₂×D₄×C₂₃	—	4	2	no	—
113	C₂×D₄×C₂₃	—	4	2	no	—
114	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
115	C₂²×C₄×C₂₃	—	2	2	no	—
116	C₂²×C₄×C₂₃	—	2	1	no	—
117	C₂²×C₄×C₂₃	—	2	2	no	—
118	C₂²×C₄×C₂₃	—	2	2	no	—
119	C₂²×C₄×C₂₃	—	2	2	no	—
120	C₂²×C₄×C₂₃	—	4	2	no	—
121	C₂²×C₄×C₂₃	—	4	2	no	—
122	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
123	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
124	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
125	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₂₃	#41	1	2	yes	—
128	C₂²×C₄×C₂₃	#41	1	2	yes	—
129	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
130	C₂²×C₄×C₂₃	#41	1	2	yes	—
131	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
132	M₁₆×C₂₃	—	2	4	no	—
133	M₁₆×C₂₃	—	2	4	no	—
134	M₁₆×C₂₃	—	2	2	no	—
135	M₁₆×C₂₃	—	2	2	no	—
136	M₁₆×C₂₃	—	2	2	no	—
137	M₁₆×C₂₃	—	2	2	no	—
138	M₁₆×C₂₃	—	4	2	no	—
139	M₁₆×C₂₃	—	4	2	no	—
140	C₂×C₈×C₂₃	—	2	4	no	—
141	C₂×C₈×C₂₃	—	2	4	no	—
142	C₂×C₈×C₂₃	—	2	1	no	—
143	C₂×C₈×C₂₃	—	2	1	no	—
144	C₂×C₈×C₂₃	—	2	2	no	—
145	C₂×C₈×C₂₃	—	2	1	no	—
146	C₂×C₈×C₂₃	—	2	1	no	—
147	C₂×C₈×C₂₃	—	2	2	no	—
148	C₂×C₈×C₂₃	—	2	2	no	—
149	C₂×C₈×C₂₃	—	2	2	no	—
150	C₂×C₈×C₂₃	—	4	2	no	—
151	C₂×C₈×C₂₃	—	4	2	no	—
152	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
153	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
154	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
155	C₄⋊C₄×C₂₃	—	2	2	no	—
156	C₄⋊C₄×C₂₃	—	2	4	no	—
157	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
158	C₂²⋊C₄×C₂₃	—	2	2	no	—
159	C₂²⋊C₄×C₂₃	—	2	4	no	—
160	C₂²⋊C₄×C₂₃	—	2	2	no	—
161	C₄²×C₂₃	—	2	2	no	—
162	C₄²×C₂₃	—	2	2	no	—
163	C₄²×C₂₃	—	2	2	no	—
164	C₄²×C₂₃	C₂×D₄	1	4	yes	—
165	#21100	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
166	C₂³×C₂₃	—	2	2	no	—
167	C₂³×C₂₃	—	4	2	no	—
168	C₂³×C₂₃	—	4	4	no	—
169	C₂³×C₂₃	—	4	2	no	—
170	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
171	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
172	C₂³×C₂₃	#123	1	2	yes	—
173	C₂³×C₂₃	#126	1	2	yes	—
174	C₂³×C₂₃	C₂×#41	1	1	yes	characteristic
175	Q₈×C₂₃	—	2	2	no	—
176	Q₈×C₂₃	—	2	2	no	—
177	Q₈×C₂₃	—	4	2	no	—
178	D₄×C₂₃	—	2	2	no	—
179	D₄×C₂₃	—	2	2	no	—
180	D₄×C₂₃	—	4	2	no	—
181	D₄×C₂₃	—	4	2	no	—
182	D₄×C₂₃	—	4	4	no	—
183	D₄×C₂₃	—	4	2	no	—
184	C₂×C₄×C₂₃	—	2	4	no	—
185	C₂×C₄×C₂₃	—	2	2	no	—
186	C₂×C₄×C₂₃	—	2	4	no	—
187	C₂×C₄×C₂₃	—	4	4	no	—
188	C₂×C₄×C₂₃	—	4	4	no	—
189	C₂×C₄×C₂₃	—	4	2	no	—
190	C₂×C₄×C₂₃	—	4	2	no	—
191	C₂×C₄×C₂₃	—	4	2	no	—
192	C₂×C₄×C₂₃	—	4	4	no	—
193	C₂×C₄×C₂₃	#117	1	2	yes	—
194	C₂×C₄×C₂₃	#117	1	2	yes	—
195	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
196	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
197	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
198	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
199	C₂×C₄×C₂₃	#121	1	2	yes	—
200	C₂×C₄×C₂₃	#121	1	2	yes	—
201	C₂×C₄×C₂₃	#123	1	2	yes	—
202	C₂×C₄×C₂₃	#124	1	2	yes	—
203	C₂×C₄×C₂₃	#124	1	2	yes	—
204	C₂×C₄×C₂₃	#126	1	2	yes	—
205	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
206	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
207	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
208	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
209	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
210	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
211	C₈×C₂₃	—	2	4	no	—
212	C₈×C₂₃	—	2	2	no	—
213	C₈×C₂₃	—	2	2	no	—
214	C₈×C₂₃	—	4	2	no	—
215	C₈×C₂₃	—	4	2	no	—
216	C₂×#424	C₂×C₂₃	1	1	yes	characteristic, complemented
217	C₂×#421	C₂×C₂₃	1	1	yes	characteristic
218	#2896	C₂×C₂₃	1	1	yes	characteristic
219	C₂×#408	C₂×C₂₃	1	1	yes	characteristic
220	C₂×#405	C₂×C₂₃	1	1	yes	characteristic, complemented
221	#1938	C₂×C₂₃	1	4	yes	complemented
222	#1937	C₂×C₂₃	1	4	yes	complemented
223	#1738	C₂×C₂₃	1	1	yes	characteristic, complemented
224	#1733	C₂×C₂₃	1	1	yes	characteristic, complemented
225	C₂²×C₂₃	—	2	2	no	—
226	C₂²×C₂₃	—	2	2	no	—
227	C₂²×C₂₃	—	2	2	no	—
228	C₂²×C₂₃	—	4	2	no	—
229	C₂²×C₂₃	—	4	2	no	—
230	C₂²×C₂₃	—	4	4	no	—
231	C₂²×C₂₃	—	4	2	no	—
232	C₂²×C₂₃	—	8	4	no	—
233	C₂²×C₂₃	#388	1	2	yes	—
234	C₂²×C₂₃	#388	1	2	yes	—
235	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
236	C₂²×C₂₃	C₂×#123	1	1	yes	characteristic
237	C₂²×C₂₃	C₂×#126	1	1	yes	characteristic
238	C₄×C₂₃	—	4	4	no	—
239	C₄×C₂₃	—	4	4	no	—
240	C₄×C₂₃	—	4	2	no	—
241	C₄×C₂₃	#388	1	2	yes	—
242	C₄×C₂₃	#388	1	2	yes	—
243	C₄×C₂₃	C₂×#117	1	1	yes	characteristic
244	C₄×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
245	C₄×C₂₃	C₂×#121	1	1	yes	characteristic
246	C₄×C₂₃	C₂×#124	1	1	yes	characteristic
247	C₂²×#41	C₂²×C₂₃	1	1	yes	characteristic
248	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
249	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
250	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
251	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
252	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
253	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
254	#424	—	2	2	no	—
255	#424	C₂²×C₂₃	1	2	yes	complemented
256	#424	C₂²×C₂₃	1	2	yes	complemented
257	C₂×C₄⋊C₈	—	2	2	no	—
258	#421	—	2	4	no	—
259	C₂×#104	C₂²×C₂₃	1	4	yes	—
260	#419	C₄×C₂₃	1	2	yes	complemented
261	#418	C₄×C₂₃	1	2	yes	complemented
262	#417	C₄×C₂₃	1	4	yes	complemented
263	#416	C₂²×C₂₃	1	4	yes	—
264	#408	—	2	2	no	—
265	#408	C₂²×C₂₃	1	2	yes	—
266	#408	C₂²×C₂₃	1	2	yes	—
267	C₂×C₂²⋊C₈	—	2	2	no	—
268	#405	—	2	4	no	—
269	C₂×#97	C₂²×C₂₃	1	2	yes	complemented
270	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	complemented
271	#341	C₂²×C₂₃	1	4	yes	complemented
272	#339	C₂²×C₂₃	1	4	yes	complemented
273	C₂×C₂₃	—	2	1	no	—
274	C₂×C₂₃	—	2	2	no	—
275	C₂×C₂₃	—	2	1	no	—
276	C₂×C₂₃	—	2	2	no	—
277	C₂×C₂₃	—	8	2	no	—
278	C₂×C₂₃	C₂×#388	1	1	yes	characteristic
279	C₂×C₂₃	#2955	1	2	yes	—
280	C₂×#41	—	2	1	no	—
281	C₂×#41	—	2	1	no	—
282	C₂×#41	—	2	2	no	—
283	C₂×#41	—	2	2	no	—
284	C₂×#41	—	2	2	no	—
285	C₂×#41	—	2	2	no	—
286	C₂×#41	C₂×C₄×C₂₃	1	2	yes	—
287	C₂×#41	C₂³×C₂₃	1	2	yes	—
288	C₂²×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
289	C₂²×D₄	—	2	1	no	—
290	C₂²×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
291	C₂³×C₄	—	2	1	no	—
292	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
293	C₂×M₁₆	—	2	2	no	—
294	C₂×M₁₆	—	2	2	no	—
295	C₂×M₁₆	—	2	1	no	—
296	C₂×M₁₆	—	2	1	no	—
297	C₂×M₁₆	—	2	2	no	—
298	C₂×M₁₆	—	2	2	no	—
299	C₂×M₁₆	—	2	2	no	—
300	C₂×M₁₆	—	2	2	no	—
301	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
302	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
303	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
304	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
305	C₂²×C₈	—	2	2	no	—
306	C₂²×C₈	—	2	2	no	—
307	C₂²×C₈	—	2	1	no	—
308	C₂²×C₈	—	2	1	no	—
309	C₂²×C₈	D₄×C₂₃	1	2	yes	—
310	C₂²×C₈	D₄×C₂₃	1	2	yes	—
311	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
312	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
313	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
314	#117	—	2	1	no	—
315	#117	—	2	1	no	—
316	#117	—	2	2	no	—
317	#117	C₂×C₄×C₂₃	1	2	yes	—
318	#117	D₄×C₂₃	1	4	yes	—
319	#117	C₂³×C₂₃	1	2	yes	—
320	C₂×C₄⋊C₄	—	2	1	no	—
321	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
322	C₂×C₂²⋊C₄	—	2	1	no	—
323	C₂×C₂²⋊C₄	—	2	1	no	—
324	C₂×C₄²	—	2	1	no	—
325	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
326	C₄⋊C₈	—	2	4	no	—
327	C₄⋊C₈	—	2	4	no	—
328	#104	—	2	8	no	—
329	#103	C₂×C₄×C₂₃	1	8	yes	—
330	#102	C₂×C₄×C₂₃	1	8	yes	—
331	C₂²⋊C₈	—	2	4	no	—
332	C₂²⋊C₈	—	2	4	no	—
333	#97	—	2	4	no	—
334	C₄×C₈	—	2	4	no	—
335	C₂₃	#21100	1	1	yes	characteristic, direct product factor, Sylow
336	C₂⁴	—	2	1	no	—
337	C₂⁴	#41×C₂₃	1	1	yes	characteristic
338	#41	—	2	4	no	—
339	#41	—	2	4	no	—
340	#41	—	2	2	no	—
341	#41	—	2	2	no	—
342	#41	—	2	2	no	—
343	#41	—	2	2	no	—
344	#41	—	4	2	no	—
345	#41	—	4	2	no	—
346	#41	—	4	2	no	—
347	#41	—	4	2	no	—
348	C₂×Q₈	—	2	1	no	—
349	C₂×Q₈	—	2	2	no	—
350	C₂×Q₈	—	2	2	no	—
351	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
352	C₂×D₄	—	2	2	no	—
353	C₂×D₄	—	2	2	no	—
354	C₂×D₄	—	2	2	no	—
355	C₂×D₄	—	2	2	no	—
356	C₂×D₄	—	2	1	no	—
357	C₂×D₄	—	4	2	no	—
358	C₂×D₄	—	4	2	no	—
359	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
360	C₂²×C₄	—	2	2	no	—
361	C₂²×C₄	—	2	1	no	—
362	C₂²×C₄	—	2	2	no	—
363	C₂²×C₄	—	2	2	no	—
364	C₂²×C₄	—	2	2	no	—
365	C₂²×C₄	—	4	2	no	—
366	C₂²×C₄	—	4	2	no	—
367	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
368	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
369	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
370	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
371	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
372	C₂²×C₄	#41×C₂₃	1	2	yes	—
373	C₂²×C₄	#41×C₂₃	1	2	yes	—
374	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
375	C₂²×C₄	#41×C₂₃	1	2	yes	—
376	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
377	M₁₆	—	2	4	no	—
378	M₁₆	—	2	4	no	—
379	M₁₆	—	2	2	no	—
380	M₁₆	—	2	2	no	—
381	M₁₆	—	2	2	no	—
382	M₁₆	—	2	2	no	—
383	M₁₆	—	4	2	no	—
384	M₁₆	—	4	2	no	—
385	C₂×C₈	—	2	4	no	—
386	C₂×C₈	—	2	4	no	—
387	C₂×C₈	—	2	1	no	—
388	C₂×C₈	—	2	2	no	—
389	C₂×C₈	—	2	1	no	—
390	C₂×C₈	—	2	1	no	—
391	C₂×C₈	—	2	1	no	—
392	C₂×C₈	—	2	2	no	—
393	C₂×C₈	—	2	2	no	—
394	C₂×C₈	—	2	2	no	—
395	C₂×C₈	—	4	2	no	—
396	C₂×C₈	—	4	2	no	—
397	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
398	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
399	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
400	C₄⋊C₄	—	2	2	no	—
401	C₄⋊C₄	—	2	4	no	—
402	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
403	C₂²⋊C₄	—	2	2	no	—
404	C₂²⋊C₄	—	2	4	no	—
405	C₂²⋊C₄	—	2	2	no	—
406	C₄²	—	2	2	no	—
407	C₄²	—	2	2	no	—
408	C₄²	—	2	2	no	—
409	C₄²	C₂×D₄×C₂₃	1	4	yes	—
410	C₂³	—	2	2	no	—
411	C₂³	—	4	2	no	—
412	C₂³	—	4	4	no	—
413	C₂³	—	4	2	no	—
414	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
415	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
416	C₂³	C₂₃×#123	1	2	yes	—
417	C₂³	C₂₃×#126	1	2	yes	—
418	C₂³	C₂×#41×C₂₃	1	1	yes	characteristic
419	Q₈	—	2	2	no	—
420	Q₈	—	2	2	no	—
421	Q₈	—	4	2	no	—
422	D₄	—	2	2	no	—
423	D₄	—	2	2	no	—
424	D₄	—	4	4	no	—
425	D₄	—	4	2	no	—
426	D₄	—	4	2	no	—
427	D₄	—	4	2	no	—
428	C₂×C₄	—	2	4	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	4	no	—
431	C₂×C₄	—	4	4	no	—
432	C₂×C₄	—	4	4	no	—
433	C₂×C₄	—	4	2	no	—
434	C₂×C₄	—	4	4	no	—
435	C₂×C₄	—	4	2	no	—
436	C₂×C₄	—	4	2	no	—
437	C₂×C₄	C₂₃×#117	1	2	yes	—
438	C₂×C₄	C₂₃×#117	1	2	yes	—
439	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
440	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
441	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
442	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
443	C₂×C₄	C₂₃×#121	1	2	yes	—
444	C₂×C₄	C₂₃×#121	1	2	yes	—
445	C₂×C₄	C₂₃×#123	1	2	yes	—
446	C₂×C₄	C₂₃×#124	1	2	yes	—
447	C₂×C₄	C₂₃×#124	1	2	yes	—
448	C₂×C₄	C₂₃×#126	1	2	yes	—
449	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
450	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
451	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
452	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
453	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
454	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
455	C₈	—	2	4	no	—
456	C₈	—	2	2	no	—
457	C₈	—	2	2	no	—
458	C₈	—	4	2	no	—
459	C₈	—	4	2	no	—
460	C₂²	—	2	2	no	—
461	C₂²	—	2	2	no	—
462	C₂²	—	2	2	no	—
463	C₂²	—	4	2	no	—
464	C₂²	—	4	2	no	—
465	C₂²	—	4	4	no	—
466	C₂²	—	4	2	no	—
467	C₂²	—	8	4	no	—
468	C₂²	C₂₃×#388	1	2	yes	—
469	C₂²	C₂₃×#388	1	2	yes	—
470	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
471	C₂²	C₂×C₂₃×#123	1	1	yes	characteristic
472	C₂²	C₂×C₂₃×#126	1	1	yes	characteristic
473	C₄	—	4	4	no	—
474	C₄	—	4	4	no	—
475	C₄	—	4	2	no	—
476	C₄	C₂₃×#388	1	2	yes	—
477	C₄	C₂₃×#388	1	2	yes	—
478	C₄	C₂×C₂₃×#117	1	1	yes	characteristic
479	C₄	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
480	C₄	C₂×C₂₃×#121	1	1	yes	characteristic
481	C₄	C₂×C₂₃×#124	1	1	yes	characteristic
482	C₂	—	2	1	no	—
483	C₂	—	2	2	no	—
484	C₂	—	2	2	no	—
485	C₂	—	2	1	no	—
486	C₂	—	8	2	no	—
487	C₂	C₂×C₂₃×#388	1	1	yes	characteristic
488	C₂	C₂₃×#2955	1	2	yes	—

	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	C₂²×#41×C₂₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂³×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#424	C₂	semidirect
5. C₂×C₂₃×#405	C₂	semidirect
6. C₂₃×#1938	C₂	semidirect
7. C₂₃×#1937	C₂	semidirect
8. C₂₃×#1738	C₂	semidirect
9. C₂₃×#1733	C₂	semidirect
18. C₂₃×#424	C₂²	semidirect
19. C₂₃×#424	C₂²	semidirect
23. C₂₃×#419	C₄	semidirect
24. C₂₃×#418	C₄	semidirect
25. C₂₃×#417	C₄	semidirect
32. C₂×C₂₃×#97	C₂²	semidirect
33. C₂×C₄×C₈×C₂₃	C₂²	semidirect
34. C₂₃×#341	C₂²	semidirect
35. C₂₃×#339	C₂²	semidirect
165. #21100	C₂₃	direct (N × H)
216. C₂×#424	C₂×C₂₃	semidirect
220. C₂×#405	C₂×C₂₃	semidirect
221. #1938	C₂×C₂₃	semidirect
222. #1937	C₂×C₂₃	semidirect
223. #1738	C₂×C₂₃	semidirect
224. #1733	C₂×C₂₃	semidirect
255. #424	C₂²×C₂₃	semidirect
256. #424	C₂²×C₂₃	semidirect
260. #419	C₄×C₂₃	semidirect
261. #418	C₄×C₂₃	semidirect
262. #417	C₄×C₂₃	semidirect
269. C₂×#97	C₂²×C₂₃	semidirect
270. C₂×C₄×C₈	C₂²×C₂₃	semidirect
271. #341	C₂²×C₂₃	semidirect
272. #339	C₂²×C₂₃	semidirect
335. C₂₃	#21100	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×#424	2
C₂×C₂₃×#421	2
C₂₃×#2896	2
C₂×C₂₃×#408	2
C₂×C₂₃×#405	2
C₂₃×#1938	2
C₂₃×#1937	2
C₂₃×#1738	2
C₂₃×#1733	2
#21100	23

Subgroups — C23×#21100 #19168726

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#21100 #19168726