Subgroups — C₂₃×#60520 #19208146

Order 5888.

2170 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#537	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#3428	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3099	C₂	1	8	yes	maximal, complemented
4	C₂₃×#3098	C₂	1	8	yes	maximal, complemented
5	C₂₃×#3095	C₂	1	4	yes	maximal, complemented
6	C₂₃×#3094	C₂	1	4	yes	maximal, complemented
7	C₂×C₂₃×#428	C₂	1	2	yes	maximal, complemented
8	C₂₃×#2896	C₂	1	2	yes	maximal, complemented
9	C₂₃×#2879	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₂₃	C₂²	1	1	yes	characteristic
11	C₂²×#41×C₂₃	C₂²	1	1	yes	characteristic
12	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
13	C₂₃×#537	—	2	4	no	—
14	C₂₃×#537	C₂²	1	8	yes	—
15	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
16	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
17	C₂×C₂₃×#121	—	2	1	no	—
18	C₂×C₂₃×#121	—	2	2	no	—
19	C₂×C₂₃×#121	C₂²	1	2	yes	—
20	C₂₃×#520	C₂²	1	4	yes	complemented
21	C₂₃×#520	C₂²	1	4	yes	complemented
22	C₂₃×#519	—	2	1	no	—
23	C₂₃×#518	—	2	1	no	—
24	C₂₃×#518	—	2	2	no	—
25	C₄×#41×C₂₃	—	2	4	no	—
26	C₂×C₂₃×#117	C₂²	1	2	yes	—
27	C₂×C₂₃×#117	C₂²	1	4	yes	—
28	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
29	C₂₃×#485	C₂²	1	16	yes	complemented
30	C₂₃×#484	C₂²	1	8	yes	complemented
31	C₂₃×#483	C₂²	1	8	yes	complemented
32	C₂₃×#482	C₂²	1	16	yes	complemented
33	C₂₃×#481	C₂²	1	8	yes	complemented
34	C₂₃×#480	C₂²	1	8	yes	complemented
35	C₂₃×#428	C₂²	1	8	yes	—
36	C₂₃×#428	C₂²	1	8	yes	complemented
37	C₂×C₂₃×#107	C₂²	1	4	yes	complemented
38	C₂×C₂₃×#106	C₂²	1	4	yes	complemented
39	C₂₃×#419	C₂²	1	4	yes	—
40	C₂₃×#418	C₂²	1	4	yes	—
41	C₂₃×#417	C₂²	1	4	yes	complemented
42	C₂₃×#417	C₂²	1	4	yes	complemented
43	C₂₃×#416	C₂²	1	8	yes	complemented
44	C₂₃×#408	C₂²	1	8	yes	complemented
45	C₂₃×#407	C₂²	1	4	yes	—
46	C₂×#41×C₂₃	—	2	2	no	—
47	C₂×#41×C₂₃	—	2	8	no	—
48	C₂×#41×C₂₃	—	2	8	no	—
49	C₂×#41×C₂₃	D₄	1	4	yes	—
50	C₂×#41×C₂₃	C₂³	1	4	yes	—
51	C₂×#41×C₂₃	C₂³	1	4	yes	—
52	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
53	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
54	C₂²×D₄×C₂₃	—	2	1	no	—
55	C₂²×D₄×C₂₃	—	2	2	no	—
56	C₂²×D₄×C₂₃	D₄	1	2	yes	—
57	C₂²×D₄×C₂₃	D₄	1	1	yes	characteristic
58	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
59	C₂³×C₄×C₂₃	—	2	1	no	—
60	C₂³×C₄×C₂₃	—	2	1	no	—
61	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
62	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
63	C₂×M₁₆×C₂₃	—	2	8	no	—
64	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
65	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
66	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
67	C₂₃×C₂²⋊Q₈	—	2	8	no	—
68	C₂₃×C₂²⋊Q₈	—	2	4	no	—
69	C₂₃×C₂²⋊Q₈	C₂³	1	8	yes	—
70	C₂₃×#121	—	2	8	no	—
71	C₂₃×#121	—	2	16	no	—
72	C₂₃×#121	—	2	4	no	—
73	C₂₃×#121	C₂³	1	8	yes	—
74	C₄×Q₈×C₂₃	—	2	4	no	—
75	C₄×Q₈×C₂₃	—	2	4	no	—
76	C₄×D₄×C₂₃	—	2	8	no	—
77	C₄×D₄×C₂₃	—	2	4	no	—
78	C₄×D₄×C₂₃	—	2	8	no	—
79	C₄×D₄×C₂₃	—	2	4	no	—
80	C₂₃×#117	—	2	4	no	—
81	C₂₃×#117	—	2	2	no	—
82	C₂₃×#117	—	2	4	no	—
83	C₂₃×#117	—	2	2	no	—
84	C₂₃×#117	C₂³	1	4	yes	—
85	C₂₃×#117	C₂³	1	4	yes	—
86	C₂₃×#117	C₂³	1	8	yes	—
87	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
88	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
89	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
90	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
91	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
92	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
93	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
94	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
95	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
96	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
97	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
98	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
99	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
100	C₂×C₄²×C₂₃	—	2	4	no	—
101	C₂×C₄²×C₂₃	—	2	2	no	—
102	C₂₃×#107	C₂³	1	16	yes	—
103	C₂₃×#106	C₂³	1	16	yes	—
104	C₂₃×#103	C₂³	1	8	yes	—
105	C₂₃×#103	C₂³	1	8	yes	—
106	C₂₃×#102	C₂³	1	8	yes	—
107	C₂₃×#102	C₂³	1	8	yes	—
108	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	—
109	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	—
110	C₂⁴×C₂₃	—	2	1	no	—
111	C₂⁴×C₂₃	C₂×D₄	1	2	yes	—
112	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
113	#41×C₂₃	—	2	16	no	—
114	#41×C₂₃	—	2	8	no	—
115	#41×C₂₃	—	2	8	no	—
116	#41×C₂₃	—	2	8	no	—
117	#41×C₂₃	—	2	8	no	—
118	#41×C₂₃	—	4	8	no	—
119	C₂×Q₈×C₂₃	—	2	4	no	—
120	C₂×Q₈×C₂₃	—	2	1	no	—
121	C₂×Q₈×C₂₃	—	2	4	no	—
122	C₂×Q₈×C₂₃	C₂×D₄	1	4	yes	—
123	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
124	C₂×D₄×C₂₃	—	2	8	no	—
125	C₂×D₄×C₂₃	—	2	4	no	—
126	C₂×D₄×C₂₃	—	2	8	no	—
127	C₂×D₄×C₂₃	—	2	4	no	—
128	C₂×D₄×C₂₃	—	2	1	no	—
129	C₂×D₄×C₂₃	—	2	4	no	—
130	C₂×D₄×C₂₃	—	4	4	no	—
131	C₂×D₄×C₂₃	—	8	4	no	—
132	C₂×D₄×C₂₃	C₂×D₄	1	8	yes	—
133	C₂×D₄×C₂₃	C₂×D₄	1	4	yes	—
134	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
135	C₂²×C₄×C₂₃	—	2	8	no	—
136	C₂²×C₄×C₂₃	—	2	2	no	—
137	C₂²×C₄×C₂₃	—	2	1	no	—
138	C₂²×C₄×C₂₃	—	2	4	no	—
139	C₂²×C₄×C₂₃	—	2	4	no	—
140	C₂²×C₄×C₂₃	—	2	2	no	—
141	C₂²×C₄×C₂₃	—	2	2	no	—
142	C₂²×C₄×C₂₃	—	2	4	no	—
143	C₂²×C₄×C₂₃	—	2	4	no	—
144	C₂²×C₄×C₂₃	—	4	4	no	—
145	C₂²×C₄×C₂₃	—	8	2	no	—
146	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
147	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
148	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
149	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
150	C₂²×C₄×C₂₃	#41	1	8	yes	—
151	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
152	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
153	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
154	M₁₆×C₂₃	—	2	8	no	—
155	M₁₆×C₂₃	—	2	8	no	—
156	C₂×C₈×C₂₃	—	2	8	no	—
157	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
158	C₄⋊C₄×C₂₃	—	2	2	no	—
159	C₄⋊C₄×C₂₃	—	2	4	no	—
160	C₄⋊C₄×C₂₃	—	2	8	no	—
161	C₄⋊C₄×C₂₃	—	2	4	no	—
162	C₄⋊C₄×C₂₃	—	2	4	no	—
163	C₄⋊C₄×C₂₃	—	2	2	no	—
164	C₄⋊C₄×C₂₃	—	2	2	no	—
165	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
166	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
167	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
168	C₂²⋊C₄×C₂₃	—	2	4	no	—
169	C₂²⋊C₄×C₂₃	—	2	8	no	—
170	C₂²⋊C₄×C₂₃	—	2	4	no	—
171	C₂²⋊C₄×C₂₃	—	2	8	no	—
172	C₂²⋊C₄×C₂₃	—	2	8	no	—
173	C₄²×C₂₃	—	2	4	no	—
174	C₄²×C₂₃	—	2	8	no	—
175	C₄²×C₂₃	—	2	4	no	—
176	#60520	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
177	C₂³×C₂₃	—	2	4	no	—
178	C₂³×C₂₃	—	2	2	no	—
179	C₂³×C₂₃	—	4	1	no	—
180	C₂³×C₂₃	—	4	1	no	—
181	C₂³×C₂₃	—	4	2	no	—
182	C₂³×C₂₃	—	4	2	no	—
183	C₂³×C₂₃	—	4	2	no	—
184	C₂³×C₂₃	—	4	2	no	—
185	C₂³×C₂₃	—	4	2	no	—
186	C₂³×C₂₃	#123	1	4	yes	—
187	C₂³×C₂₃	C₂²×D₄	1	4	yes	—
188	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
189	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
190	Q₈×C₂₃	—	2	8	no	—
191	Q₈×C₂₃	—	2	4	no	—
192	Q₈×C₂₃	—	4	2	no	—
193	D₄×C₂₃	—	2	16	no	—
194	D₄×C₂₃	—	2	8	no	—
195	D₄×C₂₃	—	2	4	no	—
196	D₄×C₂₃	—	4	8	no	—
197	D₄×C₂₃	—	4	2	no	—
198	D₄×C₂₃	—	8	8	no	—
199	C₂×C₄×C₂₃	—	2	4	no	—
200	C₂×C₄×C₂₃	—	2	8	no	—
201	C₂×C₄×C₂₃	—	2	8	no	—
202	C₂×C₄×C₂₃	—	2	4	no	—
203	C₂×C₄×C₂₃	—	2	4	no	—
204	C₂×C₄×C₂₃	—	2	4	no	—
205	C₂×C₄×C₂₃	—	2	2	no	—
206	C₂×C₄×C₂₃	—	4	4	no	—
207	C₂×C₄×C₂₃	—	4	2	no	—
208	C₂×C₄×C₂₃	—	4	8	no	—
209	C₂×C₄×C₂₃	—	8	2	no	—
210	C₂×C₄×C₂₃	—	8	4	no	—
211	C₂×C₄×C₂₃	—	8	2	no	—
212	C₂×C₄×C₂₃	—	8	4	no	—
213	C₂×C₄×C₂₃	—	8	4	no	—
214	C₂×C₄×C₂₃	#123	1	4	yes	—
215	C₂×C₄×C₂₃	#123	1	8	yes	—
216	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
217	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
218	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
219	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
220	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
221	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
222	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
223	C₈×C₂₃	—	2	8	no	—
224	C₂×#537	C₂×C₂₃	1	1	yes	characteristic
225	#3428	C₂×C₂₃	1	1	yes	characteristic, complemented
226	#3099	C₂×C₂₃	1	8	yes	complemented
227	#3098	C₂×C₂₃	1	8	yes	complemented
228	#3095	C₂×C₂₃	1	4	yes	complemented
229	#3094	C₂×C₂₃	1	4	yes	complemented
230	C₂×#428	C₂×C₂₃	1	2	yes	complemented
231	#2896	C₂×C₂₃	1	2	yes	complemented
232	#2879	C₂×C₂₃	1	1	yes	characteristic, complemented
233	C₂²×C₂₃	—	2	1	no	—
234	C₂²×C₂₃	—	2	2	no	—
235	C₂²×C₂₃	—	2	2	no	—
236	C₂²×C₂₃	—	2	1	no	—
237	C₂²×C₂₃	—	2	4	no	—
238	C₂²×C₂₃	—	2	4	no	—
239	C₂²×C₂₃	—	2	4	no	—
240	C₂²×C₂₃	—	4	4	no	—
241	C₂²×C₂₃	—	4	2	no	—
242	C₂²×C₂₃	—	4	2	no	—
243	C₂²×C₂₃	—	4	2	no	—
244	C₂²×C₂₃	—	8	4	no	—
245	C₂²×C₂₃	—	8	4	no	—
246	C₂²×C₂₃	C₂×#123	1	2	yes	—
247	C₂²×C₂₃	C₂×#123	1	4	yes	—
248	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
249	C₄×C₂₃	—	2	4	no	—
250	C₄×C₂₃	—	4	4	no	—
251	C₄×C₂₃	—	4	4	no	—
252	C₄×C₂₃	—	4	4	no	—
253	C₄×C₂₃	—	4	2	no	—
254	C₄×C₂₃	C₂×#123	1	2	yes	—
255	C₄×C₂₃	C₂×#123	1	4	yes	—
256	C₄×C₂₃	C₂²×#41	1	2	yes	—
257	C₂²×#41	C₂²×C₂₃	1	1	yes	characteristic
258	C₂²×#41	C₂²×C₂₃	1	1	yes	characteristic
259	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
260	#537	—	2	4	no	—
261	#537	C₂²×C₂₃	1	8	yes	—
262	C₂×C₂²⋊Q₈	—	2	1	no	—
263	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
264	C₂×#121	—	2	1	no	—
265	C₂×#121	—	2	2	no	—
266	C₂×#121	C₂²×C₂₃	1	2	yes	—
267	#520	C₂²×C₂₃	1	4	yes	complemented
268	#520	C₂²×C₂₃	1	4	yes	complemented
269	#519	—	2	1	no	—
270	#518	—	2	1	no	—
271	#518	—	2	2	no	—
272	C₄×#41	—	2	4	no	—
273	C₂×#117	C₂²×C₂₃	1	4	yes	—
274	C₂×#117	C₂²×C₂₃	1	2	yes	—
275	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
276	#485	C₂²×C₂₃	1	16	yes	complemented
277	#484	C₂²×C₂₃	1	8	yes	complemented
278	#483	C₂²×C₂₃	1	8	yes	complemented
279	#482	C₂²×C₂₃	1	16	yes	complemented
280	#481	C₂²×C₂₃	1	8	yes	complemented
281	#480	C₂²×C₂₃	1	8	yes	complemented
282	#428	C₂²×C₂₃	1	8	yes	—
283	#428	C₂²×C₂₃	1	8	yes	complemented
284	C₂×#107	C₂²×C₂₃	1	4	yes	complemented
285	C₂×#106	C₂²×C₂₃	1	4	yes	complemented
286	#419	C₂²×C₂₃	1	4	yes	—
287	#418	C₂²×C₂₃	1	4	yes	—
288	#417	C₂²×C₂₃	1	4	yes	complemented
289	#417	C₂²×C₂₃	1	4	yes	complemented
290	#416	C₂²×C₂₃	1	8	yes	complemented
291	#408	C₂²×C₂₃	1	8	yes	complemented
292	#407	C₂²×C₂₃	1	4	yes	—
293	C₂×C₂₃	—	2	2	no	—
294	C₂×C₂₃	—	2	4	no	—
295	C₂×C₂₃	—	4	4	no	—
296	C₂×C₂₃	—	8	2	no	—
297	C₂×C₂₃	C₂²×#123	1	1	yes	characteristic
298	C₂×C₂₃	#3585	1	1	yes	characteristic
299	C₂×C₂₃	#3586	1	1	yes	characteristic
300	C₂×#41	—	2	2	no	—
301	C₂×#41	—	2	8	no	—
302	C₂×#41	—	2	8	no	—
303	C₂×#41	D₄×C₂₃	1	4	yes	—
304	C₂×#41	C₂³×C₂₃	1	4	yes	—
305	C₂×#41	C₂³×C₂₃	1	4	yes	—
306	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
307	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
308	C₂²×D₄	—	2	1	no	—
309	C₂²×D₄	—	2	2	no	—
310	C₂²×D₄	D₄×C₂₃	1	2	yes	—
311	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
312	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
313	C₂³×C₄	—	2	1	no	—
314	C₂³×C₄	—	2	1	no	—
315	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
316	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
317	C₂×M₁₆	—	2	8	no	—
318	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
319	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
320	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
321	C₂²⋊Q₈	—	2	8	no	—
322	C₂²⋊Q₈	—	2	4	no	—
323	C₂²⋊Q₈	C₂³×C₂₃	1	8	yes	—
324	#121	—	2	8	no	—
325	#121	—	2	16	no	—
326	#121	—	2	4	no	—
327	#121	C₂³×C₂₃	1	8	yes	—
328	C₄×Q₈	—	2	4	no	—
329	C₄×Q₈	—	2	4	no	—
330	C₄×D₄	—	2	8	no	—
331	C₄×D₄	—	2	4	no	—
332	C₄×D₄	—	2	8	no	—
333	C₄×D₄	—	2	4	no	—
334	#117	—	2	4	no	—
335	#117	—	2	2	no	—
336	#117	—	2	4	no	—
337	#117	—	2	2	no	—
338	#117	C₂³×C₂₃	1	4	yes	—
339	#117	C₂³×C₂₃	1	4	yes	—
340	#117	C₂³×C₂₃	1	8	yes	—
341	C₂×C₄⋊C₄	—	2	1	no	—
342	C₂×C₄⋊C₄	—	2	2	no	—
343	C₂×C₄⋊C₄	—	2	2	no	—
344	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
345	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
346	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
347	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
348	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
349	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
350	C₂×C₂²⋊C₄	—	2	4	no	—
351	C₂×C₂²⋊C₄	—	2	2	no	—
352	C₂×C₂²⋊C₄	—	2	2	no	—
353	C₂×C₂²⋊C₄	—	2	2	no	—
354	C₂×C₄²	—	2	4	no	—
355	C₂×C₄²	—	2	2	no	—
356	#107	C₂³×C₂₃	1	16	yes	—
357	#106	C₂³×C₂₃	1	16	yes	—
358	#103	C₂³×C₂₃	1	8	yes	—
359	#103	C₂³×C₂₃	1	8	yes	—
360	#102	C₂³×C₂₃	1	8	yes	—
361	#102	C₂³×C₂₃	1	8	yes	—
362	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	—
363	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	—
364	C₂₃	#60520	1	1	yes	characteristic, direct product factor, Sylow
365	C₂⁴	—	2	1	no	—
366	C₂⁴	C₂×D₄×C₂₃	1	2	yes	—
367	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
368	#41	—	2	16	no	—
369	#41	—	2	8	no	—
370	#41	—	2	8	no	—
371	#41	—	2	8	no	—
372	#41	—	2	8	no	—
373	#41	—	4	8	no	—
374	C₂×Q₈	—	2	4	no	—
375	C₂×Q₈	—	2	1	no	—
376	C₂×Q₈	—	2	4	no	—
377	C₂×Q₈	C₂×D₄×C₂₃	1	4	yes	—
378	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
379	C₂×D₄	—	2	8	no	—
380	C₂×D₄	—	2	4	no	—
381	C₂×D₄	—	2	8	no	—
382	C₂×D₄	—	2	4	no	—
383	C₂×D₄	—	2	1	no	—
384	C₂×D₄	—	2	4	no	—
385	C₂×D₄	—	4	4	no	—
386	C₂×D₄	—	8	4	no	—
387	C₂×D₄	C₂×D₄×C₂₃	1	8	yes	—
388	C₂×D₄	C₂×D₄×C₂₃	1	4	yes	—
389	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
390	C₂²×C₄	—	2	8	no	—
391	C₂²×C₄	—	2	2	no	—
392	C₂²×C₄	—	2	1	no	—
393	C₂²×C₄	—	2	4	no	—
394	C₂²×C₄	—	2	4	no	—
395	C₂²×C₄	—	2	2	no	—
396	C₂²×C₄	—	2	2	no	—
397	C₂²×C₄	—	2	4	no	—
398	C₂²×C₄	—	2	4	no	—
399	C₂²×C₄	—	4	4	no	—
400	C₂²×C₄	—	8	2	no	—
401	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
402	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
403	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
404	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
405	C₂²×C₄	#41×C₂₃	1	8	yes	—
406	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
407	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
408	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
409	M₁₆	—	2	8	no	—
410	M₁₆	—	2	8	no	—
411	C₂×C₈	—	2	8	no	—
412	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
413	C₄⋊C₄	—	2	2	no	—
414	C₄⋊C₄	—	2	4	no	—
415	C₄⋊C₄	—	2	8	no	—
416	C₄⋊C₄	—	2	4	no	—
417	C₄⋊C₄	—	2	4	no	—
418	C₄⋊C₄	—	2	2	no	—
419	C₄⋊C₄	—	2	2	no	—
420	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
421	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
422	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
423	C₂²⋊C₄	—	2	4	no	—
424	C₂²⋊C₄	—	2	8	no	—
425	C₂²⋊C₄	—	2	4	no	—
426	C₂²⋊C₄	—	2	8	no	—
427	C₂²⋊C₄	—	2	8	no	—
428	C₄²	—	2	4	no	—
429	C₄²	—	2	8	no	—
430	C₄²	—	2	4	no	—
431	C₂³	—	2	4	no	—
432	C₂³	—	2	2	no	—
433	C₂³	—	4	1	no	—
434	C₂³	—	4	1	no	—
435	C₂³	—	4	2	no	—
436	C₂³	—	4	2	no	—
437	C₂³	—	4	2	no	—
438	C₂³	—	4	2	no	—
439	C₂³	—	4	2	no	—
440	C₂³	C₂₃×#123	1	4	yes	—
441	C₂³	C₂²×D₄×C₂₃	1	4	yes	—
442	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
443	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
444	Q₈	—	2	8	no	—
445	Q₈	—	2	4	no	—
446	Q₈	—	4	2	no	—
447	D₄	—	2	16	no	—
448	D₄	—	2	4	no	—
449	D₄	—	2	8	no	—
450	D₄	—	4	8	no	—
451	D₄	—	4	2	no	—
452	D₄	—	8	8	no	—
453	C₂×C₄	—	2	4	no	—
454	C₂×C₄	—	2	8	no	—
455	C₂×C₄	—	2	8	no	—
456	C₂×C₄	—	2	4	no	—
457	C₂×C₄	—	2	4	no	—
458	C₂×C₄	—	2	4	no	—
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	4	4	no	—
461	C₂×C₄	—	4	2	no	—
462	C₂×C₄	—	4	8	no	—
463	C₂×C₄	—	8	4	no	—
464	C₂×C₄	—	8	2	no	—
465	C₂×C₄	—	8	4	no	—
466	C₂×C₄	—	8	2	no	—
467	C₂×C₄	—	8	4	no	—
468	C₂×C₄	C₂₃×#123	1	4	yes	—
469	C₂×C₄	C₂₃×#123	1	8	yes	—
470	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
471	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
472	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
473	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
474	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
475	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
476	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
477	C₈	—	2	8	no	—
478	C₂²	—	2	1	no	—
479	C₂²	—	2	2	no	—
480	C₂²	—	2	2	no	—
481	C₂²	—	2	1	no	—
482	C₂²	—	2	4	no	—
483	C₂²	—	2	4	no	—
484	C₂²	—	2	4	no	—
485	C₂²	—	4	4	no	—
486	C₂²	—	4	2	no	—
487	C₂²	—	4	2	no	—
488	C₂²	—	4	2	no	—
489	C₂²	—	8	4	no	—
490	C₂²	—	8	4	no	—
491	C₂²	C₂×C₂₃×#123	1	2	yes	—
492	C₂²	C₂×C₂₃×#123	1	4	yes	—
493	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
494	C₄	—	2	4	no	—
495	C₄	—	4	2	no	—
496	C₄	—	4	4	no	—
497	C₄	—	4	4	no	—
498	C₄	—	4	4	no	—
499	C₄	C₂×C₂₃×#123	1	2	yes	—
500	C₄	C₂×C₂₃×#123	1	4	yes	—
501	C₄	C₂²×#41×C₂₃	1	2	yes	—
502	C₂	—	2	2	no	—
503	C₂	—	2	4	no	—
504	C₂	—	4	4	no	—
505	C₂	—	8	2	no	—
506	C₂	C₂²×C₂₃×#123	1	1	yes	characteristic
507	C₂	C₂₃×#3585	1	1	yes	characteristic
508	C₂	C₂₃×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂×C₂₃×#537	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂²×#41×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#3428	C₂	semidirect
3. C₂₃×#3099	C₂	semidirect
4. C₂₃×#3098	C₂	semidirect
5. C₂₃×#3095	C₂	semidirect
6. C₂₃×#3094	C₂	semidirect
7. C₂×C₂₃×#428	C₂	semidirect
8. C₂₃×#2896	C₂	semidirect
9. C₂₃×#2879	C₂	semidirect
20. C₂₃×#520	C₂²	semidirect
21. C₂₃×#520	C₂²	semidirect
29. C₂₃×#485	C₂²	semidirect
30. C₂₃×#484	C₂²	semidirect
31. C₂₃×#483	C₂²	semidirect
32. C₂₃×#482	C₂²	semidirect
33. C₂₃×#481	C₂²	semidirect
34. C₂₃×#480	C₂²	semidirect
36. C₂₃×#428	C₂²	semidirect
37. C₂×C₂₃×#107	C₂²	semidirect
38. C₂×C₂₃×#106	C₂²	semidirect
41. C₂₃×#417	C₂²	semidirect
42. C₂₃×#417	C₂²	semidirect
43. C₂₃×#416	C₂²	semidirect
44. C₂₃×#408	C₂²	semidirect
176. #60520	C₂₃	direct (N × H)
225. #3428	C₂×C₂₃	semidirect
226. #3099	C₂×C₂₃	semidirect
227. #3098	C₂×C₂₃	semidirect
228. #3095	C₂×C₂₃	semidirect
229. #3094	C₂×C₂₃	semidirect
230. C₂×#428	C₂×C₂₃	semidirect
231. #2896	C₂×C₂₃	semidirect
232. #2879	C₂×C₂₃	semidirect
267. #520	C₂²×C₂₃	semidirect
268. #520	C₂²×C₂₃	semidirect
276. #485	C₂²×C₂₃	semidirect
277. #484	C₂²×C₂₃	semidirect
278. #483	C₂²×C₂₃	semidirect
279. #482	C₂²×C₂₃	semidirect
280. #481	C₂²×C₂₃	semidirect
281. #480	C₂²×C₂₃	semidirect
283. #428	C₂²×C₂₃	semidirect
284. C₂×#107	C₂²×C₂₃	semidirect
285. C₂×#106	C₂²×C₂₃	semidirect
288. #417	C₂²×C₂₃	semidirect
289. #417	C₂²×C₂₃	semidirect
290. #416	C₂²×C₂₃	semidirect
291. #408	C₂²×C₂₃	semidirect
364. C₂₃	#60520	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×#537	2
C₂₃×#3428	2
C₂₃×#3099	2
C₂₃×#3098	2
C₂₃×#3095	2
C₂₃×#3094	2
C₂×C₂₃×#428	2
C₂₃×#2896	2
C₂₃×#2879	2
#60520	23

Subgroups — C23×#60520 #19208146

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#60520 #19208146