Subgroups — C₂₃×#61588 #19209214

Order 5888.

2150 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3501	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#550	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3333	C₂	1	8	yes	maximal, complemented
4	C₂₃×#3278	C₂	1	8	yes	maximal, complemented
5	C₂₃×#3208	C₂	1	2	yes	maximal, complemented
6	C₂₃×#3193	C₂	1	4	yes	maximal, complemented
7	C₂₃×#3104	C₂	1	2	yes	maximal, complemented
8	C₂₃×#3007	C₂	1	4	yes	maximal, complemented
9	C₂₃×#2975	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₂₃	C₂²	1	2	yes	—
11	C₂×C₂₃×#137	C₂²	1	8	yes	complemented
12	C₂₃×#554	—	2	4	no	—
13	C₂₃×#554	C₂²	1	4	yes	complemented
14	C₂₃×#550	—	2	4	no	—
15	C₂₃×#550	C₂²	1	8	yes	—
16	D₄×Q₈×C₂₃	C₂²	1	4	yes	complemented
17	C₂₃×#544	—	2	1	no	—
18	C₂₃×#543	—	2	2	no	—
19	C₂₃×#533	C₂²	1	2	yes	—
20	C₂×C₂₃×#127	—	2	1	no	—
21	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	4	yes	—
23	C₂×C₂₃×#121	—	2	2	no	—
24	C₂×C₂₃×#121	C₂²	1	2	yes	—
25	C₂₃×#518	—	2	1	no	—
26	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
27	C₂×C₄×D₄×C₂₃	—	2	1	no	—
28	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
29	C₂₃×#492	C₂²	1	4	yes	complemented
30	C₂₃×#486	C₂²	1	4	yes	complemented
31	C₂₃×#484	C₂²	1	4	yes	—
32	C₂₃×#480	C₂²	1	8	yes	complemented
33	C₂₃×#477	C₂²	1	4	yes	complemented
34	C₂₃×#468	C₂²	1	16	yes	complemented
35	C₂₃×#462	C₂²	1	4	yes	complemented
36	C₂₃×#460	C₂²	1	8	yes	complemented
37	C₂₃×#452	C₂²	1	16	yes	complemented
38	C₂₃×#451	C₂²	1	4	yes	—
39	C₂₃×#450	C₂²	1	8	yes	—
40	C₂₃×#448	C₂²	1	8	yes	complemented
41	C₄×Q₁₆×C₂₃	C₂²	1	8	yes	complemented
42	C₂₃×#436	C₂²	1	8	yes	complemented
43	C₂₃×#431	C₂²	1	2	yes	—
44	C₂₃×#417	C₂²	1	8	yes	complemented
45	C₂₃×#407	C₂²	1	4	yes	—
46	C₂×#41×C₂₃	—	2	4	no	—
47	C₂×#41×C₂₃	—	4	8	no	—
48	C₂×#41×C₂₃	C₂³	1	8	yes	—
49	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
50	C₂²×Q₈×C₂₃	C₂³	1	4	yes	—
51	C₂²×D₄×C₂₃	—	2	2	no	—
52	C₂²×D₄×C₂₃	—	2	2	no	—
53	C₂²×D₄×C₂₃	—	2	1	no	—
54	C₂²×D₄×C₂₃	—	2	1	no	—
55	C₂²×D₄×C₂₃	D₄	1	1	yes	characteristic
56	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
57	C₂³×C₄×C₂₃	—	2	2	no	—
58	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
59	C₂₃×#137	—	4	16	no	—
60	C₂×Q₁₆×C₂₃	C₂³	1	8	yes	—
61	C₂×SD₁₆×C₂₃	C₂³	1	16	yes	—
62	C₂×M₁₆×C₂₃	C₂³	1	8	yes	—
63	C₂₃×C₄⋊Q₈	—	2	4	no	—
64	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
65	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
66	C₂₃×#127	—	2	8	no	—
67	C₂₃×#127	—	2	2	no	—
68	C₂₃×#127	C₂³	1	4	yes	—
69	C₂₃×#125	—	2	4	no	—
70	C₂₃×#125	—	2	2	no	—
71	C₂₃×C₂²⋊Q₈	—	2	4	no	—
72	C₂₃×C₂²⋊Q₈	—	2	4	no	—
73	C₂₃×C₂²⋊Q₈	—	2	2	no	—
74	C₂₃×C₂²⋊Q₈	—	2	8	no	—
75	C₂₃×C₂²⋊Q₈	—	2	8	no	—
76	C₂₃×C₂²⋊Q₈	C₂³	1	8	yes	—
77	C₂₃×C₂²⋊Q₈	C₂³	1	4	yes	—
78	C₂₃×#121	—	2	4	no	—
79	C₂₃×#121	—	2	16	no	—
80	C₂₃×#121	C₂³	1	8	yes	—
81	C₄×Q₈×C₂₃	—	2	2	no	—
82	C₄×Q₈×C₂₃	—	2	2	no	—
83	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
84	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
85	C₄×D₄×C₂₃	—	2	4	no	—
86	C₄×D₄×C₂₃	—	2	4	no	—
87	C₄×D₄×C₂₃	—	2	8	no	—
88	C₄×D₄×C₂₃	D₄	1	4	yes	—
89	C₄×D₄×C₂₃	C₂³	1	4	yes	—
90	C₂₃×#117	—	2	2	no	—
91	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
92	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
93	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
94	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
95	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
96	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
97	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
98	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
99	C₂×C₄²×C₂₃	—	2	1	no	—
100	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
101	C₂₃×#107	C₂³	1	8	yes	—
102	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
103	C₂₃×#103	C₂³	1	8	yes	—
104	C₂₃×#103	C₂³	1	8	yes	—
105	C₂₃×#102	C₂³	1	16	yes	—
106	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
107	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	—
108	C₄×C₈×C₂₃	C₂³	1	4	yes	—
109	C₂⁴×C₂₃	—	2	2	no	—
110	C₂⁴×C₂₃	C₂×D₄	1	2	yes	—
111	#41×C₂₃	—	2	16	no	—
112	#41×C₂₃	—	4	8	no	—
113	#41×C₂₃	—	4	16	no	—
114	C₂×Q₈×C₂₃	—	2	4	no	—
115	C₂×Q₈×C₂₃	—	2	1	no	—
116	C₂×Q₈×C₂₃	—	2	2	no	—
117	C₂×Q₈×C₂₃	—	2	4	no	—
118	C₂×Q₈×C₂₃	—	4	8	no	—
119	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
120	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
121	C₂×D₄×C₂₃	—	2	8	no	—
122	C₂×D₄×C₂₃	—	2	8	no	—
123	C₂×D₄×C₂₃	—	2	4	no	—
124	C₂×D₄×C₂₃	—	2	4	no	—
125	C₂×D₄×C₂₃	—	2	2	no	—
126	C₂×D₄×C₂₃	—	2	4	no	—
127	C₂×D₄×C₂₃	—	4	4	no	—
128	C₂×D₄×C₂₃	—	4	8	no	—
129	C₂×D₄×C₂₃	—	4	4	no	—
130	C₂×D₄×C₂₃	—	8	2	no	—
131	C₂×D₄×C₂₃	—	8	4	no	—
132	C₂×D₄×C₂₃	C₂×D₄	1	4	yes	—
133	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
134	C₂²×C₄×C₂₃	—	2	1	no	—
135	C₂²×C₄×C₂₃	—	2	1	no	—
136	C₂²×C₄×C₂₃	—	2	4	no	—
137	C₂²×C₄×C₂₃	—	2	4	no	—
138	C₂²×C₄×C₂₃	—	2	8	no	—
139	C₂²×C₄×C₂₃	—	4	4	no	—
140	C₂²×C₄×C₂₃	—	4	8	no	—
141	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
142	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
143	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
144	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
146	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
147	Q₁₆×C₂₃	—	2	16	no	—
148	SD₁₆×C₂₃	—	4	16	no	—
149	M₁₆×C₂₃	—	4	8	no	—
150	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
151	C₄⋊C₄×C₂₃	—	2	4	no	—
152	C₄⋊C₄×C₂₃	—	2	4	no	—
153	C₄⋊C₄×C₂₃	—	2	8	no	—
154	C₄⋊C₄×C₂₃	—	2	1	no	—
155	C₄⋊C₄×C₂₃	—	2	4	no	—
156	C₄⋊C₄×C₂₃	—	2	2	no	—
157	C₄⋊C₄×C₂₃	—	2	2	no	—
158	C₄⋊C₄×C₂₃	—	2	2	no	—
159	C₄⋊C₄×C₂₃	—	2	4	no	—
160	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
161	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
162	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
163	C₂²⋊C₄×C₂₃	—	2	4	no	—
164	C₂²⋊C₄×C₂₃	—	2	8	no	—
165	C₂²⋊C₄×C₂₃	—	2	8	no	—
166	C₂²⋊C₄×C₂₃	C₂×D₄	1	8	yes	—
167	C₄²×C₂₃	—	2	4	no	—
168	C₄²×C₂₃	—	2	4	no	—
169	C₄²×C₂₃	C₂×D₄	1	2	yes	—
170	C₄²×C₂₃	C₂⁴	1	2	yes	—
171	#61588	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
172	C₂³×C₂₃	—	2	4	no	—
173	C₂³×C₂₃	—	2	4	no	—
174	C₂³×C₂₃	—	4	2	no	—
175	C₂³×C₂₃	—	4	2	no	—
176	C₂³×C₂₃	—	4	4	no	—
177	C₂³×C₂₃	—	4	4	no	—
178	C₂³×C₂₃	—	4	4	no	—
179	C₂³×C₂₃	C₂²×D₄	1	4	yes	—
180	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
181	Q₈×C₂₃	—	2	4	no	—
182	Q₈×C₂₃	—	2	2	no	—
183	Q₈×C₂₃	—	2	8	no	—
184	Q₈×C₂₃	—	8	2	no	—
185	Q₈×C₂₃	—	8	4	no	—
186	Q₈×C₂₃	H(5,𝔽₂)	1	4	yes	—
187	D₄×C₂₃	—	2	8	no	—
188	D₄×C₂₃	—	2	8	no	—
189	D₄×C₂₃	—	4	16	no	—
190	D₄×C₂₃	—	4	4	no	—
191	D₄×C₂₃	—	4	8	no	—
192	D₄×C₂₃	—	8	4	no	—
193	D₄×C₂₃	—	8	8	no	—
194	C₂×C₄×C₂₃	—	2	4	no	—
195	C₂×C₄×C₂₃	—	2	2	no	—
196	C₂×C₄×C₂₃	—	2	8	no	—
197	C₂×C₄×C₂₃	—	2	2	no	—
198	C₂×C₄×C₂₃	—	2	2	no	—
199	C₂×C₄×C₂₃	—	2	4	no	—
200	C₂×C₄×C₂₃	—	2	4	no	—
201	C₂×C₄×C₂₃	—	4	2	no	—
202	C₂×C₄×C₂₃	—	4	4	no	—
203	C₂×C₄×C₂₃	—	4	1	no	—
204	C₂×C₄×C₂₃	—	4	16	no	—
205	C₂×C₄×C₂₃	—	4	2	no	—
206	C₂×C₄×C₂₃	—	8	4	no	—
207	C₂×C₄×C₂₃	—	8	4	no	—
208	C₂×C₄×C₂₃	#137	1	4	yes	complemented
209	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
210	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
211	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
212	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
213	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
214	C₈×C₂₃	—	2	8	no	—
215	#3501	C₂×C₂₃	1	1	yes	characteristic, complemented
216	C₂×#550	C₂×C₂₃	1	1	yes	characteristic
217	#3333	C₂×C₂₃	1	8	yes	complemented
218	#3278	C₂×C₂₃	1	8	yes	complemented
219	#3208	C₂×C₂₃	1	2	yes	complemented
220	#3193	C₂×C₂₃	1	4	yes	complemented
221	#3104	C₂×C₂₃	1	2	yes	complemented
222	#3007	C₂×C₂₃	1	4	yes	complemented
223	#2975	C₂×C₂₃	1	1	yes	characteristic, complemented
224	C₂²×C₂₃	—	2	1	no	—
225	C₂²×C₂₃	—	2	1	no	—
226	C₂²×C₂₃	—	2	4	no	—
227	C₂²×C₂₃	—	2	4	no	—
228	C₂²×C₂₃	—	2	4	no	—
229	C₂²×C₂₃	—	4	4	no	—
230	C₂²×C₂₃	—	4	4	no	—
231	C₂²×C₂₃	—	4	4	no	—
232	C₂²×C₂₃	—	8	4	no	—
233	C₂²×C₂₃	—	8	8	no	—
234	C₂²×C₂₃	C₂×#137	1	2	yes	—
235	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
236	C₄×C₂₃	—	2	2	no	—
237	C₄×C₂₃	—	2	4	no	—
238	C₄×C₂₃	—	2	4	no	—
239	C₄×C₂₃	—	2	1	no	—
240	C₄×C₂₃	—	4	2	no	—
241	C₄×C₂₃	—	4	4	no	—
242	C₄×C₂₃	—	4	4	no	—
243	C₄×C₂₃	C₂×#137	1	4	yes	—
244	C₄×C₂₃	C₂×H(5,𝔽₂)	1	2	yes	—
245	C₂²×#41	C₂²×C₂₃	1	2	yes	—
246	C₂×#137	C₂²×C₂₃	1	8	yes	complemented
247	#554	—	2	4	no	—
248	#554	C₂²×C₂₃	1	4	yes	complemented
249	#550	—	2	4	no	—
250	#550	C₂²×C₂₃	1	8	yes	—
251	D₄×Q₈	C₂²×C₂₃	1	4	yes	complemented
252	#544	—	2	1	no	—
253	#543	—	2	2	no	—
254	#533	C₂²×C₂₃	1	2	yes	—
255	C₂×#127	—	2	1	no	—
256	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
257	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	4	yes	—
258	C₂×#121	—	2	2	no	—
259	C₂×#121	C₂²×C₂₃	1	2	yes	—
260	#518	—	2	1	no	—
261	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
262	C₂×C₄×D₄	—	2	1	no	—
263	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
264	#492	C₂²×C₂₃	1	4	yes	complemented
265	#486	C₂²×C₂₃	1	4	yes	complemented
266	#484	C₂²×C₂₃	1	4	yes	—
267	#480	C₂²×C₂₃	1	8	yes	complemented
268	#477	C₂²×C₂₃	1	4	yes	complemented
269	#468	C₂²×C₂₃	1	16	yes	complemented
270	#462	C₂²×C₂₃	1	4	yes	complemented
271	#460	C₂²×C₂₃	1	8	yes	complemented
272	#452	C₂²×C₂₃	1	16	yes	complemented
273	#451	C₂²×C₂₃	1	4	yes	—
274	#450	C₂²×C₂₃	1	8	yes	—
275	#448	C₂²×C₂₃	1	8	yes	complemented
276	C₄×Q₁₆	C₂²×C₂₃	1	8	yes	complemented
277	#436	C₂²×C₂₃	1	8	yes	complemented
278	#431	C₂²×C₂₃	1	2	yes	—
279	#417	C₂²×C₂₃	1	8	yes	complemented
280	#407	C₂²×C₂₃	1	4	yes	—
281	C₂×C₂₃	—	2	2	no	—
282	C₂×C₂₃	—	4	4	no	—
283	C₂×C₂₃	—	8	4	no	—
284	C₂×C₂₃	#3484	1	1	yes	characteristic
285	C₂×C₂₃	C₂²×#137	1	1	yes	characteristic
286	C₂×C₂₃	#3585	1	1	yes	characteristic
287	C₂×#41	—	2	4	no	—
288	C₂×#41	—	4	8	no	—
289	C₂×#41	C₂³×C₂₃	1	8	yes	—
290	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
291	C₂²×Q₈	C₂³×C₂₃	1	4	yes	—
292	C₂²×D₄	—	2	2	no	—
293	C₂²×D₄	—	2	2	no	—
294	C₂²×D₄	—	2	1	no	—
295	C₂²×D₄	—	2	1	no	—
296	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
297	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
298	C₂³×C₄	—	2	2	no	—
299	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
300	#137	—	4	16	no	—
301	C₂×Q₁₆	C₂³×C₂₃	1	8	yes	—
302	C₂×SD₁₆	C₂³×C₂₃	1	16	yes	—
303	C₂×M₁₆	C₂³×C₂₃	1	8	yes	—
304	C₄⋊Q₈	—	2	4	no	—
305	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
306	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
307	#127	—	2	8	no	—
308	#127	—	2	2	no	—
309	#127	C₂³×C₂₃	1	4	yes	—
310	#125	—	2	4	no	—
311	#125	—	2	2	no	—
312	C₂²⋊Q₈	—	2	4	no	—
313	C₂²⋊Q₈	—	2	4	no	—
314	C₂²⋊Q₈	—	2	2	no	—
315	C₂²⋊Q₈	—	2	8	no	—
316	C₂²⋊Q₈	—	2	8	no	—
317	C₂²⋊Q₈	C₂³×C₂₃	1	4	yes	—
318	C₂²⋊Q₈	C₂³×C₂₃	1	8	yes	—
319	#121	—	2	4	no	—
320	#121	—	2	16	no	—
321	#121	C₂³×C₂₃	1	8	yes	—
322	C₄×Q₈	—	2	2	no	—
323	C₄×Q₈	—	2	2	no	—
324	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
325	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
326	C₄×D₄	—	2	4	no	—
327	C₄×D₄	—	2	4	no	—
328	C₄×D₄	—	2	8	no	—
329	C₄×D₄	D₄×C₂₃	1	4	yes	—
330	C₄×D₄	C₂³×C₂₃	1	4	yes	—
331	#117	—	2	2	no	—
332	C₂×C₄⋊C₄	—	2	1	no	—
333	C₂×C₄⋊C₄	—	2	4	no	—
334	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
335	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
336	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
337	C₂×C₂²⋊C₄	—	2	4	no	—
338	C₂×C₂²⋊C₄	—	2	2	no	—
339	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
340	C₂×C₄²	—	2	1	no	—
341	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
342	#107	C₂³×C₂₃	1	8	yes	—
343	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
344	#103	C₂³×C₂₃	1	8	yes	—
345	#103	C₂³×C₂₃	1	8	yes	—
346	#102	C₂³×C₂₃	1	16	yes	—
347	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
348	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	—
349	C₄×C₈	C₂³×C₂₃	1	4	yes	—
350	C₂₃	#61588	1	1	yes	characteristic, direct product factor, Sylow
351	C₂⁴	—	2	2	no	—
352	C₂⁴	C₂×D₄×C₂₃	1	2	yes	—
353	#41	—	2	16	no	—
354	#41	—	4	8	no	—
355	#41	—	4	16	no	—
356	C₂×Q₈	—	2	4	no	—
357	C₂×Q₈	—	2	1	no	—
358	C₂×Q₈	—	2	2	no	—
359	C₂×Q₈	—	2	4	no	—
360	C₂×Q₈	—	4	8	no	—
361	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
362	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
363	C₂×D₄	—	2	8	no	—
364	C₂×D₄	—	2	8	no	—
365	C₂×D₄	—	2	4	no	—
366	C₂×D₄	—	2	4	no	—
367	C₂×D₄	—	2	2	no	—
368	C₂×D₄	—	2	4	no	—
369	C₂×D₄	—	4	4	no	—
370	C₂×D₄	—	4	8	no	—
371	C₂×D₄	—	4	4	no	—
372	C₂×D₄	—	8	2	no	—
373	C₂×D₄	—	8	4	no	—
374	C₂×D₄	C₂×D₄×C₂₃	1	4	yes	—
375	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
376	C₂²×C₄	—	2	1	no	—
377	C₂²×C₄	—	2	1	no	—
378	C₂²×C₄	—	2	4	no	—
379	C₂²×C₄	—	2	4	no	—
380	C₂²×C₄	—	2	8	no	—
381	C₂²×C₄	—	4	4	no	—
382	C₂²×C₄	—	4	8	no	—
383	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
384	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
385	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
386	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
387	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
388	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
389	Q₁₆	—	2	16	no	—
390	SD₁₆	—	4	16	no	—
391	M₁₆	—	4	8	no	—
392	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
393	C₄⋊C₄	—	2	4	no	—
394	C₄⋊C₄	—	2	4	no	—
395	C₄⋊C₄	—	2	8	no	—
396	C₄⋊C₄	—	2	1	no	—
397	C₄⋊C₄	—	2	4	no	—
398	C₄⋊C₄	—	2	2	no	—
399	C₄⋊C₄	—	2	2	no	—
400	C₄⋊C₄	—	2	2	no	—
401	C₄⋊C₄	—	2	4	no	—
402	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
403	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
404	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
405	C₂²⋊C₄	—	2	4	no	—
406	C₂²⋊C₄	—	2	8	no	—
407	C₂²⋊C₄	—	2	8	no	—
408	C₂²⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
409	C₄²	—	2	4	no	—
410	C₄²	—	2	4	no	—
411	C₄²	C₂×D₄×C₂₃	1	2	yes	—
412	C₄²	C₂⁴×C₂₃	1	2	yes	—
413	C₂³	—	2	4	no	—
414	C₂³	—	2	4	no	—
415	C₂³	—	4	2	no	—
416	C₂³	—	4	2	no	—
417	C₂³	—	4	4	no	—
418	C₂³	—	4	4	no	—
419	C₂³	—	4	4	no	—
420	C₂³	C₂²×D₄×C₂₃	1	4	yes	—
421	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
422	Q₈	—	2	4	no	—
423	Q₈	—	2	2	no	—
424	Q₈	—	2	8	no	—
425	Q₈	—	8	4	no	—
426	Q₈	—	8	2	no	—
427	Q₈	C₂₃×H(5,𝔽₂)	1	4	yes	—
428	D₄	—	2	8	no	—
429	D₄	—	2	8	no	—
430	D₄	—	4	16	no	—
431	D₄	—	4	4	no	—
432	D₄	—	4	8	no	—
433	D₄	—	8	4	no	—
434	D₄	—	8	8	no	—
435	C₂×C₄	—	2	4	no	—
436	C₂×C₄	—	2	2	no	—
437	C₂×C₄	—	2	8	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	2	4	no	—
441	C₂×C₄	—	2	4	no	—
442	C₂×C₄	—	4	2	no	—
443	C₂×C₄	—	4	4	no	—
444	C₂×C₄	—	4	1	no	—
445	C₂×C₄	—	4	16	no	—
446	C₂×C₄	—	4	2	no	—
447	C₂×C₄	—	8	4	no	—
448	C₂×C₄	—	8	4	no	—
449	C₂×C₄	C₂₃×#137	1	4	yes	complemented
450	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
451	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
452	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
453	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
454	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
455	C₈	—	2	8	no	—
456	C₂²	—	2	1	no	—
457	C₂²	—	2	1	no	—
458	C₂²	—	2	4	no	—
459	C₂²	—	2	4	no	—
460	C₂²	—	2	4	no	—
461	C₂²	—	4	4	no	—
462	C₂²	—	4	4	no	—
463	C₂²	—	4	4	no	—
464	C₂²	—	8	4	no	—
465	C₂²	—	8	8	no	—
466	C₂²	C₂×C₂₃×#137	1	2	yes	—
467	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
468	C₄	—	2	4	no	—
469	C₄	—	2	4	no	—
470	C₄	—	2	1	no	—
471	C₄	—	2	2	no	—
472	C₄	—	4	4	no	—
473	C₄	—	4	2	no	—
474	C₄	—	4	4	no	—
475	C₄	C₂×C₂₃×#137	1	4	yes	—
476	C₄	C₂×C₂₃×H(5,𝔽₂)	1	2	yes	—
477	C₂	—	2	2	no	—
478	C₂	—	4	4	no	—
479	C₂	—	8	4	no	—
480	C₂	C₂₃×#3484	1	1	yes	characteristic
481	C₂	C₂²×C₂₃×#137	1	1	yes	characteristic
482	C₂	C₂₃×#3585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂×C₂₃×#550	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄×D₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3501	C₂	semidirect
3. C₂₃×#3333	C₂	semidirect
4. C₂₃×#3278	C₂	semidirect
5. C₂₃×#3208	C₂	semidirect
6. C₂₃×#3193	C₂	semidirect
7. C₂₃×#3104	C₂	semidirect
8. C₂₃×#3007	C₂	semidirect
9. C₂₃×#2975	C₂	semidirect
11. C₂×C₂₃×#137	C₂²	semidirect
13. C₂₃×#554	C₂²	semidirect
16. D₄×Q₈×C₂₃	C₂²	semidirect
29. C₂₃×#492	C₂²	semidirect
30. C₂₃×#486	C₂²	semidirect
32. C₂₃×#480	C₂²	semidirect
33. C₂₃×#477	C₂²	semidirect
34. C₂₃×#468	C₂²	semidirect
35. C₂₃×#462	C₂²	semidirect
36. C₂₃×#460	C₂²	semidirect
37. C₂₃×#452	C₂²	semidirect
40. C₂₃×#448	C₂²	semidirect
41. C₄×Q₁₆×C₂₃	C₂²	semidirect
42. C₂₃×#436	C₂²	semidirect
44. C₂₃×#417	C₂²	semidirect
171. #61588	C₂₃	direct (N × H)
208. C₂×C₄×C₂₃	#137	semidirect
215. #3501	C₂×C₂₃	semidirect
217. #3333	C₂×C₂₃	semidirect
218. #3278	C₂×C₂₃	semidirect
219. #3208	C₂×C₂₃	semidirect
220. #3193	C₂×C₂₃	semidirect
221. #3104	C₂×C₂₃	semidirect
222. #3007	C₂×C₂₃	semidirect
223. #2975	C₂×C₂₃	semidirect
246. C₂×#137	C₂²×C₂₃	semidirect
248. #554	C₂²×C₂₃	semidirect
251. D₄×Q₈	C₂²×C₂₃	semidirect
264. #492	C₂²×C₂₃	semidirect
265. #486	C₂²×C₂₃	semidirect
267. #480	C₂²×C₂₃	semidirect
268. #477	C₂²×C₂₃	semidirect
269. #468	C₂²×C₂₃	semidirect
270. #462	C₂²×C₂₃	semidirect
271. #460	C₂²×C₂₃	semidirect
272. #452	C₂²×C₂₃	semidirect
275. #448	C₂²×C₂₃	semidirect
276. C₄×Q₁₆	C₂²×C₂₃	semidirect
277. #436	C₂²×C₂₃	semidirect
279. #417	C₂²×C₂₃	semidirect
350. C₂₃	#61588	direct (N × H)
449. C₂×C₄	C₂₃×#137	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂₃×#3501	2
C₂×C₂₃×#550	2
C₂₃×#3333	2
C₂₃×#3278	2
C₂₃×#3208	2
C₂₃×#3193	2
C₂₃×#3104	2
C₂₃×#3007	2
C₂₃×#2975	2
#61588	23

Subgroups — C23×#61588 #19209214

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#61588 #19209214