Subgroups — C₂₃×#33760 #19181386

Order 5888.

640 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3050	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#466	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2235	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2234	C₂	1	2	yes	maximal, complemented
5	C₂₃×#2232	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#2220	C₂	1	2	yes	maximal, complemented
7	C₂₃×#2218	C₂	1	2	yes	maximal, complemented
8	C₂₃×#2187	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#2186	C₂	1	2	yes	maximal, complemented
10	C₂₃×#2184	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#2112	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂₃×#135	C₂²	1	1	yes	characteristic, complemented
13	C₂²×D₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×#525	—	2	1	no	—
16	C₂×C₂₃×#121	—	2	1	no	—
17	C₂×C₂₃×SD₃₂	C₂²	1	2	yes	complemented
18	C₂×C₂₃×D₁₆	C₂²	1	2	yes	complemented
19	C₂×C₂₃×M₃₂	C₂²	1	2	yes	complemented
20	C₂₃×#467	C₂²	1	1	yes	characteristic
21	C₂₃×#467	C₂²	1	1	yes	characteristic, complemented
22	C₂₃×#466	—	2	1	no	—
23	C₂₃×#466	—	2	1	no	—
24	C₂₃×#466	C₂²	1	1	yes	characteristic
25	C₂₃×#466	C₂²	1	2	yes	—
26	C₂₃×#466	C₂²	1	2	yes	—
27	C₂₃×#466	C₂²	1	1	yes	characteristic, complemented
28	C₂₃×#465	—	2	1	no	—
29	C₂₃×#465	—	2	1	no	—
30	C₂₃×#427	C₂²	1	1	yes	characteristic, complemented
31	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
32	C₂₃×#416	—	2	1	no	—
33	C₂×C₂₃×#102	—	2	1	no	—
34	C₂₃×#367	C₂²	1	2	yes	complemented
35	C₂₃×#366	C₂²	1	2	yes	complemented
36	C₂₃×#358	C₂²	1	2	yes	complemented
37	C₂₃×#357	C₂²	1	2	yes	complemented
38	C₂₃×#357	C₂²	1	2	yes	complemented
39	C₂₃×#357	C₂²	1	2	yes	complemented
40	C₂₃×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
41	C₂₃×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
42	C₂₃×C₂²⋊C₁₆	C₂²	1	2	yes	complemented
43	C₂×#41×C₂₃	—	2	1	no	—
44	C₂²×D₄×C₂₃	—	2	1	no	—
45	C₂²×D₄×C₂₃	—	4	1	no	—
46	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
47	C₂₃×#135	—	4	2	no	—
48	C₂×Q₁₆×C₂₃	C₂³	1	1	yes	characteristic
49	C₂×SD₁₆×C₂₃	—	2	1	no	—
50	C₂×D₈×C₂₃	—	2	1	no	—
51	C₂×D₈×C₂₃	—	4	2	no	—
52	C₂×D₈×C₂₃	C₂³	1	2	yes	—
53	C₂×D₈×C₂₃	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₂₃	—	4	2	no	—
55	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
56	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
57	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
58	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
61	C₂₃×#123	—	4	1	no	—
62	C₂₃×C₂²⋊Q₈	—	2	1	no	—
63	C₂₃×C₂²⋊Q₈	—	2	1	no	—
64	C₂₃×#121	—	2	1	no	—
65	C₂₃×#121	—	2	1	no	—
66	C₂₃×#121	—	2	2	no	—
67	C₂₃×#121	—	2	2	no	—
68	C₂₃×#121	—	4	1	no	—
69	C₂₃×#121	—	4	1	no	—
70	C₂₃×#120	—	4	1	no	—
71	C₄×D₄×C₂₃	—	4	1	no	—
72	C₄×D₄×C₂₃	—	4	1	no	—
73	C₂₃×#117	—	2	1	no	—
74	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
75	C₂×C₂²⋊C₄×C₂₃	—	4	1	no	—
76	C₂₃×SD₃₂	—	4	2	no	—
77	C₂₃×D₁₆	—	4	2	no	—
78	C₂₃×M₃₂	—	4	2	no	—
79	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
80	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
81	C₂₃×#107	—	2	1	no	—
82	C₂₃×#107	C₂³	1	1	yes	characteristic
83	C₂₃×#107	C₂³	1	1	yes	characteristic
84	C₂₃×#107	C₂³	1	2	yes	—
85	C₂₃×#106	—	2	1	no	—
86	C₂₃×#103	—	2	1	no	—
87	C₂₃×#103	—	2	1	no	—
88	C₂₃×#102	—	2	2	no	—
89	C₂₃×#102	—	2	2	no	—
90	C₂₃×#102	—	2	1	no	—
91	C₂₃×#102	—	2	1	no	—
92	C₂⁴×C₂₃	—	4	1	no	—
93	C₂⁴×C₂₃	D₈	1	1	yes	characteristic
94	#41×C₂₃	—	4	2	no	—
95	#41×C₂₃	—	4	2	no	—
96	C₂×Q₈×C₂₃	—	2	1	no	—
97	C₂×D₄×C₂₃	—	2	2	no	—
98	C₂×D₄×C₂₃	—	2	1	no	—
99	C₂×D₄×C₂₃	—	4	2	no	—
100	C₂×D₄×C₂₃	—	4	2	no	—
101	C₂×D₄×C₂₃	—	4	1	no	—
102	C₂×D₄×C₂₃	—	4	1	no	—
103	C₂×D₄×C₂₃	—	4	1	no	—
104	C₂×D₄×C₂₃	—	4	2	no	—
105	C₂×D₄×C₂₃	—	4	2	no	—
106	C₂×D₄×C₂₃	—	4	1	no	—
107	C₂×D₄×C₂₃	—	8	4	no	—
108	C₂²×C₄×C₂₃	—	4	1	no	—
109	C₂²×C₄×C₂₃	—	4	1	no	—
110	C₂²×C₄×C₂₃	—	4	1	no	—
111	C₂²×C₄×C₂₃	—	4	1	no	—
112	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
113	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
114	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
115	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
118	Q₁₆×C₂₃	—	2	2	no	—
119	SD₁₆×C₂₃	—	4	2	no	—
120	D₈×C₂₃	—	2	2	no	—
121	D₈×C₂₃	—	2	2	no	—
122	D₈×C₂₃	—	2	2	no	—
123	D₈×C₂₃	—	4	2	no	—
124	C₂×C₈×C₂₃	—	2	2	no	—
125	C₂×C₈×C₂₃	—	2	2	no	—
126	C₂×C₈×C₂₃	—	2	2	no	—
127	C₂×C₈×C₂₃	—	4	1	no	—
128	C₂×C₈×C₂₃	—	4	1	no	—
129	C₂×C₈×C₂₃	—	4	1	no	—
130	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
131	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
132	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
133	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
134	C₄⋊C₄×C₂₃	—	2	2	no	—
135	C₄⋊C₄×C₂₃	—	2	1	no	—
136	C₄⋊C₄×C₂₃	—	2	1	no	—
137	C₄⋊C₄×C₂₃	—	4	1	no	—
138	C₄⋊C₄×C₂₃	—	4	1	no	—
139	C₄⋊C₄×C₂₃	—	4	1	no	—
140	C₂²⋊C₄×C₂₃	—	4	1	no	—
141	C₂²⋊C₄×C₂₃	—	4	1	no	—
142	C₂²⋊C₄×C₂₃	—	4	1	no	—
143	C₂²⋊C₄×C₂₃	—	4	2	no	—
144	C₂²⋊C₄×C₂₃	—	4	2	no	—
145	C₂²⋊C₄×C₂₃	—	4	1	no	—
146	C₂²⋊C₄×C₂₃	—	4	1	no	—
147	C₄²×C₂₃	—	4	1	no	—
148	C₁₆×C₂₃	—	2	2	no	—
149	C₁₆×C₂₃	—	2	2	no	—
150	#33760	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
151	C₂³×C₂₃	—	2	2	no	—
152	C₂³×C₂₃	—	4	1	no	—
153	C₂³×C₂₃	—	4	1	no	—
154	C₂³×C₂₃	—	4	1	no	—
155	C₂³×C₂₃	—	4	2	no	—
156	C₂³×C₂₃	—	8	2	no	—
157	C₂³×C₂₃	—	8	2	no	—
158	C₂³×C₂₃	—	8	2	no	—
159	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
160	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
161	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
162	Q₈×C₂₃	—	4	1	no	—
163	Q₈×C₂₃	—	4	1	no	—
164	D₄×C₂₃	—	4	2	no	—
165	D₄×C₂₃	—	4	1	no	—
166	D₄×C₂₃	—	4	2	no	—
167	D₄×C₂₃	—	4	1	no	—
168	D₄×C₂₃	—	8	2	no	—
169	D₄×C₂₃	—	8	2	no	—
170	D₄×C₂₃	—	8	4	no	—
171	D₄×C₂₃	—	8	4	no	—
172	D₄×C₂₃	—	8	2	no	—
173	D₄×C₂₃	—	8	2	no	—
174	C₂×C₄×C₂₃	—	2	1	no	—
175	C₂×C₄×C₂₃	—	2	1	no	—
176	C₂×C₄×C₂₃	—	2	1	no	—
177	C₂×C₄×C₂₃	—	2	1	no	—
178	C₂×C₄×C₂₃	—	2	1	no	—
179	C₂×C₄×C₂₃	—	2	1	no	—
180	C₂×C₄×C₂₃	—	4	1	no	—
181	C₂×C₄×C₂₃	—	4	1	no	—
182	C₂×C₄×C₂₃	—	4	1	no	—
183	C₂×C₄×C₂₃	—	4	1	no	—
184	C₂×C₄×C₂₃	—	4	1	no	—
185	C₂×C₄×C₂₃	—	4	2	no	—
186	C₂×C₄×C₂₃	—	4	1	no	—
187	C₂×C₄×C₂₃	—	8	2	no	—
188	C₂×C₄×C₂₃	—	8	2	no	—
189	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
190	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
191	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
192	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
193	C₈×C₂₃	—	2	1	no	—
194	C₈×C₂₃	—	2	1	no	—
195	C₈×C₂₃	—	2	1	no	—
196	C₈×C₂₃	H(5,𝔽₂)	1	2	yes	—
197	#3050	C₂×C₂₃	1	1	yes	characteristic, complemented
198	C₂×#466	C₂×C₂₃	1	1	yes	characteristic, complemented
199	#2235	C₂×C₂₃	1	1	yes	characteristic, complemented
200	#2234	C₂×C₂₃	1	2	yes	complemented
201	#2232	C₂×C₂₃	1	1	yes	characteristic, complemented
202	#2220	C₂×C₂₃	1	2	yes	complemented
203	#2218	C₂×C₂₃	1	2	yes	complemented
204	#2187	C₂×C₂₃	1	1	yes	characteristic, complemented
205	#2186	C₂×C₂₃	1	2	yes	complemented
206	#2184	C₂×C₂₃	1	1	yes	characteristic, complemented
207	#2112	C₂×C₂₃	1	1	yes	characteristic, complemented
208	C₂²×C₂₃	—	2	1	no	—
209	C₂²×C₂₃	—	2	1	no	—
210	C₂²×C₂₃	—	2	1	no	—
211	C₂²×C₂₃	—	2	1	no	—
212	C₂²×C₂₃	—	2	1	no	—
213	C₂²×C₂₃	—	2	1	no	—
214	C₂²×C₂₃	—	2	1	no	—
215	C₂²×C₂₃	—	2	1	no	—
216	C₂²×C₂₃	—	4	4	no	—
217	C₂²×C₂₃	—	8	2	no	—
218	C₂²×C₂₃	—	8	1	no	—
219	C₂²×C₂₃	—	8	1	no	—
220	C₂²×C₂₃	—	8	2	no	—
221	C₂²×C₂₃	—	8	1	no	—
222	C₂²×C₂₃	—	8	2	no	—
223	C₂²×C₂₃	—	16	4	no	—
224	C₂²×C₂₃	#509	1	2	yes	—
225	C₂²×C₂₃	C₂²×D₈	1	1	yes	characteristic
226	C₄×C₂₃	—	2	1	no	—
227	C₄×C₂₃	—	2	1	no	—
228	C₄×C₂₃	—	2	1	no	—
229	C₄×C₂₃	—	8	2	no	—
230	C₄×C₂₃	—	8	1	no	—
231	C₄×C₂₃	—	8	1	no	—
232	C₄×C₂₃	—	8	1	no	—
233	C₄×C₂₃	#534	1	1	yes	characteristic
234	C₄×C₂₃	#575	1	1	yes	characteristic
235	C₂×#135	C₂²×C₂₃	1	1	yes	characteristic, complemented
236	C₂²×D₈	C₂²×C₂₃	1	1	yes	characteristic
237	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
238	#525	—	2	1	no	—
239	C₂×#121	—	2	1	no	—
240	C₂×SD₃₂	C₂²×C₂₃	1	2	yes	complemented
241	C₂×D₁₆	C₂²×C₂₃	1	2	yes	complemented
242	C₂×M₃₂	C₂²×C₂₃	1	2	yes	complemented
243	#467	C₂²×C₂₃	1	1	yes	characteristic
244	#467	C₂²×C₂₃	1	1	yes	characteristic, complemented
245	#466	—	2	1	no	—
246	#466	—	2	1	no	—
247	#466	C₂²×C₂₃	1	1	yes	characteristic
248	#466	C₂²×C₂₃	1	2	yes	—
249	#466	C₂²×C₂₃	1	2	yes	—
250	#466	C₂²×C₂₃	1	1	yes	characteristic, complemented
251	#465	—	2	1	no	—
252	#465	—	2	1	no	—
253	#427	C₂²×C₂₃	1	1	yes	characteristic, complemented
254	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
255	#416	—	2	1	no	—
256	C₂×#102	—	2	1	no	—
257	#367	C₂²×C₂₃	1	2	yes	complemented
258	#366	C₂²×C₂₃	1	2	yes	complemented
259	#358	C₂²×C₂₃	1	2	yes	complemented
260	#357	C₂²×C₂₃	1	2	yes	complemented
261	#357	C₂²×C₂₃	1	2	yes	complemented
262	#357	C₂²×C₂₃	1	2	yes	complemented
263	C₂²⋊C₁₆	C₂²×C₂₃	1	1	yes	characteristic
264	C₂²⋊C₁₆	C₂²×C₂₃	1	1	yes	characteristic
265	C₂²⋊C₁₆	C₂²×C₂₃	1	2	yes	complemented
266	C₂×C₂₃	—	2	2	no	—
267	C₂×C₂₃	—	4	1	no	—
268	C₂×C₂₃	—	4	1	no	—
269	C₂×C₂₃	—	16	1	no	—
270	C₂×C₂₃	—	16	2	no	—
271	C₂×C₂₃	#3186	1	1	yes	characteristic
272	C₂×C₂₃	C₂×#509	1	1	yes	characteristic
273	C₂×C₂₃	#3414	1	1	yes	characteristic
274	C₂×#41	—	2	1	no	—
275	C₂²×D₄	—	2	1	no	—
276	C₂²×D₄	—	4	1	no	—
277	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
278	#135	—	4	2	no	—
279	C₂×Q₁₆	C₂³×C₂₃	1	1	yes	characteristic
280	C₂×SD₁₆	—	2	1	no	—
281	C₂×D₈	—	2	1	no	—
282	C₂×D₈	—	4	2	no	—
283	C₂×D₈	C₂³×C₂₃	1	2	yes	—
284	C₂×D₈	C₂³×C₂₃	1	1	yes	characteristic
285	C₂²×C₈	—	4	2	no	—
286	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
287	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
288	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
289	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
290	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
291	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
292	#123	—	4	1	no	—
293	C₂²⋊Q₈	—	2	1	no	—
294	C₂²⋊Q₈	—	2	1	no	—
295	#121	—	2	2	no	—
296	#121	—	2	2	no	—
297	#121	—	2	1	no	—
298	#121	—	2	1	no	—
299	#121	—	4	1	no	—
300	#121	—	4	1	no	—
301	#120	—	4	1	no	—
302	C₄×D₄	—	4	1	no	—
303	C₄×D₄	—	4	1	no	—
304	#117	—	2	1	no	—
305	C₂×C₄⋊C₄	—	2	1	no	—
306	C₂×C₂²⋊C₄	—	4	1	no	—
307	SD₃₂	—	4	2	no	—
308	D₁₆	—	4	2	no	—
309	M₃₂	—	4	2	no	—
310	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
311	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
312	#107	—	2	1	no	—
313	#107	C₂³×C₂₃	1	1	yes	characteristic
314	#107	C₂³×C₂₃	1	1	yes	characteristic
315	#107	C₂³×C₂₃	1	2	yes	—
316	#106	—	2	1	no	—
317	#103	—	2	1	no	—
318	#103	—	2	1	no	—
319	#102	—	2	2	no	—
320	#102	—	2	2	no	—
321	#102	—	2	1	no	—
322	#102	—	2	1	no	—
323	C₂₃	#33760	1	1	yes	characteristic, direct product factor, Sylow
324	C₂⁴	—	4	1	no	—
325	C₂⁴	D₈×C₂₃	1	1	yes	characteristic
326	#41	—	4	2	no	—
327	#41	—	4	2	no	—
328	C₂×Q₈	—	2	1	no	—
329	C₂×D₄	—	2	2	no	—
330	C₂×D₄	—	2	1	no	—
331	C₂×D₄	—	4	2	no	—
332	C₂×D₄	—	4	2	no	—
333	C₂×D₄	—	4	1	no	—
334	C₂×D₄	—	4	1	no	—
335	C₂×D₄	—	4	1	no	—
336	C₂×D₄	—	4	2	no	—
337	C₂×D₄	—	4	2	no	—
338	C₂×D₄	—	4	1	no	—
339	C₂×D₄	—	8	4	no	—
340	C₂²×C₄	—	4	1	no	—
341	C₂²×C₄	—	4	1	no	—
342	C₂²×C₄	—	4	1	no	—
343	C₂²×C₄	—	4	1	no	—
344	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
345	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
346	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
347	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
348	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
349	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
350	Q₁₆	—	2	2	no	—
351	SD₁₆	—	4	2	no	—
352	D₈	—	2	2	no	—
353	D₈	—	2	2	no	—
354	D₈	—	2	2	no	—
355	D₈	—	4	2	no	—
356	C₂×C₈	—	2	2	no	—
357	C₂×C₈	—	2	2	no	—
358	C₂×C₈	—	2	2	no	—
359	C₂×C₈	—	4	1	no	—
360	C₂×C₈	—	4	1	no	—
361	C₂×C₈	—	4	1	no	—
362	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
363	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
364	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
365	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
366	C₄⋊C₄	—	2	2	no	—
367	C₄⋊C₄	—	2	1	no	—
368	C₄⋊C₄	—	2	1	no	—
369	C₄⋊C₄	—	4	1	no	—
370	C₄⋊C₄	—	4	1	no	—
371	C₄⋊C₄	—	4	1	no	—
372	C₂²⋊C₄	—	4	1	no	—
373	C₂²⋊C₄	—	4	1	no	—
374	C₂²⋊C₄	—	4	1	no	—
375	C₂²⋊C₄	—	4	2	no	—
376	C₂²⋊C₄	—	4	2	no	—
377	C₂²⋊C₄	—	4	1	no	—
378	C₂²⋊C₄	—	4	1	no	—
379	C₄²	—	4	1	no	—
380	C₁₆	—	2	2	no	—
381	C₁₆	—	2	2	no	—
382	C₂³	—	2	2	no	—
383	C₂³	—	4	1	no	—
384	C₂³	—	4	1	no	—
385	C₂³	—	4	1	no	—
386	C₂³	—	4	2	no	—
387	C₂³	—	8	2	no	—
388	C₂³	—	8	2	no	—
389	C₂³	—	8	2	no	—
390	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
391	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
392	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
393	Q₈	—	4	1	no	—
394	Q₈	—	4	1	no	—
395	D₄	—	4	2	no	—
396	D₄	—	4	1	no	—
397	D₄	—	4	2	no	—
398	D₄	—	4	1	no	—
399	D₄	—	8	2	no	—
400	D₄	—	8	4	no	—
401	D₄	—	8	4	no	—
402	D₄	—	8	2	no	—
403	D₄	—	8	2	no	—
404	D₄	—	8	2	no	—
405	C₂×C₄	—	2	1	no	—
406	C₂×C₄	—	2	1	no	—
407	C₂×C₄	—	2	1	no	—
408	C₂×C₄	—	2	1	no	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	2	1	no	—
411	C₂×C₄	—	4	1	no	—
412	C₂×C₄	—	4	1	no	—
413	C₂×C₄	—	4	1	no	—
414	C₂×C₄	—	4	1	no	—
415	C₂×C₄	—	4	1	no	—
416	C₂×C₄	—	4	2	no	—
417	C₂×C₄	—	4	1	no	—
418	C₂×C₄	—	8	2	no	—
419	C₂×C₄	—	8	2	no	—
420	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
421	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
422	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
423	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
424	C₈	—	2	1	no	—
425	C₈	—	2	1	no	—
426	C₈	—	2	1	no	—
427	C₈	C₂₃×H(5,𝔽₂)	1	2	yes	—
428	C₂²	—	2	1	no	—
429	C₂²	—	2	1	no	—
430	C₂²	—	2	1	no	—
431	C₂²	—	2	1	no	—
432	C₂²	—	2	1	no	—
433	C₂²	—	2	1	no	—
434	C₂²	—	2	1	no	—
435	C₂²	—	2	1	no	—
436	C₂²	—	4	4	no	—
437	C₂²	—	8	1	no	—
438	C₂²	—	8	2	no	—
439	C₂²	—	8	2	no	—
440	C₂²	—	8	1	no	—
441	C₂²	—	8	1	no	—
442	C₂²	—	8	2	no	—
443	C₂²	—	16	4	no	—
444	C₂²	C₂₃×#509	1	2	yes	—
445	C₂²	C₂²×D₈×C₂₃	1	1	yes	characteristic
446	C₄	—	2	1	no	—
447	C₄	—	2	1	no	—
448	C₄	—	2	1	no	—
449	C₄	—	8	1	no	—
450	C₄	—	8	2	no	—
451	C₄	—	8	1	no	—
452	C₄	—	8	1	no	—
453	C₄	C₂₃×#534	1	1	yes	characteristic
454	C₄	C₂₃×#575	1	1	yes	characteristic
455	C₂	—	2	2	no	—
456	C₂	—	4	1	no	—
457	C₂	—	4	1	no	—
458	C₂	—	16	2	no	—
459	C₂	—	16	1	no	—
460	C₂	C₂₃×#3186	1	1	yes	characteristic
461	C₂	C₂×C₂₃×#509	1	1	yes	characteristic
462	C₂	C₂₃×#3414	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂³×C₄×C₂₃ ◁ C₂³×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3050	C₂	semidirect
2. C₂×C₂₃×#466	C₂	semidirect
3. C₂₃×#2235	C₂	semidirect
4. C₂₃×#2234	C₂	semidirect
5. C₂₃×#2232	C₂	semidirect
6. C₂₃×#2220	C₂	semidirect
7. C₂₃×#2218	C₂	semidirect
8. C₂₃×#2187	C₂	semidirect
9. C₂₃×#2186	C₂	semidirect
10. C₂₃×#2184	C₂	semidirect
11. C₂₃×#2112	C₂	semidirect
12. C₂×C₂₃×#135	C₂²	semidirect
17. C₂×C₂₃×SD₃₂	C₂²	semidirect
18. C₂×C₂₃×D₁₆	C₂²	semidirect
19. C₂×C₂₃×M₃₂	C₂²	semidirect
21. C₂₃×#467	C₂²	semidirect
27. C₂₃×#466	C₂²	semidirect
30. C₂₃×#427	C₂²	semidirect
34. C₂₃×#367	C₂²	semidirect
35. C₂₃×#366	C₂²	semidirect
36. C₂₃×#358	C₂²	semidirect
37. C₂₃×#357	C₂²	semidirect
38. C₂₃×#357	C₂²	semidirect
39. C₂₃×#357	C₂²	semidirect
42. C₂₃×C₂²⋊C₁₆	C₂²	semidirect
150. #33760	C₂₃	direct (N × H)
197. #3050	C₂×C₂₃	semidirect
198. C₂×#466	C₂×C₂₃	semidirect
199. #2235	C₂×C₂₃	semidirect
200. #2234	C₂×C₂₃	semidirect
201. #2232	C₂×C₂₃	semidirect
202. #2220	C₂×C₂₃	semidirect
203. #2218	C₂×C₂₃	semidirect
204. #2187	C₂×C₂₃	semidirect
205. #2186	C₂×C₂₃	semidirect
206. #2184	C₂×C₂₃	semidirect
207. #2112	C₂×C₂₃	semidirect
235. C₂×#135	C₂²×C₂₃	semidirect
240. C₂×SD₃₂	C₂²×C₂₃	semidirect
241. C₂×D₁₆	C₂²×C₂₃	semidirect
242. C₂×M₃₂	C₂²×C₂₃	semidirect
244. #467	C₂²×C₂₃	semidirect
250. #466	C₂²×C₂₃	semidirect
253. #427	C₂²×C₂₃	semidirect
257. #367	C₂²×C₂₃	semidirect
258. #366	C₂²×C₂₃	semidirect
259. #358	C₂²×C₂₃	semidirect
260. #357	C₂²×C₂₃	semidirect
261. #357	C₂²×C₂₃	semidirect
262. #357	C₂²×C₂₃	semidirect
265. C₂²⋊C₁₆	C₂²×C₂₃	semidirect
323. C₂₃	#33760	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×#3050	2
C₂×C₂₃×#466	2
C₂₃×#2235	2
C₂₃×#2234	2
C₂₃×#2232	2
C₂₃×#2220	2
C₂₃×#2218	2
C₂₃×#2187	2
C₂₃×#2186	2
C₂₃×#2184	2
C₂₃×#2112	2
#33760	23

Subgroups — C23×#33760 #19181386

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33760 #19181386