Subgroups — C₂₃×#33605 #19181231

Order 5888.

738 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#509	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₂₃×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3050	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂₃×#466	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#2220	C₂	1	2	yes	maximal, complemented
6	C₂₃×#2218	C₂	1	2	yes	maximal, complemented
7	C₂₃×#2217	C₂	1	2	yes	maximal, complemented
8	C₂₃×#2215	C₂	1	2	yes	maximal, complemented
9	C₂₃×#2159	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#2140	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#2139	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂₃×#135	C₂²	1	1	yes	characteristic
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	C₂₃×#509	—	2	4	no	—
18	C₂×C₂₃×SD₃₂	C₂²	1	2	yes	complemented
19	C₂×C₂₃×D₁₆	C₂²	1	2	yes	complemented
20	C₂×C₂₃×M₃₂	—	2	4	no	—
21	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
23	C₂×C₂₃×M₃₂	C₂²	1	2	yes	complemented
24	C₂²×C₁₆×C₂₃	C₂²	1	2	yes	complemented
25	C₂₃×#467	C₂²	1	1	yes	characteristic, complemented
26	C₂₃×#467	C₂²	1	1	yes	characteristic, complemented
27	C₂₃×#466	—	2	1	no	—
28	C₂₃×#466	—	2	1	no	—
29	C₂₃×#466	C₂²	1	2	yes	—
30	C₂₃×#466	C₂²	1	2	yes	—
31	C₂₃×#466	C₂²	1	1	yes	characteristic, complemented
32	C₂₃×#466	C₂²	1	1	yes	characteristic, complemented
33	C₂₃×#465	—	2	1	no	—
34	C₂₃×#465	—	2	1	no	—
35	C₂₃×#427	C₂²	1	1	yes	characteristic
36	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
37	C₂₃×#416	—	2	1	no	—
38	C₂×C₂₃×#102	—	2	1	no	—
39	C₂₃×#367	C₂²	1	2	yes	complemented
40	C₂₃×#366	C₂²	1	2	yes	complemented
41	C₂₃×#358	C₂²	1	2	yes	complemented
42	C₂₃×#357	C₂²	1	2	yes	complemented
43	C₂₃×#357	C₂²	1	2	yes	complemented
44	C₂₃×#357	C₂²	1	2	yes	complemented
45	C₂×#41×C₂₃	—	2	1	no	—
46	C₂²×D₄×C₂₃	—	2	1	no	—
47	C₂²×D₄×C₂₃	—	4	1	no	—
48	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
49	C₂₃×#135	—	2	2	no	—
50	C₂₃×#135	—	2	2	no	—
51	C₂×Q₁₆×C₂₃	C₂³	1	1	yes	characteristic
52	C₂×SD₁₆×C₂₃	—	2	1	no	—
53	C₂×D₈×C₂₃	—	2	2	no	—
54	C₂×D₈×C₂₃	—	2	1	no	—
55	C₂×D₈×C₂₃	—	2	2	no	—
56	C₂×D₈×C₂₃	C₂³	1	2	yes	—
57	C₂×D₈×C₂₃	C₂³	1	1	yes	characteristic
58	C₂²×C₈×C₂₃	—	2	2	no	—
59	C₂²×C₈×C₂₃	—	2	2	no	—
60	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
61	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
62	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
63	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
66	C₂₃×#123	—	4	1	no	—
67	C₂₃×C₂²⋊Q₈	—	2	1	no	—
68	C₂₃×C₂²⋊Q₈	—	2	1	no	—
69	C₂₃×#121	—	2	1	no	—
70	C₂₃×#121	—	2	1	no	—
71	C₂₃×#121	—	2	2	no	—
72	C₂₃×#121	—	2	2	no	—
73	C₂₃×#121	—	4	1	no	—
74	C₂₃×#121	—	4	1	no	—
75	C₂₃×#120	—	4	1	no	—
76	C₄×D₄×C₂₃	—	4	1	no	—
77	C₄×D₄×C₂₃	—	4	1	no	—
78	C₂₃×#117	—	2	1	no	—
79	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
80	C₂×C₂²⋊C₄×C₂₃	—	4	1	no	—
81	C₂₃×SD₃₂	—	2	4	no	—
82	C₂₃×D₁₆	—	2	4	no	—
83	C₂₃×M₃₂	—	2	2	no	—
84	C₂₃×M₃₂	—	2	4	no	—
85	C₂₃×M₃₂	D₄	1	4	yes	complemented
86	C₂×C₁₆×C₂₃	—	2	4	no	—
87	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
88	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
89	C₂₃×#107	—	2	1	no	—
90	C₂₃×#107	C₂³	1	1	yes	characteristic
91	C₂₃×#107	C₂³	1	1	yes	characteristic
92	C₂₃×#107	C₂³	1	2	yes	—
93	C₂₃×#106	—	2	1	no	—
94	C₂₃×#103	—	2	1	no	—
95	C₂₃×#103	—	2	1	no	—
96	C₂₃×#102	—	2	2	no	—
97	C₂₃×#102	—	2	2	no	—
98	C₂₃×#102	—	2	1	no	—
99	C₂₃×#102	—	2	1	no	—
100	C₂⁴×C₂₃	—	4	1	no	—
101	C₂⁴×C₂₃	D₈	1	1	yes	characteristic
102	#41×C₂₃	—	4	2	no	—
103	#41×C₂₃	—	4	2	no	—
104	C₂×Q₈×C₂₃	—	2	1	no	—
105	C₂×D₄×C₂₃	—	2	2	no	—
106	C₂×D₄×C₂₃	—	2	1	no	—
107	C₂×D₄×C₂₃	—	4	1	no	—
108	C₂×D₄×C₂₃	—	4	1	no	—
109	C₂×D₄×C₂₃	—	4	1	no	—
110	C₂×D₄×C₂₃	—	4	2	no	—
111	C₂×D₄×C₂₃	—	4	2	no	—
112	C₂×D₄×C₂₃	—	4	2	no	—
113	C₂×D₄×C₂₃	—	4	1	no	—
114	C₂×D₄×C₂₃	—	4	2	no	—
115	C₂×D₄×C₂₃	—	8	4	no	—
116	C₂²×C₄×C₂₃	—	2	2	no	—
117	C₂²×C₄×C₂₃	—	2	2	no	—
118	C₂²×C₄×C₂₃	—	4	1	no	—
119	C₂²×C₄×C₂₃	—	4	1	no	—
120	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
121	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
122	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
123	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
124	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
125	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
126	Q₁₆×C₂₃	—	2	1	no	—
127	Q₁₆×C₂₃	—	2	1	no	—
128	SD₁₆×C₂₃	—	4	2	no	—
129	D₈×C₂₃	—	2	1	no	—
130	D₈×C₂₃	—	2	2	no	—
131	D₈×C₂₃	—	2	2	no	—
132	D₈×C₂₃	—	2	1	no	—
133	D₈×C₂₃	—	4	2	no	—
134	C₂×C₈×C₂₃	—	2	2	no	—
135	C₂×C₈×C₂₃	—	2	2	no	—
136	C₂×C₈×C₂₃	—	2	2	no	—
137	C₂×C₈×C₂₃	—	2	1	no	—
138	C₂×C₈×C₂₃	—	2	1	no	—
139	C₂×C₈×C₂₃	—	2	1	no	—
140	C₂×C₈×C₂₃	—	2	1	no	—
141	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
142	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
143	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
144	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
145	C₂×C₈×C₂₃	#41	1	2	yes	—
146	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
147	C₄⋊C₄×C₂₃	—	2	2	no	—
148	C₄⋊C₄×C₂₃	—	2	1	no	—
149	C₄⋊C₄×C₂₃	—	2	1	no	—
150	C₄⋊C₄×C₂₃	—	4	1	no	—
151	C₄⋊C₄×C₂₃	—	4	1	no	—
152	C₄⋊C₄×C₂₃	—	4	1	no	—
153	C₂²⋊C₄×C₂₃	—	4	1	no	—
154	C₂²⋊C₄×C₂₃	—	4	1	no	—
155	C₂²⋊C₄×C₂₃	—	4	1	no	—
156	C₂²⋊C₄×C₂₃	—	4	2	no	—
157	C₂²⋊C₄×C₂₃	—	4	2	no	—
158	C₂²⋊C₄×C₂₃	—	4	1	no	—
159	C₂²⋊C₄×C₂₃	—	4	1	no	—
160	C₄²×C₂₃	—	4	1	no	—
161	C₁₆×C₂₃	—	2	2	no	—
162	C₁₆×C₂₃	C₂×D₄	1	4	yes	complemented
163	#33605	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
164	C₂³×C₂₃	—	2	2	no	—
165	C₂³×C₂₃	—	2	2	no	—
166	C₂³×C₂₃	—	4	1	no	—
167	C₂³×C₂₃	—	4	1	no	—
168	C₂³×C₂₃	—	4	2	no	—
169	C₂³×C₂₃	—	8	2	no	—
170	C₂³×C₂₃	—	8	2	no	—
171	C₂³×C₂₃	—	8	2	no	—
172	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
173	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
174	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
175	Q₈×C₂₃	—	4	1	no	—
176	Q₈×C₂₃	—	4	1	no	—
177	D₄×C₂₃	—	4	2	no	—
178	D₄×C₂₃	—	4	1	no	—
179	D₄×C₂₃	—	4	2	no	—
180	D₄×C₂₃	—	4	1	no	—
181	D₄×C₂₃	—	8	2	no	—
182	D₄×C₂₃	—	8	4	no	—
183	D₄×C₂₃	—	8	2	no	—
184	D₄×C₂₃	—	8	2	no	—
185	D₄×C₂₃	—	8	4	no	—
186	D₄×C₂₃	—	8	2	no	—
187	C₂×C₄×C₂₃	—	2	2	no	—
188	C₂×C₄×C₂₃	—	2	1	no	—
189	C₂×C₄×C₂₃	—	2	1	no	—
190	C₂×C₄×C₂₃	—	2	1	no	—
191	C₂×C₄×C₂₃	—	2	1	no	—
192	C₂×C₄×C₂₃	—	2	2	no	—
193	C₂×C₄×C₂₃	—	2	2	no	—
194	C₂×C₄×C₂₃	—	4	1	no	—
195	C₂×C₄×C₂₃	—	4	1	no	—
196	C₂×C₄×C₂₃	—	4	2	no	—
197	C₂×C₄×C₂₃	—	4	1	no	—
198	C₂×C₄×C₂₃	—	8	2	no	—
199	C₂×C₄×C₂₃	—	8	2	no	—
200	C₂×C₄×C₂₃	#121	1	2	yes	—
201	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
202	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
203	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
204	C₂×C₄×C₂₃	#135	1	2	yes	—
205	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
206	C₈×C₂₃	—	2	1	no	—
207	C₈×C₂₃	—	2	1	no	—
208	C₈×C₂₃	#121	1	2	yes	—
209	C₈×C₂₃	C₂²×D₄	1	1	yes	characteristic
210	C₈×C₂₃	C₂×#41	1	1	yes	characteristic
211	C₂×#509	C₂×C₂₃	1	1	yes	characteristic, complemented
212	C₂²×M₃₂	C₂×C₂₃	1	1	yes	characteristic, complemented
213	#3050	C₂×C₂₃	1	1	yes	characteristic, complemented
214	C₂×#466	C₂×C₂₃	1	1	yes	characteristic, complemented
215	#2220	C₂×C₂₃	1	2	yes	complemented
216	#2218	C₂×C₂₃	1	2	yes	complemented
217	#2217	C₂×C₂₃	1	2	yes	complemented
218	#2215	C₂×C₂₃	1	2	yes	complemented
219	#2159	C₂×C₂₃	1	1	yes	characteristic, complemented
220	#2140	C₂×C₂₃	1	1	yes	characteristic, complemented
221	#2139	C₂×C₂₃	1	1	yes	characteristic, complemented
222	C₂²×C₂₃	—	2	1	no	—
223	C₂²×C₂₃	—	2	1	no	—
224	C₂²×C₂₃	—	2	1	no	—
225	C₂²×C₂₃	—	2	1	no	—
226	C₂²×C₂₃	—	2	1	no	—
227	C₂²×C₂₃	—	2	1	no	—
228	C₂²×C₂₃	—	2	1	no	—
229	C₂²×C₂₃	—	4	4	no	—
230	C₂²×C₂₃	—	8	2	no	—
231	C₂²×C₂₃	—	8	1	no	—
232	C₂²×C₂₃	—	8	1	no	—
233	C₂²×C₂₃	—	8	2	no	—
234	C₂²×C₂₃	—	8	1	no	—
235	C₂²×C₂₃	—	8	2	no	—
236	C₂²×C₂₃	—	16	4	no	—
237	C₂²×C₂₃	#466	1	2	yes	—
238	C₂²×C₂₃	#509	1	2	yes	complemented
239	C₂²×C₂₃	C₂²×D₈	1	1	yes	characteristic
240	C₄×C₂₃	—	2	1	no	—
241	C₄×C₂₃	—	2	1	no	—
242	C₄×C₂₃	—	8	2	no	—
243	C₄×C₂₃	—	8	1	no	—
244	C₄×C₂₃	—	8	1	no	—
245	C₄×C₂₃	—	8	1	no	—
246	C₄×C₂₃	#466	1	2	yes	—
247	C₄×C₂₃	C₂×#121	1	1	yes	characteristic
248	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
249	C₂×#135	C₂²×C₂₃	1	1	yes	characteristic
250	C₂²×D₈	C₂²×C₂₃	1	1	yes	characteristic
251	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
252	#525	—	2	1	no	—
253	C₂×#121	—	2	1	no	—
254	#509	—	2	4	no	—
255	C₂×SD₃₂	C₂²×C₂₃	1	2	yes	complemented
256	C₂×D₁₆	C₂²×C₂₃	1	2	yes	complemented
257	C₂×M₃₂	—	2	4	no	—
258	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
259	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
260	C₂×M₃₂	C₂²×C₂₃	1	2	yes	complemented
261	C₂²×C₁₆	C₂²×C₂₃	1	2	yes	complemented
262	#467	C₂²×C₂₃	1	1	yes	characteristic, complemented
263	#467	C₂²×C₂₃	1	1	yes	characteristic, complemented
264	#466	—	2	1	no	—
265	#466	—	2	1	no	—
266	#466	C₂²×C₂₃	1	2	yes	—
267	#466	C₂²×C₂₃	1	2	yes	—
268	#466	C₂²×C₂₃	1	1	yes	characteristic, complemented
269	#466	C₂²×C₂₃	1	1	yes	characteristic, complemented
270	#465	—	2	1	no	—
271	#465	—	2	1	no	—
272	#427	C₂²×C₂₃	1	1	yes	characteristic
273	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
274	#416	—	2	1	no	—
275	C₂×#102	—	2	1	no	—
276	#367	C₂²×C₂₃	1	2	yes	complemented
277	#366	C₂²×C₂₃	1	2	yes	complemented
278	#358	C₂²×C₂₃	1	2	yes	complemented
279	#357	C₂²×C₂₃	1	2	yes	complemented
280	#357	C₂²×C₂₃	1	2	yes	complemented
281	#357	C₂²×C₂₃	1	2	yes	complemented
282	C₂×C₂₃	—	2	2	no	—
283	C₂×C₂₃	—	2	2	no	—
284	C₂×C₂₃	—	4	1	no	—
285	C₂×C₂₃	—	16	1	no	—
286	C₂×C₂₃	—	16	2	no	—
287	C₂×C₂₃	C₂×#466	1	1	yes	characteristic
288	C₂×C₂₃	C₂×#509	1	1	yes	characteristic
289	C₂×C₂₃	#3414	1	1	yes	characteristic
290	C₂×#41	—	2	1	no	—
291	C₂²×D₄	—	2	1	no	—
292	C₂²×D₄	—	4	1	no	—
293	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
294	#135	—	2	2	no	—
295	#135	—	2	2	no	—
296	C₂×Q₁₆	C₂³×C₂₃	1	1	yes	characteristic
297	C₂×SD₁₆	—	2	1	no	—
298	C₂×D₈	—	2	2	no	—
299	C₂×D₈	—	2	1	no	—
300	C₂×D₈	—	2	2	no	—
301	C₂×D₈	C₂³×C₂₃	1	2	yes	—
302	C₂×D₈	C₂³×C₂₃	1	1	yes	characteristic
303	C₂²×C₈	—	2	2	no	—
304	C₂²×C₈	—	2	2	no	—
305	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
306	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
307	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
308	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
309	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
310	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
311	#123	—	4	1	no	—
312	C₂²⋊Q₈	—	2	1	no	—
313	C₂²⋊Q₈	—	2	1	no	—
314	#121	—	2	2	no	—
315	#121	—	2	2	no	—
316	#121	—	2	1	no	—
317	#121	—	2	1	no	—
318	#121	—	4	1	no	—
319	#121	—	4	1	no	—
320	#120	—	4	1	no	—
321	C₄×D₄	—	4	1	no	—
322	C₄×D₄	—	4	1	no	—
323	#117	—	2	1	no	—
324	C₂×C₄⋊C₄	—	2	1	no	—
325	C₂×C₂²⋊C₄	—	4	1	no	—
326	SD₃₂	—	2	4	no	—
327	D₁₆	—	2	4	no	—
328	M₃₂	—	2	2	no	—
329	M₃₂	—	2	4	no	—
330	M₃₂	D₄×C₂₃	1	4	yes	complemented
331	C₂×C₁₆	—	2	4	no	—
332	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
333	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
334	#107	—	2	1	no	—
335	#107	C₂³×C₂₃	1	1	yes	characteristic
336	#107	C₂³×C₂₃	1	1	yes	characteristic
337	#107	C₂³×C₂₃	1	2	yes	—
338	#106	—	2	1	no	—
339	#103	—	2	1	no	—
340	#103	—	2	1	no	—
341	#102	—	2	2	no	—
342	#102	—	2	2	no	—
343	#102	—	2	1	no	—
344	#102	—	2	1	no	—
345	C₂₃	#33605	1	1	yes	characteristic, direct product factor, Sylow
346	C₂⁴	—	4	1	no	—
347	C₂⁴	D₈×C₂₃	1	1	yes	characteristic
348	#41	—	4	2	no	—
349	#41	—	4	2	no	—
350	C₂×Q₈	—	2	1	no	—
351	C₂×D₄	—	2	2	no	—
352	C₂×D₄	—	2	1	no	—
353	C₂×D₄	—	4	1	no	—
354	C₂×D₄	—	4	1	no	—
355	C₂×D₄	—	4	1	no	—
356	C₂×D₄	—	4	2	no	—
357	C₂×D₄	—	4	2	no	—
358	C₂×D₄	—	4	2	no	—
359	C₂×D₄	—	4	1	no	—
360	C₂×D₄	—	4	2	no	—
361	C₂×D₄	—	8	4	no	—
362	C₂²×C₄	—	2	2	no	—
363	C₂²×C₄	—	2	2	no	—
364	C₂²×C₄	—	4	1	no	—
365	C₂²×C₄	—	4	1	no	—
366	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
367	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
368	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
369	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
370	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
371	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
372	Q₁₆	—	2	1	no	—
373	Q₁₆	—	2	1	no	—
374	SD₁₆	—	4	2	no	—
375	D₈	—	2	1	no	—
376	D₈	—	2	2	no	—
377	D₈	—	2	2	no	—
378	D₈	—	2	1	no	—
379	D₈	—	4	2	no	—
380	C₂×C₈	—	2	2	no	—
381	C₂×C₈	—	2	2	no	—
382	C₂×C₈	—	2	2	no	—
383	C₂×C₈	—	2	1	no	—
384	C₂×C₈	—	2	1	no	—
385	C₂×C₈	—	2	1	no	—
386	C₂×C₈	—	2	1	no	—
387	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
388	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
389	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
390	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
391	C₂×C₈	#41×C₂₃	1	2	yes	—
392	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
393	C₄⋊C₄	—	2	2	no	—
394	C₄⋊C₄	—	2	1	no	—
395	C₄⋊C₄	—	2	1	no	—
396	C₄⋊C₄	—	4	1	no	—
397	C₄⋊C₄	—	4	1	no	—
398	C₄⋊C₄	—	4	1	no	—
399	C₂²⋊C₄	—	4	1	no	—
400	C₂²⋊C₄	—	4	1	no	—
401	C₂²⋊C₄	—	4	1	no	—
402	C₂²⋊C₄	—	4	2	no	—
403	C₂²⋊C₄	—	4	2	no	—
404	C₂²⋊C₄	—	4	1	no	—
405	C₂²⋊C₄	—	4	1	no	—
406	C₄²	—	4	1	no	—
407	C₁₆	—	2	2	no	—
408	C₁₆	C₂×D₄×C₂₃	1	4	yes	complemented
409	C₂³	—	2	2	no	—
410	C₂³	—	2	2	no	—
411	C₂³	—	4	1	no	—
412	C₂³	—	4	1	no	—
413	C₂³	—	4	2	no	—
414	C₂³	—	8	2	no	—
415	C₂³	—	8	2	no	—
416	C₂³	—	8	2	no	—
417	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
418	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
419	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
420	Q₈	—	4	1	no	—
421	Q₈	—	4	1	no	—
422	D₄	—	4	2	no	—
423	D₄	—	4	1	no	—
424	D₄	—	4	2	no	—
425	D₄	—	4	1	no	—
426	D₄	—	8	4	no	—
427	D₄	—	8	2	no	—
428	D₄	—	8	2	no	—
429	D₄	—	8	4	no	—
430	D₄	—	8	2	no	—
431	D₄	—	8	2	no	—
432	C₂×C₄	—	2	2	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	2	1	no	—
435	C₂×C₄	—	2	2	no	—
436	C₂×C₄	—	2	2	no	—
437	C₂×C₄	—	2	1	no	—
438	C₂×C₄	—	2	1	no	—
439	C₂×C₄	—	4	1	no	—
440	C₂×C₄	—	4	1	no	—
441	C₂×C₄	—	4	2	no	—
442	C₂×C₄	—	4	1	no	—
443	C₂×C₄	—	8	2	no	—
444	C₂×C₄	—	8	2	no	—
445	C₂×C₄	C₂₃×#121	1	2	yes	—
446	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
447	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
448	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
449	C₂×C₄	C₂₃×#135	1	2	yes	—
450	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
451	C₈	—	2	1	no	—
452	C₈	—	2	1	no	—
453	C₈	C₂₃×#121	1	2	yes	—
454	C₈	C₂²×D₄×C₂₃	1	1	yes	characteristic
455	C₈	C₂×#41×C₂₃	1	1	yes	characteristic
456	C₂²	—	2	1	no	—
457	C₂²	—	2	1	no	—
458	C₂²	—	2	1	no	—
459	C₂²	—	2	1	no	—
460	C₂²	—	2	1	no	—
461	C₂²	—	2	1	no	—
462	C₂²	—	2	1	no	—
463	C₂²	—	4	4	no	—
464	C₂²	—	8	1	no	—
465	C₂²	—	8	2	no	—
466	C₂²	—	8	2	no	—
467	C₂²	—	8	1	no	—
468	C₂²	—	8	1	no	—
469	C₂²	—	8	2	no	—
470	C₂²	—	16	4	no	—
471	C₂²	C₂₃×#466	1	2	yes	—
472	C₂²	C₂₃×#509	1	2	yes	complemented
473	C₂²	C₂²×D₈×C₂₃	1	1	yes	characteristic
474	C₄	—	2	1	no	—
475	C₄	—	2	1	no	—
476	C₄	—	8	1	no	—
477	C₄	—	8	2	no	—
478	C₄	—	8	1	no	—
479	C₄	—	8	1	no	—
480	C₄	C₂₃×#466	1	2	yes	—
481	C₄	C₂×C₂₃×#121	1	1	yes	characteristic
482	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
483	C₂	—	2	2	no	—
484	C₂	—	2	2	no	—
485	C₂	—	4	1	no	—
486	C₂	—	16	2	no	—
487	C₂	—	16	1	no	—
488	C₂	C₂×C₂₃×#466	1	1	yes	characteristic
489	C₂	C₂×C₂₃×#509	1	1	yes	characteristic
490	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. C₂×C₂₃×#509	C₂	semidirect
2. C₂²×C₂₃×M₃₂	C₂	semidirect
3. C₂₃×#3050	C₂	semidirect
4. C₂×C₂₃×#466	C₂	semidirect
5. C₂₃×#2220	C₂	semidirect
6. C₂₃×#2218	C₂	semidirect
7. C₂₃×#2217	C₂	semidirect
8. C₂₃×#2215	C₂	semidirect
9. C₂₃×#2159	C₂	semidirect
10. C₂₃×#2140	C₂	semidirect
11. C₂₃×#2139	C₂	semidirect
18. C₂×C₂₃×SD₃₂	C₂²	semidirect
19. C₂×C₂₃×D₁₆	C₂²	semidirect
23. C₂×C₂₃×M₃₂	C₂²	semidirect
24. C₂²×C₁₆×C₂₃	C₂²	semidirect
25. C₂₃×#467	C₂²	semidirect
26. C₂₃×#467	C₂²	semidirect
31. C₂₃×#466	C₂²	semidirect
32. C₂₃×#466	C₂²	semidirect
39. C₂₃×#367	C₂²	semidirect
40. C₂₃×#366	C₂²	semidirect
41. C₂₃×#358	C₂²	semidirect
42. C₂₃×#357	C₂²	semidirect
43. C₂₃×#357	C₂²	semidirect
44. C₂₃×#357	C₂²	semidirect
85. C₂₃×M₃₂	D₄	semidirect
162. C₁₆×C₂₃	C₂×D₄	semidirect
163. #33605	C₂₃	direct (N × H)
211. C₂×#509	C₂×C₂₃	semidirect
212. C₂²×M₃₂	C₂×C₂₃	semidirect
213. #3050	C₂×C₂₃	semidirect
214. C₂×#466	C₂×C₂₃	semidirect
215. #2220	C₂×C₂₃	semidirect
216. #2218	C₂×C₂₃	semidirect
217. #2217	C₂×C₂₃	semidirect
218. #2215	C₂×C₂₃	semidirect
219. #2159	C₂×C₂₃	semidirect
220. #2140	C₂×C₂₃	semidirect
221. #2139	C₂×C₂₃	semidirect
238. C₂²×C₂₃	#509	semidirect
255. C₂×SD₃₂	C₂²×C₂₃	semidirect
256. C₂×D₁₆	C₂²×C₂₃	semidirect
260. C₂×M₃₂	C₂²×C₂₃	semidirect
261. C₂²×C₁₆	C₂²×C₂₃	semidirect
262. #467	C₂²×C₂₃	semidirect
263. #467	C₂²×C₂₃	semidirect
268. #466	C₂²×C₂₃	semidirect
269. #466	C₂²×C₂₃	semidirect
276. #367	C₂²×C₂₃	semidirect
277. #366	C₂²×C₂₃	semidirect
278. #358	C₂²×C₂₃	semidirect
279. #357	C₂²×C₂₃	semidirect
280. #357	C₂²×C₂₃	semidirect
281. #357	C₂²×C₂₃	semidirect
330. M₃₂	D₄×C₂₃	semidirect
345. C₂₃	#33605	direct (N × H)
408. C₁₆	C₂×D₄×C₂₃	semidirect
472. C₂²	C₂₃×#509	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#509	2
C₂²×C₂₃×M₃₂	2
C₂₃×#3050	2
C₂×C₂₃×#466	2
C₂₃×#2220	2
C₂₃×#2218	2
C₂₃×#2217	2
C₂₃×#2215	2
C₂₃×#2159	2
C₂₃×#2140	2
C₂₃×#2139	2
#33605	23

Subgroups — C23×#33605 #19181231

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33605 #19181231