Subgroups — C₂₃×#34814 #19182440

Order 5888.

1994 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2497	C₂	1	2	yes	maximal, complemented
2	C₂₃×#2496	C₂	1	2	yes	maximal, complemented
3	C₂₃×#2495	C₂	1	2	yes	maximal, complemented
4	C₂₃×#2494	C₂	1	2	yes	maximal, complemented
5	C₂₃×#2493	C₂	1	4	yes	maximal, complemented
6	C₂₃×#2480	C₂	1	4	yes	maximal
7	C₂₃×#2401	C₂	1	4	yes	maximal, complemented
8	C₄×C₂₃×#126	C₂	1	4	yes	maximal, complemented
9	C₂₃×#2296	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#2296	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#2285	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#2280	C₂	1	1	yes	maximal, characteristic
13	C₂₃×#2280	C₂	1	1	yes	maximal, characteristic
14	C₄×C₂₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×C₄×C₂²⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
16	C₂⁴×C₄×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×#126	C₂²	1	4	yes	—
18	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
19	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
22	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
23	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
24	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
25	C₂₃×#401	C₂²	1	4	yes	complemented
26	C₂₃×#400	C₂²	1	4	yes	complemented
27	C₂₃×#400	C₂²	1	4	yes	complemented
28	C₂₃×#400	C₂²	1	4	yes	complemented
29	C₂₃×#397	C₂²	1	4	yes	—
30	C₂₃×#397	C₂²	1	4	yes	—
31	C₂₃×#397	C₂²	1	4	yes	—
32	C₂₃×#396	C₂²	1	4	yes	—
33	C₂₃×#389	C₂²	1	2	yes	complemented
34	C₂₃×#389	C₂²	1	2	yes	complemented
35	C₂₃×#389	C₂²	1	2	yes	complemented
36	C₂₃×#389	C₂²	1	2	yes	complemented
37	C₂₃×#388	C₂²	1	4	yes	—
38	C₂₃×#388	C₂²	1	4	yes	—
39	C₂₃×#388	C₂²	1	4	yes	—
40	C₂₃×#387	C₂²	1	4	yes	complemented
41	C₂₃×#387	C₂²	1	4	yes	complemented
42	C₂₃×#387	C₂²	1	4	yes	complemented
43	C₂₃×#385	C₂²	1	4	yes	—
44	C₂₃×#385	C₂²	1	4	yes	—
45	C₂₃×#384	C₂²	1	1	yes	characteristic, complemented
46	C₂₃×#384	C₂²	1	1	yes	characteristic, complemented
47	C₂₃×#383	C₂²	1	4	yes	complemented
48	C₂₃×#382	C₂²	1	1	yes	characteristic, complemented
49	C₂₃×#382	C₂²	1	1	yes	characteristic, complemented
50	C₂₃×#381	C₂²	1	2	yes	—
51	C₂₃×#381	C₂²	1	2	yes	—
52	C₂₃×#380	C₂²	1	4	yes	—
53	C₂₃×#380	C₂²	1	2	yes	—
54	C₂₃×#380	C₂²	1	2	yes	—
55	C₂₃×#380	C₂²	1	2	yes	—
56	C₂₃×#380	C₂²	1	2	yes	—
57	C₂₃×#379	C₂²	1	2	yes	—
58	C₂₃×#379	C₂²	1	2	yes	—
59	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	complemented
60	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	complemented
61	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	complemented
62	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	complemented
63	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	complemented
64	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	complemented
65	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
66	C₄×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
67	C₄×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
68	C₄×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
69	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
70	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
71	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
72	C₂₃×#376	C₂²	1	2	yes	complemented
73	C₂×C₂₃×#95	C₂²	1	2	yes	—
74	C₄³×C₂₃	C₂²	1	2	yes	complemented
75	C₂⁵×C₂₃	C₂³	1	1	yes	characteristic
76	C₂³×C₄×C₂₃	—	2	4	no	—
77	C₂³×C₄×C₂₃	—	2	4	no	—
78	C₂³×C₄×C₂₃	—	2	4	no	—
79	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
80	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
81	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
82	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
83	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
84	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
85	C₂₃×#126	—	2	16	no	—
86	C₂₃×#117	—	2	4	no	—
87	C₂₃×#117	—	2	4	no	—
88	C₂₃×#117	—	2	4	no	—
89	C₂₃×#117	—	2	4	no	—
90	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
91	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
92	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
93	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
94	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
95	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
96	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
97	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
98	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
99	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
100	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
101	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
102	C₂×C₂²⋊C₄×C₂₃	—	2	8	no	—
103	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
104	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
105	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
106	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
107	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
108	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
109	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
110	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
111	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
112	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
113	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
114	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
115	C₂×C₄²×C₂₃	—	2	4	no	—
116	C₂×C₄²×C₂₃	—	2	4	no	—
117	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
118	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
119	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
120	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
121	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
122	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
123	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
124	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
125	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
126	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
127	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
128	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
129	C₂₃×#95	C₂³	1	2	yes	—
130	C₂₃×#95	C₂³	1	2	yes	—
131	C₂₃×#95	C₂³	1	2	yes	—
132	C₂₃×#95	C₂³	1	2	yes	—
133	C₂₃×#95	C₂³	1	4	yes	—
134	C₂₃×#95	C₂³	1	4	yes	—
135	C₂₃×#95	C₂³	1	4	yes	—
136	C₂₃×#95	C₂³	1	4	yes	—
137	C₂₃×#95	C₂³	1	4	yes	—
138	C₂₃×#95	C₂³	1	4	yes	—
139	C₂₃×#95	C₂³	1	4	yes	—
140	C₂₃×#95	C₂³	1	4	yes	—
141	C₂₃×#95	C₂³	1	4	yes	—
142	C₂₃×#95	C₂³	1	4	yes	—
143	C₂₃×#95	C₂³	1	4	yes	—
144	C₂₃×#95	C₂³	1	4	yes	—
145	C₂⁴×C₂₃	—	2	4	no	—
146	C₂⁴×C₂₃	—	2	2	no	—
147	C₂⁴×C₂₃	—	2	2	no	—
148	C₂⁴×C₂₃	—	2	2	no	—
149	C₂⁴×C₂₃	—	4	2	no	—
150	C₂⁴×C₂₃	C₂⁴	1	2	yes	—
151	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
152	C₂²×C₄×C₂₃	—	2	4	no	—
153	C₂²×C₄×C₂₃	—	2	4	no	—
154	C₂²×C₄×C₂₃	—	2	2	no	—
155	C₂²×C₄×C₂₃	—	2	2	no	—
156	C₂²×C₄×C₂₃	—	2	2	no	—
157	C₂²×C₄×C₂₃	—	2	4	no	—
158	C₂²×C₄×C₂₃	—	2	4	no	—
159	C₂²×C₄×C₂₃	—	2	4	no	—
160	C₂²×C₄×C₂₃	—	2	4	no	—
161	C₂²×C₄×C₂₃	—	2	4	no	—
162	C₂²×C₄×C₂₃	—	2	4	no	—
163	C₂²×C₄×C₂₃	—	2	4	no	—
164	C₂²×C₄×C₂₃	—	2	4	no	—
165	C₂²×C₄×C₂₃	—	4	2	no	—
166	C₂²×C₄×C₂₃	—	4	2	no	—
167	C₂²×C₄×C₂₃	—	4	16	no	—
168	C₂²×C₄×C₂₃	#41	1	4	yes	—
169	C₂²×C₄×C₂₃	#41	1	4	yes	—
170	C₂²×C₄×C₂₃	#41	1	4	yes	—
171	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
172	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
173	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
174	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
175	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
176	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
177	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
178	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
179	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
180	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
181	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
182	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
183	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
184	C₄⋊C₄×C₂₃	—	2	4	no	—
185	C₄⋊C₄×C₂₃	—	2	4	no	—
186	C₄⋊C₄×C₂₃	—	2	2	no	—
187	C₄⋊C₄×C₂₃	—	2	2	no	—
188	C₄⋊C₄×C₂₃	—	2	2	no	—
189	C₄⋊C₄×C₂₃	—	2	2	no	—
190	C₄⋊C₄×C₂₃	—	2	8	no	—
191	C₄⋊C₄×C₂₃	—	2	8	no	—
192	C₄⋊C₄×C₂₃	—	2	8	no	—
193	C₄⋊C₄×C₂₃	—	4	4	no	—
194	C₄⋊C₄×C₂₃	—	4	4	no	—
195	C₄⋊C₄×C₂₃	—	4	4	no	—
196	C₂²⋊C₄×C₂₃	—	2	4	no	—
197	C₂²⋊C₄×C₂₃	—	2	4	no	—
198	C₂²⋊C₄×C₂₃	—	2	4	no	—
199	C₂²⋊C₄×C₂₃	—	2	4	no	—
200	C₂²⋊C₄×C₂₃	—	2	4	no	—
201	C₂²⋊C₄×C₂₃	—	2	8	no	—
202	C₂²⋊C₄×C₂₃	—	2	8	no	—
203	C₂²⋊C₄×C₂₃	—	2	4	no	—
204	C₂²⋊C₄×C₂₃	—	2	4	no	—
205	C₂²⋊C₄×C₂₃	—	4	4	no	—
206	C₂²⋊C₄×C₂₃	—	4	4	no	—
207	C₂²⋊C₄×C₂₃	#41	1	8	yes	—
208	C₄²×C₂₃	—	2	2	no	—
209	C₄²×C₂₃	—	2	2	no	—
210	C₄²×C₂₃	—	2	4	no	—
211	C₄²×C₂₃	—	2	4	no	—
212	C₄²×C₂₃	—	2	2	no	—
213	C₄²×C₂₃	—	2	2	no	—
214	C₄²×C₂₃	—	2	4	no	—
215	C₄²×C₂₃	—	2	4	no	—
216	C₄²×C₂₃	—	2	4	no	—
217	C₄²×C₂₃	—	2	4	no	—
218	C₄²×C₂₃	—	2	8	no	—
219	C₄²×C₂₃	—	4	4	no	—
220	#34814	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
221	C₂³×C₂₃	—	2	2	no	—
222	C₂³×C₂₃	—	2	2	no	—
223	C₂³×C₂₃	—	2	8	no	—
224	C₂³×C₂₃	—	2	2	no	—
225	C₂³×C₂₃	—	2	2	no	—
226	C₂³×C₂₃	—	2	2	no	—
227	C₂³×C₂₃	—	2	2	no	—
228	C₂³×C₂₃	—	2	2	no	—
229	C₂³×C₂₃	—	2	2	no	—
230	C₂³×C₂₃	—	4	8	no	—
231	C₂³×C₂₃	—	4	8	no	—
232	C₂³×C₂₃	—	4	4	no	—
233	C₂³×C₂₃	—	4	4	no	—
234	C₂³×C₂₃	C₂×#41	1	4	yes	—
235	C₂³×C₂₃	C₂×#41	1	2	yes	—
236	C₂³×C₂₃	C₂×#41	1	2	yes	—
237	C₂³×C₂₃	C₂×#41	1	2	yes	—
238	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
239	C₂×C₄×C₂₃	—	2	2	no	—
240	C₂×C₄×C₂₃	—	2	2	no	—
241	C₂×C₄×C₂₃	—	2	2	no	—
242	C₂×C₄×C₂₃	—	2	2	no	—
243	C₂×C₄×C₂₃	—	2	2	no	—
244	C₂×C₄×C₂₃	—	2	2	no	—
245	C₂×C₄×C₂₃	—	2	2	no	—
246	C₂×C₄×C₂₃	—	2	2	no	—
247	C₂×C₄×C₂₃	—	2	4	no	—
248	C₂×C₄×C₂₃	—	2	2	no	—
249	C₂×C₄×C₂₃	—	2	2	no	—
250	C₂×C₄×C₂₃	—	2	4	no	—
251	C₂×C₄×C₂₃	—	2	4	no	—
252	C₂×C₄×C₂₃	—	2	4	no	—
253	C₂×C₄×C₂₃	—	2	2	no	—
254	C₂×C₄×C₂₃	—	2	2	no	—
255	C₂×C₄×C₂₃	—	2	4	no	—
256	C₂×C₄×C₂₃	—	2	4	no	—
257	C₂×C₄×C₂₃	—	2	2	no	—
258	C₂×C₄×C₂₃	—	2	2	no	—
259	C₂×C₄×C₂₃	—	2	2	no	—
260	C₂×C₄×C₂₃	—	2	4	no	—
261	C₂×C₄×C₂₃	—	2	4	no	—
262	C₂×C₄×C₂₃	—	2	4	no	—
263	C₂×C₄×C₂₃	—	2	4	no	—
264	C₂×C₄×C₂₃	—	2	2	no	—
265	C₂×C₄×C₂₃	—	2	2	no	—
266	C₂×C₄×C₂₃	—	4	8	no	—
267	C₂×C₄×C₂₃	—	4	8	no	—
268	C₂×C₄×C₂₃	—	4	4	no	—
269	C₂×C₄×C₂₃	—	4	8	no	—
270	C₂×C₄×C₂₃	—	4	8	no	—
271	C₂×C₄×C₂₃	#126	1	16	yes	complemented
272	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
273	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
274	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
275	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
276	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
277	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
278	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
279	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
280	#2497	C₂×C₂₃	1	2	yes	complemented
281	#2496	C₂×C₂₃	1	2	yes	complemented
282	#2495	C₂×C₂₃	1	2	yes	complemented
283	#2494	C₂×C₂₃	1	2	yes	complemented
284	#2493	C₂×C₂₃	1	4	yes	complemented
285	#2480	C₂×C₂₃	1	4	yes	—
286	#2401	C₂×C₂₃	1	4	yes	complemented
287	C₄×#126	C₂×C₂₃	1	4	yes	complemented
288	#2296	C₂×C₂₃	1	1	yes	characteristic, complemented
289	#2296	C₂×C₂₃	1	1	yes	characteristic, complemented
290	#2285	C₂×C₂₃	1	1	yes	characteristic, complemented
291	#2280	C₂×C₂₃	1	1	yes	characteristic
292	#2280	C₂×C₂₃	1	1	yes	characteristic
293	C₄×#117	C₂×C₂₃	1	1	yes	characteristic, complemented
294	C₂×C₄×C₂²⋊C₄	C₂×C₂₃	1	1	yes	characteristic
295	C₂²×C₂₃	—	2	2	no	—
296	C₂²×C₂₃	—	2	2	no	—
297	C₂²×C₂₃	—	2	4	no	—
298	C₂²×C₂₃	—	2	4	no	—
299	C₂²×C₂₃	—	2	4	no	—
300	C₂²×C₂₃	—	2	4	no	—
301	C₂²×C₂₃	—	2	4	no	—
302	C₂²×C₂₃	—	4	16	no	—
303	C₂²×C₂₃	—	4	2	no	—
304	C₂²×C₂₃	—	4	2	no	—
305	C₂²×C₂₃	—	4	2	no	—
306	C₂²×C₂₃	—	4	2	no	—
307	C₂²×C₂₃	C₂×#126	1	4	yes	—
308	C₂²×C₂₃	C₂²×#41	1	1	yes	characteristic
309	C₂²×C₂₃	C₂²×#41	1	1	yes	characteristic
310	C₂²×C₂₃	C₂²×#41	1	1	yes	characteristic
311	C₂²×C₂₃	C₂²×#41	1	2	yes	—
312	C₂²×C₂₃	#585	1	2	yes	—
313	C₄×C₂₃	—	2	4	no	—
314	C₄×C₂₃	—	2	4	no	—
315	C₄×C₂₃	—	4	2	no	—
316	C₄×C₂₃	—	4	2	no	—
317	C₄×C₂₃	—	4	2	no	—
318	C₄×C₂₃	—	4	2	no	—
319	C₄×C₂₃	—	4	2	no	—
320	C₄×C₂₃	—	4	4	no	—
321	C₄×C₂₃	—	4	4	no	—
322	C₄×C₂₃	—	4	2	no	—
323	C₄×C₂₃	—	4	2	no	—
324	C₄×C₂₃	C₂×#126	1	4	yes	—
325	C₄×C₂₃	C₂×#126	1	4	yes	—
326	C₂⁴×C₄	C₂²×C₂₃	1	1	yes	characteristic
327	C₂×#126	C₂²×C₂₃	1	4	yes	—
328	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
329	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
330	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
331	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
332	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
333	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
334	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
335	#401	C₂²×C₂₃	1	4	yes	complemented
336	#400	C₂²×C₂₃	1	4	yes	complemented
337	#400	C₂²×C₂₃	1	4	yes	complemented
338	#400	C₂²×C₂₃	1	4	yes	complemented
339	#397	C₂²×C₂₃	1	4	yes	—
340	#397	C₂²×C₂₃	1	4	yes	—
341	#397	C₂²×C₂₃	1	4	yes	—
342	#396	C₂²×C₂₃	1	4	yes	—
343	#389	C₂²×C₂₃	1	2	yes	complemented
344	#389	C₂²×C₂₃	1	2	yes	complemented
345	#389	C₂²×C₂₃	1	2	yes	complemented
346	#389	C₂²×C₂₃	1	2	yes	complemented
347	#388	C₂²×C₂₃	1	4	yes	—
348	#388	C₂²×C₂₃	1	4	yes	—
349	#388	C₂²×C₂₃	1	4	yes	—
350	#387	C₂²×C₂₃	1	4	yes	complemented
351	#387	C₂²×C₂₃	1	4	yes	complemented
352	#387	C₂²×C₂₃	1	4	yes	complemented
353	#385	C₂²×C₂₃	1	4	yes	—
354	#385	C₂²×C₂₃	1	4	yes	—
355	#384	C₂²×C₂₃	1	1	yes	characteristic, complemented
356	#384	C₂²×C₂₃	1	1	yes	characteristic, complemented
357	#383	C₂²×C₂₃	1	4	yes	complemented
358	#382	C₂²×C₂₃	1	1	yes	characteristic, complemented
359	#382	C₂²×C₂₃	1	1	yes	characteristic, complemented
360	#381	C₂²×C₂₃	1	2	yes	—
361	#381	C₂²×C₂₃	1	2	yes	—
362	#380	C₂²×C₂₃	1	4	yes	—
363	#380	C₂²×C₂₃	1	2	yes	—
364	#380	C₂²×C₂₃	1	2	yes	—
365	#380	C₂²×C₂₃	1	2	yes	—
366	#380	C₂²×C₂₃	1	2	yes	—
367	#379	C₂²×C₂₃	1	2	yes	—
368	#379	C₂²×C₂₃	1	2	yes	—
369	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	complemented
370	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	complemented
371	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	complemented
372	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	complemented
373	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	complemented
374	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	complemented
375	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
376	C₄×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
377	C₄×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
378	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
379	C₄×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
380	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
381	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
382	#376	C₂²×C₂₃	1	2	yes	complemented
383	C₂×#95	C₂²×C₂₃	1	2	yes	—
384	C₄³	C₂²×C₂₃	1	2	yes	complemented
385	C₂×C₂₃	—	2	4	no	—
386	C₂×C₂₃	—	4	4	no	—
387	C₂×C₂₃	C₂²×#126	1	1	yes	characteristic
388	C₂×C₂₃	#3444	1	1	yes	characteristic
389	C₂×C₂₃	#3455	1	1	yes	characteristic
390	C₂×C₂₃	#3475	1	2	yes	—
391	C₂×C₂₃	#3475	1	2	yes	—
392	C₂⁵	C₂³×C₂₃	1	1	yes	characteristic
393	C₂³×C₄	—	2	4	no	—
394	C₂³×C₄	—	2	4	no	—
395	C₂³×C₄	—	2	4	no	—
396	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
397	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
398	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
399	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
400	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
401	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
402	#126	—	2	16	no	—
403	#117	—	2	4	no	—
404	#117	—	2	4	no	—
405	#117	—	2	4	no	—
406	#117	—	2	4	no	—
407	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
408	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
409	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
410	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
411	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
412	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
413	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
414	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
415	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
416	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
417	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
418	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
419	C₂×C₂²⋊C₄	—	2	8	no	—
420	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
421	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
422	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
423	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
424	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
425	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
426	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
427	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
428	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
429	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
430	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
431	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
432	C₂×C₄²	—	2	4	no	—
433	C₂×C₄²	—	2	4	no	—
434	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
435	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
436	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
437	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
438	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
439	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
440	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
441	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
442	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
443	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
444	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
445	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
446	#95	C₂³×C₂₃	1	2	yes	—
447	#95	C₂³×C₂₃	1	2	yes	—
448	#95	C₂³×C₂₃	1	2	yes	—
449	#95	C₂³×C₂₃	1	2	yes	—
450	#95	C₂³×C₂₃	1	4	yes	—
451	#95	C₂³×C₂₃	1	4	yes	—
452	#95	C₂³×C₂₃	1	4	yes	—
453	#95	C₂³×C₂₃	1	4	yes	—
454	#95	C₂³×C₂₃	1	4	yes	—
455	#95	C₂³×C₂₃	1	4	yes	—
456	#95	C₂³×C₂₃	1	4	yes	—
457	#95	C₂³×C₂₃	1	4	yes	—
458	#95	C₂³×C₂₃	1	4	yes	—
459	#95	C₂³×C₂₃	1	4	yes	—
460	#95	C₂³×C₂₃	1	4	yes	—
461	#95	C₂³×C₂₃	1	4	yes	—
462	C₂₃	#34814	1	1	yes	characteristic, direct product factor, Sylow
463	C₂⁴	—	2	4	no	—
464	C₂⁴	—	2	2	no	—
465	C₂⁴	—	2	2	no	—
466	C₂⁴	—	2	2	no	—
467	C₂⁴	—	4	2	no	—
468	C₂⁴	C₂⁴×C₂₃	1	2	yes	—
469	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
470	C₂²×C₄	—	2	4	no	—
471	C₂²×C₄	—	2	4	no	—
472	C₂²×C₄	—	2	2	no	—
473	C₂²×C₄	—	2	2	no	—
474	C₂²×C₄	—	2	2	no	—
475	C₂²×C₄	—	2	4	no	—
476	C₂²×C₄	—	2	4	no	—
477	C₂²×C₄	—	2	4	no	—
478	C₂²×C₄	—	2	4	no	—
479	C₂²×C₄	—	2	4	no	—
480	C₂²×C₄	—	2	4	no	—
481	C₂²×C₄	—	2	4	no	—
482	C₂²×C₄	—	2	4	no	—
483	C₂²×C₄	—	4	2	no	—
484	C₂²×C₄	—	4	2	no	—
485	C₂²×C₄	—	4	16	no	—
486	C₂²×C₄	#41×C₂₃	1	4	yes	—
487	C₂²×C₄	#41×C₂₃	1	4	yes	—
488	C₂²×C₄	#41×C₂₃	1	4	yes	—
489	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
490	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
491	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
492	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
493	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
494	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
495	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
496	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
497	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
498	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
499	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
500	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
501	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
502	C₄⋊C₄	—	2	4	no	—
503	C₄⋊C₄	—	2	4	no	—
504	C₄⋊C₄	—	2	2	no	—
505	C₄⋊C₄	—	2	2	no	—
506	C₄⋊C₄	—	2	2	no	—
507	C₄⋊C₄	—	2	2	no	—
508	C₄⋊C₄	—	2	8	no	—
509	C₄⋊C₄	—	2	8	no	—
510	C₄⋊C₄	—	2	8	no	—
511	C₄⋊C₄	—	4	4	no	—
512	C₄⋊C₄	—	4	4	no	—
513	C₄⋊C₄	—	4	4	no	—
514	C₂²⋊C₄	—	2	4	no	—
515	C₂²⋊C₄	—	2	4	no	—
516	C₂²⋊C₄	—	2	4	no	—
517	C₂²⋊C₄	—	2	4	no	—
518	C₂²⋊C₄	—	2	4	no	—
519	C₂²⋊C₄	—	2	8	no	—
520	C₂²⋊C₄	—	2	8	no	—
521	C₂²⋊C₄	—	2	4	no	—
522	C₂²⋊C₄	—	2	4	no	—
523	C₂²⋊C₄	—	4	4	no	—
524	C₂²⋊C₄	—	4	4	no	—
525	C₂²⋊C₄	#41×C₂₃	1	8	yes	—
526	C₄²	—	2	2	no	—
527	C₄²	—	2	2	no	—
528	C₄²	—	2	4	no	—
529	C₄²	—	2	4	no	—
530	C₄²	—	2	2	no	—
531	C₄²	—	2	2	no	—
532	C₄²	—	2	4	no	—
533	C₄²	—	2	4	no	—
534	C₄²	—	2	4	no	—
535	C₄²	—	2	4	no	—
536	C₄²	—	2	8	no	—
537	C₄²	—	4	4	no	—
538	C₂³	—	2	2	no	—
539	C₂³	—	2	2	no	—
540	C₂³	—	2	8	no	—
541	C₂³	—	2	2	no	—
542	C₂³	—	2	2	no	—
543	C₂³	—	2	2	no	—
544	C₂³	—	2	2	no	—
545	C₂³	—	2	2	no	—
546	C₂³	—	2	2	no	—
547	C₂³	—	4	8	no	—
548	C₂³	—	4	8	no	—
549	C₂³	—	4	4	no	—
550	C₂³	—	4	4	no	—
551	C₂³	C₂×#41×C₂₃	1	4	yes	—
552	C₂³	C₂×#41×C₂₃	1	2	yes	—
553	C₂³	C₂×#41×C₂₃	1	2	yes	—
554	C₂³	C₂×#41×C₂₃	1	2	yes	—
555	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
556	C₂×C₄	—	2	2	no	—
557	C₂×C₄	—	2	2	no	—
558	C₂×C₄	—	2	2	no	—
559	C₂×C₄	—	2	2	no	—
560	C₂×C₄	—	2	2	no	—
561	C₂×C₄	—	2	2	no	—
562	C₂×C₄	—	2	2	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	2	4	no	—
565	C₂×C₄	—	2	2	no	—
566	C₂×C₄	—	2	2	no	—
567	C₂×C₄	—	2	4	no	—
568	C₂×C₄	—	2	4	no	—
569	C₂×C₄	—	2	4	no	—
570	C₂×C₄	—	2	2	no	—
571	C₂×C₄	—	2	2	no	—
572	C₂×C₄	—	2	4	no	—
573	C₂×C₄	—	2	4	no	—
574	C₂×C₄	—	2	2	no	—
575	C₂×C₄	—	2	2	no	—
576	C₂×C₄	—	2	2	no	—
577	C₂×C₄	—	2	4	no	—
578	C₂×C₄	—	2	4	no	—
579	C₂×C₄	—	2	4	no	—
580	C₂×C₄	—	2	4	no	—
581	C₂×C₄	—	2	2	no	—
582	C₂×C₄	—	2	2	no	—
583	C₂×C₄	—	4	8	no	—
584	C₂×C₄	—	4	8	no	—
585	C₂×C₄	—	4	4	no	—
586	C₂×C₄	—	4	8	no	—
587	C₂×C₄	—	4	8	no	—
588	C₂×C₄	C₂₃×#126	1	16	yes	complemented
589	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
590	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
591	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
592	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
593	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
594	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
595	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
596	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
597	C₂²	—	2	2	no	—
598	C₂²	—	2	2	no	—
599	C₂²	—	2	4	no	—
600	C₂²	—	2	4	no	—
601	C₂²	—	2	4	no	—
602	C₂²	—	2	4	no	—
603	C₂²	—	2	4	no	—
604	C₂²	—	4	16	no	—
605	C₂²	—	4	2	no	—
606	C₂²	—	4	2	no	—
607	C₂²	—	4	2	no	—
608	C₂²	—	4	2	no	—
609	C₂²	C₂×C₂₃×#126	1	4	yes	—
610	C₂²	C₂²×#41×C₂₃	1	1	yes	characteristic
611	C₂²	C₂²×#41×C₂₃	1	1	yes	characteristic
612	C₂²	C₂²×#41×C₂₃	1	1	yes	characteristic
613	C₂²	C₂²×#41×C₂₃	1	2	yes	—
614	C₂²	C₂₃×#585	1	2	yes	—
615	C₄	—	2	4	no	—
616	C₄	—	2	4	no	—
617	C₄	—	4	2	no	—
618	C₄	—	4	2	no	—
619	C₄	—	4	2	no	—
620	C₄	—	4	2	no	—
621	C₄	—	4	2	no	—
622	C₄	—	4	2	no	—
623	C₄	—	4	2	no	—
624	C₄	—	4	4	no	—
625	C₄	—	4	4	no	—
626	C₄	C₂×C₂₃×#126	1	4	yes	—
627	C₄	C₂×C₂₃×#126	1	4	yes	—
628	C₂	—	2	4	no	—
629	C₂	—	4	4	no	—
630	C₂	C₂²×C₂₃×#126	1	1	yes	characteristic
631	C₂	C₂₃×#3444	1	1	yes	characteristic
632	C₂	C₂₃×#3455	1	1	yes	characteristic
633	C₂	C₂₃×#3475	1	2	yes	—
634	C₂	C₂₃×#3475	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₂₃
socle	C₂³×C₂₃	C₂⁵
prime radical	C₂⁵×C₂₃	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#2497	C₂	semidirect
2. C₂₃×#2496	C₂	semidirect
3. C₂₃×#2495	C₂	semidirect
4. C₂₃×#2494	C₂	semidirect
5. C₂₃×#2493	C₂	semidirect
7. C₂₃×#2401	C₂	semidirect
8. C₄×C₂₃×#126	C₂	semidirect
9. C₂₃×#2296	C₂	semidirect
10. C₂₃×#2296	C₂	semidirect
11. C₂₃×#2285	C₂	semidirect
14. C₄×C₂₃×#117	C₂	semidirect
25. C₂₃×#401	C₂²	semidirect
26. C₂₃×#400	C₂²	semidirect
27. C₂₃×#400	C₂²	semidirect
28. C₂₃×#400	C₂²	semidirect
33. C₂₃×#389	C₂²	semidirect
34. C₂₃×#389	C₂²	semidirect
35. C₂₃×#389	C₂²	semidirect
36. C₂₃×#389	C₂²	semidirect
40. C₂₃×#387	C₂²	semidirect
41. C₂₃×#387	C₂²	semidirect
42. C₂₃×#387	C₂²	semidirect
45. C₂₃×#384	C₂²	semidirect
46. C₂₃×#384	C₂²	semidirect
47. C₂₃×#383	C₂²	semidirect
48. C₂₃×#382	C₂²	semidirect
49. C₂₃×#382	C₂²	semidirect
59. C₄×C₄⋊C₄×C₂₃	C₂²	semidirect
60. C₄×C₄⋊C₄×C₂₃	C₂²	semidirect
61. C₄×C₄⋊C₄×C₂₃	C₂²	semidirect
62. C₄×C₄⋊C₄×C₂₃	C₂²	semidirect
63. C₄×C₄⋊C₄×C₂₃	C₂²	semidirect
64. C₄×C₄⋊C₄×C₂₃	C₂²	semidirect
72. C₂₃×#376	C₂²	semidirect
74. C₄³×C₂₃	C₂²	semidirect
220. #34814	C₂₃	direct (N × H)
271. C₂×C₄×C₂₃	#126	semidirect
280. #2497	C₂×C₂₃	semidirect
281. #2496	C₂×C₂₃	semidirect
282. #2495	C₂×C₂₃	semidirect
283. #2494	C₂×C₂₃	semidirect
284. #2493	C₂×C₂₃	semidirect
286. #2401	C₂×C₂₃	semidirect
287. C₄×#126	C₂×C₂₃	semidirect
288. #2296	C₂×C₂₃	semidirect
289. #2296	C₂×C₂₃	semidirect
290. #2285	C₂×C₂₃	semidirect
293. C₄×#117	C₂×C₂₃	semidirect
335. #401	C₂²×C₂₃	semidirect
336. #400	C₂²×C₂₃	semidirect
337. #400	C₂²×C₂₃	semidirect
338. #400	C₂²×C₂₃	semidirect
343. #389	C₂²×C₂₃	semidirect
344. #389	C₂²×C₂₃	semidirect
345. #389	C₂²×C₂₃	semidirect
346. #389	C₂²×C₂₃	semidirect
350. #387	C₂²×C₂₃	semidirect
351. #387	C₂²×C₂₃	semidirect
352. #387	C₂²×C₂₃	semidirect
355. #384	C₂²×C₂₃	semidirect
356. #384	C₂²×C₂₃	semidirect
357. #383	C₂²×C₂₃	semidirect
358. #382	C₂²×C₂₃	semidirect
359. #382	C₂²×C₂₃	semidirect
369. C₄×C₄⋊C₄	C₂²×C₂₃	semidirect
370. C₄×C₄⋊C₄	C₂²×C₂₃	semidirect
371. C₄×C₄⋊C₄	C₂²×C₂₃	semidirect
372. C₄×C₄⋊C₄	C₂²×C₂₃	semidirect
373. C₄×C₄⋊C₄	C₂²×C₂₃	semidirect
374. C₄×C₄⋊C₄	C₂²×C₂₃	semidirect
382. #376	C₂²×C₂₃	semidirect
384. C₄³	C₂²×C₂₃	semidirect
462. C₂₃	#34814	direct (N × H)
588. C₂×C₄	C₂₃×#126	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#2497	2
C₂₃×#2496	2
C₂₃×#2495	2
C₂₃×#2494	2
C₂₃×#2493	2
C₂₃×#2480	2
C₂₃×#2401	2
C₄×C₂₃×#126	2
C₂₃×#2296	2
C₂₃×#2285	2
C₂₃×#2280	2
C₄×C₂₃×#117	2
C₂×C₄×C₂²⋊C₄×C₂₃	2
#34814	23

Subgroups — C23×#34814 #19182440

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#34814 #19182440