Subgroups — C₂₃×#34065 #19181691

Order 5888.

3134 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#525	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#41×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₂₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2324	C₂	1	4	yes	maximal, complemented
5	C₂₃×#2323	C₂	1	4	yes	maximal, complemented
6	C₂₃×#2322	C₂	1	4	yes	maximal, complemented
7	C₂₃×#2321	C₂	1	4	yes	maximal, complemented
8	C₂₃×#2297	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#2296	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#2293	C₂	1	4	yes	maximal, complemented
11	C₂₃×#2290	C₂	1	4	yes	maximal, complemented
12	C₂₃×#2280	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×C₂₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂²×#41×C₂₃	C₂²	1	1	yes	characteristic
15	C₂⁴×C₄×C₂₃	C₂²	1	1	yes	characteristic
16	C₂₃×#525	C₄	1	16	yes	complemented
17	C₂×C₂₃×#123	C₂²	1	2	yes	—
18	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
19	C₂×C₂₃×#121	C₂²	1	2	yes	—
20	C₂×C₂₃×#120	C₂²	1	2	yes	—
21	C₄×#41×C₂₃	—	2	8	no	—
22	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	complemented
23	C₂×C₄×D₄×C₂₃	C₂²	1	4	yes	—
24	C₂×C₄×D₄×C₂₃	C₂²	1	4	yes	—
25	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	complemented
26	C₂×C₂₃×#117	—	2	8	no	—
27	C₂×C₂₃×#117	C₂²	1	2	yes	—
28	C₂×C₂₃×#117	C₂²	1	2	yes	—
29	C₂×C₂₃×#117	C₂²	1	4	yes	—
30	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
31	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
32	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
33	C₂×C₂₃×#117	C₂²	1	2	yes	complemented
34	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
35	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
36	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
37	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
38	C₂²×C₄²×C₂₃	C₂²	1	2	yes	complemented
39	C₂₃×#391	C₂²	1	2	yes	complemented
40	C₂₃×#390	C₂²	1	8	yes	—
41	C₂₃×#390	C₂²	1	2	yes	complemented
42	C₂₃×#389	C₂²	1	8	yes	complemented
43	C₂₃×#389	C₂²	1	4	yes	complemented
44	C₂₃×#388	C₂²	1	8	yes	—
45	C₂₃×#388	C₂²	1	8	yes	—
46	C₂₃×#388	C₂²	1	8	yes	complemented
47	C₂₃×#387	C₂²	1	8	yes	complemented
48	C₂₃×#386	C₂²	1	8	yes	—
49	C₂₃×#386	C₂²	1	4	yes	—
50	C₂₃×#385	C₂²	1	4	yes	—
51	C₂₃×#385	C₂²	1	8	yes	complemented
52	C₂₃×#381	C₂²	1	2	yes	—
53	C₂₃×#380	C₂²	1	4	yes	—
54	C₂₃×#379	C₂²	1	2	yes	—
55	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	complemented
56	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	complemented
57	C₄×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
58	C₄×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
59	C₄×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	complemented
60	C₂₃×#376	C₂²	1	2	yes	complemented
61	C₄³×C₂₃	C₂²	1	2	yes	complemented
62	C₂⁵×C₂₃	C₂³	1	1	yes	characteristic
63	C₂×#41×C₂₃	—	2	4	no	—
64	C₂×#41×C₂₃	—	2	4	no	—
65	C₂×#41×C₂₃	C₂×C₄	1	8	yes	—
66	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
67	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
68	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
69	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
70	C₂³×C₄×C₂₃	—	2	4	no	—
71	C₂³×C₄×C₂₃	—	2	4	no	—
72	C₂³×C₄×C₂₃	C₂×C₄	1	8	yes	complemented
73	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
74	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
75	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
76	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
77	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
78	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
79	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
80	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
81	C₂₃×#123	C₂×C₄	1	16	yes	complemented
82	C₂₃×C₂²⋊Q₈	C₂×C₄	1	16	yes	complemented
83	C₂₃×#121	C₂×C₄	1	16	yes	complemented
84	C₂₃×#120	C₂×C₄	1	16	yes	complemented
85	C₄×Q₈×C₂₃	—	2	8	no	—
86	C₄×D₄×C₂₃	—	2	16	no	—
87	C₄×D₄×C₂₃	—	2	8	no	—
88	C₄×D₄×C₂₃	C₂×C₄	1	32	yes	complemented
89	C₂₃×#117	—	2	4	no	—
90	C₂₃×#117	—	2	4	no	—
91	C₂₃×#117	—	2	8	no	—
92	C₂₃×#117	—	2	8	no	—
93	C₂₃×#117	—	2	16	no	—
94	C₂₃×#117	—	2	8	no	—
95	C₂₃×#117	C₂×C₄	1	8	yes	complemented
96	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
97	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
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×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
103	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
104	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
105	C₂×C₂²⋊C₄×C₂₃	—	2	8	no	—
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	4	yes	—
111	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
112	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
113	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
114	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
115	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
116	C₂×C₄²×C₂₃	—	2	4	no	—
117	C₂×C₄²×C₂₃	—	2	8	no	—
118	C₂×C₄²×C₂₃	D₄	1	8	yes	complemented
119	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
120	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
121	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
122	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
123	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
124	C₂×C₄²×C₂₃	C₂³	1	8	yes	—
125	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
126	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
127	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
128	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
129	C₂₃×#95	C₂³	1	8	yes	—
130	C₂₃×#95	C₂³	1	8	yes	—
131	C₂₃×#95	C₂³	1	8	yes	—
132	C₂₃×#95	C₂³	1	8	yes	—
133	C₂⁴×C₂₃	—	2	2	no	—
134	C₂⁴×C₂₃	—	2	2	no	—
135	C₂⁴×C₂₃	—	4	4	no	—
136	C₂⁴×C₂₃	C₂²×C₄	1	4	yes	—
137	C₂⁴×C₂₃	C₂⁴	1	2	yes	—
138	C₂⁴×C₂₃	C₂⁴	1	2	yes	—
139	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
140	#41×C₂₃	—	2	32	no	—
141	C₂×Q₈×C₂₃	—	2	2	no	—
142	C₂×Q₈×C₂₃	—	2	2	no	—
143	C₂×Q₈×C₂₃	C₂²×C₄	1	4	yes	—
144	C₂×D₄×C₂₃	—	2	4	no	—
145	C₂×D₄×C₂₃	—	2	4	no	—
146	C₂×D₄×C₂₃	—	2	2	no	—
147	C₂×D₄×C₂₃	—	2	2	no	—
148	C₂×D₄×C₂₃	—	4	8	no	—
149	C₂×D₄×C₂₃	C₂²×C₄	1	8	yes	—
150	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
151	C₂×D₄×C₂₃	C₂²×C₄	1	16	yes	—
152	C₂²×C₄×C₂₃	—	2	2	no	—
153	C₂²×C₄×C₂₃	—	2	2	no	—
154	C₂²×C₄×C₂₃	—	2	32	no	—
155	C₂²×C₄×C₂₃	—	2	4	no	—
156	C₂²×C₄×C₂₃	—	2	4	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	8	no	—
163	C₂²×C₄×C₂₃	—	4	4	no	—
164	C₂²×C₄×C₂₃	—	4	8	no	—
165	C₂²×C₄×C₂₃	—	4	4	no	—
166	C₂²×C₄×C₂₃	C₂²×C₄	1	8	yes	—
167	C₂²×C₄×C₂₃	C₂²×C₄	1	8	yes	—
168	C₂²×C₄×C₂₃	C₂²×C₄	1	8	yes	—
169	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
170	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
171	C₂²×C₄×C₂₃	C₂×D₄	1	8	yes	—
172	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
173	C₂²×C₄×C₂₃	#41	1	4	yes	—
174	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
175	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
176	C₂²×C₄×C₂₃	C₂⁴	1	4	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	8	no	—
185	C₄⋊C₄×C₂₃	—	2	8	no	—
186	C₄⋊C₄×C₂₃	—	2	8	no	—
187	C₄⋊C₄×C₂₃	—	2	4	no	—
188	C₄⋊C₄×C₂₃	—	2	4	no	—
189	C₄⋊C₄×C₂₃	—	4	8	no	—
190	C₄⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
191	C₄⋊C₄×C₂₃	C₂²×C₄	1	16	yes	—
192	C₂²⋊C₄×C₂₃	—	2	4	no	—
193	C₂²⋊C₄×C₂₃	—	2	4	no	—
194	C₂²⋊C₄×C₂₃	—	2	8	no	—
195	C₂²⋊C₄×C₂₃	—	2	8	no	—
196	C₂²⋊C₄×C₂₃	—	2	8	no	—
197	C₂²⋊C₄×C₂₃	—	4	8	no	—
198	C₂²⋊C₄×C₂₃	—	4	8	no	—
199	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
200	C₂²⋊C₄×C₂₃	C₂²×C₄	1	16	yes	—
201	C₂²⋊C₄×C₂₃	C₂²×C₄	1	16	yes	—
202	C₄²×C₂₃	—	2	4	no	—
203	C₄²×C₂₃	—	2	4	no	—
204	C₄²×C₂₃	—	2	4	no	—
205	C₄²×C₂₃	—	2	8	no	—
206	C₄²×C₂₃	—	2	4	no	—
207	C₄²×C₂₃	—	4	8	no	—
208	C₄²×C₂₃	C₂²×C₄	1	8	yes	—
209	C₄²×C₂₃	C₂×D₄	1	4	yes	—
210	C₄²×C₂₃	C₂×D₄	1	4	yes	—
211	C₄²×C₂₃	C₂×D₄	1	8	yes	—
212	#34065	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
213	C₂³×C₂₃	—	2	4	no	—
214	C₂³×C₂₃	—	2	8	no	—
215	C₂³×C₂₃	—	2	2	no	—
216	C₂³×C₂₃	—	2	2	no	—
217	C₂³×C₂₃	—	2	2	no	—
218	C₂³×C₂₃	—	2	2	no	—
219	C₂³×C₂₃	—	2	4	no	—
220	C₂³×C₂₃	—	2	8	no	—
221	C₂³×C₂₃	—	2	4	no	—
222	C₂³×C₂₃	—	4	8	no	—
223	C₂³×C₂₃	—	4	8	no	—
224	C₂³×C₂₃	—	4	8	no	—
225	C₂³×C₂₃	C₂³×C₄	1	4	yes	—
226	C₂³×C₂₃	C₂³×C₄	1	4	yes	—
227	C₂³×C₂₃	C₂³×C₄	1	2	yes	—
228	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
229	C₂³×C₂₃	C₂×#41	1	2	yes	—
230	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
231	Q₈×C₂₃	—	2	8	no	—
232	D₄×C₂₃	—	2	16	no	—
233	D₄×C₂₃	—	2	8	no	—
234	D₄×C₂₃	—	4	16	no	—
235	C₂×C₄×C₂₃	—	2	8	no	—
236	C₂×C₄×C₂₃	—	2	2	no	—
237	C₂×C₄×C₂₃	—	2	2	no	—
238	C₂×C₄×C₂₃	—	2	2	no	—
239	C₂×C₄×C₂₃	—	2	2	no	—
240	C₂×C₄×C₂₃	—	2	4	no	—
241	C₂×C₄×C₂₃	—	2	8	no	—
242	C₂×C₄×C₂₃	—	2	4	no	—
243	C₂×C₄×C₂₃	—	2	16	no	—
244	C₂×C₄×C₂₃	—	2	16	no	—
245	C₂×C₄×C₂₃	—	2	16	no	—
246	C₂×C₄×C₂₃	—	2	4	no	—
247	C₂×C₄×C₂₃	—	2	4	no	—
248	C₂×C₄×C₂₃	—	2	4	no	—
249	C₂×C₄×C₂₃	—	2	8	no	—
250	C₂×C₄×C₂₃	—	2	4	no	—
251	C₂×C₄×C₂₃	—	2	8	no	—
252	C₂×C₄×C₂₃	—	2	4	no	—
253	C₂×C₄×C₂₃	—	4	8	no	—
254	C₂×C₄×C₂₃	—	4	16	no	—
255	C₂×C₄×C₂₃	—	4	8	no	—
256	C₂×C₄×C₂₃	C₄×D₄	1	16	yes	—
257	C₂×C₄×C₂₃	C₂³×C₄	1	8	yes	—
258	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
259	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
260	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
261	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
262	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
263	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
264	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
265	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
266	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
267	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
268	C₂×#525	C₂×C₂₃	1	1	yes	characteristic
269	C₂×C₄×#41	C₂×C₂₃	1	1	yes	characteristic, complemented
270	C₂²×#117	C₂×C₂₃	1	1	yes	characteristic, complemented
271	#2324	C₂×C₂₃	1	4	yes	complemented
272	#2323	C₂×C₂₃	1	4	yes	complemented
273	#2322	C₂×C₂₃	1	4	yes	complemented
274	#2321	C₂×C₂₃	1	4	yes	complemented
275	#2297	C₂×C₂₃	1	1	yes	characteristic, complemented
276	#2296	C₂×C₂₃	1	1	yes	characteristic, complemented
277	#2293	C₂×C₂₃	1	4	yes	complemented
278	#2290	C₂×C₂₃	1	4	yes	complemented
279	#2280	C₂×C₂₃	1	1	yes	characteristic, complemented
280	C₄×#117	C₂×C₂₃	1	1	yes	characteristic, complemented
281	C₂²×C₂₃	—	2	4	no	—
282	C₂²×C₂₃	—	2	4	no	—
283	C₂²×C₂₃	—	2	4	no	—
284	C₂²×C₂₃	—	2	4	no	—
285	C₂²×C₂₃	—	2	4	no	—
286	C₂²×C₂₃	—	2	4	no	—
287	C₂²×C₂₃	—	2	8	no	—
288	C₂²×C₂₃	—	2	4	no	—
289	C₂²×C₂₃	—	4	16	no	—
290	C₂²×C₂₃	—	4	4	no	—
291	C₂²×C₂₃	—	4	4	no	—
292	C₂²×C₂₃	—	4	4	no	—
293	C₂²×C₂₃	—	4	4	no	—
294	C₂²×C₂₃	C₂×C₄×D₄	1	4	yes	—
295	C₂²×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
296	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
297	C₂²×C₂₃	C₂²×#41	1	1	yes	characteristic
298	C₂²×C₂₃	#585	1	4	yes	—
299	C₄×C₂₃	—	2	4	no	—
300	C₄×C₂₃	—	2	4	no	—
301	C₄×C₂₃	—	2	8	no	—
302	C₄×C₂₃	—	2	4	no	—
303	C₄×C₂₃	—	2	4	no	—
304	C₄×C₂₃	—	4	4	no	—
305	C₄×C₂₃	—	4	4	no	—
306	C₄×C₂₃	—	4	4	no	—
307	C₄×C₂₃	C₂×C₄×D₄	1	4	yes	—
308	C₄×C₂₃	C₂×C₄×D₄	1	4	yes	—
309	C₂²×#41	C₂²×C₂₃	1	1	yes	characteristic
310	C₂⁴×C₄	C₂²×C₂₃	1	1	yes	characteristic
311	#525	C₄×C₂₃	1	16	yes	complemented
312	C₂×#123	C₂²×C₂₃	1	2	yes	—
313	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
314	C₂×#121	C₂²×C₂₃	1	2	yes	—
315	C₂×#120	C₂²×C₂₃	1	2	yes	—
316	C₄×#41	—	2	8	no	—
317	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	complemented
318	C₂×C₄×D₄	C₂²×C₂₃	1	4	yes	—
319	C₂×C₄×D₄	C₂²×C₂₃	1	4	yes	—
320	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	complemented
321	C₂×#117	—	2	8	no	—
322	C₂×#117	C₂²×C₂₃	1	4	yes	—
323	C₂×#117	C₂²×C₂₃	1	2	yes	—
324	C₂×#117	C₂²×C₂₃	1	2	yes	—
325	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
326	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
327	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
328	C₂×#117	C₂²×C₂₃	1	2	yes	complemented
329	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
330	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
331	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
332	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
333	C₂²×C₄²	C₂²×C₂₃	1	2	yes	complemented
334	#391	C₂²×C₂₃	1	2	yes	complemented
335	#390	C₂²×C₂₃	1	8	yes	—
336	#390	C₂²×C₂₃	1	2	yes	complemented
337	#389	C₂²×C₂₃	1	8	yes	complemented
338	#389	C₂²×C₂₃	1	4	yes	complemented
339	#388	C₂²×C₂₃	1	8	yes	—
340	#388	C₂²×C₂₃	1	8	yes	—
341	#388	C₂²×C₂₃	1	8	yes	complemented
342	#387	C₂²×C₂₃	1	8	yes	complemented
343	#386	C₂²×C₂₃	1	4	yes	—
344	#386	C₂²×C₂₃	1	8	yes	—
345	#385	C₂²×C₂₃	1	4	yes	—
346	#385	C₂²×C₂₃	1	8	yes	complemented
347	#381	C₂²×C₂₃	1	2	yes	—
348	#380	C₂²×C₂₃	1	4	yes	—
349	#379	C₂²×C₂₃	1	2	yes	—
350	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	complemented
351	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	complemented
352	C₄×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
353	C₄×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
354	C₄×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	complemented
355	#376	C₂²×C₂₃	1	2	yes	complemented
356	C₄³	C₂²×C₂₃	1	2	yes	complemented
357	C₂×C₂₃	—	2	4	no	—
358	C₂×C₂₃	—	4	4	no	—
359	C₂×C₂₃	—	4	4	no	—
360	C₂×C₂₃	C₂²×C₄×D₄	1	1	yes	characteristic
361	C₂×C₂₃	#3428	1	2	yes	—
362	C₂×C₂₃	#3449	1	2	yes	—
363	C₂×C₂₃	#3455	1	2	yes	—
364	C₂⁵	C₂³×C₂₃	1	1	yes	characteristic
365	C₂×#41	—	2	4	no	—
366	C₂×#41	—	2	4	no	—
367	C₂×#41	C₂×C₄×C₂₃	1	8	yes	—
368	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
369	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
370	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
371	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
372	C₂³×C₄	—	2	4	no	—
373	C₂³×C₄	—	2	4	no	—
374	C₂³×C₄	C₂×C₄×C₂₃	1	8	yes	complemented
375	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
376	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
377	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
378	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
379	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
380	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
381	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
382	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
383	#123	C₂×C₄×C₂₃	1	16	yes	complemented
384	C₂²⋊Q₈	C₂×C₄×C₂₃	1	16	yes	complemented
385	#121	C₂×C₄×C₂₃	1	16	yes	complemented
386	#120	C₂×C₄×C₂₃	1	16	yes	complemented
387	C₄×Q₈	—	2	8	no	—
388	C₄×D₄	—	2	16	no	—
389	C₄×D₄	—	2	8	no	—
390	C₄×D₄	C₂×C₄×C₂₃	1	32	yes	complemented
391	#117	—	2	4	no	—
392	#117	—	2	4	no	—
393	#117	—	2	8	no	—
394	#117	—	2	8	no	—
395	#117	—	2	16	no	—
396	#117	—	2	8	no	—
397	#117	C₂×C₄×C₂₃	1	8	yes	complemented
398	C₂×C₄⋊C₄	—	2	8	no	—
399	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
400	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
401	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
402	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
403	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
404	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
405	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
406	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
407	C₂×C₂²⋊C₄	—	2	8	no	—
408	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
409	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
410	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
411	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
412	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
413	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
414	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
415	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
416	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
417	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
418	C₂×C₄²	—	2	4	no	—
419	C₂×C₄²	—	2	8	no	—
420	C₂×C₄²	D₄×C₂₃	1	8	yes	complemented
421	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
422	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
423	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
424	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
425	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
426	C₂×C₄²	C₂³×C₂₃	1	8	yes	—
427	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
428	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
429	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
430	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
431	#95	C₂³×C₂₃	1	8	yes	—
432	#95	C₂³×C₂₃	1	8	yes	—
433	#95	C₂³×C₂₃	1	8	yes	—
434	#95	C₂³×C₂₃	1	8	yes	—
435	C₂₃	#34065	1	1	yes	characteristic, direct product factor, Sylow
436	C₂⁴	—	2	2	no	—
437	C₂⁴	—	2	2	no	—
438	C₂⁴	—	4	4	no	—
439	C₂⁴	C₂²×C₄×C₂₃	1	4	yes	—
440	C₂⁴	C₂⁴×C₂₃	1	2	yes	—
441	C₂⁴	C₂⁴×C₂₃	1	2	yes	—
442	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
443	#41	—	2	32	no	—
444	C₂×Q₈	—	2	2	no	—
445	C₂×Q₈	—	2	2	no	—
446	C₂×Q₈	C₂²×C₄×C₂₃	1	4	yes	—
447	C₂×D₄	—	2	4	no	—
448	C₂×D₄	—	2	4	no	—
449	C₂×D₄	—	2	2	no	—
450	C₂×D₄	—	2	2	no	—
451	C₂×D₄	—	4	8	no	—
452	C₂×D₄	C₂²×C₄×C₂₃	1	8	yes	—
453	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
454	C₂×D₄	C₂²×C₄×C₂₃	1	16	yes	—
455	C₂²×C₄	—	2	2	no	—
456	C₂²×C₄	—	2	2	no	—
457	C₂²×C₄	—	2	32	no	—
458	C₂²×C₄	—	2	4	no	—
459	C₂²×C₄	—	2	4	no	—
460	C₂²×C₄	—	2	4	no	—
461	C₂²×C₄	—	2	4	no	—
462	C₂²×C₄	—	2	4	no	—
463	C₂²×C₄	—	2	4	no	—
464	C₂²×C₄	—	2	4	no	—
465	C₂²×C₄	—	2	8	no	—
466	C₂²×C₄	—	4	4	no	—
467	C₂²×C₄	—	4	8	no	—
468	C₂²×C₄	—	4	4	no	—
469	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
470	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
471	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
472	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
473	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
474	C₂²×C₄	C₂×D₄×C₂₃	1	8	yes	—
475	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
476	C₂²×C₄	#41×C₂₃	1	4	yes	—
477	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
478	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
479	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
480	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
481	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
482	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
483	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
484	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
485	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
486	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
487	C₄⋊C₄	—	2	8	no	—
488	C₄⋊C₄	—	2	8	no	—
489	C₄⋊C₄	—	2	8	no	—
490	C₄⋊C₄	—	2	4	no	—
491	C₄⋊C₄	—	2	4	no	—
492	C₄⋊C₄	—	4	8	no	—
493	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
494	C₄⋊C₄	C₂²×C₄×C₂₃	1	16	yes	—
495	C₂²⋊C₄	—	2	4	no	—
496	C₂²⋊C₄	—	2	4	no	—
497	C₂²⋊C₄	—	2	8	no	—
498	C₂²⋊C₄	—	2	8	no	—
499	C₂²⋊C₄	—	2	8	no	—
500	C₂²⋊C₄	—	4	8	no	—
501	C₂²⋊C₄	—	4	8	no	—
502	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
503	C₂²⋊C₄	C₂²×C₄×C₂₃	1	16	yes	—
504	C₂²⋊C₄	C₂²×C₄×C₂₃	1	16	yes	—
505	C₄²	—	2	4	no	—
506	C₄²	—	2	4	no	—
507	C₄²	—	2	4	no	—
508	C₄²	—	2	8	no	—
509	C₄²	—	2	4	no	—
510	C₄²	—	4	8	no	—
511	C₄²	C₂²×C₄×C₂₃	1	8	yes	—
512	C₄²	C₂×D₄×C₂₃	1	4	yes	—
513	C₄²	C₂×D₄×C₂₃	1	4	yes	—
514	C₄²	C₂×D₄×C₂₃	1	8	yes	—
515	C₂³	—	2	4	no	—
516	C₂³	—	2	2	no	—
517	C₂³	—	2	2	no	—
518	C₂³	—	2	2	no	—
519	C₂³	—	2	2	no	—
520	C₂³	—	2	4	no	—
521	C₂³	—	2	8	no	—
522	C₂³	—	2	4	no	—
523	C₂³	—	2	8	no	—
524	C₂³	—	4	8	no	—
525	C₂³	—	4	8	no	—
526	C₂³	—	4	8	no	—
527	C₂³	C₂³×C₄×C₂₃	1	4	yes	—
528	C₂³	C₂³×C₄×C₂₃	1	4	yes	—
529	C₂³	C₂³×C₄×C₂₃	1	2	yes	—
530	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
531	C₂³	C₂×#41×C₂₃	1	2	yes	—
532	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
533	Q₈	—	2	8	no	—
534	D₄	—	2	16	no	—
535	D₄	—	2	8	no	—
536	D₄	—	4	16	no	—
537	C₂×C₄	—	2	2	no	—
538	C₂×C₄	—	2	2	no	—
539	C₂×C₄	—	2	2	no	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	2	4	no	—
542	C₂×C₄	—	2	8	no	—
543	C₂×C₄	—	2	4	no	—
544	C₂×C₄	—	2	8	no	—
545	C₂×C₄	—	2	16	no	—
546	C₂×C₄	—	2	16	no	—
547	C₂×C₄	—	2	16	no	—
548	C₂×C₄	—	2	4	no	—
549	C₂×C₄	—	2	4	no	—
550	C₂×C₄	—	2	4	no	—
551	C₂×C₄	—	2	8	no	—
552	C₂×C₄	—	2	4	no	—
553	C₂×C₄	—	2	8	no	—
554	C₂×C₄	—	2	4	no	—
555	C₂×C₄	—	4	8	no	—
556	C₂×C₄	—	4	16	no	—
557	C₂×C₄	—	4	8	no	—
558	C₂×C₄	C₄×D₄×C₂₃	1	16	yes	—
559	C₂×C₄	C₂³×C₄×C₂₃	1	8	yes	—
560	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
561	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
562	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
563	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
564	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
565	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
566	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
567	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
568	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
569	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
570	C₂²	—	2	4	no	—
571	C₂²	—	2	4	no	—
572	C₂²	—	2	4	no	—
573	C₂²	—	2	4	no	—
574	C₂²	—	2	4	no	—
575	C₂²	—	2	4	no	—
576	C₂²	—	2	8	no	—
577	C₂²	—	2	4	no	—
578	C₂²	—	4	16	no	—
579	C₂²	—	4	4	no	—
580	C₂²	—	4	4	no	—
581	C₂²	—	4	4	no	—
582	C₂²	—	4	4	no	—
583	C₂²	C₂×C₄×D₄×C₂₃	1	4	yes	—
584	C₂²	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
585	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
586	C₂²	C₂²×#41×C₂₃	1	1	yes	characteristic
587	C₂²	C₂₃×#585	1	4	yes	—
588	C₄	—	2	4	no	—
589	C₄	—	2	4	no	—
590	C₄	—	2	8	no	—
591	C₄	—	2	4	no	—
592	C₄	—	2	4	no	—
593	C₄	—	4	4	no	—
594	C₄	—	4	4	no	—
595	C₄	—	4	4	no	—
596	C₄	C₂×C₄×D₄×C₂₃	1	4	yes	—
597	C₄	C₂×C₄×D₄×C₂₃	1	4	yes	—
598	C₂	—	2	4	no	—
599	C₂	—	4	4	no	—
600	C₂	—	4	4	no	—
601	C₂	C₂²×C₄×D₄×C₂₃	1	1	yes	characteristic
602	C₂	C₂₃×#3428	1	2	yes	—
603	C₂	C₂₃×#3449	1	2	yes	—
604	C₂	C₂₃×#3455	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₂₃
Frattini, cosocle	C₂³	C₂⁵×C₂₃
socle	C₂³×C₂₃	C₂⁵
prime radical	C₂×C₂₃×#525	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×#41×C₂₃	C₂	semidirect
3. C₂²×C₂₃×#117	C₂	semidirect
4. C₂₃×#2324	C₂	semidirect
5. C₂₃×#2323	C₂	semidirect
6. C₂₃×#2322	C₂	semidirect
7. C₂₃×#2321	C₂	semidirect
8. C₂₃×#2297	C₂	semidirect
9. C₂₃×#2296	C₂	semidirect
10. C₂₃×#2293	C₂	semidirect
11. C₂₃×#2290	C₂	semidirect
12. C₂₃×#2280	C₂	semidirect
13. C₄×C₂₃×#117	C₂	semidirect
16. C₂₃×#525	C₄	semidirect
22. C₂×C₄×Q₈×C₂₃	C₂²	semidirect
25. C₂×C₄×D₄×C₂₃	C₂²	semidirect
33. C₂×C₂₃×#117	C₂²	semidirect
38. C₂²×C₄²×C₂₃	C₂²	semidirect
39. C₂₃×#391	C₂²	semidirect
41. C₂₃×#390	C₂²	semidirect
42. C₂₃×#389	C₂²	semidirect
43. C₂₃×#389	C₂²	semidirect
46. C₂₃×#388	C₂²	semidirect
47. C₂₃×#387	C₂²	semidirect
51. C₂₃×#385	C₂²	semidirect
55. C₄×C₄⋊C₄×C₂₃	C₂²	semidirect
56. C₄×C₄⋊C₄×C₂₃	C₂²	semidirect
59. C₄×C₂²⋊C₄×C₂₃	C₂²	semidirect
60. C₂₃×#376	C₂²	semidirect
61. C₄³×C₂₃	C₂²	semidirect
72. C₂³×C₄×C₂₃	C₂×C₄	semidirect
81. C₂₃×#123	C₂×C₄	semidirect
82. C₂₃×C₂²⋊Q₈	C₂×C₄	semidirect
83. C₂₃×#121	C₂×C₄	semidirect
84. C₂₃×#120	C₂×C₄	semidirect
88. C₄×D₄×C₂₃	C₂×C₄	semidirect
95. C₂₃×#117	C₂×C₄	semidirect
118. C₂×C₄²×C₂₃	D₄	semidirect
212. #34065	C₂₃	direct (N × H)
269. C₂×C₄×#41	C₂×C₂₃	semidirect
270. C₂²×#117	C₂×C₂₃	semidirect
271. #2324	C₂×C₂₃	semidirect
272. #2323	C₂×C₂₃	semidirect
273. #2322	C₂×C₂₃	semidirect
274. #2321	C₂×C₂₃	semidirect
275. #2297	C₂×C₂₃	semidirect
276. #2296	C₂×C₂₃	semidirect
277. #2293	C₂×C₂₃	semidirect
278. #2290	C₂×C₂₃	semidirect
279. #2280	C₂×C₂₃	semidirect
280. C₄×#117	C₂×C₂₃	semidirect
311. #525	C₄×C₂₃	semidirect
317. C₂×C₄×Q₈	C₂²×C₂₃	semidirect
320. C₂×C₄×D₄	C₂²×C₂₃	semidirect
328. C₂×#117	C₂²×C₂₃	semidirect
333. C₂²×C₄²	C₂²×C₂₃	semidirect
334. #391	C₂²×C₂₃	semidirect
336. #390	C₂²×C₂₃	semidirect
337. #389	C₂²×C₂₃	semidirect
338. #389	C₂²×C₂₃	semidirect
341. #388	C₂²×C₂₃	semidirect
342. #387	C₂²×C₂₃	semidirect
346. #385	C₂²×C₂₃	semidirect
350. C₄×C₄⋊C₄	C₂²×C₂₃	semidirect
351. C₄×C₄⋊C₄	C₂²×C₂₃	semidirect
354. C₄×C₂²⋊C₄	C₂²×C₂₃	semidirect
355. #376	C₂²×C₂₃	semidirect
356. C₄³	C₂²×C₂₃	semidirect
374. C₂³×C₄	C₂×C₄×C₂₃	semidirect
383. #123	C₂×C₄×C₂₃	semidirect
384. C₂²⋊Q₈	C₂×C₄×C₂₃	semidirect
385. #121	C₂×C₄×C₂₃	semidirect
386. #120	C₂×C₄×C₂₃	semidirect
390. C₄×D₄	C₂×C₄×C₂₃	semidirect
397. #117	C₂×C₄×C₂₃	semidirect
420. C₂×C₄²	D₄×C₂₃	semidirect
435. C₂₃	#34065	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×#525	2
C₂×C₄×#41×C₂₃	2
C₂²×C₂₃×#117	2
C₂₃×#2324	2
C₂₃×#2323	2
C₂₃×#2322	2
C₂₃×#2321	2
C₂₃×#2297	2
C₂₃×#2296	2
C₂₃×#2293	2
C₂₃×#2290	2
C₂₃×#2280	2
C₄×C₂₃×#117	2
#34065	23

Subgroups — C23×#34065 #19181691

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#34065 #19181691