Subgroups — C₂₃×#34201 #19181827

Order 5888.

3118 conjugacy classes of subgroups.

The subgroup lattice has 630 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×C₂₃×#520	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₂₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2363	C₂	1	4	yes	maximal, complemented
5	C₂₃×#2362	C₂	1	4	yes	maximal, complemented
6	C₂₃×#2354	C₂	1	4	yes	maximal, complemented
7	C₂₃×#2352	C₂	1	4	yes	maximal, complemented
8	C₂₃×#2321	C₂	1	4	yes	maximal, complemented
9	C₂₃×#2317	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#2311	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₂₃×#123	C₂	1	4	yes	maximal, complemented
12	C₂₃×#2280	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#2274	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	C₂₃×#520	—	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	4	yes	—
28	C₂×C₂₃×#117	C₂²	1	4	yes	—
29	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
30	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
31	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
32	C₂×C₂₃×#117	C₂²	1	2	yes	complemented
33	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
34	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
35	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	complemented
36	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
37	C₂²×C₄²×C₂₃	C₂²	1	2	yes	—
38	C₂₃×#391	C₂²	1	2	yes	complemented
39	C₂₃×#390	C₂²	1	8	yes	—
40	C₂₃×#390	C₂²	1	2	yes	complemented
41	C₂₃×#389	C₂²	1	4	yes	complemented
42	C₂₃×#388	C₂²	1	8	yes	—
43	C₂₃×#388	C₂²	1	8	yes	—
44	C₂₃×#388	C₂²	1	8	yes	complemented
45	C₂₃×#387	C₂²	1	8	yes	complemented
46	C₂₃×#386	C₂²	1	4	yes	—
47	C₂₃×#386	C₂²	1	8	yes	—
48	C₂₃×#385	C₂²	1	4	yes	—
49	C₂₃×#384	C₂²	1	2	yes	complemented
50	C₂₃×#382	C₂²	1	2	yes	complemented
51	C₂₃×#381	C₂²	1	2	yes	—
52	C₂₃×#380	C₂²	1	2	yes	—
53	C₂₃×#380	C₂²	1	2	yes	—
54	C₂₃×#380	C₂²	1	4	yes	complemented
55	C₂₃×#379	C₂²	1	2	yes	—
56	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	complemented
57	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	complemented
58	C₄×C₄⋊C₄×C₂₃	C₂²	1	8	yes	complemented
59	C₄×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
60	C₄×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
61	C₄×C₂²⋊C₄×C₂₃	C₂²	1	8	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	2	yes	—
75	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
76	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
77	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
78	C₂³×C₄×C₂₃	C₂³	1	2	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	16	no	—
92	C₂₃×#117	—	2	16	no	—
93	C₂₃×#117	—	2	8	no	—
94	C₂₃×#117	C₂×C₄	1	8	yes	complemented
95	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
96	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
97	C₂×C₄⋊C₄×C₂₃	D₄	1	8	yes	complemented
98	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
99	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
100	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
101	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
102	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
103	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
104	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
105	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
106	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
107	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
108	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
109	C₂×C₂²⋊C₄×C₂₃	—	2	8	no	—
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	8	yes	—
113	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
114	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
115	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
116	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
117	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
118	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
119	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
120	C₂×C₄²×C₂₃	—	2	8	no	—
121	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
122	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
123	C₂×C₄²×C₂₃	C₂³	1	8	yes	—
124	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
125	C₂×C₄²×C₂₃	C₂³	1	8	yes	—
126	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
127	C₂₃×#95	C₂³	1	4	yes	—
128	C₂₃×#95	C₂³	1	4	yes	—
129	C₂₃×#95	C₂³	1	8	yes	—
130	C₂₃×#95	C₂³	1	8	yes	—
131	C₂₃×#95	C₂³	1	8	yes	—
132	C₂⁴×C₂₃	—	2	2	no	—
133	C₂⁴×C₂₃	—	2	2	no	—
134	C₂⁴×C₂₃	—	4	2	no	—
135	C₂⁴×C₂₃	—	4	2	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	4	no	—
149	C₂×D₄×C₂₃	—	4	4	no	—
150	C₂×D₄×C₂₃	C₂²×C₄	1	8	yes	—
151	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
152	C₂×D₄×C₂₃	C₂²×C₄	1	16	yes	—
153	C₂²×C₄×C₂₃	—	2	2	no	—
154	C₂²×C₄×C₂₃	—	2	2	no	—
155	C₂²×C₄×C₂₃	—	2	32	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	4	no	—
163	C₂²×C₄×C₂₃	—	2	8	no	—
164	C₂²×C₄×C₂₃	—	4	4	no	—
165	C₂²×C₄×C₂₃	—	4	4	no	—
166	C₂²×C₄×C₂₃	—	4	8	no	—
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	8	yes	—
170	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
171	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
172	C₂²×C₄×C₂₃	C₂×D₄	1	8	yes	—
173	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
174	C₂²×C₄×C₂₃	#41	1	4	yes	—
175	C₂²×C₄×C₂₃	C₂⁴	1	2	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	4	yes	—
181	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
182	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
183	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
184	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
185	C₄⋊C₄×C₂₃	—	2	2	no	—
186	C₄⋊C₄×C₂₃	—	2	4	no	—
187	C₄⋊C₄×C₂₃	—	2	4	no	—
188	C₄⋊C₄×C₂₃	—	2	8	no	—
189	C₄⋊C₄×C₂₃	—	2	2	no	—
190	C₄⋊C₄×C₂₃	—	2	4	no	—
191	C₄⋊C₄×C₂₃	—	4	8	no	—
192	C₄⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
193	C₄⋊C₄×C₂₃	C₂²×C₄	1	16	yes	—
194	C₄⋊C₄×C₂₃	C₂×D₄	1	8	yes	—
195	C₄⋊C₄×C₂₃	C₂×D₄	1	8	yes	—
196	C₂²⋊C₄×C₂₃	—	2	4	no	—
197	C₂²⋊C₄×C₂₃	—	2	4	no	—
198	C₂²⋊C₄×C₂₃	—	2	16	no	—
199	C₂²⋊C₄×C₂₃	—	2	8	no	—
200	C₂²⋊C₄×C₂₃	—	4	4	no	—
201	C₂²⋊C₄×C₂₃	—	4	4	no	—
202	C₂²⋊C₄×C₂₃	—	4	8	no	—
203	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
204	C₂²⋊C₄×C₂₃	C₂²×C₄	1	16	yes	—
205	C₂²⋊C₄×C₂₃	C₂²×C₄	1	16	yes	—
206	C₄²×C₂₃	—	2	16	no	—
207	C₄²×C₂₃	—	2	8	no	—
208	C₄²×C₂₃	—	2	4	no	—
209	C₄²×C₂₃	—	2	4	no	—
210	C₄²×C₂₃	—	4	8	no	—
211	C₄²×C₂₃	C₂²×C₄	1	8	yes	—
212	#34201	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
213	C₂³×C₂₃	—	2	2	no	—
214	C₂³×C₂₃	—	2	2	no	—
215	C₂³×C₂₃	—	2	8	no	—
216	C₂³×C₂₃	—	2	2	no	—
217	C₂³×C₂₃	—	2	2	no	—
218	C₂³×C₂₃	—	2	2	no	—
219	C₂³×C₂₃	—	2	2	no	—
220	C₂³×C₂₃	—	2	4	no	—
221	C₂³×C₂₃	—	2	2	no	—
222	C₂³×C₂₃	—	2	2	no	—
223	C₂³×C₂₃	—	2	4	no	—
224	C₂³×C₂₃	—	2	2	no	—
225	C₂³×C₂₃	—	2	2	no	—
226	C₂³×C₂₃	—	4	8	no	—
227	C₂³×C₂₃	—	4	4	no	—
228	C₂³×C₂₃	—	4	4	no	—
229	C₂³×C₂₃	—	4	8	no	—
230	C₂³×C₂₃	C₂³×C₄	1	4	yes	—
231	C₂³×C₂₃	C₂³×C₄	1	4	yes	—
232	C₂³×C₂₃	C₂³×C₄	1	2	yes	—
233	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
234	C₂³×C₂₃	C₂×#41	1	2	yes	—
235	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
236	Q₈×C₂₃	—	2	8	no	—
237	D₄×C₂₃	—	2	16	no	—
238	D₄×C₂₃	—	2	8	no	—
239	D₄×C₂₃	—	4	16	no	—
240	C₂×C₄×C₂₃	—	2	8	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	4	no	—
246	C₂×C₄×C₂₃	—	2	8	no	—
247	C₂×C₄×C₂₃	—	2	4	no	—
248	C₂×C₄×C₂₃	—	2	16	no	—
249	C₂×C₄×C₂₃	—	2	16	no	—
250	C₂×C₄×C₂₃	—	2	16	no	—
251	C₂×C₄×C₂₃	—	2	2	no	—
252	C₂×C₄×C₂₃	—	2	4	no	—
253	C₂×C₄×C₂₃	—	2	2	no	—
254	C₂×C₄×C₂₃	—	2	4	no	—
255	C₂×C₄×C₂₃	—	2	4	no	—
256	C₂×C₄×C₂₃	—	2	4	no	—
257	C₂×C₄×C₂₃	—	2	4	no	—
258	C₂×C₄×C₂₃	—	2	8	no	—
259	C₂×C₄×C₂₃	—	2	4	no	—
260	C₂×C₄×C₂₃	—	4	8	no	—
261	C₂×C₄×C₂₃	—	4	8	no	—
262	C₂×C₄×C₂₃	—	8	8	no	—
263	C₂×C₄×C₂₃	C₄×D₄	1	16	yes	—
264	C₂×C₄×C₂₃	C₂³×C₄	1	8	yes	—
265	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
266	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
267	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
268	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
269	C₂×C₄×C₂₃	C₂²×D₄	1	8	yes	—
270	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
271	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
272	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
273	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
274	C₂×#525	C₂×C₂₃	1	1	yes	characteristic
275	C₂×#520	C₂×C₂₃	1	1	yes	characteristic, complemented
276	C₂²×#117	C₂×C₂₃	1	1	yes	characteristic, complemented
277	#2363	C₂×C₂₃	1	4	yes	complemented
278	#2362	C₂×C₂₃	1	4	yes	complemented
279	#2354	C₂×C₂₃	1	4	yes	complemented
280	#2352	C₂×C₂₃	1	4	yes	complemented
281	#2321	C₂×C₂₃	1	4	yes	complemented
282	#2317	C₂×C₂₃	1	1	yes	characteristic, complemented
283	#2311	C₂×C₂₃	1	1	yes	characteristic, complemented
284	C₄×#123	C₂×C₂₃	1	4	yes	complemented
285	#2280	C₂×C₂₃	1	1	yes	characteristic, complemented
286	#2274	C₂×C₂₃	1	1	yes	characteristic, complemented
287	C₂²×C₂₃	—	2	4	no	—
288	C₂²×C₂₃	—	2	2	no	—
289	C₂²×C₂₃	—	2	2	no	—
290	C₂²×C₂₃	—	2	4	no	—
291	C₂²×C₂₃	—	2	4	no	—
292	C₂²×C₂₃	—	2	4	no	—
293	C₂²×C₂₃	—	2	4	no	—
294	C₂²×C₂₃	—	2	8	no	—
295	C₂²×C₂₃	—	2	4	no	—
296	C₂²×C₂₃	—	4	16	no	—
297	C₂²×C₂₃	—	4	2	no	—
298	C₂²×C₂₃	—	4	2	no	—
299	C₂²×C₂₃	—	4	2	no	—
300	C₂²×C₂₃	—	4	2	no	—
301	C₂²×C₂₃	—	4	4	no	—
302	C₂²×C₂₃	—	4	2	no	—
303	C₂²×C₂₃	—	4	2	no	—
304	C₂²×C₂₃	C₂×C₄×D₄	1	4	yes	—
305	C₂²×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
306	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
307	C₂²×C₂₃	C₂²×#41	1	1	yes	characteristic
308	C₂²×C₂₃	#585	1	2	yes	—
309	C₂²×C₂₃	#585	1	2	yes	—
310	C₄×C₂₃	—	2	4	no	—
311	C₄×C₂₃	—	2	4	no	—
312	C₄×C₂₃	—	2	8	no	—
313	C₄×C₂₃	—	2	8	no	—
314	C₄×C₂₃	—	4	2	no	—
315	C₄×C₂₃	—	4	2	no	—
316	C₄×C₂₃	—	4	4	no	—
317	C₄×C₂₃	—	4	4	no	—
318	C₄×C₂₃	C₂×C₄×D₄	1	4	yes	—
319	C₄×C₂₃	C₂×C₄×D₄	1	4	yes	—
320	C₂²×#41	C₂²×C₂₃	1	1	yes	characteristic
321	C₂⁴×C₄	C₂²×C₂₃	1	1	yes	characteristic
322	#525	C₄×C₂₃	1	16	yes	complemented
323	C₂×#123	C₂²×C₂₃	1	2	yes	—
324	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
325	C₂×#121	C₂²×C₂₃	1	2	yes	—
326	C₂×#120	C₂²×C₂₃	1	2	yes	—
327	#520	—	2	8	no	—
328	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	complemented
329	C₂×C₄×D₄	C₂²×C₂₃	1	4	yes	—
330	C₂×C₄×D₄	C₂²×C₂₃	1	4	yes	—
331	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	complemented
332	C₂×#117	—	2	8	no	—
333	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
334	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
335	C₂×#117	C₂²×C₂₃	1	4	yes	—
336	C₂×#117	C₂²×C₂₃	1	4	yes	—
337	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
338	C₂×#117	C₂²×C₂₃	1	2	yes	complemented
339	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
340	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
341	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	complemented
342	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
343	C₂²×C₄²	C₂²×C₂₃	1	2	yes	—
344	#391	C₂²×C₂₃	1	2	yes	complemented
345	#390	C₂²×C₂₃	1	8	yes	—
346	#390	C₂²×C₂₃	1	2	yes	complemented
347	#389	C₂²×C₂₃	1	4	yes	complemented
348	#388	C₂²×C₂₃	1	8	yes	—
349	#388	C₂²×C₂₃	1	8	yes	—
350	#388	C₂²×C₂₃	1	8	yes	complemented
351	#387	C₂²×C₂₃	1	8	yes	complemented
352	#386	C₂²×C₂₃	1	4	yes	—
353	#386	C₂²×C₂₃	1	8	yes	—
354	#385	C₂²×C₂₃	1	4	yes	—
355	#384	C₂²×C₂₃	1	2	yes	complemented
356	#382	C₂²×C₂₃	1	2	yes	complemented
357	#381	C₂²×C₂₃	1	2	yes	—
358	#380	C₂²×C₂₃	1	2	yes	—
359	#380	C₂²×C₂₃	1	2	yes	—
360	#380	C₂²×C₂₃	1	4	yes	complemented
361	#379	C₂²×C₂₃	1	2	yes	—
362	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	complemented
363	C₄×C₄⋊C₄	C₂²×C₂₃	1	8	yes	complemented
364	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	complemented
365	C₄×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
366	C₄×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
367	C₄×C₂²⋊C₄	C₂²×C₂₃	1	8	yes	complemented
368	C₂×C₂₃	—	2	4	no	—
369	C₂×C₂₃	—	4	4	no	—
370	C₂×C₂₃	—	4	4	no	—
371	C₂×C₂₃	C₂²×C₄×D₄	1	1	yes	characteristic
372	C₂×C₂₃	#3428	1	2	yes	—
373	C₂×C₂₃	#3485	1	1	yes	characteristic
374	C₂×C₂₃	#3487	1	1	yes	characteristic
375	C₂×C₂₃	#3493	1	2	yes	—
376	C₂⁵	C₂³×C₂₃	1	1	yes	characteristic
377	C₂×#41	—	2	4	no	—
378	C₂×#41	—	2	4	no	—
379	C₂×#41	C₂×C₄×C₂₃	1	8	yes	—
380	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
381	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
382	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
383	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
384	C₂³×C₄	—	2	4	no	—
385	C₂³×C₄	—	2	4	no	—
386	C₂³×C₄	C₂×C₄×C₂₃	1	8	yes	complemented
387	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
388	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
389	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
390	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
391	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
392	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
393	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
394	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
395	#123	C₂×C₄×C₂₃	1	16	yes	complemented
396	C₂²⋊Q₈	C₂×C₄×C₂₃	1	16	yes	complemented
397	#121	C₂×C₄×C₂₃	1	16	yes	complemented
398	#120	C₂×C₄×C₂₃	1	16	yes	complemented
399	C₄×Q₈	—	2	8	no	—
400	C₄×D₄	—	2	16	no	—
401	C₄×D₄	—	2	8	no	—
402	C₄×D₄	C₂×C₄×C₂₃	1	32	yes	complemented
403	#117	—	2	4	no	—
404	#117	—	2	4	no	—
405	#117	—	2	16	no	—
406	#117	—	2	16	no	—
407	#117	—	2	8	no	—
408	#117	C₂×C₄×C₂₃	1	8	yes	complemented
409	C₂×C₄⋊C₄	—	2	4	no	—
410	C₂×C₄⋊C₄	—	2	8	no	—
411	C₂×C₄⋊C₄	D₄×C₂₃	1	8	yes	complemented
412	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
413	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
414	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
415	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
416	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
417	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
418	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
419	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
420	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
421	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
422	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
423	C₂×C₂²⋊C₄	—	2	8	no	—
424	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	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	8	yes	—
431	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
432	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
433	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
434	C₂×C₄²	—	2	8	no	—
435	C₂×C₄²	C₂³×C₂₃	1	8	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	8	yes	—
440	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
441	#95	C₂³×C₂₃	1	4	yes	—
442	#95	C₂³×C₂₃	1	4	yes	—
443	#95	C₂³×C₂₃	1	8	yes	—
444	#95	C₂³×C₂₃	1	8	yes	—
445	#95	C₂³×C₂₃	1	8	yes	—
446	C₂₃	#34201	1	1	yes	characteristic, direct product factor, Sylow
447	C₂⁴	—	2	2	no	—
448	C₂⁴	—	2	2	no	—
449	C₂⁴	—	4	2	no	—
450	C₂⁴	—	4	2	no	—
451	C₂⁴	C₂²×C₄×C₂₃	1	4	yes	—
452	C₂⁴	C₂⁴×C₂₃	1	2	yes	—
453	C₂⁴	C₂⁴×C₂₃	1	2	yes	—
454	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
455	#41	—	2	32	no	—
456	C₂×Q₈	—	2	2	no	—
457	C₂×Q₈	—	2	2	no	—
458	C₂×Q₈	C₂²×C₄×C₂₃	1	4	yes	—
459	C₂×D₄	—	2	4	no	—
460	C₂×D₄	—	2	4	no	—
461	C₂×D₄	—	2	2	no	—
462	C₂×D₄	—	2	2	no	—
463	C₂×D₄	—	4	4	no	—
464	C₂×D₄	—	4	4	no	—
465	C₂×D₄	C₂²×C₄×C₂₃	1	8	yes	—
466	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
467	C₂×D₄	C₂²×C₄×C₂₃	1	16	yes	—
468	C₂²×C₄	—	2	2	no	—
469	C₂²×C₄	—	2	2	no	—
470	C₂²×C₄	—	2	32	no	—
471	C₂²×C₄	—	2	4	no	—
472	C₂²×C₄	—	2	4	no	—
473	C₂²×C₄	—	2	4	no	—
474	C₂²×C₄	—	2	4	no	—
475	C₂²×C₄	—	2	4	no	—
476	C₂²×C₄	—	2	4	no	—
477	C₂²×C₄	—	2	4	no	—
478	C₂²×C₄	—	2	8	no	—
479	C₂²×C₄	—	4	4	no	—
480	C₂²×C₄	—	4	4	no	—
481	C₂²×C₄	—	4	8	no	—
482	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
483	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
484	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
485	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
486	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
487	C₂²×C₄	C₂×D₄×C₂₃	1	8	yes	—
488	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
489	C₂²×C₄	#41×C₂₃	1	4	yes	—
490	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
491	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
492	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
493	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
494	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
495	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
496	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
497	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
498	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
499	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
500	C₄⋊C₄	—	2	2	no	—
501	C₄⋊C₄	—	2	4	no	—
502	C₄⋊C₄	—	2	4	no	—
503	C₄⋊C₄	—	2	8	no	—
504	C₄⋊C₄	—	2	2	no	—
505	C₄⋊C₄	—	2	4	no	—
506	C₄⋊C₄	—	4	8	no	—
507	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
508	C₄⋊C₄	C₂²×C₄×C₂₃	1	16	yes	—
509	C₄⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
510	C₄⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
511	C₂²⋊C₄	—	2	4	no	—
512	C₂²⋊C₄	—	2	4	no	—
513	C₂²⋊C₄	—	2	16	no	—
514	C₂²⋊C₄	—	2	8	no	—
515	C₂²⋊C₄	—	4	4	no	—
516	C₂²⋊C₄	—	4	4	no	—
517	C₂²⋊C₄	—	4	8	no	—
518	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
519	C₂²⋊C₄	C₂²×C₄×C₂₃	1	16	yes	—
520	C₂²⋊C₄	C₂²×C₄×C₂₃	1	16	yes	—
521	C₄²	—	2	16	no	—
522	C₄²	—	2	8	no	—
523	C₄²	—	2	4	no	—
524	C₄²	—	2	4	no	—
525	C₄²	—	4	8	no	—
526	C₄²	C₂²×C₄×C₂₃	1	8	yes	—
527	C₂³	—	2	2	no	—
528	C₂³	—	2	2	no	—
529	C₂³	—	2	2	no	—
530	C₂³	—	2	2	no	—
531	C₂³	—	2	2	no	—
532	C₂³	—	2	2	no	—
533	C₂³	—	2	4	no	—
534	C₂³	—	2	2	no	—
535	C₂³	—	2	2	no	—
536	C₂³	—	2	4	no	—
537	C₂³	—	2	2	no	—
538	C₂³	—	2	2	no	—
539	C₂³	—	2	8	no	—
540	C₂³	—	4	8	no	—
541	C₂³	—	4	4	no	—
542	C₂³	—	4	4	no	—
543	C₂³	—	4	8	no	—
544	C₂³	C₂³×C₄×C₂₃	1	4	yes	—
545	C₂³	C₂³×C₄×C₂₃	1	4	yes	—
546	C₂³	C₂³×C₄×C₂₃	1	2	yes	—
547	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
548	C₂³	C₂×#41×C₂₃	1	2	yes	—
549	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
550	Q₈	—	2	8	no	—
551	D₄	—	2	16	no	—
552	D₄	—	2	8	no	—
553	D₄	—	4	16	no	—
554	C₂×C₄	—	2	2	no	—
555	C₂×C₄	—	2	2	no	—
556	C₂×C₄	—	2	2	no	—
557	C₂×C₄	—	2	2	no	—
558	C₂×C₄	—	2	4	no	—
559	C₂×C₄	—	2	8	no	—
560	C₂×C₄	—	2	4	no	—
561	C₂×C₄	—	2	8	no	—
562	C₂×C₄	—	2	16	no	—
563	C₂×C₄	—	2	16	no	—
564	C₂×C₄	—	2	16	no	—
565	C₂×C₄	—	2	2	no	—
566	C₂×C₄	—	2	4	no	—
567	C₂×C₄	—	2	2	no	—
568	C₂×C₄	—	2	4	no	—
569	C₂×C₄	—	2	4	no	—
570	C₂×C₄	—	2	4	no	—
571	C₂×C₄	—	2	4	no	—
572	C₂×C₄	—	2	8	no	—
573	C₂×C₄	—	2	4	no	—
574	C₂×C₄	—	4	8	no	—
575	C₂×C₄	—	4	8	no	—
576	C₂×C₄	—	8	8	no	—
577	C₂×C₄	C₄×D₄×C₂₃	1	16	yes	—
578	C₂×C₄	C₂³×C₄×C₂₃	1	8	yes	—
579	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
580	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
581	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
582	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
583	C₂×C₄	C₂²×D₄×C₂₃	1	8	yes	—
584	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
585	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
586	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
587	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
588	C₂²	—	2	4	no	—
589	C₂²	—	2	2	no	—
590	C₂²	—	2	2	no	—
591	C₂²	—	2	4	no	—
592	C₂²	—	2	4	no	—
593	C₂²	—	2	4	no	—
594	C₂²	—	2	4	no	—
595	C₂²	—	2	8	no	—
596	C₂²	—	2	4	no	—
597	C₂²	—	4	16	no	—
598	C₂²	—	4	2	no	—
599	C₂²	—	4	2	no	—
600	C₂²	—	4	2	no	—
601	C₂²	—	4	2	no	—
602	C₂²	—	4	2	no	—
603	C₂²	—	4	2	no	—
604	C₂²	—	4	4	no	—
605	C₂²	C₂×C₄×D₄×C₂₃	1	4	yes	—
606	C₂²	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
607	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
608	C₂²	C₂²×#41×C₂₃	1	1	yes	characteristic
609	C₂²	C₂₃×#585	1	2	yes	—
610	C₂²	C₂₃×#585	1	2	yes	—
611	C₄	—	2	4	no	—
612	C₄	—	2	4	no	—
613	C₄	—	2	8	no	—
614	C₄	—	2	8	no	—
615	C₄	—	4	2	no	—
616	C₄	—	4	4	no	—
617	C₄	—	4	2	no	—
618	C₄	—	4	4	no	—
619	C₄	C₂×C₄×D₄×C₂₃	1	4	yes	—
620	C₄	C₂×C₄×D₄×C₂₃	1	4	yes	—
621	C₂	—	2	4	no	—
622	C₂	—	4	4	no	—
623	C₂	—	4	4	no	—
624	C₂	C₂²×C₄×D₄×C₂₃	1	1	yes	characteristic
625	C₂	C₂₃×#3428	1	2	yes	—
626	C₂	C₂₃×#3485	1	1	yes	characteristic
627	C₂	C₂₃×#3487	1	1	yes	characteristic
628	C₂	C₂₃×#3493	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#520	C₂	semidirect
3. C₂²×C₂₃×#117	C₂	semidirect
4. C₂₃×#2363	C₂	semidirect
5. C₂₃×#2362	C₂	semidirect
6. C₂₃×#2354	C₂	semidirect
7. C₂₃×#2352	C₂	semidirect
8. C₂₃×#2321	C₂	semidirect
9. C₂₃×#2317	C₂	semidirect
10. C₂₃×#2311	C₂	semidirect
11. C₄×C₂₃×#123	C₂	semidirect
12. C₂₃×#2280	C₂	semidirect
13. C₂₃×#2274	C₂	semidirect
16. C₂₃×#525	C₄	semidirect
22. C₂×C₄×Q₈×C₂₃	C₂²	semidirect
25. C₂×C₄×D₄×C₂₃	C₂²	semidirect
32. C₂×C₂₃×#117	C₂²	semidirect
35. C₂²×C₄⋊C₄×C₂₃	C₂²	semidirect
38. C₂₃×#391	C₂²	semidirect
40. C₂₃×#390	C₂²	semidirect
41. C₂₃×#389	C₂²	semidirect
44. C₂₃×#388	C₂²	semidirect
45. C₂₃×#387	C₂²	semidirect
49. C₂₃×#384	C₂²	semidirect
50. C₂₃×#382	C₂²	semidirect
54. C₂₃×#380	C₂²	semidirect
56. C₄×C₄⋊C₄×C₂₃	C₂²	semidirect
57. C₄×C₄⋊C₄×C₂₃	C₂²	semidirect
58. C₄×C₄⋊C₄×C₂₃	C₂²	semidirect
61. C₄×C₂²⋊C₄×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
94. C₂₃×#117	C₂×C₄	semidirect
97. C₂×C₄⋊C₄×C₂₃	D₄	semidirect
212. #34201	C₂₃	direct (N × H)
275. C₂×#520	C₂×C₂₃	semidirect
276. C₂²×#117	C₂×C₂₃	semidirect
277. #2363	C₂×C₂₃	semidirect
278. #2362	C₂×C₂₃	semidirect
279. #2354	C₂×C₂₃	semidirect
280. #2352	C₂×C₂₃	semidirect
281. #2321	C₂×C₂₃	semidirect
282. #2317	C₂×C₂₃	semidirect
283. #2311	C₂×C₂₃	semidirect
284. C₄×#123	C₂×C₂₃	semidirect
285. #2280	C₂×C₂₃	semidirect
286. #2274	C₂×C₂₃	semidirect
322. #525	C₄×C₂₃	semidirect
328. C₂×C₄×Q₈	C₂²×C₂₃	semidirect
331. C₂×C₄×D₄	C₂²×C₂₃	semidirect
338. C₂×#117	C₂²×C₂₃	semidirect
341. C₂²×C₄⋊C₄	C₂²×C₂₃	semidirect
344. #391	C₂²×C₂₃	semidirect
346. #390	C₂²×C₂₃	semidirect
347. #389	C₂²×C₂₃	semidirect
350. #388	C₂²×C₂₃	semidirect
351. #387	C₂²×C₂₃	semidirect
355. #384	C₂²×C₂₃	semidirect
356. #382	C₂²×C₂₃	semidirect
360. #380	C₂²×C₂₃	semidirect
362. C₄×C₄⋊C₄	C₂²×C₂₃	semidirect
363. C₄×C₄⋊C₄	C₂²×C₂₃	semidirect
364. C₄×C₄⋊C₄	C₂²×C₂₃	semidirect
367. C₄×C₂²⋊C₄	C₂²×C₂₃	semidirect
386. C₂³×C₄	C₂×C₄×C₂₃	semidirect
395. #123	C₂×C₄×C₂₃	semidirect
396. C₂²⋊Q₈	C₂×C₄×C₂₃	semidirect
397. #121	C₂×C₄×C₂₃	semidirect
398. #120	C₂×C₄×C₂₃	semidirect
402. C₄×D₄	C₂×C₄×C₂₃	semidirect
408. #117	C₂×C₄×C₂₃	semidirect
411. C₂×C₄⋊C₄	D₄×C₂₃	semidirect
446. C₂₃	#34201	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×C₂₃×#520	2
C₂²×C₂₃×#117	2
C₂₃×#2363	2
C₂₃×#2362	2
C₂₃×#2354	2
C₂₃×#2352	2
C₂₃×#2321	2
C₂₃×#2317	2
C₂₃×#2311	2
C₄×C₂₃×#123	2
C₂₃×#2280	2
C₂₃×#2274	2
#34201	23

Subgroups — C23×#34201 #19181827

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#34201 #19181827