Subgroups — C₂₃×#34240 #19181866

Order 5888.

2682 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×#124	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2372	C₂	1	4	yes	maximal, complemented
3	C₂₃×#2367	C₂	1	4	yes	maximal, complemented
4	C₂₃×#2342	C₂	1	4	yes	maximal, complemented
5	Q₈×C₂²⋊C₄×C₂₃	C₂	1	2	yes	maximal, complemented
6	D₄×C₂²⋊C₄×C₂₃	C₂	1	2	yes	maximal
7	C₂₃×#2337	C₂	1	4	yes	maximal, complemented
8	C₂₃×#2312	C₂	1	2	yes	maximal, complemented
9	C₄×C₂₃×#124	C₂	1	4	yes	maximal, complemented
10	C₂₃×#2283	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#2273	C₂	1	2	yes	maximal, complemented
12	C₄×C₂₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂³×Q₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂³×D₄×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₂₃×#124	C₄	1	16	yes	complemented
16	C₂×C₂₃×#124	C₂²	1	2	yes	—
17	C₂×C₂₃×#124	C₂²	1	2	yes	—
18	C₂×C₂₃×#124	C₂²	1	2	yes	—
19	C₂×C₂₃×#124	C₂²	1	2	yes	—
20	C₂×C₄×Q₈×C₂₃	C₂²	1	4	yes	complemented
21	C₂×C₄×D₄×C₂₃	C₂²	1	4	yes	—
22	C₂×C₂₃×#117	C₂²	1	2	yes	—
23	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
24	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
25	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
26	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
27	C₂₃×#391	C₂²	1	4	yes	complemented
28	C₂₃×#391	C₂²	1	8	yes	complemented
29	C₂₃×#390	C₂²	1	4	yes	—
30	C₂₃×#388	C₂²	1	8	yes	complemented
31	C₂₃×#388	C₂²	1	8	yes	complemented
32	C₂₃×#388	C₂²	1	8	yes	complemented
33	C₂₃×#388	C₂²	1	8	yes	complemented
34	C₂₃×#386	C₂²	1	8	yes	—
35	C₂₃×#385	C₂²	1	8	yes	complemented
36	C₂₃×#383	C₂²	1	2	yes	complemented
37	C₂₃×#382	C₂²	1	2	yes	complemented
38	C₂₃×#381	C₂²	1	2	yes	—
39	C₂₃×#380	C₂²	1	2	yes	—
40	C₂₃×#380	C₂²	1	4	yes	—
41	C₂₃×#379	C₂²	1	4	yes	—
42	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	complemented
43	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	complemented
44	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	complemented
45	C₄×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
46	C₄×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
47	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
48	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
49	C₄×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
50	C₄×C₂²⋊C₄×C₂₃	C₂²	1	8	yes	complemented
51	C₄×C₂²⋊C₄×C₂₃	C₂²	1	8	yes	complemented
52	C₄×C₂²⋊C₄×C₂₃	C₂²	1	8	yes	complemented
53	C₂₃×#376	C₂²	1	2	yes	complemented
54	C₂₃×#376	C₂²	1	2	yes	complemented
55	C₄³×C₂₃	C₂²	1	2	yes	complemented
56	C₂⁵×C₂₃	C₂³	1	2	yes	—
57	C₂²×Q₈×C₂₃	—	4	4	no	—
58	C₂²×Q₈×C₂₃	C₂×C₄	1	8	yes	complemented
59	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
60	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
61	C₂²×D₄×C₂₃	—	4	4	no	—
62	C₂²×D₄×C₂₃	C₂×C₄	1	8	yes	—
63	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
64	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
65	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
67	C₂³×C₄×C₂₃	C₂³	1	8	yes	—
68	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
69	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
70	C₂₃×#124	C₂×C₄	1	16	yes	complemented
71	C₂₃×#124	C₂×C₄	1	16	yes	complemented
72	C₂₃×#124	C₂×C₄	1	16	yes	complemented
73	C₂₃×#124	C₂×C₄	1	16	yes	complemented
74	C₄×Q₈×C₂₃	—	4	8	no	—
75	C₄×D₄×C₂₃	—	4	8	no	—
76	C₂₃×#117	—	2	8	no	—
77	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
78	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
79	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
80	C₂×C₂²⋊C₄×C₂₃	—	2	16	no	—
81	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	16	yes	complemented
82	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	16	yes	complemented
83	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
84	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
85	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
86	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
87	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
88	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
89	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
90	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
91	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
92	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
93	C₂×C₄²×C₂₃	C₂×C₄	1	8	yes	complemented
94	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
95	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
96	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
97	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
98	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
99	C₂×C₄²×C₂₃	C₂³	1	8	yes	—
100	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
101	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
102	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
103	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
104	C₂₃×#95	C₂³	1	2	yes	—
105	C₂₃×#95	C₂³	1	4	yes	—
106	C₂₃×#95	C₂³	1	2	yes	—
107	C₂₃×#95	C₂³	1	4	yes	—
108	C₂₃×#95	C₂³	1	4	yes	—
109	C₂₃×#95	C₂³	1	4	yes	—
110	C₂₃×#95	C₂³	1	8	yes	—
111	C₂₃×#95	C₂³	1	8	yes	—
112	C₂₃×#95	C₂³	1	4	yes	—
113	C₂⁴×C₂₃	—	2	8	no	—
114	C₂⁴×C₂₃	—	4	4	no	—
115	C₂⁴×C₂₃	—	4	4	no	—
116	C₂⁴×C₂₃	C₂²×C₄	1	8	yes	—
117	C₂⁴×C₂₃	C₂⁴	1	4	yes	—
118	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
119	C₂×Q₈×C₂₃	—	2	8	no	—
120	C₂×Q₈×C₂₃	—	4	4	no	—
121	C₂×Q₈×C₂₃	—	4	16	no	—
122	C₂×Q₈×C₂₃	C₂²×C₄	1	8	yes	—
123	C₂×Q₈×C₂₃	C₂²×C₄	1	8	yes	—
124	C₂×D₄×C₂₃	—	2	8	no	—
125	C₂×D₄×C₂₃	—	4	16	no	—
126	C₂×D₄×C₂₃	—	4	4	no	—
127	C₂×D₄×C₂₃	C₂²×C₄	1	8	yes	—
128	C₂×D₄×C₂₃	C₂²×C₄	1	8	yes	—
129	C₂²×C₄×C₂₃	—	2	4	no	—
130	C₂²×C₄×C₂₃	—	2	4	no	—
131	C₂²×C₄×C₂₃	—	2	16	no	—
132	C₂²×C₄×C₂₃	—	4	2	no	—
133	C₂²×C₄×C₂₃	—	4	4	no	—
134	C₂²×C₄×C₂₃	—	4	8	no	—
135	C₂²×C₄×C₂₃	—	4	8	no	—
136	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
137	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
138	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
139	C₂²×C₄×C₂₃	C₂²×C₄	1	8	yes	—
140	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
141	C₂²×C₄×C₂₃	C₂⁴	1	8	yes	—
142	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
143	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
144	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
145	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
146	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
147	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
148	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
149	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
150	C₄⋊C₄×C₂₃	—	2	8	no	—
151	C₄⋊C₄×C₂₃	—	2	8	no	—
152	C₄⋊C₄×C₂₃	—	4	8	no	—
153	C₂²⋊C₄×C₂₃	—	2	8	no	—
154	C₂²⋊C₄×C₂₃	—	2	4	no	—
155	C₂²⋊C₄×C₂₃	—	2	16	no	—
156	C₂²⋊C₄×C₂₃	—	4	8	no	—
157	C₂²⋊C₄×C₂₃	—	4	8	no	—
158	C₂²⋊C₄×C₂₃	C₂²×C₄	1	16	yes	—
159	C₂²⋊C₄×C₂₃	C₂²×C₄	1	16	yes	—
160	C₂²⋊C₄×C₂₃	C₂²×C₄	1	16	yes	—
161	C₂²⋊C₄×C₂₃	C₂²×C₄	1	16	yes	—
162	C₂²⋊C₄×C₂₃	#41	1	8	yes	—
163	C₄²×C₂₃	—	2	8	no	—
164	C₄²×C₂₃	—	2	8	no	—
165	C₄²×C₂₃	—	4	2	no	—
166	C₄²×C₂₃	—	4	2	no	—
167	C₄²×C₂₃	—	4	4	no	—
168	C₄²×C₂₃	—	4	8	no	—
169	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
170	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
171	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
172	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
173	#34240	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
174	C₂³×C₂₃	—	2	2	no	—
175	C₂³×C₂₃	—	2	2	no	—
176	C₂³×C₂₃	—	2	4	no	—
177	C₂³×C₂₃	—	2	4	no	—
178	C₂³×C₂₃	—	2	8	no	—
179	C₂³×C₂₃	—	2	8	no	—
180	C₂³×C₂₃	—	2	16	no	—
181	C₂³×C₂₃	—	4	8	no	—
182	C₂³×C₂₃	—	4	8	no	—
183	C₂³×C₂₃	—	8	16	no	—
184	C₂³×C₂₃	C₂³×C₄	1	8	yes	—
185	C₂³×C₂₃	C₂³×C₄	1	2	yes	—
186	C₂³×C₂₃	C₂×#41	1	4	yes	—
187	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
188	Q₈×C₂₃	—	2	16	no	—
189	Q₈×C₂₃	—	4	8	no	—
190	D₄×C₂₃	—	2	16	no	—
191	D₄×C₂₃	—	4	8	no	—
192	C₂×C₄×C₂₃	—	2	8	no	—
193	C₂×C₄×C₂₃	—	2	2	no	—
194	C₂×C₄×C₂₃	—	2	2	no	—
195	C₂×C₄×C₂₃	—	2	4	no	—
196	C₂×C₄×C₂₃	—	2	8	no	—
197	C₂×C₄×C₂₃	—	2	2	no	—
198	C₂×C₄×C₂₃	—	2	2	no	—
199	C₂×C₄×C₂₃	—	2	2	no	—
200	C₂×C₄×C₂₃	—	2	4	no	—
201	C₂×C₄×C₂₃	—	2	8	no	—
202	C₂×C₄×C₂₃	—	2	4	no	—
203	C₂×C₄×C₂₃	—	2	8	no	—
204	C₂×C₄×C₂₃	—	2	4	no	—
205	C₂×C₄×C₂₃	—	2	8	no	—
206	C₂×C₄×C₂₃	—	2	4	no	—
207	C₂×C₄×C₂₃	—	2	8	no	—
208	C₂×C₄×C₂₃	—	4	4	no	—
209	C₂×C₄×C₂₃	—	4	8	no	—
210	C₂×C₄×C₂₃	—	8	16	no	—
211	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
212	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
213	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
214	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
215	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
216	C₂×C₄×C₂₃	C₂³×C₄	1	8	yes	—
217	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
218	C₂×C₄×C₂₃	C₂×#41	1	8	yes	—
219	C₂×C₄×C₂₃	H(5,𝔽₂)	1	4	yes	—
220	C₂²×#124	C₂×C₂₃	1	1	yes	characteristic
221	#2372	C₂×C₂₃	1	4	yes	complemented
222	#2367	C₂×C₂₃	1	4	yes	complemented
223	#2342	C₂×C₂₃	1	4	yes	complemented
224	Q₈×C₂²⋊C₄	C₂×C₂₃	1	2	yes	complemented
225	D₄×C₂²⋊C₄	C₂×C₂₃	1	2	yes	—
226	#2337	C₂×C₂₃	1	4	yes	complemented
227	#2312	C₂×C₂₃	1	2	yes	complemented
228	C₄×#124	C₂×C₂₃	1	4	yes	complemented
229	#2283	C₂×C₂₃	1	1	yes	characteristic, complemented
230	#2273	C₂×C₂₃	1	2	yes	complemented
231	C₄×#117	C₂×C₂₃	1	1	yes	characteristic, complemented
232	C₂²×C₂₃	—	2	2	no	—
233	C₂²×C₂₃	—	2	2	no	—
234	C₂²×C₂₃	—	2	8	no	—
235	C₂²×C₂₃	—	2	8	no	—
236	C₂²×C₂₃	—	2	8	no	—
237	C₂²×C₂₃	—	2	8	no	—
238	C₂²×C₂₃	—	4	8	no	—
239	C₂²×C₂₃	—	4	8	no	—
240	C₂²×C₂₃	—	8	16	no	—
241	C₂²×C₂₃	C₄×#41	1	4	yes	—
242	C₂²×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
243	C₂²×C₂₃	C₂²×#41	1	2	yes	—
244	C₂²×C₂₃	C₂×H(5,𝔽₂)	1	2	yes	—
245	C₂²×C₂₃	#585	1	2	yes	—
246	C₄×C₂₃	—	2	2	no	—
247	C₄×C₂₃	—	2	2	no	—
248	C₄×C₂₃	—	2	2	no	—
249	C₄×C₂₃	—	2	8	no	—
250	C₄×C₂₃	—	2	2	no	—
251	C₄×C₂₃	—	2	2	no	—
252	C₄×C₂₃	—	2	8	no	—
253	C₄×C₂₃	—	4	4	no	—
254	C₄×C₂₃	—	4	8	no	—
255	C₄×C₂₃	#518	1	4	yes	—
256	C₂³×Q₈	C₂²×C₂₃	1	1	yes	characteristic
257	C₂³×D₄	C₂²×C₂₃	1	1	yes	characteristic
258	C₂×#124	C₄×C₂₃	1	16	yes	complemented
259	C₂×#124	C₂²×C₂₃	1	2	yes	—
260	C₂×#124	C₂²×C₂₃	1	2	yes	—
261	C₂×#124	C₂²×C₂₃	1	2	yes	—
262	C₂×#124	C₂²×C₂₃	1	2	yes	—
263	C₂×C₄×Q₈	C₂²×C₂₃	1	4	yes	complemented
264	C₂×C₄×D₄	C₂²×C₂₃	1	4	yes	—
265	C₂×#117	C₂²×C₂₃	1	2	yes	—
266	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
267	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
268	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
269	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
270	#391	C₂²×C₂₃	1	8	yes	complemented
271	#391	C₂²×C₂₃	1	4	yes	complemented
272	#390	C₂²×C₂₃	1	4	yes	—
273	#388	C₂²×C₂₃	1	8	yes	complemented
274	#388	C₂²×C₂₃	1	8	yes	complemented
275	#388	C₂²×C₂₃	1	8	yes	complemented
276	#388	C₂²×C₂₃	1	8	yes	complemented
277	#386	C₂²×C₂₃	1	8	yes	—
278	#385	C₂²×C₂₃	1	8	yes	complemented
279	#383	C₂²×C₂₃	1	2	yes	complemented
280	#382	C₂²×C₂₃	1	2	yes	complemented
281	#381	C₂²×C₂₃	1	2	yes	—
282	#380	C₂²×C₂₃	1	4	yes	—
283	#380	C₂²×C₂₃	1	2	yes	—
284	#379	C₂²×C₂₃	1	4	yes	—
285	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	complemented
286	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	complemented
287	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	complemented
288	C₄×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
289	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
290	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
291	C₄×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
292	C₄×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
293	C₄×C₂²⋊C₄	C₂²×C₂₃	1	8	yes	complemented
294	C₄×C₂²⋊C₄	C₂²×C₂₃	1	8	yes	complemented
295	C₄×C₂²⋊C₄	C₂²×C₂₃	1	8	yes	complemented
296	#376	C₂²×C₂₃	1	2	yes	complemented
297	#376	C₂²×C₂₃	1	2	yes	complemented
298	C₄³	C₂²×C₂₃	1	2	yes	complemented
299	C₂×C₂₃	—	2	4	no	—
300	C₂×C₂₃	—	4	8	no	—
301	C₂×C₂₃	C₂×C₄×#41	1	1	yes	characteristic
302	C₂×C₂₃	C₂×#518	1	1	yes	characteristic
303	C₂×C₂₃	#3428	1	1	yes	characteristic
304	C₂×C₂₃	#3514	1	4	yes	—
305	C₂⁵	C₂³×C₂₃	1	2	yes	—
306	C₂²×Q₈	—	4	4	no	—
307	C₂²×Q₈	C₂×C₄×C₂₃	1	8	yes	complemented
308	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
309	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
310	C₂²×D₄	—	4	4	no	—
311	C₂²×D₄	C₂×C₄×C₂₃	1	8	yes	—
312	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
313	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
314	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
315	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
316	C₂³×C₄	C₂³×C₂₃	1	8	yes	—
317	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
318	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
319	#124	C₂×C₄×C₂₃	1	16	yes	complemented
320	#124	C₂×C₄×C₂₃	1	16	yes	complemented
321	#124	C₂×C₄×C₂₃	1	16	yes	complemented
322	#124	C₂×C₄×C₂₃	1	16	yes	complemented
323	C₄×Q₈	—	4	8	no	—
324	C₄×D₄	—	4	8	no	—
325	#117	—	2	8	no	—
326	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
327	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
328	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
329	C₂×C₂²⋊C₄	—	2	16	no	—
330	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	16	yes	complemented
331	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	16	yes	complemented
332	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
333	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
334	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
335	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
336	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
337	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
338	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
339	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
340	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
341	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
342	C₂×C₄²	C₂×C₄×C₂₃	1	8	yes	complemented
343	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
344	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
345	C₂×C₄²	C₂³×C₂₃	1	8	yes	—
346	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
347	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
348	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
349	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
350	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
351	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
352	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
353	#95	C₂³×C₂₃	1	2	yes	—
354	#95	C₂³×C₂₃	1	4	yes	—
355	#95	C₂³×C₂₃	1	2	yes	—
356	#95	C₂³×C₂₃	1	4	yes	—
357	#95	C₂³×C₂₃	1	4	yes	—
358	#95	C₂³×C₂₃	1	4	yes	—
359	#95	C₂³×C₂₃	1	8	yes	—
360	#95	C₂³×C₂₃	1	8	yes	—
361	#95	C₂³×C₂₃	1	4	yes	—
362	C₂₃	#34240	1	1	yes	characteristic, direct product factor, Sylow
363	C₂⁴	—	2	8	no	—
364	C₂⁴	—	4	4	no	—
365	C₂⁴	—	4	4	no	—
366	C₂⁴	C₂²×C₄×C₂₃	1	8	yes	—
367	C₂⁴	C₂⁴×C₂₃	1	4	yes	—
368	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
369	C₂×Q₈	—	2	8	no	—
370	C₂×Q₈	—	4	4	no	—
371	C₂×Q₈	—	4	16	no	—
372	C₂×Q₈	C₂²×C₄×C₂₃	1	8	yes	—
373	C₂×Q₈	C₂²×C₄×C₂₃	1	8	yes	—
374	C₂×D₄	—	2	8	no	—
375	C₂×D₄	—	4	16	no	—
376	C₂×D₄	—	4	4	no	—
377	C₂×D₄	C₂²×C₄×C₂₃	1	8	yes	—
378	C₂×D₄	C₂²×C₄×C₂₃	1	8	yes	—
379	C₂²×C₄	—	2	4	no	—
380	C₂²×C₄	—	2	4	no	—
381	C₂²×C₄	—	2	16	no	—
382	C₂²×C₄	—	4	4	no	—
383	C₂²×C₄	—	4	2	no	—
384	C₂²×C₄	—	4	8	no	—
385	C₂²×C₄	—	4	8	no	—
386	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
387	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
388	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
389	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
390	C₂²×C₄	C₂⁴×C₂₃	1	8	yes	—
391	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
392	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
393	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
394	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
395	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
396	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
397	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
398	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
399	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
400	C₄⋊C₄	—	2	8	no	—
401	C₄⋊C₄	—	2	8	no	—
402	C₄⋊C₄	—	4	8	no	—
403	C₂²⋊C₄	—	2	8	no	—
404	C₂²⋊C₄	—	2	4	no	—
405	C₂²⋊C₄	—	2	16	no	—
406	C₂²⋊C₄	—	4	8	no	—
407	C₂²⋊C₄	—	4	8	no	—
408	C₂²⋊C₄	C₂²×C₄×C₂₃	1	16	yes	—
409	C₂²⋊C₄	C₂²×C₄×C₂₃	1	16	yes	—
410	C₂²⋊C₄	C₂²×C₄×C₂₃	1	16	yes	—
411	C₂²⋊C₄	C₂²×C₄×C₂₃	1	16	yes	—
412	C₂²⋊C₄	#41×C₂₃	1	8	yes	—
413	C₄²	—	2	8	no	—
414	C₄²	—	2	8	no	—
415	C₄²	—	4	8	no	—
416	C₄²	—	4	2	no	—
417	C₄²	—	4	2	no	—
418	C₄²	—	4	4	no	—
419	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
420	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
421	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
422	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
423	C₂³	—	2	2	no	—
424	C₂³	—	2	2	no	—
425	C₂³	—	2	4	no	—
426	C₂³	—	2	4	no	—
427	C₂³	—	2	8	no	—
428	C₂³	—	2	8	no	—
429	C₂³	—	2	16	no	—
430	C₂³	—	4	8	no	—
431	C₂³	—	4	8	no	—
432	C₂³	—	8	16	no	—
433	C₂³	C₂³×C₄×C₂₃	1	8	yes	—
434	C₂³	C₂³×C₄×C₂₃	1	2	yes	—
435	C₂³	C₂×#41×C₂₃	1	4	yes	—
436	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
437	Q₈	—	2	16	no	—
438	Q₈	—	4	8	no	—
439	D₄	—	2	16	no	—
440	D₄	—	4	8	no	—
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	2	4	no	—
444	C₂×C₄	—	2	8	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	4	no	—
449	C₂×C₄	—	2	8	no	—
450	C₂×C₄	—	2	8	no	—
451	C₂×C₄	—	2	4	no	—
452	C₂×C₄	—	2	8	no	—
453	C₂×C₄	—	2	4	no	—
454	C₂×C₄	—	2	8	no	—
455	C₂×C₄	—	2	4	no	—
456	C₂×C₄	—	2	8	no	—
457	C₂×C₄	—	4	8	no	—
458	C₂×C₄	—	4	4	no	—
459	C₂×C₄	—	8	16	no	—
460	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
461	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
462	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
463	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
464	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
465	C₂×C₄	C₂³×C₄×C₂₃	1	8	yes	—
466	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
467	C₂×C₄	C₂×#41×C₂₃	1	8	yes	—
468	C₂×C₄	C₂₃×H(5,𝔽₂)	1	4	yes	—
469	C₂²	—	2	2	no	—
470	C₂²	—	2	2	no	—
471	C₂²	—	2	8	no	—
472	C₂²	—	2	8	no	—
473	C₂²	—	2	8	no	—
474	C₂²	—	2	8	no	—
475	C₂²	—	4	8	no	—
476	C₂²	—	4	8	no	—
477	C₂²	—	8	16	no	—
478	C₂²	C₄×#41×C₂₃	1	4	yes	—
479	C₂²	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
480	C₂²	C₂²×#41×C₂₃	1	2	yes	—
481	C₂²	C₂×C₂₃×H(5,𝔽₂)	1	2	yes	—
482	C₂²	C₂₃×#585	1	2	yes	—
483	C₄	—	2	2	no	—
484	C₄	—	2	2	no	—
485	C₄	—	2	2	no	—
486	C₄	—	2	8	no	—
487	C₄	—	2	2	no	—
488	C₄	—	2	2	no	—
489	C₄	—	2	8	no	—
490	C₄	—	4	8	no	—
491	C₄	—	4	4	no	—
492	C₄	C₂₃×#518	1	4	yes	—
493	C₂	—	2	4	no	—
494	C₂	—	4	8	no	—
495	C₂	C₂×C₄×#41×C₂₃	1	1	yes	characteristic
496	C₂	C₂×C₂₃×#518	1	1	yes	characteristic
497	C₂	C₂₃×#3428	1	1	yes	characteristic
498	C₂	C₂₃×#3514	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₂₃
Frattini, cosocle	C₂³	C₂⁵×C₂₃
prime radical	C₂³×D₄×C₂₃	C₂²
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. C₂₃×#2372	C₂	semidirect
3. C₂₃×#2367	C₂	semidirect
4. C₂₃×#2342	C₂	semidirect
5. Q₈×C₂²⋊C₄×C₂₃	C₂	semidirect
7. C₂₃×#2337	C₂	semidirect
8. C₂₃×#2312	C₂	semidirect
9. C₄×C₂₃×#124	C₂	semidirect
10. C₂₃×#2283	C₂	semidirect
11. C₂₃×#2273	C₂	semidirect
12. C₄×C₂₃×#117	C₂	semidirect
15. C₂×C₂₃×#124	C₄	semidirect
20. C₂×C₄×Q₈×C₂₃	C₂²	semidirect
27. C₂₃×#391	C₂²	semidirect
28. C₂₃×#391	C₂²	semidirect
30. C₂₃×#388	C₂²	semidirect
31. C₂₃×#388	C₂²	semidirect
32. C₂₃×#388	C₂²	semidirect
33. C₂₃×#388	C₂²	semidirect
35. C₂₃×#385	C₂²	semidirect
36. C₂₃×#383	C₂²	semidirect
37. C₂₃×#382	C₂²	semidirect
42. C₄×C₄⋊C₄×C₂₃	C₂²	semidirect
43. C₄×C₄⋊C₄×C₂₃	C₂²	semidirect
44. C₄×C₄⋊C₄×C₂₃	C₂²	semidirect
50. C₄×C₂²⋊C₄×C₂₃	C₂²	semidirect
51. C₄×C₂²⋊C₄×C₂₃	C₂²	semidirect
52. C₄×C₂²⋊C₄×C₂₃	C₂²	semidirect
53. C₂₃×#376	C₂²	semidirect
54. C₂₃×#376	C₂²	semidirect
55. C₄³×C₂₃	C₂²	semidirect
58. C₂²×Q₈×C₂₃	C₂×C₄	semidirect
70. C₂₃×#124	C₂×C₄	semidirect
71. C₂₃×#124	C₂×C₄	semidirect
72. C₂₃×#124	C₂×C₄	semidirect
73. C₂₃×#124	C₂×C₄	semidirect
81. C₂×C₂²⋊C₄×C₂₃	C₂×C₄	semidirect
82. C₂×C₂²⋊C₄×C₂₃	C₂×C₄	semidirect
93. C₂×C₄²×C₂₃	C₂×C₄	semidirect
173. #34240	C₂₃	direct (N × H)
221. #2372	C₂×C₂₃	semidirect
222. #2367	C₂×C₂₃	semidirect
223. #2342	C₂×C₂₃	semidirect
224. Q₈×C₂²⋊C₄	C₂×C₂₃	semidirect
226. #2337	C₂×C₂₃	semidirect
227. #2312	C₂×C₂₃	semidirect
228. C₄×#124	C₂×C₂₃	semidirect
229. #2283	C₂×C₂₃	semidirect
230. #2273	C₂×C₂₃	semidirect
231. C₄×#117	C₂×C₂₃	semidirect
258. C₂×#124	C₄×C₂₃	semidirect
263. C₂×C₄×Q₈	C₂²×C₂₃	semidirect
270. #391	C₂²×C₂₃	semidirect
271. #391	C₂²×C₂₃	semidirect
273. #388	C₂²×C₂₃	semidirect
274. #388	C₂²×C₂₃	semidirect
275. #388	C₂²×C₂₃	semidirect
276. #388	C₂²×C₂₃	semidirect
278. #385	C₂²×C₂₃	semidirect
279. #383	C₂²×C₂₃	semidirect
280. #382	C₂²×C₂₃	semidirect
285. C₄×C₄⋊C₄	C₂²×C₂₃	semidirect
286. C₄×C₄⋊C₄	C₂²×C₂₃	semidirect
287. C₄×C₄⋊C₄	C₂²×C₂₃	semidirect
293. C₄×C₂²⋊C₄	C₂²×C₂₃	semidirect
294. C₄×C₂²⋊C₄	C₂²×C₂₃	semidirect
295. C₄×C₂²⋊C₄	C₂²×C₂₃	semidirect
296. #376	C₂²×C₂₃	semidirect
297. #376	C₂²×C₂₃	semidirect
298. C₄³	C₂²×C₂₃	semidirect
307. C₂²×Q₈	C₂×C₄×C₂₃	semidirect
319. #124	C₂×C₄×C₂₃	semidirect
320. #124	C₂×C₄×C₂₃	semidirect
321. #124	C₂×C₄×C₂₃	semidirect
322. #124	C₂×C₄×C₂₃	semidirect
330. C₂×C₂²⋊C₄	C₂×C₄×C₂₃	semidirect
331. C₂×C₂²⋊C₄	C₂×C₄×C₂₃	semidirect
342. C₂×C₄²	C₂×C₄×C₂₃	semidirect
362. C₂₃	#34240	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₂₃×#124	2
C₂₃×#2372	2
C₂₃×#2367	2
C₂₃×#2342	2
Q₈×C₂²⋊C₄×C₂₃	2
D₄×C₂²⋊C₄×C₂₃	2
C₂₃×#2337	2
C₂₃×#2312	2
C₄×C₂₃×#124	2
C₂₃×#2283	2
C₂₃×#2273	2
C₄×C₂₃×#117	2
#34240	23

Subgroups — C23×#34240 #19181866

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#34240 #19181866