Subgroups — C₂₃×#25424 #19173050

Order 5888.

1020 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#495	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#470	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#469	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×#441	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#2426	C₂	1	2	yes	maximal, complemented
6	C₂₃×#2042	C₂	1	2	yes	maximal, complemented
7	C₂₃×#2000	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1972	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1941	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1880	C₂	1	2	yes	maximal, complemented
11	C₂₃×#1753	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
13	C₂²×SD₁₆×C₂₃	C₂²	1	2	yes	—
14	C₂²×D₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×#124	C₂²	1	2	yes	—
17	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
18	C₂×C₂₃×#121	C₂²	1	2	yes	—
19	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	—
20	C₂₃×#495	—	2	4	no	—
21	C₂₃×#470	—	2	4	no	—
22	C₂₃×#469	—	2	4	no	—
23	C₂₃×#441	—	2	4	no	—
24	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
25	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
26	C₂×C₂₃×#103	C₂²	1	2	yes	—
27	C₂×C₂₃×#103	C₂²	1	2	yes	—
28	C₂×C₂₃×#102	C₂²	1	2	yes	—
29	C₂×C₂₃×#102	C₂²	1	2	yes	—
30	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
31	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
33	C₂₃×#396	—	2	2	no	—
34	C₂₃×#394	—	2	2	no	—
35	C₂₃×#390	—	2	2	no	—
36	C₂₃×#390	C₂²	1	2	yes	—
37	C₂₃×#389	C₂²	1	2	yes	—
38	C₂₃×#388	—	2	2	no	—
39	C₄×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
40	C₂₃×#340	C₂²	1	2	yes	—
41	C₂₃×#336	C₂²	1	1	yes	characteristic
42	C₂₃×#336	C₂²	1	1	yes	characteristic
43	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
44	C₂²×D₄×C₂₃	—	2	2	no	—
45	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
46	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
47	C₂×Q₁₆×C₂₃	—	2	2	no	—
48	C₂×Q₁₆×C₂₃	—	2	2	no	—
49	C₂×Q₁₆×C₂₃	D₄	1	4	yes	complemented
50	C₂×SD₁₆×C₂₃	—	2	4	no	—
51	C₂×SD₁₆×C₂₃	—	2	4	no	—
52	C₂×SD₁₆×C₂₃	—	4	2	no	—
53	C₂×D₈×C₂₃	—	2	2	no	—
54	C₂×D₈×C₂₃	—	2	2	no	—
55	C₂×D₈×C₂₃	—	2	2	no	—
56	C₂×M₁₆×C₂₃	—	2	2	no	—
57	C₂×M₁₆×C₂₃	—	2	2	no	—
58	C₂×M₁₆×C₂₃	—	2	2	no	—
59	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
63	C₂₃×#124	—	2	4	no	—
64	C₂₃×#124	—	4	2	no	—
65	C₂₃×C₂²⋊Q₈	—	2	4	no	—
66	C₂₃×C₂²⋊Q₈	—	4	2	no	—
67	C₂₃×#121	—	2	4	no	—
68	C₂₃×#121	—	4	2	no	—
69	C₄×Q₈×C₂₃	—	2	4	no	—
70	C₄×Q₈×C₂₃	—	4	2	no	—
71	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
72	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
73	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
74	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
75	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
76	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
77	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
78	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
79	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
80	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
81	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
82	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
83	C₂×C₄²×C₂₃	—	2	2	no	—
84	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
85	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
86	C₂₃×#106	—	2	2	no	—
87	C₂₃×#106	D₄	1	4	yes	—
88	C₂₃×#103	—	2	4	no	—
89	C₂₃×#103	—	2	4	no	—
90	C₂₃×#102	—	2	4	no	—
91	C₂₃×#102	—	4	2	no	—
92	C₂₃×C₂²⋊C₈	—	2	2	no	—
93	C₂₃×#97	—	2	2	no	—
94	C₄×C₈×C₂₃	—	2	2	no	—
95	C₂₃×#95	—	2	2	no	—
96	C₂₃×#95	D₄	1	2	yes	—
97	C₂⁴×C₂₃	—	2	2	no	—
98	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
99	C₂×Q₈×C₂₃	—	2	2	no	—
100	C₂×Q₈×C₂₃	—	2	2	no	—
101	C₂×Q₈×C₂₃	—	2	2	no	—
102	C₂×Q₈×C₂₃	—	2	2	no	—
103	C₂×Q₈×C₂₃	—	2	2	no	—
104	C₂×Q₈×C₂₃	C₂×D₄	1	4	yes	—
105	C₂×D₄×C₂₃	—	2	2	no	—
106	C₂×D₄×C₂₃	—	2	2	no	—
107	C₂×D₄×C₂₃	—	2	2	no	—
108	C₂×D₄×C₂₃	—	2	2	no	—
109	C₂×D₄×C₂₃	—	2	2	no	—
110	C₂×D₄×C₂₃	—	2	2	no	—
111	C₂×D₄×C₂₃	—	4	4	no	—
112	C₂×D₄×C₂₃	—	4	4	no	—
113	C₂×D₄×C₂₃	—	8	2	no	—
114	C₂²×C₄×C₂₃	—	2	2	no	—
115	C₂²×C₄×C₂₃	—	2	2	no	—
116	C₂²×C₄×C₂₃	—	2	1	no	—
117	C₂²×C₄×C₂₃	—	2	1	no	—
118	C₂²×C₄×C₂₃	—	2	1	no	—
119	C₂²×C₄×C₂₃	—	2	1	no	—
120	C₂²×C₄×C₂₃	—	2	1	no	—
121	C₂²×C₄×C₂₃	—	2	1	no	—
122	C₂²×C₄×C₂₃	—	2	2	no	—
123	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
124	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
125	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
127	Q₁₆×C₂₃	—	2	8	no	—
128	SD₁₆×C₂₃	—	4	8	no	—
129	D₈×C₂₃	—	4	4	no	—
130	M₁₆×C₂₃	—	4	4	no	—
131	C₂×C₈×C₂₃	—	2	1	no	—
132	C₂×C₈×C₂₃	—	2	1	no	—
133	C₂×C₈×C₂₃	—	2	1	no	—
134	C₂×C₈×C₂₃	—	2	1	no	—
135	C₂×C₈×C₂₃	—	2	1	no	—
136	C₂×C₈×C₂₃	—	2	1	no	—
137	C₂×C₈×C₂₃	—	2	1	no	—
138	C₂×C₈×C₂₃	—	2	1	no	—
139	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
140	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
141	C₂×C₈×C₂₃	#41	1	2	yes	—
142	C₂×C₈×C₂₃	#41	1	2	yes	—
143	C₄⋊C₄×C₂₃	—	2	2	no	—
144	C₄⋊C₄×C₂₃	—	2	2	no	—
145	C₄⋊C₄×C₂₃	—	2	2	no	—
146	C₄⋊C₄×C₂₃	—	4	4	no	—
147	C₄⋊C₄×C₂₃	—	4	4	no	—
148	C₄⋊C₄×C₂₃	—	8	2	no	—
149	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
150	C₂²⋊C₄×C₂₃	—	4	2	no	—
151	C₂²⋊C₄×C₂₃	—	4	4	no	—
152	C₂²⋊C₄×C₂₃	—	4	4	no	—
153	C₂²⋊C₄×C₂₃	—	4	4	no	—
154	C₂²⋊C₄×C₂₃	—	4	4	no	—
155	C₂²⋊C₄×C₂₃	—	4	4	no	—
156	C₂²⋊C₄×C₂₃	—	8	2	no	—
157	C₄²×C₂₃	—	2	1	no	—
158	C₄²×C₂₃	—	2	1	no	—
159	C₄²×C₂₃	—	2	1	no	—
160	C₄²×C₂₃	—	2	1	no	—
161	C₄²×C₂₃	—	4	4	no	—
162	#25424	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
163	C₂³×C₂₃	—	2	1	no	—
164	C₂³×C₂₃	—	2	1	no	—
165	C₂³×C₂₃	—	2	1	no	—
166	C₂³×C₂₃	—	2	2	no	—
167	C₂³×C₂₃	—	2	2	no	—
168	C₂³×C₂₃	—	4	2	no	—
169	C₂³×C₂₃	—	4	2	no	—
170	C₂³×C₂₃	—	4	2	no	—
171	C₂³×C₂₃	—	4	2	no	—
172	C₂³×C₂₃	—	4	2	no	—
173	C₂³×C₂₃	—	4	2	no	—
174	C₂³×C₂₃	—	4	2	no	—
175	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
176	Q₈×C₂₃	—	2	4	no	—
177	Q₈×C₂₃	—	2	4	no	—
178	Q₈×C₂₃	—	4	2	no	—
179	Q₈×C₂₃	—	4	2	no	—
180	D₄×C₂₃	—	4	2	no	—
181	D₄×C₂₃	—	4	2	no	—
182	D₄×C₂₃	—	4	2	no	—
183	D₄×C₂₃	—	4	2	no	—
184	D₄×C₂₃	—	8	8	no	—
185	C₂×C₄×C₂₃	—	2	1	no	—
186	C₂×C₄×C₂₃	—	2	2	no	—
187	C₂×C₄×C₂₃	—	2	2	no	—
188	C₂×C₄×C₂₃	—	2	4	no	—
189	C₂×C₄×C₂₃	—	2	1	no	—
190	C₂×C₄×C₂₃	—	2	4	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	—	2	2	no	—
195	C₂×C₄×C₂₃	—	2	2	no	—
196	C₂×C₄×C₂₃	—	4	2	no	—
197	C₂×C₄×C₂₃	—	4	2	no	—
198	C₂×C₄×C₂₃	—	4	2	no	—
199	C₂×C₄×C₂₃	—	4	2	no	—
200	C₂×C₄×C₂₃	—	4	1	no	—
201	C₂×C₄×C₂₃	—	4	1	no	—
202	C₂×C₄×C₂₃	—	4	1	no	—
203	C₂×C₄×C₂₃	—	4	1	no	—
204	C₂×C₄×C₂₃	—	4	2	no	—
205	C₂×C₄×C₂₃	—	4	2	no	—
206	C₂×C₄×C₂₃	—	4	2	no	—
207	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
208	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
209	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
210	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
211	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
212	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
213	C₈×C₂₃	—	2	2	no	—
214	C₈×C₂₃	—	4	1	no	—
215	C₈×C₂₃	—	4	1	no	—
216	C₈×C₂₃	#121	1	4	yes	—
217	C₂×#495	C₂×C₂₃	1	1	yes	characteristic, complemented
218	C₂×#470	C₂×C₂₃	1	1	yes	characteristic, complemented
219	C₂×#469	C₂×C₂₃	1	1	yes	characteristic
220	C₂×#441	C₂×C₂₃	1	1	yes	characteristic, complemented
221	#2426	C₂×C₂₃	1	2	yes	complemented
222	#2042	C₂×C₂₃	1	2	yes	complemented
223	#2000	C₂×C₂₃	1	2	yes	complemented
224	#1972	C₂×C₂₃	1	1	yes	characteristic, complemented
225	#1941	C₂×C₂₃	1	1	yes	characteristic, complemented
226	#1880	C₂×C₂₃	1	2	yes	complemented
227	#1753	C₂×C₂₃	1	1	yes	characteristic, complemented
228	C₂²×C₂₃	—	4	1	no	—
229	C₂²×C₂₃	—	4	1	no	—
230	C₂²×C₂₃	—	4	1	no	—
231	C₂²×C₂₃	—	4	2	no	—
232	C₂²×C₂₃	—	4	2	no	—
233	C₂²×C₂₃	—	8	2	no	—
234	C₂²×C₂₃	—	8	2	no	—
235	C₂²×C₂₃	—	8	2	no	—
236	C₂²×C₂₃	—	8	2	no	—
237	C₂²×C₂₃	—	8	2	no	—
238	C₂²×C₂₃	—	8	2	no	—
239	C₂²×C₂₃	—	8	2	no	—
240	C₂²×C₂₃	#532	1	1	yes	characteristic
241	C₂²×C₂₃	D₄²	1	1	yes	characteristic
242	C₂²×C₂₃	D₄²	1	1	yes	characteristic
243	C₂²×C₂₃	#575	1	2	yes	—
244	C₂²×C₂₃	#577	1	2	yes	—
245	C₄×C₂₃	—	4	1	no	—
246	C₄×C₂₃	—	4	4	no	—
247	C₄×C₂₃	—	4	2	no	—
248	C₄×C₂₃	—	4	2	no	—
249	C₄×C₂₃	—	4	4	no	—
250	C₄×C₂₃	—	8	2	no	—
251	C₄×C₂₃	C₂×#121	1	1	yes	characteristic
252	C₄×C₂₃	#540	1	1	yes	characteristic
253	C₄×C₂₃	#546	1	1	yes	characteristic
254	C₄×C₂₃	#550	1	1	yes	characteristic
255	C₂²×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
256	C₂²×SD₁₆	C₂²×C₂₃	1	2	yes	—
257	C₂²×D₈	C₂²×C₂₃	1	1	yes	characteristic
258	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
259	C₂×#124	C₂²×C₂₃	1	2	yes	—
260	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
261	C₂×#121	C₂²×C₂₃	1	2	yes	—
262	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	—
263	#495	—	2	4	no	—
264	#470	—	2	4	no	—
265	#469	—	2	4	no	—
266	#441	—	2	4	no	—
267	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
268	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
269	C₂×#103	C₂²×C₂₃	1	2	yes	—
270	C₂×#103	C₂²×C₂₃	1	2	yes	—
271	C₂×#102	C₂²×C₂₃	1	2	yes	—
272	C₂×#102	C₂²×C₂₃	1	2	yes	—
273	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
274	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
275	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
276	#396	—	2	2	no	—
277	#394	—	2	2	no	—
278	#390	—	2	2	no	—
279	#390	C₂²×C₂₃	1	2	yes	—
280	#389	C₂²×C₂₃	1	2	yes	—
281	#388	—	2	2	no	—
282	C₄×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
283	#340	C₂²×C₂₃	1	2	yes	—
284	#336	C₂²×C₂₃	1	1	yes	characteristic
285	#336	C₂²×C₂₃	1	1	yes	characteristic
286	C₂×C₂₃	—	8	1	no	—
287	C₂×C₂₃	—	16	2	no	—
288	C₂×C₂₃	#2428	1	1	yes	characteristic
289	C₂×C₂₃	#3173	1	1	yes	characteristic
290	C₂×C₂₃	#3183	1	1	yes	characteristic
291	C₂×C₂₃	#3271	1	2	yes	—
292	C₂×C₂₃	#3275	1	2	yes	—
293	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
294	C₂²×D₄	—	2	2	no	—
295	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
296	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
297	C₂×Q₁₆	—	2	2	no	—
298	C₂×Q₁₆	—	2	2	no	—
299	C₂×Q₁₆	D₄×C₂₃	1	4	yes	complemented
300	C₂×SD₁₆	—	2	4	no	—
301	C₂×SD₁₆	—	2	4	no	—
302	C₂×SD₁₆	—	4	2	no	—
303	C₂×D₈	—	2	2	no	—
304	C₂×D₈	—	2	2	no	—
305	C₂×D₈	—	2	2	no	—
306	C₂×M₁₆	—	2	2	no	—
307	C₂×M₁₆	—	2	2	no	—
308	C₂×M₁₆	—	2	2	no	—
309	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
310	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
311	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
312	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
313	#124	—	2	4	no	—
314	#124	—	4	2	no	—
315	C₂²⋊Q₈	—	2	4	no	—
316	C₂²⋊Q₈	—	4	2	no	—
317	#121	—	2	4	no	—
318	#121	—	4	2	no	—
319	C₄×Q₈	—	2	4	no	—
320	C₄×Q₈	—	4	2	no	—
321	C₂×C₄⋊C₄	—	2	2	no	—
322	C₂×C₄⋊C₄	—	2	2	no	—
323	C₂×C₄⋊C₄	—	2	2	no	—
324	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
325	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
326	C₂×C₂²⋊C₄	—	2	2	no	—
327	C₂×C₂²⋊C₄	—	2	2	no	—
328	C₂×C₂²⋊C₄	—	2	2	no	—
329	C₂×C₂²⋊C₄	—	2	2	no	—
330	C₂×C₂²⋊C₄	—	2	2	no	—
331	C₂×C₂²⋊C₄	—	2	2	no	—
332	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
333	C₂×C₄²	—	2	2	no	—
334	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
335	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
336	#106	—	2	2	no	—
337	#106	D₄×C₂₃	1	4	yes	—
338	#103	—	2	4	no	—
339	#103	—	2	4	no	—
340	#102	—	2	4	no	—
341	#102	—	4	2	no	—
342	C₂²⋊C₈	—	2	2	no	—
343	#97	—	2	2	no	—
344	C₄×C₈	—	2	2	no	—
345	#95	—	2	2	no	—
346	#95	D₄×C₂₃	1	2	yes	—
347	C₂₃	#25424	1	1	yes	characteristic, direct product factor, Sylow
348	C₂⁴	—	2	2	no	—
349	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
350	C₂×Q₈	—	2	2	no	—
351	C₂×Q₈	—	2	2	no	—
352	C₂×Q₈	—	2	2	no	—
353	C₂×Q₈	—	2	2	no	—
354	C₂×Q₈	—	2	2	no	—
355	C₂×Q₈	C₂×D₄×C₂₃	1	4	yes	—
356	C₂×D₄	—	2	2	no	—
357	C₂×D₄	—	2	2	no	—
358	C₂×D₄	—	2	2	no	—
359	C₂×D₄	—	2	2	no	—
360	C₂×D₄	—	2	2	no	—
361	C₂×D₄	—	2	2	no	—
362	C₂×D₄	—	4	4	no	—
363	C₂×D₄	—	4	4	no	—
364	C₂×D₄	—	8	2	no	—
365	C₂²×C₄	—	2	2	no	—
366	C₂²×C₄	—	2	2	no	—
367	C₂²×C₄	—	2	1	no	—
368	C₂²×C₄	—	2	1	no	—
369	C₂²×C₄	—	2	1	no	—
370	C₂²×C₄	—	2	1	no	—
371	C₂²×C₄	—	2	1	no	—
372	C₂²×C₄	—	2	1	no	—
373	C₂²×C₄	—	2	2	no	—
374	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
375	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
376	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
377	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
378	Q₁₆	—	2	8	no	—
379	SD₁₆	—	4	8	no	—
380	D₈	—	4	4	no	—
381	M₁₆	—	4	4	no	—
382	C₂×C₈	—	2	1	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	—	2	1	no	—
388	C₂×C₈	—	2	1	no	—
389	C₂×C₈	—	2	1	no	—
390	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
391	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
392	C₂×C₈	#41×C₂₃	1	2	yes	—
393	C₂×C₈	#41×C₂₃	1	2	yes	—
394	C₄⋊C₄	—	2	2	no	—
395	C₄⋊C₄	—	2	2	no	—
396	C₄⋊C₄	—	2	2	no	—
397	C₄⋊C₄	—	4	4	no	—
398	C₄⋊C₄	—	4	4	no	—
399	C₄⋊C₄	—	8	2	no	—
400	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
401	C₂²⋊C₄	—	4	2	no	—
402	C₂²⋊C₄	—	4	4	no	—
403	C₂²⋊C₄	—	4	4	no	—
404	C₂²⋊C₄	—	4	4	no	—
405	C₂²⋊C₄	—	4	4	no	—
406	C₂²⋊C₄	—	4	4	no	—
407	C₂²⋊C₄	—	8	2	no	—
408	C₄²	—	2	1	no	—
409	C₄²	—	2	1	no	—
410	C₄²	—	2	1	no	—
411	C₄²	—	2	1	no	—
412	C₄²	—	4	4	no	—
413	C₂³	—	2	2	no	—
414	C₂³	—	2	2	no	—
415	C₂³	—	2	1	no	—
416	C₂³	—	2	1	no	—
417	C₂³	—	2	1	no	—
418	C₂³	—	4	2	no	—
419	C₂³	—	4	2	no	—
420	C₂³	—	4	2	no	—
421	C₂³	—	4	2	no	—
422	C₂³	—	4	2	no	—
423	C₂³	—	4	2	no	—
424	C₂³	—	4	2	no	—
425	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
426	Q₈	—	2	4	no	—
427	Q₈	—	2	4	no	—
428	Q₈	—	4	2	no	—
429	Q₈	—	4	2	no	—
430	D₄	—	4	2	no	—
431	D₄	—	4	2	no	—
432	D₄	—	4	2	no	—
433	D₄	—	4	2	no	—
434	D₄	—	8	8	no	—
435	C₂×C₄	—	2	2	no	—
436	C₂×C₄	—	2	2	no	—
437	C₂×C₄	—	2	2	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	2	1	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	2	4	no	—
443	C₂×C₄	—	2	1	no	—
444	C₂×C₄	—	2	4	no	—
445	C₂×C₄	—	2	1	no	—
446	C₂×C₄	—	4	2	no	—
447	C₂×C₄	—	4	2	no	—
448	C₂×C₄	—	4	2	no	—
449	C₂×C₄	—	4	2	no	—
450	C₂×C₄	—	4	1	no	—
451	C₂×C₄	—	4	1	no	—
452	C₂×C₄	—	4	1	no	—
453	C₂×C₄	—	4	1	no	—
454	C₂×C₄	—	4	2	no	—
455	C₂×C₄	—	4	2	no	—
456	C₂×C₄	—	4	2	no	—
457	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
458	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
459	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
460	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
461	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
462	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
463	C₈	—	2	2	no	—
464	C₈	—	4	1	no	—
465	C₈	—	4	1	no	—
466	C₈	C₂₃×#121	1	4	yes	—
467	C₂²	—	4	1	no	—
468	C₂²	—	4	1	no	—
469	C₂²	—	4	1	no	—
470	C₂²	—	4	2	no	—
471	C₂²	—	4	2	no	—
472	C₂²	—	8	2	no	—
473	C₂²	—	8	2	no	—
474	C₂²	—	8	2	no	—
475	C₂²	—	8	2	no	—
476	C₂²	—	8	2	no	—
477	C₂²	—	8	2	no	—
478	C₂²	—	8	2	no	—
479	C₂²	C₂₃×#532	1	1	yes	characteristic
480	C₂²	D₄²×C₂₃	1	1	yes	characteristic
481	C₂²	D₄²×C₂₃	1	1	yes	characteristic
482	C₂²	C₂₃×#575	1	2	yes	—
483	C₂²	C₂₃×#577	1	2	yes	—
484	C₄	—	4	1	no	—
485	C₄	—	4	4	no	—
486	C₄	—	4	2	no	—
487	C₄	—	4	2	no	—
488	C₄	—	4	4	no	—
489	C₄	—	8	2	no	—
490	C₄	C₂×C₂₃×#121	1	1	yes	characteristic
491	C₄	C₂₃×#540	1	1	yes	characteristic
492	C₄	C₂₃×#546	1	1	yes	characteristic
493	C₄	C₂₃×#550	1	1	yes	characteristic
494	C₂	—	8	1	no	—
495	C₂	—	16	2	no	—
496	C₂	C₂₃×#2428	1	1	yes	characteristic
497	C₂	C₂₃×#3173	1	1	yes	characteristic
498	C₂	C₂₃×#3183	1	1	yes	characteristic
499	C₂	C₂₃×#3271	1	2	yes	—
500	C₂	C₂₃×#3275	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂×C₂₃×#469	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₄×C₂²⋊C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#495	C₂	semidirect
2. C₂×C₂₃×#470	C₂	semidirect
4. C₂×C₂₃×#441	C₂	semidirect
5. C₂₃×#2426	C₂	semidirect
6. C₂₃×#2042	C₂	semidirect
7. C₂₃×#2000	C₂	semidirect
8. C₂₃×#1972	C₂	semidirect
9. C₂₃×#1941	C₂	semidirect
10. C₂₃×#1880	C₂	semidirect
11. C₂₃×#1753	C₂	semidirect
49. C₂×Q₁₆×C₂₃	D₄	semidirect
162. #25424	C₂₃	direct (N × H)
217. C₂×#495	C₂×C₂₃	semidirect
218. C₂×#470	C₂×C₂₃	semidirect
220. C₂×#441	C₂×C₂₃	semidirect
221. #2426	C₂×C₂₃	semidirect
222. #2042	C₂×C₂₃	semidirect
223. #2000	C₂×C₂₃	semidirect
224. #1972	C₂×C₂₃	semidirect
225. #1941	C₂×C₂₃	semidirect
226. #1880	C₂×C₂₃	semidirect
227. #1753	C₂×C₂₃	semidirect
299. C₂×Q₁₆	D₄×C₂₃	semidirect
347. C₂₃	#25424	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×#495	2
C₂×C₂₃×#470	2
C₂×C₂₃×#469	2
C₂×C₂₃×#441	2
C₂₃×#2426	2
C₂₃×#2042	2
C₂₃×#2000	2
C₂₃×#1972	2
C₂₃×#1941	2
C₂₃×#1880	2
C₂₃×#1753	2
#25424	23

Subgroups — C23×#25424 #19173050

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#25424 #19173050