Subgroups — C₂₃×#18777 #19166403

Order 5888.

1138 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#3229	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3101	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#3078	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1651	C₂	1	4	yes	maximal, complemented
6	C₂₃×#1637	C₂	1	4	yes	maximal, complemented
7	C₂₃×#1546	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1486	C₂	1	1	yes	maximal, characteristic
9	C₂₃×#1479	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×Q₈×C₂₃	—	2	1	no	—
11	D₄×Q₈×C₂₃	—	2	2	no	—
12	D₄×Q₈×C₂₃	—	2	2	no	—
13	D₄×Q₈×C₂₃	—	2	2	no	—
14	D₄×Q₈×C₂₃	C₂²	1	4	yes	—
15	C₂₃×#533	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×C₄⋊Q₈	—	2	1	no	—
17	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
19	C₂×C₂₃×C₂²⋊Q₈	—	2	2	no	—
20	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₂₃	—	2	1	no	—
22	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
23	C₂₃×#498	C₂²	1	2	yes	complemented
24	C₂₃×#487	C₂²	1	2	yes	complemented
25	C₂₃×#483	—	2	2	no	—
26	C₂₃×#483	—	2	1	no	—
27	C₂₃×#479	—	2	2	no	—
28	C₂₃×#477	C₂²	1	2	yes	complemented
29	C₂₃×#476	C₂²	1	2	yes	—
30	C₂₃×#475	—	2	2	no	—
31	C₂₃×#475	C₂²	1	2	yes	complemented
32	C₂₃×#461	C₂²	1	2	yes	—
33	C₂₃×#450	—	2	1	no	—
34	C₄×SD₁₆×C₂₃	—	2	2	no	—
35	C₂₃×#431	C₂²	1	1	yes	characteristic
36	C₂₃×#431	C₂²	1	1	yes	characteristic
37	C₂₃×#428	—	2	1	no	—
38	C₂₃×#423	C₂²	1	1	yes	characteristic
39	C₂₃×#419	—	2	1	no	—
40	C₂₃×#332	C₂²	1	4	yes	complemented
41	C₂₃×Q₈⋊C₈	C₂²	1	4	yes	complemented
42	C₂₃×D₄⋊C₈	C₂²	1	4	yes	—
43	C₂²×Q₈×C₂₃	—	2	4	no	—
44	C₂²×Q₈×C₂₃	—	2	2	no	—
45	C₂²×Q₈×C₂₃	—	2	4	no	—
46	C₂²×Q₈×C₂₃	—	2	1	no	—
47	C₂²×Q₈×C₂₃	—	4	4	no	—
48	C₂²×Q₈×C₂₃	—	4	4	no	—
49	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
50	C₂²×D₄×C₂₃	D₄	1	1	yes	characteristic
51	C₂³×C₄×C₂₃	—	2	2	no	—
52	C₂³×C₄×C₂₃	—	2	1	no	—
53	C₂×SD₁₆×C₂₃	—	2	2	no	—
54	C₂×M₁₆×C₂₃	—	2	1	no	—
55	C₂₃×C₄⋊Q₈	—	2	4	no	—
56	C₂₃×C₄⋊Q₈	—	2	4	no	—
57	C₂₃×C₄⋊Q₈	—	2	1	no	—
58	C₂₃×C₄⋊Q₈	—	2	1	no	—
59	C₂₃×C₄⋊Q₈	D₄	1	2	yes	—
60	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
61	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
62	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
63	C₂₃×#125	—	2	1	no	—
64	C₂₃×C₂²⋊Q₈	—	2	4	no	—
65	C₂₃×C₂²⋊Q₈	—	2	8	no	—
66	C₂₃×C₂²⋊Q₈	—	2	4	no	—
67	C₂₃×C₂²⋊Q₈	—	2	8	no	—
68	C₂₃×C₂²⋊Q₈	—	2	1	no	—
69	C₂₃×C₂²⋊Q₈	—	2	1	no	—
70	C₄×Q₈×C₂₃	—	2	1	no	—
71	C₄×Q₈×C₂₃	—	2	1	no	—
72	C₄×Q₈×C₂₃	—	2	1	no	—
73	C₄×Q₈×C₂₃	—	2	1	no	—
74	C₄×Q₈×C₂₃	D₄	1	2	yes	—
75	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
76	C₄×D₄×C₂₃	—	2	4	no	—
77	C₄×D₄×C₂₃	—	2	4	no	—
78	C₄×D₄×C₂₃	—	2	1	no	—
79	C₄×D₄×C₂₃	—	2	1	no	—
80	C₄×D₄×C₂₃	D₄	1	2	yes	—
81	C₄×D₄×C₂₃	C₂³	1	2	yes	—
82	C₂₃×#117	—	2	1	no	—
83	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
84	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
85	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
86	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
88	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
89	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
90	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
91	C₂×C₄²×C₂₃	—	2	1	no	—
92	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
93	C₂₃×#107	—	2	2	no	—
94	C₂₃×#106	—	2	2	no	—
95	C₂₃×#106	—	2	2	no	—
96	C₂₃×#106	—	2	4	no	—
97	C₂₃×C₄⋊C₈	—	2	1	no	—
98	C₂₃×C₄⋊C₈	—	2	1	no	—
99	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
100	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
101	C₂₃×#103	—	2	2	no	—
102	C₂₃×#103	—	2	2	no	—
103	C₂₃×#103	—	2	2	no	—
104	C₂₃×#103	—	2	4	no	—
105	C₂₃×#102	—	2	2	no	—
106	C₂₃×C₂²⋊C₈	—	2	1	no	—
107	C₂₃×C₂²⋊C₈	—	2	1	no	—
108	C₄×C₈×C₂₃	—	2	1	no	—
109	C₄×C₈×C₂₃	C₂³	1	2	yes	—
110	C₂⁴×C₂₃	—	2	1	no	—
111	C₂×Q₈×C₂₃	—	2	8	no	—
112	C₂×Q₈×C₂₃	—	2	1	no	—
113	C₂×Q₈×C₂₃	—	2	2	no	—
114	C₂×Q₈×C₂₃	—	2	2	no	—
115	C₂×Q₈×C₂₃	—	2	2	no	—
116	C₂×Q₈×C₂₃	—	2	1	no	—
117	C₂×Q₈×C₂₃	—	2	1	no	—
118	C₂×Q₈×C₂₃	—	4	8	no	—
119	C₂×Q₈×C₂₃	—	4	8	no	—
120	C₂×Q₈×C₂₃	—	4	16	no	—
121	C₂×Q₈×C₂₃	—	4	8	no	—
122	C₂×Q₈×C₂₃	—	4	2	no	—
123	C₂×Q₈×C₂₃	—	4	2	no	—
124	C₂×Q₈×C₂₃	SD₁₆	1	4	yes	complemented
125	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
126	C₂×D₄×C₂₃	—	2	4	no	—
127	C₂×D₄×C₂₃	—	2	1	no	—
128	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
129	C₂²×C₄×C₂₃	—	2	1	no	—
130	C₂²×C₄×C₂₃	—	2	8	no	—
131	C₂²×C₄×C₂₃	—	2	1	no	—
132	C₂²×C₄×C₂₃	—	2	2	no	—
133	C₂²×C₄×C₂₃	—	2	1	no	—
134	C₂²×C₄×C₂₃	—	2	2	no	—
135	C₂²×C₄×C₂₃	—	2	2	no	—
136	C₂²×C₄×C₂₃	—	4	8	no	—
137	C₂²×C₄×C₂₃	—	4	2	no	—
138	C₂²×C₄×C₂₃	—	4	2	no	—
139	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
140	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
141	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
142	SD₁₆×C₂₃	—	4	4	no	—
143	M₁₆×C₂₃	—	4	2	no	—
144	C₂×C₈×C₂₃	—	2	2	no	—
145	C₂×C₈×C₂₃	—	2	2	no	—
146	C₂×C₈×C₂₃	—	2	2	no	—
147	C₄⋊C₄×C₂₃	—	2	8	no	—
148	C₄⋊C₄×C₂₃	—	2	2	no	—
149	C₄⋊C₄×C₂₃	—	2	2	no	—
150	C₄⋊C₄×C₂₃	—	2	1	no	—
151	C₄⋊C₄×C₂₃	—	2	1	no	—
152	C₄⋊C₄×C₂₃	—	2	2	no	—
153	C₄⋊C₄×C₂₃	—	2	2	no	—
154	C₄⋊C₄×C₂₃	—	2	2	no	—
155	C₄⋊C₄×C₂₃	—	2	1	no	—
156	C₄⋊C₄×C₂₃	—	2	1	no	—
157	C₄⋊C₄×C₂₃	—	2	1	no	—
158	C₄⋊C₄×C₂₃	—	2	1	no	—
159	C₄⋊C₄×C₂₃	—	4	1	no	—
160	C₄⋊C₄×C₂₃	—	4	1	no	—
161	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
162	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
163	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
164	C₂²⋊C₄×C₂₃	—	2	8	no	—
165	C₂²⋊C₄×C₂₃	—	2	2	no	—
166	C₂²⋊C₄×C₂₃	—	2	2	no	—
167	C₂²⋊C₄×C₂₃	—	4	1	no	—
168	C₄²×C₂₃	—	2	2	no	—
169	C₄²×C₂₃	—	2	2	no	—
170	C₄²×C₂₃	—	4	1	no	—
171	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
172	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
173	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
174	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
175	#18777	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
176	C₂³×C₂₃	—	2	4	no	—
177	C₂³×C₂₃	—	2	2	no	—
178	C₂³×C₂₃	—	4	2	no	—
179	C₂³×C₂₃	—	4	2	no	—
180	C₂³×C₂₃	#120	1	1	yes	characteristic
181	Q₈×C₂₃	—	2	2	no	—
182	Q₈×C₂₃	—	2	4	no	—
183	Q₈×C₂₃	—	4	16	no	—
184	Q₈×C₂₃	—	4	2	no	—
185	Q₈×C₂₃	—	4	2	no	—
186	Q₈×C₂₃	—	4	2	no	—
187	Q₈×C₂₃	—	4	4	no	—
188	Q₈×C₂₃	—	4	2	no	—
189	Q₈×C₂₃	C₂×SD₁₆	1	4	yes	—
190	D₄×C₂₃	—	2	4	no	—
191	D₄×C₂₃	—	2	2	no	—
192	D₄×C₂₃	#137	1	4	yes	—
193	C₂×C₄×C₂₃	—	2	2	no	—
194	C₂×C₄×C₂₃	—	2	4	no	—
195	C₂×C₄×C₂₃	—	2	2	no	—
196	C₂×C₄×C₂₃	—	2	2	no	—
197	C₂×C₄×C₂₃	—	2	4	no	—
198	C₂×C₄×C₂₃	—	2	2	no	—
199	C₂×C₄×C₂₃	—	4	1	no	—
200	C₂×C₄×C₂₃	—	4	16	no	—
201	C₂×C₄×C₂₃	—	4	2	no	—
202	C₂×C₄×C₂₃	—	4	4	no	—
203	C₂×C₄×C₂₃	—	4	2	no	—
204	C₂×C₄×C₂₃	—	4	2	no	—
205	C₂×C₄×C₂₃	—	4	2	no	—
206	C₂×C₄×C₂₃	—	4	4	no	—
207	C₂×C₄×C₂₃	—	4	1	no	—
208	C₂×C₄×C₂₃	—	4	1	no	—
209	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
210	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
211	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
212	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
213	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
214	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
215	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
216	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
217	C₈×C₂₃	—	4	2	no	—
218	C₈×C₂₃	—	4	2	no	—
219	C₈×C₂₃	—	4	2	no	—
220	C₂×D₄×Q₈	C₂×C₂₃	1	1	yes	characteristic
221	#3229	C₂×C₂₃	1	1	yes	characteristic, complemented
222	#3101	C₂×C₂₃	1	1	yes	characteristic
223	#3078	C₂×C₂₃	1	1	yes	characteristic, complemented
224	#1651	C₂×C₂₃	1	4	yes	complemented
225	#1637	C₂×C₂₃	1	4	yes	complemented
226	#1546	C₂×C₂₃	1	1	yes	characteristic, complemented
227	#1486	C₂×C₂₃	1	1	yes	characteristic
228	#1479	C₂×C₂₃	1	1	yes	characteristic, complemented
229	C₂²×C₂₃	—	2	1	no	—
230	C₂²×C₂₃	—	2	1	no	—
231	C₂²×C₂₃	—	2	4	no	—
232	C₂²×C₂₃	—	4	8	no	—
233	C₂²×C₂₃	—	4	2	no	—
234	C₂²×C₂₃	—	4	2	no	—
235	C₂²×C₂₃	#450	1	2	yes	—
236	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
237	C₄×C₂₃	—	2	4	no	—
238	C₄×C₂₃	—	2	1	no	—
239	C₄×C₂₃	—	2	1	no	—
240	C₄×C₂₃	—	2	1	no	—
241	C₄×C₂₃	—	4	4	no	—
242	C₄×C₂₃	—	4	2	no	—
243	C₄×C₂₃	—	4	2	no	—
244	C₄×C₂₃	—	8	1	no	—
245	C₄×C₂₃	—	8	1	no	—
246	C₄×C₂₃	#450	1	2	yes	—
247	C₄×C₂₃	C₂²×SD₁₆	1	1	yes	characteristic
248	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
249	C₄×C₂₃	C₂×#137	1	2	yes	—
250	C₂³×Q₈	—	2	1	no	—
251	D₄×Q₈	—	2	2	no	—
252	D₄×Q₈	—	2	2	no	—
253	D₄×Q₈	—	2	2	no	—
254	D₄×Q₈	C₂²×C₂₃	1	4	yes	—
255	#533	C₂²×C₂₃	1	1	yes	characteristic
256	C₂×C₄⋊Q₈	—	2	1	no	—
257	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
258	C₂×C₂²⋊Q₈	—	2	1	no	—
259	C₂×C₂²⋊Q₈	—	2	2	no	—
260	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
261	C₂×C₄×D₄	—	2	1	no	—
262	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
263	#498	C₂²×C₂₃	1	2	yes	complemented
264	#487	C₂²×C₂₃	1	2	yes	complemented
265	#483	—	2	1	no	—
266	#483	—	2	2	no	—
267	#479	—	2	2	no	—
268	#477	C₂²×C₂₃	1	2	yes	complemented
269	#476	C₂²×C₂₃	1	2	yes	—
270	#475	—	2	2	no	—
271	#475	C₂²×C₂₃	1	2	yes	complemented
272	#461	C₂²×C₂₃	1	2	yes	—
273	#450	—	2	1	no	—
274	C₄×SD₁₆	—	2	2	no	—
275	#431	C₂²×C₂₃	1	1	yes	characteristic
276	#431	C₂²×C₂₃	1	1	yes	characteristic
277	#428	—	2	1	no	—
278	#423	C₂²×C₂₃	1	1	yes	characteristic
279	#419	—	2	1	no	—
280	#332	C₂²×C₂₃	1	4	yes	complemented
281	Q₈⋊C₈	C₂²×C₂₃	1	4	yes	complemented
282	D₄⋊C₈	C₂²×C₂₃	1	4	yes	—
283	C₂×C₂₃	—	2	2	no	—
284	C₂×C₂₃	—	4	4	no	—
285	C₂×C₂₃	C₂×#450	1	1	yes	characteristic
286	C₂×C₂₃	#3004	1	1	yes	characteristic
287	C₂×C₂₃	#3020	1	1	yes	characteristic
288	C₂²×Q₈	—	2	2	no	—
289	C₂²×Q₈	—	2	4	no	—
290	C₂²×Q₈	—	2	4	no	—
291	C₂²×Q₈	—	2	1	no	—
292	C₂²×Q₈	—	4	4	no	—
293	C₂²×Q₈	—	4	4	no	—
294	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
295	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
296	C₂³×C₄	—	2	2	no	—
297	C₂³×C₄	—	2	1	no	—
298	C₂×SD₁₆	—	2	2	no	—
299	C₂×M₁₆	—	2	1	no	—
300	C₄⋊Q₈	—	2	4	no	—
301	C₄⋊Q₈	—	2	4	no	—
302	C₄⋊Q₈	—	2	1	no	—
303	C₄⋊Q₈	—	2	1	no	—
304	C₄⋊Q₈	D₄×C₂₃	1	2	yes	—
305	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
306	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
307	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
308	#125	—	2	1	no	—
309	C₂²⋊Q₈	—	2	4	no	—
310	C₂²⋊Q₈	—	2	8	no	—
311	C₂²⋊Q₈	—	2	4	no	—
312	C₂²⋊Q₈	—	2	8	no	—
313	C₂²⋊Q₈	—	2	1	no	—
314	C₂²⋊Q₈	—	2	1	no	—
315	C₄×Q₈	—	2	1	no	—
316	C₄×Q₈	—	2	1	no	—
317	C₄×Q₈	—	2	1	no	—
318	C₄×Q₈	—	2	1	no	—
319	C₄×Q₈	D₄×C₂₃	1	2	yes	—
320	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
321	C₄×D₄	—	2	4	no	—
322	C₄×D₄	—	2	4	no	—
323	C₄×D₄	—	2	1	no	—
324	C₄×D₄	—	2	1	no	—
325	C₄×D₄	D₄×C₂₃	1	2	yes	—
326	C₄×D₄	C₂³×C₂₃	1	2	yes	—
327	#117	—	2	1	no	—
328	C₂×C₄⋊C₄	—	2	2	no	—
329	C₂×C₄⋊C₄	—	2	1	no	—
330	C₂×C₄⋊C₄	—	2	1	no	—
331	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
332	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
333	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
334	C₂×C₂²⋊C₄	—	2	2	no	—
335	C₂×C₂²⋊C₄	—	2	1	no	—
336	C₂×C₄²	—	2	1	no	—
337	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
338	#107	—	2	2	no	—
339	#106	—	2	2	no	—
340	#106	—	2	2	no	—
341	#106	—	2	4	no	—
342	C₄⋊C₈	—	2	1	no	—
343	C₄⋊C₈	—	2	1	no	—
344	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
345	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
346	#103	—	2	2	no	—
347	#103	—	2	2	no	—
348	#103	—	2	2	no	—
349	#103	—	2	4	no	—
350	#102	—	2	2	no	—
351	C₂²⋊C₈	—	2	1	no	—
352	C₂²⋊C₈	—	2	1	no	—
353	C₄×C₈	—	2	1	no	—
354	C₄×C₈	C₂³×C₂₃	1	2	yes	—
355	C₂₃	#18777	1	1	yes	characteristic, direct product factor, Sylow
356	C₂⁴	—	2	1	no	—
357	C₂×Q₈	—	2	8	no	—
358	C₂×Q₈	—	2	1	no	—
359	C₂×Q₈	—	2	2	no	—
360	C₂×Q₈	—	2	2	no	—
361	C₂×Q₈	—	2	2	no	—
362	C₂×Q₈	—	2	1	no	—
363	C₂×Q₈	—	2	1	no	—
364	C₂×Q₈	—	4	8	no	—
365	C₂×Q₈	—	4	8	no	—
366	C₂×Q₈	—	4	16	no	—
367	C₂×Q₈	—	4	8	no	—
368	C₂×Q₈	—	4	2	no	—
369	C₂×Q₈	—	4	2	no	—
370	C₂×Q₈	SD₁₆×C₂₃	1	4	yes	complemented
371	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
372	C₂×D₄	—	2	4	no	—
373	C₂×D₄	—	2	1	no	—
374	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
375	C₂²×C₄	—	2	1	no	—
376	C₂²×C₄	—	2	8	no	—
377	C₂²×C₄	—	2	1	no	—
378	C₂²×C₄	—	2	2	no	—
379	C₂²×C₄	—	2	1	no	—
380	C₂²×C₄	—	2	2	no	—
381	C₂²×C₄	—	2	2	no	—
382	C₂²×C₄	—	4	8	no	—
383	C₂²×C₄	—	4	2	no	—
384	C₂²×C₄	—	4	2	no	—
385	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
386	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
387	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
388	SD₁₆	—	4	4	no	—
389	M₁₆	—	4	2	no	—
390	C₂×C₈	—	2	2	no	—
391	C₂×C₈	—	2	2	no	—
392	C₂×C₈	—	2	2	no	—
393	C₄⋊C₄	—	2	8	no	—
394	C₄⋊C₄	—	2	2	no	—
395	C₄⋊C₄	—	2	2	no	—
396	C₄⋊C₄	—	2	1	no	—
397	C₄⋊C₄	—	2	1	no	—
398	C₄⋊C₄	—	2	2	no	—
399	C₄⋊C₄	—	2	2	no	—
400	C₄⋊C₄	—	2	2	no	—
401	C₄⋊C₄	—	2	1	no	—
402	C₄⋊C₄	—	2	1	no	—
403	C₄⋊C₄	—	2	1	no	—
404	C₄⋊C₄	—	2	1	no	—
405	C₄⋊C₄	—	4	1	no	—
406	C₄⋊C₄	—	4	1	no	—
407	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
408	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
409	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
410	C₂²⋊C₄	—	2	8	no	—
411	C₂²⋊C₄	—	2	2	no	—
412	C₂²⋊C₄	—	2	2	no	—
413	C₂²⋊C₄	—	4	1	no	—
414	C₄²	—	2	2	no	—
415	C₄²	—	2	2	no	—
416	C₄²	—	4	1	no	—
417	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
418	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
419	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
420	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
421	C₂³	—	2	4	no	—
422	C₂³	—	2	2	no	—
423	C₂³	—	4	2	no	—
424	C₂³	—	4	2	no	—
425	C₂³	C₂₃×#120	1	1	yes	characteristic
426	Q₈	—	2	2	no	—
427	Q₈	—	2	4	no	—
428	Q₈	—	4	16	no	—
429	Q₈	—	4	2	no	—
430	Q₈	—	4	2	no	—
431	Q₈	—	4	2	no	—
432	Q₈	—	4	4	no	—
433	Q₈	—	4	2	no	—
434	Q₈	C₂×SD₁₆×C₂₃	1	4	yes	—
435	D₄	—	2	4	no	—
436	D₄	—	2	2	no	—
437	D₄	C₂₃×#137	1	4	yes	—
438	C₂×C₄	—	2	4	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	2	4	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	4	1	no	—
445	C₂×C₄	—	4	16	no	—
446	C₂×C₄	—	4	2	no	—
447	C₂×C₄	—	4	4	no	—
448	C₂×C₄	—	4	2	no	—
449	C₂×C₄	—	4	2	no	—
450	C₂×C₄	—	4	2	no	—
451	C₂×C₄	—	4	4	no	—
452	C₂×C₄	—	4	1	no	—
453	C₂×C₄	—	4	1	no	—
454	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
455	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
456	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
457	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
458	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
459	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
460	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
461	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
462	C₈	—	4	2	no	—
463	C₈	—	4	2	no	—
464	C₈	—	4	2	no	—
465	C₂²	—	2	1	no	—
466	C₂²	—	2	1	no	—
467	C₂²	—	2	4	no	—
468	C₂²	—	4	8	no	—
469	C₂²	—	4	2	no	—
470	C₂²	—	4	2	no	—
471	C₂²	C₂₃×#450	1	2	yes	—
472	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
473	C₄	—	2	4	no	—
474	C₄	—	2	1	no	—
475	C₄	—	2	1	no	—
476	C₄	—	2	1	no	—
477	C₄	—	4	4	no	—
478	C₄	—	4	2	no	—
479	C₄	—	4	2	no	—
480	C₄	—	8	1	no	—
481	C₄	—	8	1	no	—
482	C₄	C₂₃×#450	1	2	yes	—
483	C₄	C₂²×SD₁₆×C₂₃	1	1	yes	characteristic
484	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
485	C₄	C₂×C₂₃×#137	1	2	yes	—
486	C₂	—	2	2	no	—
487	C₂	—	4	4	no	—
488	C₂	C₂×C₂₃×#450	1	1	yes	characteristic
489	C₂	C₂₃×#3004	1	1	yes	characteristic
490	C₂	C₂₃×#3020	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂²×D₄×C₂₃	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#3229	C₂	semidirect
4. C₂₃×#3078	C₂	semidirect
5. C₂₃×#1651	C₂	semidirect
6. C₂₃×#1637	C₂	semidirect
7. C₂₃×#1546	C₂	semidirect
9. C₂₃×#1479	C₂	semidirect
23. C₂₃×#498	C₂²	semidirect
24. C₂₃×#487	C₂²	semidirect
28. C₂₃×#477	C₂²	semidirect
31. C₂₃×#475	C₂²	semidirect
40. C₂₃×#332	C₂²	semidirect
41. C₂₃×Q₈⋊C₈	C₂²	semidirect
124. C₂×Q₈×C₂₃	SD₁₆	semidirect
175. #18777	C₂₃	direct (N × H)
221. #3229	C₂×C₂₃	semidirect
223. #3078	C₂×C₂₃	semidirect
224. #1651	C₂×C₂₃	semidirect
225. #1637	C₂×C₂₃	semidirect
226. #1546	C₂×C₂₃	semidirect
228. #1479	C₂×C₂₃	semidirect
263. #498	C₂²×C₂₃	semidirect
264. #487	C₂²×C₂₃	semidirect
268. #477	C₂²×C₂₃	semidirect
271. #475	C₂²×C₂₃	semidirect
280. #332	C₂²×C₂₃	semidirect
281. Q₈⋊C₈	C₂²×C₂₃	semidirect
355. C₂₃	#18777	direct (N × H)
370. C₂×Q₈	SD₁₆×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×Q₈×C₂₃	2
C₂₃×#3229	2
C₂₃×#3101	2
C₂₃×#3078	2
C₂₃×#1651	2
C₂₃×#1637	2
C₂₃×#1546	2
C₂₃×#1486	2
C₂₃×#1479	2
#18777	23

Subgroups — C23×#18777 #19166403

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18777 #19166403