Subgroups — C₂₃×#21713 #19169339

Order 5888.

802 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#446	C₂	1	2	yes	maximal
2	C₄×C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1985	C₂	1	2	yes	maximal
4	C₂₃×#1940	C₂	1	2	yes	maximal
5	C₂₃×#1939	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#1939	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#1842	C₂	1	1	yes	maximal, characteristic
8	C₂₃×#1771	C₂	1	2	yes	maximal
9	C₂₃×#1770	C₂	1	1	yes	maximal, characteristic
10	C₈×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
11	C₄×C₂₃×#97	C₂	1	1	yes	maximal, characteristic
12	C₂×C₂₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	—
14	C₂₃×#446	—	2	8	no	—
15	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
18	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
21	C₂×C₂₃×#97	C₂²	1	2	yes	—
22	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
23	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
24	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
25	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
29	C₂₃×#391	C₄	1	2	yes	—
30	C₂₃×#389	C₄	1	1	yes	characteristic
31	C₂₃×#389	C₄	1	1	yes	characteristic
32	C₂₃×#389	C₄	1	2	yes	—
33	C₂₃×#384	C₄	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
35	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
36	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
37	C₄³×C₂₃	C₂²	1	1	yes	characteristic
38	C₂₃×#336	C₂²	1	2	yes	—
39	C₂₃×#336	C₂²	1	2	yes	—
40	C₂₃×#336	C₂²	1	2	yes	—
41	C₂₃×#336	C₂²	1	2	yes	—
42	C₂²×Q₈×C₂₃	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
44	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
45	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
49	C₂₃×C₄⋊Q₈	—	2	4	no	—
50	C₄×Q₈×C₂₃	—	2	4	no	—
51	C₄×Q₈×C₂₃	—	2	4	no	—
52	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
56	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
57	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
58	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
59	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
60	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
66	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
67	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
70	C₂₃×C₄⋊C₈	—	2	4	no	—
71	C₂₃×C₄⋊C₈	—	2	4	no	—
72	C₂₃×C₄⋊C₈	—	2	4	no	—
73	C₂₃×C₄⋊C₈	—	2	2	no	—
74	C₂₃×C₄⋊C₈	—	2	2	no	—
75	C₂₃×C₄⋊C₈	—	2	4	no	—
76	C₂₃×#97	—	2	4	no	—
77	C₂₃×#97	—	2	2	no	—
78	C₂₃×#97	—	2	2	no	—
79	C₂₃×#97	D₄	1	4	yes	—
80	C₂₃×#97	Q₈	1	4	yes	complemented
81	C₄×C₈×C₂₃	—	2	4	no	—
82	C₄×C₈×C₂₃	—	2	2	no	—
83	C₄×C₈×C₂₃	Q₈	1	4	yes	complemented
84	C₂₃×#95	C₂×C₄	1	2	yes	—
85	C₂₃×#95	C₂×C₄	1	2	yes	—
86	C₂₃×#95	C₂×C₄	1	2	yes	—
87	C₂₃×#95	C₂×C₄	1	2	yes	—
88	C₂×Q₈×C₂₃	—	2	4	no	—
89	C₂×Q₈×C₂₃	—	2	4	no	—
90	C₂×Q₈×C₂₃	—	4	2	no	—
91	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
92	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
94	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
100	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
102	C₂×C₈×C₂₃	—	2	1	no	—
103	C₂×C₈×C₂₃	—	2	1	no	—
104	C₂×C₈×C₂₃	—	2	1	no	—
105	C₂×C₈×C₂₃	—	2	1	no	—
106	C₂×C₈×C₂₃	—	2	1	no	—
107	C₂×C₈×C₂₃	—	2	1	no	—
108	C₂×C₈×C₂₃	—	2	2	no	—
109	C₂×C₈×C₂₃	—	2	2	no	—
110	C₂×C₈×C₂₃	—	2	2	no	—
111	C₂×C₈×C₂₃	—	2	2	no	—
112	C₂×C₈×C₂₃	—	2	2	no	—
113	C₂×C₈×C₂₃	—	2	2	no	—
114	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
115	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
116	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
117	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
118	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
119	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
120	C₄⋊C₄×C₂₃	—	2	4	no	—
121	C₄⋊C₄×C₂₃	—	2	4	no	—
122	C₄⋊C₄×C₂₃	—	2	4	no	—
123	C₄⋊C₄×C₂₃	—	2	4	no	—
124	C₄⋊C₄×C₂₃	—	2	2	no	—
125	C₄⋊C₄×C₂₃	—	2	2	no	—
126	C₄⋊C₄×C₂₃	—	2	2	no	—
127	C₄⋊C₄×C₂₃	—	4	2	no	—
128	C₄⋊C₄×C₂₃	M₁₆	1	4	yes	—
129	C₄²×C₂₃	—	2	2	no	—
130	C₄²×C₂₃	—	2	2	no	—
131	C₄²×C₂₃	—	2	2	no	—
132	C₄²×C₂₃	—	2	2	no	—
133	C₄²×C₂₃	—	2	2	no	—
134	C₄²×C₂₃	—	2	2	no	—
135	C₄²×C₂₃	—	4	1	no	—
136	C₄²×C₂₃	—	4	1	no	—
137	C₄²×C₂₃	C₂×D₄	1	2	yes	—
138	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
139	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
140	C₄²×C₂₃	#41	1	2	yes	—
141	C₄²×C₂₃	#41	1	2	yes	—
142	C₄²×C₂₃	#41	1	2	yes	—
143	#21713	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
145	Q₈×C₂₃	—	4	8	no	—
146	C₂×C₄×C₂₃	—	2	2	no	—
147	C₂×C₄×C₂₃	—	2	1	no	—
148	C₂×C₄×C₂₃	—	2	1	no	—
149	C₂×C₄×C₂₃	—	2	1	no	—
150	C₂×C₄×C₂₃	—	2	1	no	—
151	C₂×C₄×C₂₃	—	2	1	no	—
152	C₂×C₄×C₂₃	—	2	1	no	—
153	C₂×C₄×C₂₃	—	2	1	no	—
154	C₂×C₄×C₂₃	—	2	1	no	—
155	C₂×C₄×C₂₃	—	2	1	no	—
156	C₂×C₄×C₂₃	—	2	1	no	—
157	C₂×C₄×C₂₃	—	2	2	no	—
158	C₂×C₄×C₂₃	—	2	2	no	—
159	C₂×C₄×C₂₃	—	2	2	no	—
160	C₂×C₄×C₂₃	—	2	2	no	—
161	C₂×C₄×C₂₃	—	2	2	no	—
162	C₂×C₄×C₂₃	—	2	2	no	—
163	C₂×C₄×C₂₃	—	2	2	no	—
164	C₂×C₄×C₂₃	—	2	2	no	—
165	C₂×C₄×C₂₃	—	2	2	no	—
166	C₂×C₄×C₂₃	—	2	2	no	—
167	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
168	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
169	C₂×C₄×C₂₃	C₄×Q₈	1	2	yes	—
170	C₂×C₄×C₂₃	C₄×Q₈	1	2	yes	—
171	C₂×C₄×C₂₃	C₄×Q₈	1	2	yes	—
172	C₂×C₄×C₂₃	C₄×Q₈	1	2	yes	—
173	C₂×C₄×C₂₃	C₂×M₁₆	1	4	yes	—
174	C₂×C₄×C₂₃	C₂×M₁₆	1	2	yes	—
175	C₂×C₄×C₂₃	#131	1	2	yes	—
176	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
177	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
178	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
179	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
180	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
181	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
182	C₈×C₂₃	—	2	2	no	—
183	C₈×C₂₃	—	2	2	no	—
184	C₈×C₂₃	—	2	2	no	—
185	C₈×C₂₃	—	4	2	no	—
186	C₈×C₂₃	—	4	2	no	—
187	C₈×C₂₃	C₄⋊Q₈	1	4	yes	complemented
188	C₂×#446	C₂×C₂₃	1	2	yes	—
189	C₄×C₄⋊Q₈	C₂×C₂₃	1	1	yes	characteristic
190	#1985	C₂×C₂₃	1	2	yes	—
191	#1940	C₂×C₂₃	1	2	yes	—
192	#1939	C₂×C₂₃	1	1	yes	characteristic
193	#1939	C₂×C₂₃	1	1	yes	characteristic
194	#1842	C₂×C₂₃	1	1	yes	characteristic
195	#1771	C₂×C₂₃	1	2	yes	—
196	#1770	C₂×C₂₃	1	1	yes	characteristic
197	C₈×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic
198	C₄×#97	C₂×C₂₃	1	1	yes	characteristic
199	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
200	C₂²×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
201	C₂²×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
202	C₂²×C₂₃	C₂²×M₁₆	1	1	yes	characteristic
203	C₂²×C₂₃	C₂×#131	1	1	yes	characteristic
204	C₂²×C₂₃	#568	1	1	yes	characteristic
205	C₂²×C₂₃	#568	1	1	yes	characteristic
206	C₄×C₂₃	—	2	2	no	—
207	C₄×C₂₃	—	2	4	no	—
208	C₄×C₂₃	—	2	2	no	—
209	C₄×C₂₃	—	2	2	no	—
210	C₄×C₂₃	—	2	2	no	—
211	C₄×C₂₃	—	2	2	no	—
212	C₄×C₂₃	—	4	2	no	—
213	C₄×C₂₃	—	4	2	no	—
214	C₄×C₂₃	—	4	2	no	—
215	C₄×C₂₃	#446	1	4	yes	complemented
216	C₄×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
217	C₄×C₂₃	#532	1	1	yes	characteristic
218	C₄×C₂₃	#533	1	1	yes	characteristic
219	C₄×C₂₃	#533	1	1	yes	characteristic
220	C₂×C₄⋊Q₈	C₄×C₂₃	1	1	yes	characteristic
221	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	—
222	#446	—	2	8	no	—
223	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
224	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
225	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
226	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
227	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
228	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
229	C₂×#97	C₂²×C₂₃	1	2	yes	—
230	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
231	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
232	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
233	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
234	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
235	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
236	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
237	#391	C₄×C₂₃	1	2	yes	—
238	#389	C₄×C₂₃	1	1	yes	characteristic
239	#389	C₄×C₂₃	1	1	yes	characteristic
240	#389	C₄×C₂₃	1	2	yes	—
241	#384	C₄×C₂₃	1	1	yes	characteristic
242	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
243	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
244	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
245	C₄³	C₂²×C₂₃	1	1	yes	characteristic
246	#336	C₂²×C₂₃	1	2	yes	—
247	#336	C₂²×C₂₃	1	2	yes	—
248	#336	C₂²×C₂₃	1	2	yes	—
249	#336	C₂²×C₂₃	1	2	yes	—
250	C₂×C₂₃	C₄×C₄⋊Q₈	1	1	yes	characteristic
251	C₂×C₂₃	D₄×M₁₆	1	1	yes	characteristic
252	C₂×C₂₃	#2935	1	1	yes	characteristic
253	C₂×C₂₃	C₂×#446	1	1	yes	characteristic
254	C₂×C₂₃	#2961	1	1	yes	characteristic
255	C₂×C₂₃	#2962	1	1	yes	characteristic
256	C₂×C₂₃	Q₈×M₁₆	1	1	yes	characteristic
257	C₂²×Q₈	C₂×C₄×C₂₃	1	2	yes	—
258	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
259	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
260	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
261	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
262	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
263	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
264	C₄⋊Q₈	—	2	4	no	—
265	C₄×Q₈	—	2	4	no	—
266	C₄×Q₈	—	2	4	no	—
267	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
268	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
269	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
270	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
271	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
272	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
273	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
274	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
275	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
276	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
277	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
278	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
279	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
280	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
281	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
282	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
283	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
284	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
285	C₄⋊C₈	—	2	4	no	—
286	C₄⋊C₈	—	2	4	no	—
287	C₄⋊C₈	—	2	4	no	—
288	C₄⋊C₈	—	2	2	no	—
289	C₄⋊C₈	—	2	2	no	—
290	C₄⋊C₈	—	2	4	no	—
291	#97	—	2	4	no	—
292	#97	—	2	2	no	—
293	#97	—	2	2	no	—
294	#97	D₄×C₂₃	1	4	yes	—
295	#97	Q₈×C₂₃	1	4	yes	complemented
296	C₄×C₈	—	2	4	no	—
297	C₄×C₈	—	2	2	no	—
298	C₄×C₈	Q₈×C₂₃	1	4	yes	complemented
299	#95	C₂×C₄×C₂₃	1	2	yes	—
300	#95	C₂×C₄×C₂₃	1	2	yes	—
301	#95	C₂×C₄×C₂₃	1	2	yes	—
302	#95	C₂×C₄×C₂₃	1	2	yes	—
303	C₂₃	#21713	1	1	yes	characteristic, direct product factor, Sylow
304	C₂×Q₈	—	2	4	no	—
305	C₂×Q₈	—	2	4	no	—
306	C₂×Q₈	—	4	2	no	—
307	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
308	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
309	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
310	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
311	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
312	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
313	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
314	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
315	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
316	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
317	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
318	C₂×C₈	—	2	1	no	—
319	C₂×C₈	—	2	1	no	—
320	C₂×C₈	—	2	1	no	—
321	C₂×C₈	—	2	1	no	—
322	C₂×C₈	—	2	1	no	—
323	C₂×C₈	—	2	1	no	—
324	C₂×C₈	—	2	2	no	—
325	C₂×C₈	—	2	2	no	—
326	C₂×C₈	—	2	2	no	—
327	C₂×C₈	—	2	2	no	—
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	—	2	2	no	—
330	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
331	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
332	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
333	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
334	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
335	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
336	C₄⋊C₄	—	2	4	no	—
337	C₄⋊C₄	—	2	4	no	—
338	C₄⋊C₄	—	2	4	no	—
339	C₄⋊C₄	—	2	4	no	—
340	C₄⋊C₄	—	2	2	no	—
341	C₄⋊C₄	—	2	2	no	—
342	C₄⋊C₄	—	2	2	no	—
343	C₄⋊C₄	—	4	2	no	—
344	C₄⋊C₄	M₁₆×C₂₃	1	4	yes	—
345	C₄²	—	2	2	no	—
346	C₄²	—	2	2	no	—
347	C₄²	—	2	2	no	—
348	C₄²	—	2	2	no	—
349	C₄²	—	2	2	no	—
350	C₄²	—	2	2	no	—
351	C₄²	—	4	1	no	—
352	C₄²	—	4	1	no	—
353	C₄²	C₂×D₄×C₂₃	1	2	yes	—
354	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
355	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
356	C₄²	#41×C₂₃	1	2	yes	—
357	C₄²	#41×C₂₃	1	2	yes	—
358	C₄²	#41×C₂₃	1	2	yes	—
359	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
360	Q₈	—	4	8	no	—
361	C₂×C₄	—	2	1	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	1	no	—
364	C₂×C₄	—	2	1	no	—
365	C₂×C₄	—	2	1	no	—
366	C₂×C₄	—	2	1	no	—
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	2	1	no	—
370	C₂×C₄	—	2	1	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	—	2	2	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	2	2	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	2	2	no	—
381	C₂×C₄	—	2	2	no	—
382	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
383	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
384	C₂×C₄	C₄×Q₈×C₂₃	1	2	yes	—
385	C₂×C₄	C₄×Q₈×C₂₃	1	2	yes	—
386	C₂×C₄	C₄×Q₈×C₂₃	1	2	yes	—
387	C₂×C₄	C₄×Q₈×C₂₃	1	2	yes	—
388	C₂×C₄	C₂×M₁₆×C₂₃	1	4	yes	—
389	C₂×C₄	C₂×M₁₆×C₂₃	1	2	yes	—
390	C₂×C₄	C₂₃×#131	1	2	yes	—
391	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
392	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
393	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
394	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
395	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
396	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
397	C₈	—	2	2	no	—
398	C₈	—	2	2	no	—
399	C₈	—	2	2	no	—
400	C₈	—	4	2	no	—
401	C₈	—	4	2	no	—
402	C₈	C₂₃×C₄⋊Q₈	1	4	yes	complemented
403	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
404	C₂²	C₂×C₄×Q₈×C₂₃	1	1	yes	characteristic
405	C₂²	C₂×C₄×Q₈×C₂₃	1	1	yes	characteristic
406	C₂²	C₂²×M₁₆×C₂₃	1	1	yes	characteristic
407	C₂²	C₂×C₂₃×#131	1	1	yes	characteristic
408	C₂²	C₂₃×#568	1	1	yes	characteristic
409	C₂²	C₂₃×#568	1	1	yes	characteristic
410	C₄	—	2	2	no	—
411	C₄	—	2	4	no	—
412	C₄	—	2	2	no	—
413	C₄	—	2	2	no	—
414	C₄	—	2	2	no	—
415	C₄	—	2	2	no	—
416	C₄	—	4	2	no	—
417	C₄	—	4	2	no	—
418	C₄	—	4	2	no	—
419	C₄	C₂₃×#446	1	4	yes	complemented
420	C₄	C₂×C₂₃×C₄⋊Q₈	1	1	yes	characteristic
421	C₄	C₂₃×#532	1	1	yes	characteristic
422	C₄	C₂₃×#533	1	1	yes	characteristic
423	C₄	C₂₃×#533	1	1	yes	characteristic
424	C₂	C₄×C₂₃×C₄⋊Q₈	1	1	yes	characteristic
425	C₂	D₄×M₁₆×C₂₃	1	1	yes	characteristic
426	C₂	C₂₃×#2935	1	1	yes	characteristic
427	C₂	C₂×C₂₃×#446	1	1	yes	characteristic
428	C₂	C₂₃×#2961	1	1	yes	characteristic
429	C₂	C₂₃×#2962	1	1	yes	characteristic
430	C₂	Q₈×M₁₆×C₂₃	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
80. C₂₃×#97	Q₈	semidirect
83. C₄×C₈×C₂₃	Q₈	semidirect
143. #21713	C₂₃	direct (N × H)
187. C₈×C₂₃	C₄⋊Q₈	semidirect
215. C₄×C₂₃	#446	semidirect
295. #97	Q₈×C₂₃	semidirect
298. C₄×C₈	Q₈×C₂₃	semidirect
303. C₂₃	#21713	direct (N × H)
402. C₈	C₂₃×C₄⋊Q₈	semidirect
419. C₄	C₂₃×#446	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#446	2
C₄×C₂₃×C₄⋊Q₈	2
C₂₃×#1985	2
C₂₃×#1940	2
C₂₃×#1939	2
C₂₃×#1842	2
C₂₃×#1771	2
C₂₃×#1770	2
C₈×C₄⋊C₄×C₂₃	2
C₄×C₂₃×#97	2
#21713	23

Subgroups — C23×#21713 #19169339

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#21713 #19169339