Subgroups — C₂₃×#17778 #19165404

Order 5888.

868 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3509	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2991	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2990	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1561	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1559	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1554	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1550	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1548	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1548	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1544	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1542	C₂	1	2	yes	maximal, complemented
12	C₂₃×#557	—	2	1	no	—
13	C₂₃×#557	—	2	1	no	—
14	C₂₃×#556	—	2	1	no	—
15	C₂₃×#556	—	2	1	no	—
16	C₂₃×#552	—	2	2	no	—
17	C₂₃×#552	—	2	2	no	—
18	C₂₃×#541	C₄	1	4	yes	—
19	C₂₃×#536	C₄	1	2	yes	—
20	C₂₃×#533	C₂²	1	2	yes	—
21	C₂₃×#533	C₂²	1	2	yes	—
22	C₂₃×#533	C₂²	1	1	yes	characteristic
23	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
24	C₂₃×#519	C₄	1	2	yes	—
25	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
26	C₂₃×#446	—	2	1	no	—
27	C₈×Q₈×C₂₃	—	2	1	no	—
28	C₂₃×#436	—	2	2	no	—
29	C₂₃×#436	—	2	1	no	—
30	C₂₃×#435	—	2	2	no	—
31	C₈×D₄×C₂₃	—	2	1	no	—
32	C₂₃×#432	C₂²	1	2	yes	—
33	C₂₃×#431	C₂²	1	1	yes	characteristic
34	C₂₃×#431	C₂²	1	2	yes	—
35	C₂₃×#431	C₂²	1	1	yes	characteristic
36	C₂₃×#423	C₂²	1	1	yes	characteristic
37	C₂₃×#423	C₂²	1	1	yes	characteristic
38	C₂₃×#423	C₂²	1	2	yes	—
39	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
40	C₂₃×#333	C₂²	1	2	yes	—
41	C₂₃×#333	C₂²	1	2	yes	—
42	C₂₃×#333	C₂²	1	2	yes	—
43	C₂₃×#333	C₂²	1	2	yes	—
44	C₂₃×#332	C₂²	1	1	yes	characteristic
45	C₂₃×#332	C₂²	1	2	yes	—
46	C₂₃×#332	C₂²	1	1	yes	characteristic
47	C₂₃×#332	C₂²	1	1	yes	characteristic
48	C₂₃×#332	C₂²	1	1	yes	characteristic
49	C₂₃×#332	C₂²	1	2	yes	—
50	C₂×#41×C₂₃	C₂×C₄	1	1	yes	characteristic
51	C₂²×Q₈×C₂₃	C₂×C₄	1	2	yes	—
52	C₂×M₁₆×C₂₃	—	2	1	no	—
53	C₂×M₁₆×C₂₃	—	2	1	no	—
54	C₂×M₁₆×C₂₃	—	2	2	no	—
55	C₂²×C₈×C₂₃	—	2	2	no	—
56	C₂₃×C₄⋊Q₈	—	2	2	no	—
57	C₂₃×C₄⋊Q₈	—	2	1	no	—
58	C₂₃×C₄⋊Q₈	—	2	1	no	—
59	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
60	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
61	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
62	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
63	C₂₃×#126	—	2	4	no	—
64	C₂₃×#126	—	2	4	no	—
65	C₂₃×#125	—	2	4	no	—
66	C₂₃×#125	—	2	4	no	—
67	C₂₃×#125	—	2	2	no	—
68	C₂₃×#125	—	2	2	no	—
69	C₂₃×#125	C₂×C₄	1	2	yes	—
70	C₂₃×#124	—	2	2	no	—
71	C₂₃×#123	—	2	4	no	—
72	C₂₃×C₂²⋊Q₈	—	2	1	no	—
73	C₂₃×C₂²⋊Q₈	—	2	1	no	—
74	C₂₃×C₂²⋊Q₈	—	2	2	no	—
75	C₂₃×C₂²⋊Q₈	—	2	2	no	—
76	C₂₃×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
77	C₂₃×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
78	C₄×Q₈×C₂₃	—	2	2	no	—
79	C₄×Q₈×C₂₃	—	2	2	no	—
80	C₄×Q₈×C₂₃	—	2	2	no	—
81	C₄×Q₈×C₂₃	—	2	2	no	—
82	C₄×Q₈×C₂₃	—	2	1	no	—
83	C₄×Q₈×C₂₃	—	2	1	no	—
84	C₄×Q₈×C₂₃	C₂×C₄	1	4	yes	—
85	C₄×Q₈×C₂₃	C₂×C₄	1	4	yes	—
86	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
87	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
88	C₄×D₄×C₂₃	D₄	1	2	yes	—
89	C₄×D₄×C₂₃	D₄	1	2	yes	—
90	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
91	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
92	C₂₃×#117	—	2	2	no	—
93	C₂₃×#117	—	2	2	no	—
94	C₂₃×#117	—	2	4	no	—
95	C₂₃×#117	C₂×C₄	1	2	yes	—
96	C₂₃×#117	C₂×C₄	1	1	yes	characteristic
97	C₂₃×#117	C₂×C₄	1	2	yes	—
98	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
99	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
100	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
101	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
102	C₂₃×C₄⋊C₈	—	2	2	no	—
103	C₂₃×C₄⋊C₈	—	2	1	no	—
104	C₂₃×C₄⋊C₈	—	2	2	no	—
105	C₂₃×C₄⋊C₈	—	2	1	no	—
106	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
107	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
108	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
109	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
110	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
111	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
112	C₂₃×C₂²⋊C₈	—	2	1	no	—
113	C₂₃×C₂²⋊C₈	—	2	1	no	—
114	C₂₃×C₂²⋊C₈	—	2	2	no	—
115	C₂₃×C₂²⋊C₈	—	2	2	no	—
116	C₂₃×#97	—	2	2	no	—
117	C₄×C₈×C₂₃	—	2	1	no	—
118	C₄×C₈×C₂₃	—	2	2	no	—
119	C₄×C₈×C₂₃	—	2	1	no	—
120	#41×C₂₃	—	4	2	no	—
121	C₂×Q₈×C₂₃	—	2	2	no	—
122	C₂×Q₈×C₂₃	—	2	1	no	—
123	C₂×Q₈×C₂₃	—	2	1	no	—
124	C₂×Q₈×C₂₃	—	4	4	no	—
125	C₂×Q₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
126	C₂×Q₈×C₂₃	C₂²×C₄	1	4	yes	—
127	C₂×D₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
128	C₂×D₄×C₂₃	C₂²⋊C₄	1	2	yes	—
129	C₂²×C₄×C₂₃	—	2	2	no	—
130	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
131	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
132	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
134	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
135	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
136	M₁₆×C₂₃	—	8	1	no	—
137	M₁₆×C₂₃	—	8	1	no	—
138	M₁₆×C₂₃	—	8	2	no	—
139	C₂×C₈×C₂₃	—	2	1	no	—
140	C₂×C₈×C₂₃	—	2	2	no	—
141	C₂×C₈×C₂₃	—	2	1	no	—
142	C₂×C₈×C₂₃	—	2	2	no	—
143	C₂×C₈×C₂₃	—	2	1	no	—
144	C₂×C₈×C₂₃	—	2	1	no	—
145	C₂×C₈×C₂₃	—	8	2	no	—
146	C₄⋊C₄×C₂₃	—	2	2	no	—
147	C₄⋊C₄×C₂₃	—	2	2	no	—
148	C₄⋊C₄×C₂₃	—	2	4	no	—
149	C₄⋊C₄×C₂₃	—	2	4	no	—
150	C₄⋊C₄×C₂₃	—	2	2	no	—
151	C₄⋊C₄×C₂₃	—	2	2	no	—
152	C₄⋊C₄×C₂₃	—	2	4	no	—
153	C₄⋊C₄×C₂₃	—	2	4	no	—
154	C₄⋊C₄×C₂₃	—	2	2	no	—
155	C₄⋊C₄×C₂₃	—	2	2	no	—
156	C₄⋊C₄×C₂₃	—	2	1	no	—
157	C₄⋊C₄×C₂₃	—	2	1	no	—
158	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
159	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
160	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
161	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
162	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
163	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
164	C₂²⋊C₄×C₂₃	—	2	2	no	—
165	C₂²⋊C₄×C₂₃	—	2	4	no	—
166	C₂²⋊C₄×C₂₃	—	2	4	no	—
167	C₂²⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
168	C₂²⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
169	C₂²⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
170	C₄²×C₂₃	—	2	4	no	—
171	C₄²×C₂₃	—	2	4	no	—
172	C₄²×C₂₃	—	2	2	no	—
173	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
174	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
175	C₄²×C₂₃	C₂×D₄	1	2	yes	—
176	C₄²×C₂₃	C₂×D₄	1	2	yes	—
177	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
178	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
179	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
180	#17778	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
181	C₂³×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
182	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
183	Q₈×C₂₃	—	2	2	no	—
184	Q₈×C₂₃	—	4	4	no	—
185	Q₈×C₂₃	—	4	1	no	—
186	Q₈×C₂₃	—	4	1	no	—
187	Q₈×C₂₃	—	4	1	no	—
188	Q₈×C₂₃	—	4	1	no	—
189	Q₈×C₂₃	—	4	4	no	—
190	D₄×C₂₃	—	2	2	no	—
191	D₄×C₂₃	—	4	2	no	—
192	C₂×C₄×C₂₃	—	2	2	no	—
193	C₂×C₄×C₂₃	—	2	1	no	—
194	C₂×C₄×C₂₃	—	2	1	no	—
195	C₂×C₄×C₂₃	—	2	2	no	—
196	C₂×C₄×C₂₃	—	2	1	no	—
197	C₂×C₄×C₂₃	—	2	1	no	—
198	C₂×C₄×C₂₃	—	2	2	no	—
199	C₂×C₄×C₂₃	—	2	2	no	—
200	C₂×C₄×C₂₃	—	2	2	no	—
201	C₂×C₄×C₂₃	—	2	2	no	—
202	C₂×C₄×C₂₃	—	2	2	no	—
203	C₂×C₄×C₂₃	—	2	4	no	—
204	C₂×C₄×C₂₃	—	4	2	no	—
205	C₂×C₄×C₂₃	—	4	4	no	—
206	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
207	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
208	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
209	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
210	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
211	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
212	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
213	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
214	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
215	C₈×C₂₃	—	4	1	no	—
216	C₈×C₂₃	—	4	2	no	—
217	C₈×C₂₃	—	4	1	no	—
218	C₈×C₂₃	—	4	2	no	—
219	C₈×C₂₃	—	4	1	no	—
220	C₈×C₂₃	—	4	1	no	—
221	#3509	C₂×C₂₃	1	1	yes	characteristic
222	#2991	C₂×C₂₃	1	1	yes	characteristic
223	#2990	C₂×C₂₃	1	1	yes	characteristic
224	#1561	C₂×C₂₃	1	1	yes	characteristic, complemented
225	#1559	C₂×C₂₃	1	2	yes	complemented
226	#1554	C₂×C₂₃	1	2	yes	complemented
227	#1550	C₂×C₂₃	1	2	yes	complemented
228	#1548	C₂×C₂₃	1	1	yes	characteristic, complemented
229	#1548	C₂×C₂₃	1	1	yes	characteristic, complemented
230	#1544	C₂×C₂₃	1	1	yes	characteristic, complemented
231	#1542	C₂×C₂₃	1	2	yes	complemented
232	C₂²×C₂₃	—	2	1	no	—
233	C₂²×C₂₃	—	2	2	no	—
234	C₂²×C₂₃	—	2	2	no	—
235	C₂²×C₂₃	—	2	2	no	—
236	C₂²×C₂₃	—	2	1	no	—
237	C₂²×C₂₃	—	2	1	no	—
238	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
239	C₄×C₂₃	—	2	1	no	—
240	C₄×C₂₃	—	2	2	no	—
241	C₄×C₂₃	—	2	2	no	—
242	C₄×C₂₃	—	2	2	no	—
243	C₄×C₂₃	—	2	1	no	—
244	C₄×C₂₃	—	2	1	no	—
245	C₄×C₂₃	—	4	4	no	—
246	C₄×C₂₃	—	4	2	no	—
247	C₄×C₂₃	—	4	2	no	—
248	C₄×C₂₃	#413	1	2	yes	—
249	C₄×C₂₃	#577	1	1	yes	characteristic
250	C₄×C₂₃	#577	1	1	yes	characteristic
251	C₄×C₂₃	#578	1	1	yes	characteristic
252	C₄×C₂₃	#578	1	1	yes	characteristic
253	#557	—	2	1	no	—
254	#557	—	2	1	no	—
255	#556	—	2	1	no	—
256	#556	—	2	1	no	—
257	#552	—	2	2	no	—
258	#552	—	2	2	no	—
259	#541	C₄×C₂₃	1	4	yes	—
260	#536	C₄×C₂₃	1	2	yes	—
261	#533	C₂²×C₂₃	1	2	yes	—
262	#533	C₂²×C₂₃	1	2	yes	—
263	#533	C₂²×C₂₃	1	1	yes	characteristic
264	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
265	#519	C₄×C₂₃	1	2	yes	—
266	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
267	#446	—	2	1	no	—
268	C₈×Q₈	—	2	1	no	—
269	#436	—	2	2	no	—
270	#436	—	2	1	no	—
271	#435	—	2	2	no	—
272	C₈×D₄	—	2	1	no	—
273	#432	C₂²×C₂₃	1	2	yes	—
274	#431	C₂²×C₂₃	1	2	yes	—
275	#431	C₂²×C₂₃	1	1	yes	characteristic
276	#431	C₂²×C₂₃	1	1	yes	characteristic
277	#423	C₂²×C₂₃	1	1	yes	characteristic
278	#423	C₂²×C₂₃	1	1	yes	characteristic
279	#423	C₂²×C₂₃	1	2	yes	—
280	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
281	#333	C₂²×C₂₃	1	2	yes	—
282	#333	C₂²×C₂₃	1	2	yes	—
283	#333	C₂²×C₂₃	1	2	yes	—
284	#333	C₂²×C₂₃	1	2	yes	—
285	#332	C₂²×C₂₃	1	1	yes	characteristic
286	#332	C₂²×C₂₃	1	2	yes	—
287	#332	C₂²×C₂₃	1	1	yes	characteristic
288	#332	C₂²×C₂₃	1	1	yes	characteristic
289	#332	C₂²×C₂₃	1	1	yes	characteristic
290	#332	C₂²×C₂₃	1	2	yes	—
291	C₂×C₂₃	—	4	1	no	—
292	C₂×C₂₃	—	4	2	no	—
293	C₂×C₂₃	C₂×#413	1	1	yes	characteristic
294	C₂×C₂₃	#2898	1	1	yes	characteristic
295	C₂×C₂₃	#2898	1	1	yes	characteristic
296	C₂×#41	C₂×C₄×C₂₃	1	1	yes	characteristic
297	C₂²×Q₈	C₂×C₄×C₂₃	1	2	yes	—
298	C₂×M₁₆	—	2	1	no	—
299	C₂×M₁₆	—	2	1	no	—
300	C₂×M₁₆	—	2	2	no	—
301	C₂²×C₈	—	2	2	no	—
302	C₄⋊Q₈	—	2	2	no	—
303	C₄⋊Q₈	—	2	1	no	—
304	C₄⋊Q₈	—	2	1	no	—
305	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
306	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
307	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
308	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
309	#126	—	2	4	no	—
310	#126	—	2	4	no	—
311	#125	—	2	4	no	—
312	#125	—	2	4	no	—
313	#125	—	2	2	no	—
314	#125	—	2	2	no	—
315	#125	C₂×C₄×C₂₃	1	2	yes	—
316	#124	—	2	2	no	—
317	#123	—	2	4	no	—
318	C₂²⋊Q₈	—	2	1	no	—
319	C₂²⋊Q₈	—	2	1	no	—
320	C₂²⋊Q₈	—	2	2	no	—
321	C₂²⋊Q₈	—	2	2	no	—
322	C₂²⋊Q₈	C₂×C₄×C₂₃	1	4	yes	—
323	C₂²⋊Q₈	C₂×C₄×C₂₃	1	4	yes	—
324	C₄×Q₈	—	2	2	no	—
325	C₄×Q₈	—	2	2	no	—
326	C₄×Q₈	—	2	2	no	—
327	C₄×Q₈	—	2	2	no	—
328	C₄×Q₈	—	2	1	no	—
329	C₄×Q₈	—	2	1	no	—
330	C₄×Q₈	C₂×C₄×C₂₃	1	4	yes	—
331	C₄×Q₈	C₂×C₄×C₂₃	1	4	yes	—
332	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
333	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
334	C₄×D₄	D₄×C₂₃	1	2	yes	—
335	C₄×D₄	D₄×C₂₃	1	2	yes	—
336	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
337	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
338	#117	—	2	2	no	—
339	#117	—	2	2	no	—
340	#117	—	2	4	no	—
341	#117	C₂×C₄×C₂₃	1	2	yes	—
342	#117	C₂×C₄×C₂₃	1	1	yes	characteristic
343	#117	C₂×C₄×C₂₃	1	2	yes	—
344	C₂×C₄⋊C₄	—	2	1	no	—
345	C₂×C₄⋊C₄	—	2	1	no	—
346	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
347	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
348	C₄⋊C₈	—	2	2	no	—
349	C₄⋊C₈	—	2	1	no	—
350	C₄⋊C₈	—	2	2	no	—
351	C₄⋊C₈	—	2	1	no	—
352	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
353	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
354	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
355	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
356	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
357	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
358	C₂²⋊C₈	—	2	1	no	—
359	C₂²⋊C₈	—	2	1	no	—
360	C₂²⋊C₈	—	2	2	no	—
361	C₂²⋊C₈	—	2	2	no	—
362	#97	—	2	2	no	—
363	C₄×C₈	—	2	1	no	—
364	C₄×C₈	—	2	2	no	—
365	C₄×C₈	—	2	1	no	—
366	C₂₃	#17778	1	1	yes	characteristic, direct product factor, Sylow
367	#41	—	4	2	no	—
368	C₂×Q₈	—	2	2	no	—
369	C₂×Q₈	—	2	1	no	—
370	C₂×Q₈	—	2	1	no	—
371	C₂×Q₈	—	4	4	no	—
372	C₂×Q₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
373	C₂×Q₈	C₂²×C₄×C₂₃	1	4	yes	—
374	C₂×D₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
375	C₂×D₄	C₂²⋊C₄×C₂₃	1	2	yes	—
376	C₂²×C₄	—	2	2	no	—
377	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
378	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
379	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
380	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
381	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
382	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
383	M₁₆	—	8	1	no	—
384	M₁₆	—	8	1	no	—
385	M₁₆	—	8	2	no	—
386	C₂×C₈	—	2	1	no	—
387	C₂×C₈	—	2	1	no	—
388	C₂×C₈	—	2	1	no	—
389	C₂×C₈	—	2	2	no	—
390	C₂×C₈	—	2	1	no	—
391	C₂×C₈	—	2	2	no	—
392	C₂×C₈	—	8	2	no	—
393	C₄⋊C₄	—	2	2	no	—
394	C₄⋊C₄	—	2	2	no	—
395	C₄⋊C₄	—	2	4	no	—
396	C₄⋊C₄	—	2	4	no	—
397	C₄⋊C₄	—	2	2	no	—
398	C₄⋊C₄	—	2	2	no	—
399	C₄⋊C₄	—	2	4	no	—
400	C₄⋊C₄	—	2	4	no	—
401	C₄⋊C₄	—	2	2	no	—
402	C₄⋊C₄	—	2	2	no	—
403	C₄⋊C₄	—	2	1	no	—
404	C₄⋊C₄	—	2	1	no	—
405	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
406	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
407	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
408	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
409	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
410	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
411	C₂²⋊C₄	—	2	2	no	—
412	C₂²⋊C₄	—	2	4	no	—
413	C₂²⋊C₄	—	2	4	no	—
414	C₂²⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
415	C₂²⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
416	C₂²⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
417	C₄²	—	2	4	no	—
418	C₄²	—	2	4	no	—
419	C₄²	—	2	2	no	—
420	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
421	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
422	C₄²	C₂×D₄×C₂₃	1	2	yes	—
423	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
424	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
425	C₄²	C₂×D₄×C₂₃	1	2	yes	—
426	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
427	C₂³	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
428	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
429	Q₈	—	2	2	no	—
430	Q₈	—	4	4	no	—
431	Q₈	—	4	1	no	—
432	Q₈	—	4	1	no	—
433	Q₈	—	4	1	no	—
434	Q₈	—	4	1	no	—
435	Q₈	—	4	4	no	—
436	D₄	—	2	2	no	—
437	D₄	—	4	2	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	2	1	no	—
440	C₂×C₄	—	2	1	no	—
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	2	1	no	—
443	C₂×C₄	—	2	1	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	4	no	—
450	C₂×C₄	—	4	2	no	—
451	C₂×C₄	—	4	4	no	—
452	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
453	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
454	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
455	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
456	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
457	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
458	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
459	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
460	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
461	C₈	—	4	1	no	—
462	C₈	—	4	2	no	—
463	C₈	—	4	2	no	—
464	C₈	—	4	1	no	—
465	C₈	—	4	1	no	—
466	C₈	—	4	1	no	—
467	C₂²	—	2	1	no	—
468	C₂²	—	2	2	no	—
469	C₂²	—	2	2	no	—
470	C₂²	—	2	2	no	—
471	C₂²	—	2	1	no	—
472	C₂²	—	2	1	no	—
473	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
474	C₄	—	2	1	no	—
475	C₄	—	2	2	no	—
476	C₄	—	2	2	no	—
477	C₄	—	2	2	no	—
478	C₄	—	2	1	no	—
479	C₄	—	2	1	no	—
480	C₄	—	4	4	no	—
481	C₄	—	4	2	no	—
482	C₄	—	4	2	no	—
483	C₄	C₂₃×#413	1	2	yes	—
484	C₄	C₂₃×#577	1	1	yes	characteristic
485	C₄	C₂₃×#577	1	1	yes	characteristic
486	C₄	C₂₃×#578	1	1	yes	characteristic
487	C₄	C₂₃×#578	1	1	yes	characteristic
488	C₂	—	4	1	no	—
489	C₂	—	4	2	no	—
490	C₂	C₂×C₂₃×#413	1	1	yes	characteristic
491	C₂	C₂₃×#2898	1	1	yes	characteristic
492	C₂	C₂₃×#2898	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂×#41×C₂₃	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄×#41×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂₃×#1561	C₂	semidirect
5. C₂₃×#1559	C₂	semidirect
6. C₂₃×#1554	C₂	semidirect
7. C₂₃×#1550	C₂	semidirect
8. C₂₃×#1548	C₂	semidirect
9. C₂₃×#1548	C₂	semidirect
10. C₂₃×#1544	C₂	semidirect
11. C₂₃×#1542	C₂	semidirect
180. #17778	C₂₃	direct (N × H)
224. #1561	C₂×C₂₃	semidirect
225. #1559	C₂×C₂₃	semidirect
226. #1554	C₂×C₂₃	semidirect
227. #1550	C₂×C₂₃	semidirect
228. #1548	C₂×C₂₃	semidirect
229. #1548	C₂×C₂₃	semidirect
230. #1544	C₂×C₂₃	semidirect
231. #1542	C₂×C₂₃	semidirect
366. C₂₃	#17778	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#3509	2
C₂₃×#2991	2
C₂₃×#2990	2
C₂₃×#1561	2
C₂₃×#1559	2
C₂₃×#1554	2
C₂₃×#1550	2
C₂₃×#1548	2
C₂₃×#1544	2
C₂₃×#1542	2
#17778	23

Subgroups — C23×#17778 #19165404

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#17778 #19165404