Subgroups — C₂²×C₂₃×#451 #19207976

Order 5888.

4982 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂³×Q₁₆×C₂₃	C₂	1	2	yes	maximal, complemented
3	C₂²×C₂₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×#451	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₂₃×#103	C₂	1	2	yes	maximal, complemented
6	C₂²×C₂₃×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
7	C₂³×Q₈×C₂₃	—	2	16	no	—
8	C₂³×Q₈×C₂₃	—	2	6	no	—
9	C₂³×Q₈×C₂₃	—	2	1	no	—
10	C₂³×Q₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂³×Q₈×C₂₃	C₂²	1	12	yes	—
12	C₂³×Q₈×C₂₃	C₂²	1	2	yes	—
13	C₂⁴×C₄×C₂₃	—	2	1	no	—
14	C₂⁴×C₄×C₂₃	C₂²	1	1	yes	characteristic
15	C₂²×Q₁₆×C₂₃	—	2	16	no	—
16	C₂²×Q₁₆×C₂₃	C₂²	1	24	yes	complemented
17	C₂³×C₈×C₂₃	C₂²	1	2	yes	—
18	C₂×C₂₃×C₂²⋊Q₈	—	2	6	no	—
19	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	12	yes	—
20	C₂²×C₄⋊C₄×C₂₃	—	2	1	no	—
21	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
22	C₂²×C₂²⋊C₄×C₂₃	—	2	1	no	—
23	C₂₃×#451	C₂²	1	64	yes	direct product factor
24	C₂×C₂₃×#103	C₂²	1	24	yes	complemented
25	C₂×C₂₃×C₂²⋊C₈	C₂²	1	12	yes	—
26	C₂⁵×C₂₃	D₄	1	1	yes	characteristic
27	C₂²×Q₈×C₂₃	—	2	96	no	—
28	C₂²×Q₈×C₂₃	—	2	16	no	—
29	C₂²×Q₈×C₂₃	—	2	96	no	—
30	C₂²×Q₈×C₂₃	—	2	8	no	—
31	C₂²×Q₈×C₂₃	—	2	12	no	—
32	C₂²×Q₈×C₂₃	—	2	24	no	—
33	C₂²×Q₈×C₂₃	—	2	6	no	—
34	C₂²×Q₈×C₂₃	—	2	3	no	—
35	C₂²×Q₈×C₂₃	—	2	8	no	—
36	C₂²×Q₈×C₂₃	D₄	1	16	yes	—
37	C₂²×Q₈×C₂₃	C₂³	1	6	yes	—
38	C₂²×Q₈×C₂₃	C₂³	1	16	yes	—
39	C₂²×Q₈×C₂₃	C₂³	1	12	yes	—
40	C₂³×C₄×C₂₃	—	2	16	no	—
41	C₂³×C₄×C₂₃	—	2	12	no	—
42	C₂³×C₄×C₂₃	—	2	2	no	—
43	C₂³×C₄×C₂₃	—	2	8	no	—
44	C₂³×C₄×C₂₃	—	2	1	no	—
45	C₂³×C₄×C₂₃	—	2	1	no	—
46	C₂³×C₄×C₂₃	D₄	1	6	yes	—
47	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
48	C₂³×C₄×C₂₃	C₂³	1	6	yes	—
49	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
50	C₂×Q₁₆×C₂₃	—	2	96	no	—
51	C₂×Q₁₆×C₂₃	C₂³	1	32	yes	complemented
52	C₂²×C₈×C₂₃	—	2	8	no	—
53	C₂²×C₈×C₂₃	C₂³	1	12	yes	—
54	C₂₃×C₂²⋊Q₈	—	2	24	no	—
55	C₂₃×C₂²⋊Q₈	C₂³	1	16	yes	—
56	C₂×C₄⋊C₄×C₂₃	—	2	12	no	—
57	C₂×C₄⋊C₄×C₂₃	—	2	3	no	—
58	C₂×C₄⋊C₄×C₂₃	C₂³	1	6	yes	—
59	C₂×C₂²⋊C₄×C₂₃	—	2	12	no	—
60	C₂₃×#103	C₂³	1	32	yes	complemented
61	C₂₃×C₂²⋊C₈	C₂³	1	16	yes	—
62	C₂⁴×C₂₃	—	2	4	no	—
63	C₂⁴×C₂₃	—	2	4	no	—
64	C₂⁴×C₂₃	Q₁₆	1	8	yes	complemented
65	C₂⁴×C₂₃	C₂×D₄	1	6	yes	—
66	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
67	C₂×Q₈×C₂₃	—	2	192	no	—
68	C₂×Q₈×C₂₃	—	2	96	no	—
69	C₂×Q₈×C₂₃	—	2	64	no	—
70	C₂×Q₈×C₂₃	—	2	48	no	—
71	C₂×Q₈×C₂₃	—	2	24	no	—
72	C₂×Q₈×C₂₃	—	2	16	no	—
73	C₂×Q₈×C₂₃	—	2	12	no	—
74	C₂×Q₈×C₂₃	—	2	6	no	—
75	C₂×Q₈×C₂₃	—	2	24	no	—
76	C₂×Q₈×C₂₃	—	4	12	no	—
77	C₂×Q₈×C₂₃	C₂×D₄	1	48	yes	—
78	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
79	C₂×Q₈×C₂₃	C₂⁴	1	8	yes	—
80	C₂²×C₄×C₂₃	—	2	96	no	—
81	C₂²×C₄×C₂₃	—	2	16	no	—
82	C₂²×C₄×C₂₃	—	2	4	no	—
83	C₂²×C₄×C₂₃	—	2	16	no	—
84	C₂²×C₄×C₂₃	—	2	12	no	—
85	C₂²×C₄×C₂₃	—	2	24	no	—
86	C₂²×C₄×C₂₃	—	2	24	no	—
87	C₂²×C₄×C₂₃	—	2	6	no	—
88	C₂²×C₄×C₂₃	—	2	6	no	—
89	C₂²×C₄×C₂₃	—	4	4	no	—
90	C₂²×C₄×C₂₃	—	4	4	no	—
91	C₂²×C₄×C₂₃	C₂×D₄	1	6	yes	—
92	C₂²×C₄×C₂₃	C₂×D₄	1	12	yes	—
93	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
94	C₂²×C₄×C₂₃	C₂×D₄	1	8	yes	—
95	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
96	C₂²×C₄×C₂₃	C₂⁴	1	3	yes	—
97	Q₁₆×C₂₃	—	2	64	no	—
98	C₂×C₈×C₂₃	—	2	24	no	—
99	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
100	C₄⋊C₄×C₂₃	—	2	16	no	—
101	C₄⋊C₄×C₂₃	—	2	6	no	—
102	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
103	C₂²⋊C₄×C₂₃	—	2	16	no	—
104	C₂²×#451	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₂³×C₂₃	—	2	12	no	—
106	C₂³×C₂₃	—	2	12	no	—
107	C₂³×C₂₃	—	2	32	no	—
108	C₂³×C₂₃	#120	1	4	yes	—
109	C₂³×C₂₃	C₂×Q₁₆	1	4	yes	—
110	C₂³×C₂₃	C₂×Q₁₆	1	24	yes	complemented
111	C₂³×C₂₃	#137	1	4	yes	—
112	C₂³×C₂₃	C₂²×D₄	1	4	yes	—
113	C₂³×C₂₃	C₂²×D₄	1	3	yes	—
114	Q₈×C₂₃	—	2	64	no	—
115	Q₈×C₂₃	—	2	48	no	—
116	Q₈×C₂₃	—	2	8	no	—
117	Q₈×C₂₃	—	2	8	no	—
118	Q₈×C₂₃	—	4	12	no	—
119	Q₈×C₂₃	C₂²×D₄	1	16	yes	—
120	C₂×C₄×C₂₃	—	2	64	no	—
121	C₂×C₄×C₂₃	—	2	48	no	—
122	C₂×C₄×C₂₃	—	2	12	no	—
123	C₂×C₄×C₂₃	—	2	8	no	—
124	C₂×C₄×C₂₃	—	2	24	no	—
125	C₂×C₄×C₂₃	—	2	8	no	—
126	C₂×C₄×C₂₃	—	2	4	no	—
127	C₂×C₄×C₂₃	—	2	4	no	—
128	C₂×C₄×C₂₃	—	4	12	no	—
129	C₂×C₄×C₂₃	—	4	12	no	—
130	C₂×C₄×C₂₃	#120	1	12	yes	—
131	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
132	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
133	C₂×C₄×C₂₃	C₂²×D₄	1	12	yes	—
134	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
135	C₈×C₂₃	—	2	8	no	—
136	C₂⁴×Q₈	C₂×C₂₃	1	1	yes	characteristic
137	C₂³×Q₁₆	C₂×C₂₃	1	2	yes	complemented
138	C₂²×C₂²⋊Q₈	C₂×C₂₃	1	1	yes	characteristic
139	C₂×#451	C₂×C₂₃	1	24	yes	direct product factor
140	C₂²×#103	C₂×C₂₃	1	2	yes	complemented
141	C₂²×C₂²⋊C₈	C₂×C₂₃	1	1	yes	characteristic
142	C₂²×C₂₃	—	2	4	no	—
143	C₂²×C₂₃	—	2	48	no	—
144	C₂²×C₂₃	—	2	4	no	—
145	C₂²×C₂₃	#451	1	16	yes	direct product factor
146	C₂²×C₂₃	C₂×#120	1	6	yes	—
147	C₂²×C₂₃	C₂²×Q₁₆	1	6	yes	—
148	C₂²×C₂₃	C₂²×Q₁₆	1	8	yes	complemented
149	C₂²×C₂₃	C₂×#137	1	6	yes	—
150	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
151	C₄×C₂₃	—	2	16	no	—
152	C₄×C₂₃	—	2	4	no	—
153	C₄×C₂₃	—	4	4	no	—
154	C₄×C₂₃	—	4	4	no	—
155	C₄×C₂₃	C₂×#120	1	6	yes	—
156	C₄×C₂₃	C₂³×D₄	1	2	yes	—
157	C₂³×Q₈	—	2	16	no	—
158	C₂³×Q₈	—	2	1	no	—
159	C₂³×Q₈	—	2	6	no	—
160	C₂³×Q₈	C₂²×C₂₃	1	12	yes	—
161	C₂³×Q₈	C₂²×C₂₃	1	1	yes	characteristic
162	C₂³×Q₈	C₂²×C₂₃	1	2	yes	—
163	C₂⁴×C₄	—	2	1	no	—
164	C₂⁴×C₄	C₂²×C₂₃	1	1	yes	characteristic
165	C₂²×Q₁₆	—	2	16	no	—
166	C₂²×Q₁₆	C₂²×C₂₃	1	24	yes	complemented
167	C₂³×C₈	C₂²×C₂₃	1	2	yes	—
168	C₂×C₂²⋊Q₈	—	2	6	no	—
169	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	12	yes	—
170	C₂²×C₄⋊C₄	—	2	1	no	—
171	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
172	C₂²×C₂²⋊C₄	—	2	1	no	—
173	#451	C₂²×C₂₃	1	64	yes	direct product factor
174	C₂×#103	C₂²×C₂₃	1	24	yes	complemented
175	C₂×C₂²⋊C₈	C₂²×C₂₃	1	12	yes	—
176	C₂×C₂₃	—	2	8	no	—
177	C₂×C₂₃	C₂×#451	1	12	yes	direct product factor
178	C₂×C₂₃	C₂²×#120	1	1	yes	characteristic
179	C₂×C₂₃	C₂³×Q₁₆	1	1	yes	characteristic
180	C₂×C₂₃	C₂²×#137	1	1	yes	characteristic
181	C₂⁵	D₄×C₂₃	1	1	yes	characteristic
182	C₂²×Q₈	—	2	96	no	—
183	C₂²×Q₈	—	2	16	no	—
184	C₂²×Q₈	—	2	96	no	—
185	C₂²×Q₈	—	2	8	no	—
186	C₂²×Q₈	—	2	12	no	—
187	C₂²×Q₈	—	2	24	no	—
188	C₂²×Q₈	—	2	6	no	—
189	C₂²×Q₈	—	2	3	no	—
190	C₂²×Q₈	—	2	8	no	—
191	C₂²×Q₈	D₄×C₂₃	1	16	yes	—
192	C₂²×Q₈	C₂³×C₂₃	1	6	yes	—
193	C₂²×Q₈	C₂³×C₂₃	1	16	yes	—
194	C₂²×Q₈	C₂³×C₂₃	1	12	yes	—
195	C₂³×C₄	—	2	16	no	—
196	C₂³×C₄	—	2	12	no	—
197	C₂³×C₄	—	2	2	no	—
198	C₂³×C₄	—	2	8	no	—
199	C₂³×C₄	—	2	1	no	—
200	C₂³×C₄	—	2	1	no	—
201	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
202	C₂³×C₄	D₄×C₂₃	1	6	yes	—
203	C₂³×C₄	C₂³×C₂₃	1	6	yes	—
204	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
205	C₂×Q₁₆	—	2	96	no	—
206	C₂×Q₁₆	C₂³×C₂₃	1	32	yes	complemented
207	C₂²×C₈	—	2	8	no	—
208	C₂²×C₈	C₂³×C₂₃	1	12	yes	—
209	C₂²⋊Q₈	—	2	24	no	—
210	C₂²⋊Q₈	C₂³×C₂₃	1	16	yes	—
211	C₂×C₄⋊C₄	—	2	12	no	—
212	C₂×C₄⋊C₄	—	2	3	no	—
213	C₂×C₄⋊C₄	C₂³×C₂₃	1	6	yes	—
214	C₂×C₂²⋊C₄	—	2	12	no	—
215	#103	C₂³×C₂₃	1	32	yes	complemented
216	C₂²⋊C₈	C₂³×C₂₃	1	16	yes	—
217	C₂₃	C₂²×#451	1	1	yes	characteristic, direct product factor, Sylow
218	C₂⁴	—	2	4	no	—
219	C₂⁴	—	2	4	no	—
220	C₂⁴	Q₁₆×C₂₃	1	8	yes	complemented
221	C₂⁴	C₂×D₄×C₂₃	1	6	yes	—
222	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
223	C₂×Q₈	—	2	192	no	—
224	C₂×Q₈	—	2	96	no	—
225	C₂×Q₈	—	2	64	no	—
226	C₂×Q₈	—	2	48	no	—
227	C₂×Q₈	—	2	16	no	—
228	C₂×Q₈	—	2	12	no	—
229	C₂×Q₈	—	2	24	no	—
230	C₂×Q₈	—	2	6	no	—
231	C₂×Q₈	—	2	24	no	—
232	C₂×Q₈	—	4	12	no	—
233	C₂×Q₈	C₂×D₄×C₂₃	1	48	yes	—
234	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
235	C₂×Q₈	C₂⁴×C₂₃	1	8	yes	—
236	C₂²×C₄	—	2	96	no	—
237	C₂²×C₄	—	2	16	no	—
238	C₂²×C₄	—	2	4	no	—
239	C₂²×C₄	—	2	16	no	—
240	C₂²×C₄	—	2	12	no	—
241	C₂²×C₄	—	2	24	no	—
242	C₂²×C₄	—	2	24	no	—
243	C₂²×C₄	—	2	6	no	—
244	C₂²×C₄	—	2	6	no	—
245	C₂²×C₄	—	4	4	no	—
246	C₂²×C₄	—	4	4	no	—
247	C₂²×C₄	C₂×D₄×C₂₃	1	6	yes	—
248	C₂²×C₄	C₂×D₄×C₂₃	1	12	yes	—
249	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
250	C₂²×C₄	C₂×D₄×C₂₃	1	8	yes	—
251	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
252	C₂²×C₄	C₂⁴×C₂₃	1	3	yes	—
253	Q₁₆	—	2	64	no	—
254	C₂×C₈	—	2	24	no	—
255	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
256	C₄⋊C₄	—	2	16	no	—
257	C₄⋊C₄	—	2	6	no	—
258	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
259	C₂²⋊C₄	—	2	16	no	—
260	C₂³	—	2	12	no	—
261	C₂³	—	2	32	no	—
262	C₂³	—	2	12	no	—
263	C₂³	C₂₃×#120	1	4	yes	—
264	C₂³	C₂×Q₁₆×C₂₃	1	4	yes	—
265	C₂³	C₂×Q₁₆×C₂₃	1	24	yes	complemented
266	C₂³	C₂₃×#137	1	4	yes	—
267	C₂³	C₂²×D₄×C₂₃	1	4	yes	—
268	C₂³	C₂²×D₄×C₂₃	1	3	yes	—
269	Q₈	—	2	64	no	—
270	Q₈	—	2	8	no	—
271	Q₈	—	2	48	no	—
272	Q₈	—	2	8	no	—
273	Q₈	—	4	12	no	—
274	Q₈	C₂²×D₄×C₂₃	1	16	yes	—
275	C₂×C₄	—	2	64	no	—
276	C₂×C₄	—	2	8	no	—
277	C₂×C₄	—	2	48	no	—
278	C₂×C₄	—	2	12	no	—
279	C₂×C₄	—	2	8	no	—
280	C₂×C₄	—	2	24	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	4	12	no	—
284	C₂×C₄	—	4	12	no	—
285	C₂×C₄	C₂₃×#120	1	12	yes	—
286	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
287	C₂×C₄	C₂²×D₄×C₂₃	1	12	yes	—
288	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
289	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
290	C₈	—	2	8	no	—
291	C₂²	—	2	4	no	—
292	C₂²	—	2	48	no	—
293	C₂²	—	2	4	no	—
294	C₂²	C₂₃×#451	1	16	yes	direct product factor
295	C₂²	C₂×C₂₃×#120	1	6	yes	—
296	C₂²	C₂²×Q₁₆×C₂₃	1	6	yes	—
297	C₂²	C₂²×Q₁₆×C₂₃	1	8	yes	complemented
298	C₂²	C₂×C₂₃×#137	1	6	yes	—
299	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
300	C₄	—	2	16	no	—
301	C₄	—	2	4	no	—
302	C₄	—	4	4	no	—
303	C₄	—	4	4	no	—
304	C₄	C₂×C₂₃×#120	1	6	yes	—
305	C₄	C₂³×D₄×C₂₃	1	2	yes	—
306	C₂	—	2	8	no	—
307	C₂	C₂×C₂₃×#451	1	12	yes	direct product factor
308	C₂	C₂²×C₂₃×#120	1	1	yes	characteristic
309	C₂	C₂³×Q₁₆×C₂₃	1	1	yes	characteristic
310	C₂	C₂²×C₂₃×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₂₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂⁵×C₂₃	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₂₃ ◁ C₂⁴×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×Q₁₆×C₂₃	C₂	semidirect
4. C₂×C₂₃×#451	C₂	direct (N × H)
5. C₂²×C₂₃×#103	C₂	semidirect
16. C₂²×Q₁₆×C₂₃	C₂²	semidirect
23. C₂₃×#451	C₂²	direct (N × H)
24. C₂×C₂₃×#103	C₂²	semidirect
51. C₂×Q₁₆×C₂₃	C₂³	semidirect
60. C₂₃×#103	C₂³	semidirect
64. C₂⁴×C₂₃	Q₁₆	semidirect
104. C₂²×#451	C₂₃	direct (N × H)
110. C₂³×C₂₃	C₂×Q₁₆	semidirect
137. C₂³×Q₁₆	C₂×C₂₃	semidirect
139. C₂×#451	C₂×C₂₃	direct (N × H)
140. C₂²×#103	C₂×C₂₃	semidirect
145. C₂²×C₂₃	#451	direct (N × H)
148. C₂²×C₂₃	C₂²×Q₁₆	semidirect
166. C₂²×Q₁₆	C₂²×C₂₃	semidirect
173. #451	C₂²×C₂₃	direct (N × H)
174. C₂×#103	C₂²×C₂₃	semidirect
177. C₂×C₂₃	C₂×#451	direct (N × H)
206. C₂×Q₁₆	C₂³×C₂₃	semidirect
215. #103	C₂³×C₂₃	semidirect
217. C₂₃	C₂²×#451	direct (N × H)
220. C₂⁴	Q₁₆×C₂₃	semidirect
265. C₂³	C₂×Q₁₆×C₂₃	semidirect
294. C₂²	C₂₃×#451	direct (N × H)
297. C₂²	C₂²×Q₁₆×C₂₃	semidirect
307. C₂	C₂×C₂₃×#451	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×Q₈×C₂₃	2
C₂³×Q₁₆×C₂₃	2
C₂²×C₂₃×C₂²⋊Q₈	2
C₂×C₂₃×#451	2
C₂²×C₂₃×#103	2
C₂²×C₂₃×C₂²⋊C₈	2
C₂²×#451	23

Subgroups — C22×C23×#451 #19207976

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂₃×#451 #19207976