Subgroups — C₂²×C₂₃×#466 #19208061

Order 5888.

3694 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁴×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₂₃×#121	C₂	1	2	yes	maximal, complemented
4	C₂×C₂₃×#466	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₂₃×#107	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₂₃×#102	C₂	1	2	yes	maximal, complemented
7	C₂³×D₄×C₂₃	—	2	2	no	—
8	C₂³×D₄×C₂₃	C₂²	1	2	yes	—
9	C₂⁴×C₄×C₂₃	C₂²	1	1	yes	characteristic
10	C₂²×D₈×C₂₃	—	2	8	no	—
11	C₂²×D₈×C₂₃	C₂²	1	12	yes	complemented
12	C₂³×C₈×C₂₃	—	2	8	no	—
13	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₂₃	C₂²	1	12	yes	complemented
16	C₂×C₂₃×#121	—	2	12	no	—
17	C₂×C₂₃×#121	C₂²	1	24	yes	complemented
18	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
19	C₂²×C₂²⋊C₄×C₂₃	—	2	2	no	—
20	C₂₃×#466	C₂²	1	64	yes	direct product factor
21	C₂×C₂₃×#107	C₂²	1	12	yes	complemented
22	C₂×C₂₃×#102	C₂²	1	24	yes	complemented
23	C₂⁵×C₂₃	—	2	2	no	—
24	C₂⁵×C₂₃	D₄	1	1	yes	characteristic
25	C₂²×D₄×C₂₃	—	2	24	no	—
26	C₂²×D₄×C₂₃	—	2	8	no	—
27	C₂²×D₄×C₂₃	—	2	6	no	—
28	C₂²×D₄×C₂₃	—	2	8	no	—
29	C₂²×D₄×C₂₃	—	4	16	no	—
30	C₂²×D₄×C₂₃	C₂³	1	12	yes	—
31	C₂³×C₄×C₂₃	—	2	4	no	—
32	C₂³×C₄×C₂₃	—	2	4	no	—
33	C₂³×C₄×C₂₃	—	2	2	no	—
34	C₂³×C₄×C₂₃	D₄	1	6	yes	—
35	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₂₃	C₂³	1	6	yes	—
37	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
38	C₂×D₈×C₂₃	—	2	48	no	—
39	C₂×D₈×C₂₃	C₂³	1	16	yes	complemented
40	C₂²×C₈×C₂₃	—	2	4	no	—
41	C₂²×C₈×C₂₃	—	2	48	no	—
42	C₂²×C₈×C₂₃	D₄	1	8	yes	complemented
43	C₂²×C₈×C₂₃	C₂³	1	6	yes	—
44	C₂²×C₈×C₂₃	C₂³	1	6	yes	—
45	C₂²×C₈×C₂₃	C₂³	1	16	yes	complemented
46	C₂₃×#121	—	2	48	no	—
47	C₂₃×#121	C₂³	1	32	yes	complemented
48	C₂×C₄⋊C₄×C₂₃	—	2	6	no	—
49	C₂×C₄⋊C₄×C₂₃	C₂³	1	12	yes	—
50	C₂×C₂²⋊C₄×C₂₃	—	2	24	no	—
51	C₂₃×#107	C₂³	1	16	yes	complemented
52	C₂₃×#102	C₂³	1	32	yes	complemented
53	C₂⁴×C₂₃	—	2	4	no	—
54	C₂⁴×C₂₃	—	2	4	no	—
55	C₂⁴×C₂₃	—	2	12	no	—
56	C₂⁴×C₂₃	—	4	8	no	—
57	C₂⁴×C₂₃	—	4	8	no	—
58	C₂⁴×C₂₃	—	4	8	no	—
59	C₂⁴×C₂₃	D₈	1	8	yes	complemented
60	C₂⁴×C₂₃	C₂×D₄	1	6	yes	—
61	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
62	C₂×D₄×C₂₃	—	2	32	no	—
63	C₂×D₄×C₂₃	—	2	24	no	—
64	C₂×D₄×C₂₃	—	2	12	no	—
65	C₂×D₄×C₂₃	—	2	24	no	—
66	C₂×D₄×C₂₃	—	4	96	no	—
67	C₂×D₄×C₂₃	—	4	24	no	—
68	C₂×D₄×C₂₃	C₂⁴	1	8	yes	—
69	C₂²×C₄×C₂₃	—	2	4	no	—
70	C₂²×C₄×C₂₃	—	2	24	no	—
71	C₂²×C₄×C₂₃	—	2	12	no	—
72	C₂²×C₄×C₂₃	—	2	12	no	—
73	C₂²×C₄×C₂₃	—	2	12	no	—
74	C₂²×C₄×C₂₃	—	4	8	no	—
75	C₂²×C₄×C₂₃	C₂×D₄	1	6	yes	—
76	C₂²×C₄×C₂₃	C₂×D₄	1	12	yes	—
77	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
78	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
79	C₂²×C₄×C₂₃	#41	1	4	yes	—
80	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
81	C₂²×C₄×C₂₃	C₂⁴	1	3	yes	—
82	D₈×C₂₃	—	2	32	no	—
83	C₂×C₈×C₂₃	—	2	12	no	—
84	C₂×C₈×C₂₃	—	2	32	no	—
85	C₂×C₈×C₂₃	C₂×D₄	1	24	yes	complemented
86	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
87	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
88	C₄⋊C₄×C₂₃	—	2	12	no	—
89	C₄⋊C₄×C₂₃	C₂⁴	1	8	yes	—
90	C₂²⋊C₄×C₂₃	—	2	32	no	—
91	C₂²×#466	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂³×C₂₃	—	2	12	no	—
93	C₂³×C₂₃	—	2	12	no	—
94	C₂³×C₂₃	—	2	32	no	—
95	C₂³×C₂₃	—	2	8	no	—
96	C₂³×C₂₃	—	4	24	no	—
97	C₂³×C₂₃	—	4	24	no	—
98	C₂³×C₂₃	—	4	24	no	—
99	C₂³×C₂₃	—	8	32	no	—
100	C₂³×C₂₃	#121	1	4	yes	—
101	C₂³×C₂₃	C₂×D₈	1	4	yes	—
102	C₂³×C₂₃	C₂×D₈	1	24	yes	complemented
103	C₂³×C₂₃	#135	1	4	yes	—
104	C₂³×C₂₃	C₂²×D₄	1	4	yes	—
105	C₂³×C₂₃	C₂²×D₄	1	3	yes	—
106	D₄×C₂₃	—	2	8	no	—
107	D₄×C₂₃	—	2	8	no	—
108	D₄×C₂₃	—	4	64	no	—
109	D₄×C₂₃	—	4	24	no	—
110	C₂×C₄×C₂₃	—	2	12	no	—
111	C₂×C₄×C₂₃	—	2	24	no	—
112	C₂×C₄×C₂₃	—	2	4	no	—
113	C₂×C₄×C₂₃	—	2	4	no	—
114	C₂×C₄×C₂₃	—	2	8	no	—
115	C₂×C₄×C₂₃	—	4	24	no	—
116	C₂×C₄×C₂₃	#121	1	12	yes	—
117	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
118	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
119	C₂×C₄×C₂₃	C₂²×D₄	1	6	yes	—
120	C₂×C₄×C₂₃	C₂×#41	1	6	yes	—
121	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
122	C₈×C₂₃	—	2	4	no	—
123	C₈×C₂₃	C₂²×D₄	1	8	yes	complemented
124	C₂³×D₈	C₂×C₂₃	1	1	yes	characteristic, complemented
125	C₂⁴×C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
126	C₂²×#121	C₂×C₂₃	1	2	yes	complemented
127	C₂×#466	C₂×C₂₃	1	24	yes	direct product factor
128	C₂²×#107	C₂×C₂₃	1	1	yes	characteristic, complemented
129	C₂²×#102	C₂×C₂₃	1	2	yes	complemented
130	C₂²×C₂₃	—	2	4	no	—
131	C₂²×C₂₃	—	2	48	no	—
132	C₂²×C₂₃	—	2	4	no	—
133	C₂²×C₂₃	—	4	8	no	—
134	C₂²×C₂₃	—	4	8	no	—
135	C₂²×C₂₃	—	4	8	no	—
136	C₂²×C₂₃	—	8	48	no	—
137	C₂²×C₂₃	#466	1	16	yes	direct product factor
138	C₂²×C₂₃	C₂×#121	1	6	yes	—
139	C₂²×C₂₃	C₂²×D₈	1	6	yes	—
140	C₂²×C₂₃	C₂²×D₈	1	8	yes	complemented
141	C₂²×C₂₃	C₂×#135	1	6	yes	—
142	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
143	C₄×C₂₃	—	2	4	no	—
144	C₄×C₂₃	—	4	8	no	—
145	C₄×C₂₃	C₂×#121	1	6	yes	—
146	C₄×C₂₃	C₂³×D₄	1	1	yes	characteristic
147	C₄×C₂₃	C₂²×#41	1	1	yes	characteristic
148	C₂³×D₄	—	2	2	no	—
149	C₂³×D₄	C₂²×C₂₃	1	2	yes	—
150	C₂⁴×C₄	C₂²×C₂₃	1	1	yes	characteristic
151	C₂²×D₈	—	2	8	no	—
152	C₂²×D₈	C₂²×C₂₃	1	12	yes	complemented
153	C₂³×C₈	—	2	8	no	—
154	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
155	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
156	C₂³×C₈	C₂²×C₂₃	1	12	yes	complemented
157	C₂×#121	—	2	12	no	—
158	C₂×#121	C₂²×C₂₃	1	24	yes	complemented
159	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
160	C₂²×C₂²⋊C₄	—	2	2	no	—
161	#466	C₂²×C₂₃	1	64	yes	direct product factor
162	C₂×#107	C₂²×C₂₃	1	12	yes	complemented
163	C₂×#102	C₂²×C₂₃	1	24	yes	complemented
164	C₂×C₂₃	—	2	8	no	—
165	C₂×C₂₃	—	8	8	no	—
166	C₂×C₂₃	C₂×#466	1	12	yes	direct product factor
167	C₂×C₂₃	C₂²×#121	1	1	yes	characteristic
168	C₂×C₂₃	C₂³×D₈	1	1	yes	characteristic
169	C₂×C₂₃	C₂²×#135	1	1	yes	characteristic
170	C₂⁵	—	2	2	no	—
171	C₂⁵	D₄×C₂₃	1	1	yes	characteristic
172	C₂²×D₄	—	2	24	no	—
173	C₂²×D₄	—	2	8	no	—
174	C₂²×D₄	—	2	6	no	—
175	C₂²×D₄	—	2	8	no	—
176	C₂²×D₄	—	4	16	no	—
177	C₂²×D₄	C₂³×C₂₃	1	12	yes	—
178	C₂³×C₄	—	2	4	no	—
179	C₂³×C₄	—	2	4	no	—
180	C₂³×C₄	—	2	2	no	—
181	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
182	C₂³×C₄	D₄×C₂₃	1	6	yes	—
183	C₂³×C₄	C₂³×C₂₃	1	6	yes	—
184	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
185	C₂×D₈	—	2	48	no	—
186	C₂×D₈	C₂³×C₂₃	1	16	yes	complemented
187	C₂²×C₈	—	2	4	no	—
188	C₂²×C₈	—	2	48	no	—
189	C₂²×C₈	D₄×C₂₃	1	8	yes	complemented
190	C₂²×C₈	C₂³×C₂₃	1	6	yes	—
191	C₂²×C₈	C₂³×C₂₃	1	6	yes	—
192	C₂²×C₈	C₂³×C₂₃	1	16	yes	complemented
193	#121	—	2	48	no	—
194	#121	C₂³×C₂₃	1	32	yes	complemented
195	C₂×C₄⋊C₄	—	2	6	no	—
196	C₂×C₄⋊C₄	C₂³×C₂₃	1	12	yes	—
197	C₂×C₂²⋊C₄	—	2	24	no	—
198	#107	C₂³×C₂₃	1	16	yes	complemented
199	#102	C₂³×C₂₃	1	32	yes	complemented
200	C₂₃	C₂²×#466	1	1	yes	characteristic, direct product factor, Sylow
201	C₂⁴	—	2	4	no	—
202	C₂⁴	—	2	4	no	—
203	C₂⁴	—	2	12	no	—
204	C₂⁴	—	4	8	no	—
205	C₂⁴	—	4	8	no	—
206	C₂⁴	—	4	8	no	—
207	C₂⁴	D₈×C₂₃	1	8	yes	complemented
208	C₂⁴	C₂×D₄×C₂₃	1	6	yes	—
209	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
210	C₂×D₄	—	2	32	no	—
211	C₂×D₄	—	2	24	no	—
212	C₂×D₄	—	2	12	no	—
213	C₂×D₄	—	2	24	no	—
214	C₂×D₄	—	4	96	no	—
215	C₂×D₄	—	4	24	no	—
216	C₂×D₄	C₂⁴×C₂₃	1	8	yes	—
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	24	no	—
219	C₂²×C₄	—	2	12	no	—
220	C₂²×C₄	—	2	12	no	—
221	C₂²×C₄	—	2	12	no	—
222	C₂²×C₄	—	4	8	no	—
223	C₂²×C₄	C₂×D₄×C₂₃	1	6	yes	—
224	C₂²×C₄	C₂×D₄×C₂₃	1	12	yes	—
225	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
226	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
227	C₂²×C₄	#41×C₂₃	1	4	yes	—
228	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
229	C₂²×C₄	C₂⁴×C₂₃	1	3	yes	—
230	D₈	—	2	32	no	—
231	C₂×C₈	—	2	12	no	—
232	C₂×C₈	—	2	32	no	—
233	C₂×C₈	C₂×D₄×C₂₃	1	24	yes	complemented
234	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
235	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
236	C₄⋊C₄	—	2	12	no	—
237	C₄⋊C₄	C₂⁴×C₂₃	1	8	yes	—
238	C₂²⋊C₄	—	2	32	no	—
239	C₂³	—	2	12	no	—
240	C₂³	—	2	32	no	—
241	C₂³	—	2	12	no	—
242	C₂³	—	2	8	no	—
243	C₂³	—	4	24	no	—
244	C₂³	—	4	24	no	—
245	C₂³	—	4	24	no	—
246	C₂³	—	8	32	no	—
247	C₂³	C₂₃×#121	1	4	yes	—
248	C₂³	C₂×D₈×C₂₃	1	4	yes	—
249	C₂³	C₂×D₈×C₂₃	1	24	yes	complemented
250	C₂³	C₂₃×#135	1	4	yes	—
251	C₂³	C₂²×D₄×C₂₃	1	4	yes	—
252	C₂³	C₂²×D₄×C₂₃	1	3	yes	—
253	D₄	—	2	8	no	—
254	D₄	—	2	8	no	—
255	D₄	—	4	64	no	—
256	D₄	—	4	24	no	—
257	C₂×C₄	—	2	12	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	24	no	—
261	C₂×C₄	—	2	8	no	—
262	C₂×C₄	—	4	24	no	—
263	C₂×C₄	C₂₃×#121	1	12	yes	—
264	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
265	C₂×C₄	C₂²×D₄×C₂₃	1	6	yes	—
266	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
267	C₂×C₄	C₂×#41×C₂₃	1	6	yes	—
268	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
269	C₈	—	2	4	no	—
270	C₈	C₂²×D₄×C₂₃	1	8	yes	complemented
271	C₂²	—	2	4	no	—
272	C₂²	—	2	48	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	4	8	no	—
275	C₂²	—	4	8	no	—
276	C₂²	—	4	8	no	—
277	C₂²	—	8	48	no	—
278	C₂²	C₂₃×#466	1	16	yes	direct product factor
279	C₂²	C₂×C₂₃×#121	1	6	yes	—
280	C₂²	C₂²×D₈×C₂₃	1	6	yes	—
281	C₂²	C₂²×D₈×C₂₃	1	8	yes	complemented
282	C₂²	C₂×C₂₃×#135	1	6	yes	—
283	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
284	C₄	—	2	4	no	—
285	C₄	—	4	8	no	—
286	C₄	C₂×C₂₃×#121	1	6	yes	—
287	C₄	C₂³×D₄×C₂₃	1	1	yes	characteristic
288	C₄	C₂²×#41×C₂₃	1	1	yes	characteristic
289	C₂	—	2	8	no	—
290	C₂	—	8	8	no	—
291	C₂	C₂×C₂₃×#466	1	12	yes	direct product factor
292	C₂	C₂²×C₂₃×#121	1	1	yes	characteristic
293	C₂	C₂³×D₈×C₂₃	1	1	yes	characteristic
294	C₂	C₂²×C₂₃×#135	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₂₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
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
1. C₂³×D₈×C₂₃	C₂	semidirect
2. C₂⁴×C₈×C₂₃	C₂	semidirect
3. C₂²×C₂₃×#121	C₂	semidirect
4. C₂×C₂₃×#466	C₂	direct (N × H)
5. C₂²×C₂₃×#107	C₂	semidirect
6. C₂²×C₂₃×#102	C₂	semidirect
11. C₂²×D₈×C₂₃	C₂²	semidirect
15. C₂³×C₈×C₂₃	C₂²	semidirect
17. C₂×C₂₃×#121	C₂²	semidirect
20. C₂₃×#466	C₂²	direct (N × H)
21. C₂×C₂₃×#107	C₂²	semidirect
22. C₂×C₂₃×#102	C₂²	semidirect
39. C₂×D₈×C₂₃	C₂³	semidirect
42. C₂²×C₈×C₂₃	D₄	semidirect
45. C₂²×C₈×C₂₃	C₂³	semidirect
47. C₂₃×#121	C₂³	semidirect
51. C₂₃×#107	C₂³	semidirect
52. C₂₃×#102	C₂³	semidirect
59. C₂⁴×C₂₃	D₈	semidirect
85. C₂×C₈×C₂₃	C₂×D₄	semidirect
91. C₂²×#466	C₂₃	direct (N × H)
102. C₂³×C₂₃	C₂×D₈	semidirect
123. C₈×C₂₃	C₂²×D₄	semidirect
124. C₂³×D₈	C₂×C₂₃	semidirect
125. C₂⁴×C₈	C₂×C₂₃	semidirect
126. C₂²×#121	C₂×C₂₃	semidirect
127. C₂×#466	C₂×C₂₃	direct (N × H)
128. C₂²×#107	C₂×C₂₃	semidirect
129. C₂²×#102	C₂×C₂₃	semidirect
137. C₂²×C₂₃	#466	direct (N × H)
140. C₂²×C₂₃	C₂²×D₈	semidirect
152. C₂²×D₈	C₂²×C₂₃	semidirect
156. C₂³×C₈	C₂²×C₂₃	semidirect
158. C₂×#121	C₂²×C₂₃	semidirect
161. #466	C₂²×C₂₃	direct (N × H)
162. C₂×#107	C₂²×C₂₃	semidirect
163. C₂×#102	C₂²×C₂₃	semidirect
166. C₂×C₂₃	C₂×#466	direct (N × H)
186. C₂×D₈	C₂³×C₂₃	semidirect
189. C₂²×C₈	D₄×C₂₃	semidirect
192. C₂²×C₈	C₂³×C₂₃	semidirect
194. #121	C₂³×C₂₃	semidirect
198. #107	C₂³×C₂₃	semidirect
199. #102	C₂³×C₂₃	semidirect
200. C₂₃	C₂²×#466	direct (N × H)
207. C₂⁴	D₈×C₂₃	semidirect
233. C₂×C₈	C₂×D₄×C₂₃	semidirect
249. C₂³	C₂×D₈×C₂₃	semidirect
270. C₈	C₂²×D₄×C₂₃	semidirect
278. C₂²	C₂₃×#466	direct (N × H)
281. C₂²	C₂²×D₈×C₂₃	semidirect
291. C₂	C₂×C₂₃×#466	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×D₈×C₂₃	2
C₂⁴×C₈×C₂₃	2
C₂²×C₂₃×#121	2
C₂×C₂₃×#466	2
C₂²×C₂₃×#107	2
C₂²×C₂₃×#102	2
C₂²×#466	23

Subgroups — C22×C23×#466 #19208061

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂₃×#466 #19208061