Subgroups — C₂×C₂₃×#1623 #19166132

Order 5888.

1126 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	C₂×Q₈²×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#492	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#460	C₂	1	2	yes	maximal
4	C₂₃×#1623	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₂₃×#333	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂₃×Q₈⋊C₈	C₂	1	2	yes	maximal, complemented
7	C₂²×SD₁₆×C₂₃	—	2	2	no	—
8	C₂²×SD₁₆×C₂₃	—	2	2	no	—
9	Q₈²×C₂₃	—	2	2	no	—
10	Q₈²×C₂₃	—	2	4	no	—
11	Q₈²×C₂₃	C₂²	1	4	yes	complemented
12	C₂×C₂₃×C₄⋊Q₈	—	2	2	no	—
13	C₂×C₂₃×C₄⋊Q₈	—	2	2	no	—
14	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
16	C₂×C₄×Q₈×C₂₃	—	2	2	no	—
17	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	—
18	C₂₃×#492	—	2	2	no	—
19	C₂₃×#492	C₂²	1	4	yes	—
20	C₂₃×#460	—	2	4	no	—
21	C₂₃×#460	C₂²	1	8	yes	—
22	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
23	C₂×C₂₃×#102	—	2	2	no	—
24	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
25	C₂₃×#333	C₂²	1	4	yes	complemented
26	C₂₃×Q₈⋊C₈	C₂²	1	8	yes	complemented
27	C₂²×Q₈×C₂₃	—	2	2	no	—
28	C₂²×Q₈×C₂₃	D₄	1	2	yes	—
29	C₂²×Q₈×C₂₃	D₄	1	2	yes	—
30	C₂²×D₄×C₂₃	—	2	1	no	—
31	C₂²×D₄×C₂₃	—	2	2	no	—
32	C₂×SD₁₆×C₂₃	—	2	8	no	—
33	C₂×SD₁₆×C₂₃	—	2	8	no	—
34	C₂×SD₁₆×C₂₃	—	4	8	no	—
35	C₂×SD₁₆×C₂₃	—	4	8	no	—
36	C₂²×C₈×C₂₃	—	2	1	no	—
37	C₂²×C₈×C₂₃	—	2	2	no	—
38	C₂₃×C₄⋊Q₈	—	2	8	no	—
39	C₂₃×C₄⋊Q₈	—	2	8	no	—
40	C₂₃×C₄⋊Q₈	—	2	8	no	—
41	C₂₃×C₄⋊Q₈	—	2	8	no	—
42	C₂₃×C₄⋊Q₈	—	2	2	no	—
43	C₂₃×C₄⋊Q₈	D₄	1	2	yes	—
44	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
45	C₂₃×#127	—	2	2	no	—
46	C₂₃×#127	—	2	1	no	—
47	C₂₃×#127	C₂³	1	2	yes	—
48	C₄×Q₈×C₂₃	—	2	8	no	—
49	C₄×Q₈×C₂₃	—	2	8	no	—
50	C₄×Q₈×C₂₃	—	2	2	no	—
51	C₄×Q₈×C₂₃	—	2	2	no	—
52	C₄×Q₈×C₂₃	D₄	1	4	yes	—
53	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
54	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
55	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
56	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
57	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
58	C₂×C₄²×C₂₃	—	2	2	no	—
59	C₂×C₄²×C₂₃	—	2	2	no	—
60	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
61	C₂₃×C₄⋊C₈	—	2	2	no	—
62	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
63	C₂₃×#102	—	2	8	no	—
64	C₄×C₈×C₂₃	—	2	1	no	—
65	C₄×C₈×C₂₃	C₂³	1	2	yes	—
66	C₂⁴×C₂₃	—	4	1	no	—
67	C₂×Q₈×C₂₃	—	2	8	no	—
68	C₂×Q₈×C₂₃	—	2	2	no	—
69	C₂×Q₈×C₂₃	—	2	2	no	—
70	C₂×Q₈×C₂₃	—	2	4	no	—
71	C₂×Q₈×C₂₃	—	4	4	no	—
72	C₂×Q₈×C₂₃	—	8	4	no	—
73	C₂×Q₈×C₂₃	SD₁₆	1	8	yes	complemented
74	C₂×Q₈×C₂₃	C₂×D₄	1	4	yes	—
75	C₂×Q₈×C₂₃	C₂×D₄	1	4	yes	—
76	C₂×D₄×C₂₃	—	2	4	no	—
77	C₂×D₄×C₂₃	—	2	2	no	—
78	C₂×D₄×C₂₃	—	4	4	no	—
79	C₂×D₄×C₂₃	—	4	1	no	—
80	C₂×D₄×C₂₃	—	4	2	no	—
81	C₂×D₄×C₂₃	—	4	4	no	—
82	C₂×D₄×C₂₃	—	4	4	no	—
83	C₂²×C₄×C₂₃	—	2	2	no	—
84	C₂²×C₄×C₂₃	—	2	2	no	—
85	C₂²×C₄×C₂₃	—	2	2	no	—
86	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
88	SD₁₆×C₂₃	—	4	16	no	—
89	SD₁₆×C₂₃	—	4	16	no	—
90	C₂×C₈×C₂₃	—	2	4	no	—
91	C₂×C₈×C₂₃	—	2	2	no	—
92	C₂×C₈×C₂₃	—	2	4	no	—
93	C₂×C₈×C₂₃	—	4	4	no	—
94	C₄⋊C₄×C₂₃	—	2	16	no	—
95	C₄⋊C₄×C₂₃	—	2	4	no	—
96	C₄⋊C₄×C₂₃	—	2	4	no	—
97	C₄⋊C₄×C₂₃	—	2	2	no	—
98	C₄⋊C₄×C₂₃	—	2	4	no	—
99	C₄⋊C₄×C₂₃	—	2	4	no	—
100	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
101	C₄²×C₂₃	—	2	8	no	—
102	C₄²×C₂₃	—	2	4	no	—
103	C₄²×C₂₃	—	2	4	no	—
104	C₄²×C₂₃	C₂×D₄	1	2	yes	—
105	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
107	C₂×#1623	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂³×C₂₃	—	4	2	no	—
109	C₂³×C₂₃	—	8	4	no	—
110	C₂³×C₂₃	—	8	2	no	—
111	C₂³×C₂₃	#120	1	1	yes	characteristic
112	Q₈×C₂₃	—	2	4	no	—
113	Q₈×C₂₃	—	2	8	no	—
114	Q₈×C₂₃	—	4	4	no	—
115	Q₈×C₂₃	—	8	2	no	—
116	Q₈×C₂₃	—	8	8	no	—
117	Q₈×C₂₃	C₂×SD₁₆	1	8	yes	complemented
118	D₄×C₂₃	—	4	4	no	—
119	D₄×C₂₃	—	4	4	no	—
120	D₄×C₂₃	—	4	4	no	—
121	D₄×C₂₃	—	8	2	no	—
122	D₄×C₂₃	—	8	4	no	—
123	C₂×C₄×C₂₃	—	2	4	no	—
124	C₂×C₄×C₂₃	—	2	8	no	—
125	C₂×C₄×C₂₃	—	2	2	no	—
126	C₂×C₄×C₂₃	—	2	4	no	—
127	C₂×C₄×C₂₃	—	2	4	no	—
128	C₂×C₄×C₂₃	—	4	4	no	—
129	C₂×C₄×C₂₃	—	4	4	no	—
130	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
131	C₂×C₄×C₂₃	#120	1	2	yes	—
132	C₂×C₄×C₂₃	C₂×SD₁₆	1	4	yes	—
133	C₂×C₄×C₂₃	#137	1	4	yes	—
134	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
136	C₈×C₂₃	—	2	4	no	—
137	C₈×C₂₃	—	4	4	no	—
138	C₂×Q₈²	C₂×C₂₃	1	1	yes	characteristic, complemented
139	C₂×#492	C₂×C₂₃	1	1	yes	characteristic
140	C₂×#460	C₂×C₂₃	1	2	yes	—
141	#1623	C₂×C₂₃	1	8	yes	direct product factor
142	C₂×#333	C₂×C₂₃	1	1	yes	characteristic, complemented
143	C₂×Q₈⋊C₈	C₂×C₂₃	1	2	yes	complemented
144	C₂²×C₂₃	—	8	4	no	—
145	C₂²×C₂₃	—	8	2	no	—
146	C₂²×C₂₃	—	16	4	no	—
147	C₂²×C₂₃	#448	1	4	yes	—
148	C₂²×C₂₃	#453	1	2	yes	—
149	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
150	C₄×C₂₃	—	2	8	no	—
151	C₄×C₂₃	—	2	1	no	—
152	C₄×C₂₃	—	2	1	no	—
153	C₄×C₂₃	—	4	4	no	—
154	C₄×C₂₃	—	4	4	no	—
155	C₄×C₂₃	#448	1	4	yes	—
156	C₄×C₂₃	C₂²×SD₁₆	1	2	yes	—
157	C₄×C₂₃	C₂×#137	1	2	yes	—
158	C₂²×SD₁₆	—	2	2	no	—
159	C₂²×SD₁₆	—	2	2	no	—
160	Q₈²	—	2	2	no	—
161	Q₈²	—	2	4	no	—
162	Q₈²	C₂²×C₂₃	1	4	yes	complemented
163	C₂×C₄⋊Q₈	—	2	2	no	—
164	C₂×C₄⋊Q₈	—	2	2	no	—
165	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
166	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
167	C₂×C₄×Q₈	—	2	2	no	—
168	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	—
169	#492	—	2	2	no	—
170	#492	C₂²×C₂₃	1	4	yes	—
171	#460	—	2	4	no	—
172	#460	C₂²×C₂₃	1	8	yes	—
173	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
174	C₂×#102	—	2	2	no	—
175	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
176	#333	C₂²×C₂₃	1	4	yes	complemented
177	Q₈⋊C₈	C₂²×C₂₃	1	8	yes	complemented
178	C₂×C₂₃	—	16	2	no	—
179	C₂×C₂₃	#1623	1	4	yes	direct product factor
180	C₂×C₂₃	C₂×#448	1	2	yes	—
181	C₂×C₂₃	C₂×#453	1	1	yes	characteristic
182	C₂²×Q₈	—	2	2	no	—
183	C₂²×Q₈	D₄×C₂₃	1	2	yes	—
184	C₂²×Q₈	D₄×C₂₃	1	2	yes	—
185	C₂²×D₄	—	2	1	no	—
186	C₂²×D₄	—	2	2	no	—
187	C₂×SD₁₆	—	2	8	no	—
188	C₂×SD₁₆	—	2	8	no	—
189	C₂×SD₁₆	—	4	8	no	—
190	C₂×SD₁₆	—	4	8	no	—
191	C₂²×C₈	—	2	1	no	—
192	C₂²×C₈	—	2	2	no	—
193	C₄⋊Q₈	—	2	8	no	—
194	C₄⋊Q₈	—	2	8	no	—
195	C₄⋊Q₈	—	2	8	no	—
196	C₄⋊Q₈	—	2	8	no	—
197	C₄⋊Q₈	—	2	2	no	—
198	C₄⋊Q₈	D₄×C₂₃	1	2	yes	—
199	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
200	#127	—	2	2	no	—
201	#127	—	2	1	no	—
202	#127	C₂³×C₂₃	1	2	yes	—
203	C₄×Q₈	—	2	8	no	—
204	C₄×Q₈	—	2	8	no	—
205	C₄×Q₈	—	2	2	no	—
206	C₄×Q₈	—	2	2	no	—
207	C₄×Q₈	D₄×C₂₃	1	4	yes	—
208	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
209	C₂×C₄⋊C₄	—	2	4	no	—
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	—	2	2	no	—
212	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
213	C₂×C₄²	—	2	2	no	—
214	C₂×C₄²	—	2	2	no	—
215	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
216	C₄⋊C₈	—	2	2	no	—
217	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
218	#102	—	2	8	no	—
219	C₄×C₈	—	2	1	no	—
220	C₄×C₈	C₂³×C₂₃	1	2	yes	—
221	C₂₃	C₂×#1623	1	1	yes	characteristic, direct product factor, Sylow
222	C₂⁴	—	4	1	no	—
223	C₂×Q₈	—	2	8	no	—
224	C₂×Q₈	—	2	2	no	—
225	C₂×Q₈	—	2	2	no	—
226	C₂×Q₈	—	2	4	no	—
227	C₂×Q₈	—	4	4	no	—
228	C₂×Q₈	—	8	4	no	—
229	C₂×Q₈	SD₁₆×C₂₃	1	8	yes	complemented
230	C₂×Q₈	C₂×D₄×C₂₃	1	4	yes	—
231	C₂×Q₈	C₂×D₄×C₂₃	1	4	yes	—
232	C₂×D₄	—	2	4	no	—
233	C₂×D₄	—	2	2	no	—
234	C₂×D₄	—	4	4	no	—
235	C₂×D₄	—	4	1	no	—
236	C₂×D₄	—	4	2	no	—
237	C₂×D₄	—	4	4	no	—
238	C₂×D₄	—	4	4	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
243	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
244	SD₁₆	—	4	16	no	—
245	SD₁₆	—	4	16	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	4	no	—
249	C₂×C₈	—	4	4	no	—
250	C₄⋊C₄	—	2	16	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
257	C₄²	—	2	8	no	—
258	C₄²	—	2	4	no	—
259	C₄²	—	2	4	no	—
260	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
261	C₄²	C₂×D₄×C₂₃	1	2	yes	—
262	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
263	C₂³	—	4	2	no	—
264	C₂³	—	8	4	no	—
265	C₂³	—	8	2	no	—
266	C₂³	C₂₃×#120	1	1	yes	characteristic
267	Q₈	—	2	4	no	—
268	Q₈	—	2	8	no	—
269	Q₈	—	4	4	no	—
270	Q₈	—	8	2	no	—
271	Q₈	—	8	8	no	—
272	Q₈	C₂×SD₁₆×C₂₃	1	8	yes	complemented
273	D₄	—	4	4	no	—
274	D₄	—	4	4	no	—
275	D₄	—	4	4	no	—
276	D₄	—	8	2	no	—
277	D₄	—	8	4	no	—
278	C₂×C₄	—	2	8	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
286	C₂×C₄	C₂₃×#120	1	2	yes	—
287	C₂×C₄	C₂×SD₁₆×C₂₃	1	4	yes	—
288	C₂×C₄	C₂₃×#137	1	4	yes	—
289	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
290	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
291	C₈	—	2	4	no	—
292	C₈	—	4	4	no	—
293	C₂²	—	8	4	no	—
294	C₂²	—	8	2	no	—
295	C₂²	—	16	4	no	—
296	C₂²	C₂₃×#448	1	4	yes	—
297	C₂²	C₂₃×#453	1	2	yes	—
298	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
299	C₄	—	2	8	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	4	4	no	—
303	C₄	—	4	4	no	—
304	C₄	C₂₃×#448	1	4	yes	—
305	C₄	C₂²×SD₁₆×C₂₃	1	2	yes	—
306	C₄	C₂×C₂₃×#137	1	2	yes	—
307	C₂	—	16	2	no	—
308	C₂	C₂₃×#1623	1	4	yes	direct product factor
309	C₂	C₂×C₂₃×#448	1	2	yes	—
310	C₂	C₂×C₂₃×#453	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂×C₂₃×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈²×C₂₃	C₂	semidirect
4. C₂₃×#1623	C₂	direct (N × H)
5. C₂×C₂₃×#333	C₂	semidirect
6. C₂×C₂₃×Q₈⋊C₈	C₂	semidirect
11. Q₈²×C₂₃	C₂²	semidirect
25. C₂₃×#333	C₂²	semidirect
26. C₂₃×Q₈⋊C₈	C₂²	semidirect
73. C₂×Q₈×C₂₃	SD₁₆	semidirect
107. C₂×#1623	C₂₃	direct (N × H)
117. Q₈×C₂₃	C₂×SD₁₆	semidirect
138. C₂×Q₈²	C₂×C₂₃	semidirect
141. #1623	C₂×C₂₃	direct (N × H)
142. C₂×#333	C₂×C₂₃	semidirect
143. C₂×Q₈⋊C₈	C₂×C₂₃	semidirect
162. Q₈²	C₂²×C₂₃	semidirect
176. #333	C₂²×C₂₃	semidirect
177. Q₈⋊C₈	C₂²×C₂₃	semidirect
179. C₂×C₂₃	#1623	direct (N × H)
221. C₂₃	C₂×#1623	direct (N × H)
229. C₂×Q₈	SD₁₆×C₂₃	semidirect
272. Q₈	C₂×SD₁₆×C₂₃	semidirect
308. C₂	C₂₃×#1623	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×Q₈²×C₂₃	2
C₂×C₂₃×#492	2
C₂×C₂₃×#460	2
C₂₃×#1623	2
C₂×C₂₃×#333	2
C₂×C₂₃×Q₈⋊C₈	2
C₂×#1623	23

Subgroups — C2×C23×#1623 #19166132

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#1623 #19166132