Subgroups — C₂₃×#12733 #19160359

Order 5888.

954 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×#135	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#358	C₂	1	2	yes	maximal, complemented
3	C₂×C₂₃×#357	C₂	1	2	yes	maximal, complemented
4	C₂×C₂₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1808	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#41×C₂₃	—	2	1	no	—
7	C₂×C₂₃×#135	—	2	4	no	—
8	C₂×C₂₃×#135	—	2	4	no	—
9	C₂×C₂₃×#135	—	2	2	no	—
10	C₂×C₂₃×#135	C₄	1	4	yes	complemented
11	C₂²×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
12	C₂²×SD₁₆×C₂₃	—	2	1	no	—
13	C₂²×D₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₂₃	C₂²	1	2	yes	—
16	C₂×C₂₃×#107	C₂²	1	2	yes	—
17	C₂₃×#380	—	2	1	no	—
18	C₂₃×#358	—	2	4	no	—
19	C₂₃×#357	—	2	4	no	—
20	C₂₃×C₂²⋊C₁₆	—	2	2	no	—
21	C₂₃×#340	—	2	1	no	—
22	C₂×#41×C₂₃	—	2	8	no	—
23	C₂×#41×C₂₃	—	2	4	no	—
24	C₂×#41×C₂₃	—	4	4	no	—
25	C₂×#41×C₂₃	—	4	2	no	—
26	C₂²×Q₈×C₂₃	—	2	1	no	—
27	C₂²×D₄×C₂₃	—	2	1	no	—
28	C₂²×D₄×C₂₃	—	4	1	no	—
29	C₂³×C₄×C₂₃	—	4	1	no	—
30	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
31	C₂₃×#135	—	2	16	no	—
32	C₂₃×#135	—	2	8	no	—
33	C₂₃×#135	—	2	8	no	—
34	C₂×Q₁₆×C₂₃	—	2	2	no	—
35	C₂×Q₁₆×C₂₃	—	2	1	no	—
36	C₂×Q₁₆×C₂₃	C₂×C₄	1	2	yes	—
37	C₂×Q₁₆×C₂₃	D₄	1	4	yes	—
38	C₂×SD₁₆×C₂₃	—	2	8	no	—
39	C₂×SD₁₆×C₂₃	—	2	4	no	—
40	C₂×D₈×C₂₃	—	2	2	no	—
41	C₂×D₈×C₂₃	—	2	1	no	—
42	C₂×D₈×C₂₃	C₂×C₄	1	2	yes	—
43	C₂×D₈×C₂₃	D₄	1	4	yes	—
44	C₂²×C₈×C₂₃	—	2	2	no	—
45	C₂²×C₈×C₂₃	—	2	2	no	—
46	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
47	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₂₃	D₄	1	2	yes	—
49	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
51	C₂×C₂²⋊C₄×C₂₃	—	4	1	no	—
52	C₂×C₁₆×C₂₃	—	2	2	no	—
53	C₂×C₁₆×C₂₃	—	2	2	no	—
54	C₂×C₁₆×C₂₃	—	2	2	no	—
55	C₂₃×#107	—	2	2	no	—
56	C₂₃×#107	—	2	2	no	—
57	C₂₃×#95	—	4	1	no	—
58	C₂⁴×C₂₃	—	4	1	no	—
59	C₂⁴×C₂₃	D₈	1	1	yes	characteristic
60	#41×C₂₃	—	2	16	no	—
61	#41×C₂₃	—	4	8	no	—
62	#41×C₂₃	—	4	16	no	—
63	C₂×Q₈×C₂₃	—	2	2	no	—
64	C₂×Q₈×C₂₃	—	2	4	no	—
65	C₂×Q₈×C₂₃	—	4	1	no	—
66	C₂×Q₈×C₂₃	—	4	2	no	—
67	C₂×D₄×C₂₃	—	2	2	no	—
68	C₂×D₄×C₂₃	—	2	4	no	—
69	C₂×D₄×C₂₃	—	4	8	no	—
70	C₂×D₄×C₂₃	—	4	4	no	—
71	C₂×D₄×C₂₃	—	4	1	no	—
72	C₂×D₄×C₂₃	—	4	2	no	—
73	C₂²×C₄×C₂₃	—	2	2	no	—
74	C₂²×C₄×C₂₃	—	2	2	no	—
75	C₂²×C₄×C₂₃	—	4	8	no	—
76	C₂²×C₄×C₂₃	—	4	1	no	—
77	C₂²×C₄×C₂₃	—	4	4	no	—
78	C₂²×C₄×C₂₃	—	4	2	no	—
79	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
80	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
81	C₂²×C₄×C₂₃	SD₁₆	1	2	yes	—
82	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
83	Q₁₆×C₂₃	—	2	2	no	—
84	Q₁₆×C₂₃	—	2	4	no	—
85	Q₁₆×C₂₃	C₂²⋊C₄	1	4	yes	complemented
86	SD₁₆×C₂₃	—	2	16	no	—
87	D₈×C₂₃	—	2	2	no	—
88	D₈×C₂₃	—	2	4	no	—
89	D₈×C₂₃	C₂²⋊C₄	1	4	yes	complemented
90	C₂×C₈×C₂₃	—	2	2	no	—
91	C₂×C₈×C₂₃	—	2	2	no	—
92	C₂×C₈×C₂₃	—	2	2	no	—
93	C₂×C₈×C₂₃	—	2	4	no	—
94	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
95	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
99	C₄⋊C₄×C₂₃	—	4	2	no	—
100	C₄⋊C₄×C₂₃	—	4	2	no	—
101	C₂²⋊C₄×C₂₃	—	8	2	no	—
102	C₁₆×C₂₃	—	4	2	no	—
103	#12733	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂³×C₂₃	—	2	2	no	—
105	C₂³×C₂₃	—	2	2	no	—
106	C₂³×C₂₃	—	2	1	no	—
107	C₂³×C₂₃	—	2	1	no	—
108	C₂³×C₂₃	—	4	2	no	—
109	C₂³×C₂₃	—	4	4	no	—
110	C₂³×C₂₃	—	8	2	no	—
111	C₂³×C₂₃	—	8	1	no	—
112	C₂³×C₂₃	—	8	1	no	—
113	C₂³×C₂₃	#102	1	2	yes	—
114	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
115	Q₈×C₂₃	—	2	4	no	—
116	Q₈×C₂₃	—	4	2	no	—
117	Q₈×C₂₃	—	4	4	no	—
118	D₄×C₂₃	—	2	4	no	—
119	D₄×C₂₃	—	4	16	no	—
120	D₄×C₂₃	—	4	2	no	—
121	D₄×C₂₃	—	4	4	no	—
122	C₂×C₄×C₂₃	—	2	2	no	—
123	C₂×C₄×C₂₃	—	2	2	no	—
124	C₂×C₄×C₂₃	—	2	2	no	—
125	C₂×C₄×C₂₃	—	2	4	no	—
126	C₂×C₄×C₂₃	—	4	16	no	—
127	C₂×C₄×C₂₃	—	4	2	no	—
128	C₂×C₄×C₂₃	—	4	4	no	—
129	C₂×C₄×C₂₃	—	8	2	no	—
130	C₂×C₄×C₂₃	—	8	2	no	—
131	C₂×C₄×C₂₃	—	8	2	no	—
132	C₂×C₄×C₂₃	#102	1	2	yes	—
133	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₂₃	#120	1	2	yes	—
135	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
136	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
137	C₈×C₂₃	—	2	2	no	—
138	C₈×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₈×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₈×C₂₃	#120	1	2	yes	—
141	C₂²×#135	C₂×C₂₃	1	1	yes	characteristic
142	C₂×#358	C₂×C₂₃	1	2	yes	complemented
143	C₂×#357	C₂×C₂₃	1	2	yes	complemented
144	C₂×C₂²⋊C₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
145	#1808	C₂×C₂₃	1	1	yes	characteristic, complemented
146	C₂²×C₂₃	—	2	2	no	—
147	C₂²×C₂₃	—	2	2	no	—
148	C₂²×C₂₃	—	2	2	no	—
149	C₂²×C₂₃	—	4	2	no	—
150	C₂²×C₂₃	—	4	2	no	—
151	C₂²×C₂₃	—	4	4	no	—
152	C₂²×C₂₃	—	8	2	no	—
153	C₂²×C₂₃	—	8	2	no	—
154	C₂²×C₂₃	—	8	4	no	—
155	C₂²×C₂₃	—	8	4	no	—
156	C₂²×C₂₃	C₂×#102	1	1	yes	characteristic
157	C₂²×C₂₃	#447	1	2	yes	—
158	C₂²×C₂₃	#508	1	2	yes	—
159	C₂²×C₂₃	#509	1	1	yes	characteristic
160	C₂²×C₂₃	#510	1	1	yes	characteristic
161	C₄×C₂₃	—	2	2	no	—
162	C₄×C₂₃	—	4	4	no	—
163	C₄×C₂₃	—	16	2	no	—
164	C₄×C₂₃	#379	1	1	yes	characteristic
165	C₄×C₂₃	#418	1	1	yes	characteristic
166	C₄×C₂₃	#450	1	2	yes	—
167	C₂²×#41	—	2	1	no	—
168	C₂×#135	—	2	4	no	—
169	C₂×#135	—	2	4	no	—
170	C₂×#135	—	2	2	no	—
171	C₂×#135	C₄×C₂₃	1	4	yes	complemented
172	C₂²×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
173	C₂²×SD₁₆	—	2	1	no	—
174	C₂²×D₈	C₂²×C₂₃	1	1	yes	characteristic
175	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
176	C₂²×C₁₆	C₂²×C₂₃	1	2	yes	—
177	C₂×#107	C₂²×C₂₃	1	2	yes	—
178	#380	—	2	1	no	—
179	#358	—	2	4	no	—
180	#357	—	2	4	no	—
181	C₂²⋊C₁₆	—	2	2	no	—
182	#340	—	2	1	no	—
183	C₂×C₂₃	—	4	2	no	—
184	C₂×C₂₃	—	8	4	no	—
185	C₂×C₂₃	#1786	1	1	yes	characteristic
186	C₂×C₂₃	#2138	1	1	yes	characteristic
187	C₂×C₂₃	#2139	1	1	yes	characteristic
188	C₂×C₂₃	#2186	1	2	yes	—
189	C₂×C₂₃	#2189	1	2	yes	—
190	C₂×#41	—	2	8	no	—
191	C₂×#41	—	2	4	no	—
192	C₂×#41	—	4	4	no	—
193	C₂×#41	—	4	2	no	—
194	C₂²×Q₈	—	2	1	no	—
195	C₂²×D₄	—	2	1	no	—
196	C₂²×D₄	—	4	1	no	—
197	C₂³×C₄	—	4	1	no	—
198	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
199	#135	—	2	16	no	—
200	#135	—	2	8	no	—
201	#135	—	2	8	no	—
202	C₂×Q₁₆	—	2	2	no	—
203	C₂×Q₁₆	—	2	1	no	—
204	C₂×Q₁₆	C₂×C₄×C₂₃	1	2	yes	—
205	C₂×Q₁₆	D₄×C₂₃	1	4	yes	—
206	C₂×SD₁₆	—	2	8	no	—
207	C₂×SD₁₆	—	2	4	no	—
208	C₂×D₈	—	2	2	no	—
209	C₂×D₈	—	2	1	no	—
210	C₂×D₈	C₂×C₄×C₂₃	1	2	yes	—
211	C₂×D₈	D₄×C₂₃	1	4	yes	—
212	C₂²×C₈	—	2	2	no	—
213	C₂²×C₈	—	2	2	no	—
214	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
215	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
216	C₂²×C₈	D₄×C₂₃	1	2	yes	—
217	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
218	C₂×C₄⋊C₄	—	2	2	no	—
219	C₂×C₂²⋊C₄	—	4	1	no	—
220	C₂×C₁₆	—	2	2	no	—
221	C₂×C₁₆	—	2	2	no	—
222	C₂×C₁₆	—	2	2	no	—
223	#107	—	2	2	no	—
224	#107	—	2	2	no	—
225	#95	—	4	1	no	—
226	C₂₃	#12733	1	1	yes	characteristic, direct product factor, Sylow
227	C₂⁴	—	4	1	no	—
228	C₂⁴	D₈×C₂₃	1	1	yes	characteristic
229	#41	—	2	16	no	—
230	#41	—	4	8	no	—
231	#41	—	4	16	no	—
232	C₂×Q₈	—	2	2	no	—
233	C₂×Q₈	—	2	4	no	—
234	C₂×Q₈	—	4	1	no	—
235	C₂×Q₈	—	4	2	no	—
236	C₂×D₄	—	2	2	no	—
237	C₂×D₄	—	2	4	no	—
238	C₂×D₄	—	4	8	no	—
239	C₂×D₄	—	4	4	no	—
240	C₂×D₄	—	4	1	no	—
241	C₂×D₄	—	4	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	4	8	no	—
245	C₂²×C₄	—	4	1	no	—
246	C₂²×C₄	—	4	4	no	—
247	C₂²×C₄	—	4	2	no	—
248	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
249	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
250	C₂²×C₄	SD₁₆×C₂₃	1	2	yes	—
251	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
252	Q₁₆	—	2	2	no	—
253	Q₁₆	—	2	4	no	—
254	Q₁₆	C₂²⋊C₄×C₂₃	1	4	yes	complemented
255	SD₁₆	—	2	16	no	—
256	D₈	—	2	2	no	—
257	D₈	—	2	4	no	—
258	D₈	C₂²⋊C₄×C₂₃	1	4	yes	complemented
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	4	no	—
263	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
264	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
265	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
266	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
267	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₂²⋊C₄	—	8	2	no	—
271	C₁₆	—	4	2	no	—
272	C₂³	—	2	2	no	—
273	C₂³	—	2	2	no	—
274	C₂³	—	2	1	no	—
275	C₂³	—	2	1	no	—
276	C₂³	—	4	2	no	—
277	C₂³	—	4	4	no	—
278	C₂³	—	8	2	no	—
279	C₂³	—	8	1	no	—
280	C₂³	—	8	1	no	—
281	C₂³	C₂₃×#102	1	2	yes	—
282	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
283	Q₈	—	2	4	no	—
284	Q₈	—	4	2	no	—
285	Q₈	—	4	4	no	—
286	D₄	—	2	4	no	—
287	D₄	—	4	16	no	—
288	D₄	—	4	2	no	—
289	D₄	—	4	4	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	4	16	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	8	2	no	—
298	C₂×C₄	—	8	2	no	—
299	C₂×C₄	—	8	2	no	—
300	C₂×C₄	C₂₃×#102	1	2	yes	—
301	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
302	C₂×C₄	C₂₃×#120	1	2	yes	—
303	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
304	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
305	C₈	—	2	2	no	—
306	C₈	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
307	C₈	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
308	C₈	C₂₃×#120	1	2	yes	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	4	2	no	—
313	C₂²	—	4	2	no	—
314	C₂²	—	4	4	no	—
315	C₂²	—	8	4	no	—
316	C₂²	—	8	4	no	—
317	C₂²	—	8	2	no	—
318	C₂²	—	8	2	no	—
319	C₂²	C₂×C₂₃×#102	1	1	yes	characteristic
320	C₂²	C₂₃×#447	1	2	yes	—
321	C₂²	C₂₃×#508	1	2	yes	—
322	C₂²	C₂₃×#509	1	1	yes	characteristic
323	C₂²	C₂₃×#510	1	1	yes	characteristic
324	C₄	—	2	2	no	—
325	C₄	—	4	4	no	—
326	C₄	—	16	2	no	—
327	C₄	C₂₃×#379	1	1	yes	characteristic
328	C₄	C₂₃×#418	1	1	yes	characteristic
329	C₄	C₂₃×#450	1	2	yes	—
330	C₂	—	4	2	no	—
331	C₂	—	8	4	no	—
332	C₂	C₂₃×#1786	1	1	yes	characteristic
333	C₂	C₂₃×#2138	1	1	yes	characteristic
334	C₂	C₂₃×#2139	1	1	yes	characteristic
335	C₂	C₂₃×#2186	1	2	yes	—
336	C₂	C₂₃×#2189	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
prime radical	C₂²×C₂₃×#135	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂³×C₄×C₂₃ ◁ C₂³×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#358	C₂	semidirect
3. C₂×C₂₃×#357	C₂	semidirect
4. C₂×C₂₃×C₂²⋊C₁₆	C₂	semidirect
5. C₂₃×#1808	C₂	semidirect
10. C₂×C₂₃×#135	C₄	semidirect
85. Q₁₆×C₂₃	C₂²⋊C₄	semidirect
89. D₈×C₂₃	C₂²⋊C₄	semidirect
103. #12733	C₂₃	direct (N × H)
142. C₂×#358	C₂×C₂₃	semidirect
143. C₂×#357	C₂×C₂₃	semidirect
144. C₂×C₂²⋊C₁₆	C₂×C₂₃	semidirect
145. #1808	C₂×C₂₃	semidirect
171. C₂×#135	C₄×C₂₃	semidirect
226. C₂₃	#12733	direct (N × H)
254. Q₁₆	C₂²⋊C₄×C₂₃	semidirect
258. D₈	C₂²⋊C₄×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₂₃×#135	2
C₂×C₂₃×#358	2
C₂×C₂₃×#357	2
C₂×C₂₃×C₂²⋊C₁₆	2
C₂₃×#1808	2
#12733	23

Subgroups — C23×#12733 #19160359

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12733 #19160359