Subgroups — #15683643

Order 2944.

394 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13093027	C₂	1	1	yes	maximal, characteristic, complemented
2	#13092892	C₂	1	1	yes	maximal, characteristic
3	#13092759	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092755	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092719	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092715	C₂	1	1	yes	maximal, characteristic, complemented
7	#13092620	C₂	1	1	yes	maximal, characteristic, complemented
8	#13092620	C₂	1	1	yes	maximal, characteristic, complemented
9	#13092568	C₂	1	1	yes	maximal, characteristic
10	#13092518	C₂	1	1	yes	maximal, characteristic, complemented
11	#13092514	C₂	1	1	yes	maximal, characteristic, complemented
12	#13092508	C₂	1	1	yes	maximal, characteristic, complemented
13	SD₁₆×Dic₂₃	C₂	1	1	yes	maximal, characteristic, complemented
14	#13092475	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#450	C₂	1	1	yes	maximal, characteristic
16	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊Dic₂₃	—	2	1	no	—
21	C₂×C₂²⋊Dic₂₃	—	2	1	no	—
22	Q₈×Dic₂₃	C₂²	1	1	yes	characteristic, complemented
23	#10625115	C₂²	1	1	yes	characteristic, complemented
24	D₄×Dic₂₃	—	2	1	no	—
25	D₄×Dic₂₃	—	2	1	no	—
26	D₄×Dic₂₃	C₂²	1	1	yes	characteristic
27	D₄×Dic₂₃	C₂²	1	1	yes	characteristic
28	D₄×Dic₂₃	C₂²	1	1	yes	characteristic
29	D₄×Dic₂₃	C₂²	1	1	yes	characteristic
30	C₂×#67273	C₂²	1	1	yes	characteristic
31	C₂×#67273	C₂²	1	1	yes	characteristic
32	#10625097	C₂²	1	1	yes	characteristic
33	#10625097	C₂²	1	1	yes	characteristic
34	#10625095	C₂²	1	1	yes	characteristic
35	#10625061	—	2	1	no	—
36	#10625061	C₂²	1	1	yes	characteristic, complemented
37	#10625060	—	2	1	no	—
38	#10625058	C₂²	1	1	yes	characteristic, complemented
39	#10625050	—	2	1	no	—
40	#10625050	—	2	1	no	—
41	C₂₃×#102	C₂²	1	1	yes	characteristic
42	C₂₃×#102	C₂²	1	1	yes	characteristic
43	C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
44	#10625017	C₂²	1	1	yes	characteristic, complemented
45	#10625017	C₂²	1	1	yes	characteristic, complemented
46	#10625014	C₂²	1	1	yes	characteristic
47	#10625014	C₂²	1	1	yes	characteristic
48	C₂²⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
49	#10625000	C₂²	1	1	yes	characteristic, complemented
50	#10624999	C₂²	1	1	yes	characteristic, complemented
51	#10624998	C₂²	1	1	yes	characteristic, complemented
52	#10624998	C₂²	1	1	yes	characteristic, complemented
53	#10624997	C₂²	1	1	yes	characteristic, complemented
54	Dic₂₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
55	Dic₂₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
56	C₈×Dic₂₃	C₂²	1	1	yes	characteristic, complemented
57	#10624990	C₂²	1	1	yes	characteristic, complemented
58	#10624989	C₂²	1	1	yes	characteristic, complemented
59	C₂⁴×C₂₃	—	2	1	no	—
60	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
61	C₂×D₄×C₂₃	—	2	2	no	—
62	C₂×D₄×C₂₃	—	2	2	no	—
63	C₂×D₄×C₂₃	—	2	1	no	—
64	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
65	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
66	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
67	C₂²×Dic₂₃	—	2	1	no	—
68	C₂²×Dic₂₃	—	2	1	no	—
69	C₂²×Dic₂₃	—	2	1	no	—
70	C₂²×Dic₂₃	—	2	1	no	—
71	C₂×Dic₄₆	C₂³	1	1	yes	characteristic
72	SD₁₆×C₂₃	—	2	2	no	—
73	SD₁₆×C₂₃	—	2	2	no	—
74	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₂₃	—	2	1	no	—
77	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
78	C₂²⋊C₄×C₂₃	—	2	1	no	—
79	C₂²⋊Dic₂₃	—	2	1	no	—
80	C₂²⋊Dic₂₃	—	2	1	no	—
81	C₂²⋊Dic₂₃	—	2	1	no	—
82	C₂²⋊Dic₂₃	—	2	1	no	—
83	C₂²⋊Dic₂₃	—	2	1	no	—
84	C₂²⋊Dic₂₃	—	2	1	no	—
85	C₂²⋊Dic₂₃	D₄	1	1	yes	characteristic
86	C₂²⋊Dic₂₃	D₄	1	1	yes	characteristic
87	#67273	—	2	2	no	—
88	#67273	—	2	2	no	—
89	C₄⋊Dic₂₃	—	2	1	no	—
90	C₄⋊Dic₂₃	—	2	1	no	—
91	C₄⋊Dic₂₃	D₄	1	1	yes	characteristic
92	C₄⋊Dic₂₃	D₄	1	1	yes	characteristic
93	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
94	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
95	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
96	Dic₂₃⋊C₄	—	2	1	no	—
97	Dic₂₃⋊C₄	—	2	1	no	—
98	Dic₂₃⋊C₄	—	2	1	no	—
99	Dic₂₃⋊C₄	—	2	1	no	—
100	Dic₂₃⋊C₄	—	2	1	no	—
101	C₄×Dic₂₃	—	2	1	no	—
102	C₄×Dic₂₃	—	2	1	no	—
103	C₄×Dic₂₃	—	2	1	no	—
104	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
105	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
106	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
107	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
108	C₂³×C₂₃	—	2	4	no	—
109	C₂³×C₂₃	—	2	1	no	—
110	C₂³×C₂₃	—	2	1	no	—
111	C₂³×C₂₃	—	8	1	no	—
112	C₂³×C₂₃	—	8	1	no	—
113	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
114	Q₈×C₂₃	—	2	1	no	—
115	Q₈×C₂₃	—	2	1	no	—
116	D₄×C₂₃	—	2	4	no	—
117	D₄×C₂₃	—	2	1	no	—
118	D₄×C₂₃	—	2	1	no	—
119	D₄×C₂₃	#41	1	2	yes	—
120	D₄×C₂₃	#41	1	2	yes	—
121	C₂×C₄×C₂₃	—	2	1	no	—
122	C₂×C₄×C₂₃	—	2	1	no	—
123	C₂×C₄×C₂₃	—	2	1	no	—
124	C₂×C₄×C₂₃	—	2	1	no	—
125	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
127	C₂×Dic₂₃	—	2	1	no	—
128	C₂×Dic₂₃	—	2	1	no	—
129	C₂×Dic₂₃	—	2	1	no	—
130	C₂×Dic₂₃	—	2	1	no	—
131	C₂×Dic₂₃	—	8	1	no	—
132	C₂×Dic₂₃	—	8	1	no	—
133	C₂×Dic₂₃	—	8	1	no	—
134	C₂×Dic₂₃	—	8	1	no	—
135	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
136	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
137	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
138	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
139	Dic₄₆	—	2	1	no	—
140	Dic₄₆	—	2	1	no	—
141	C₈×C₂₃	—	2	1	no	—
142	C₈×C₂₃	—	2	1	no	—
143	C₂₃⋊C₈	—	2	1	no	—
144	C₂₃⋊C₈	—	2	1	no	—
145	#3123	—	23	1	no	maximal, Sylow
146	C₂²×C₂₃	—	2	1	no	—
147	C₂²×C₂₃	—	2	2	no	—
148	C₂²×C₂₃	—	2	2	no	—
149	C₂²×C₂₃	—	2	1	no	—
150	C₂²×C₂₃	—	2	1	no	—
151	C₂²×C₂₃	—	4	1	no	—
152	C₂²×C₂₃	—	4	1	no	—
153	C₂²×C₂₃	—	4	1	no	—
154	C₂²×C₂₃	—	4	1	no	—
155	C₂²×C₂₃	—	8	4	no	—
156	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
157	C₄×C₂₃	—	2	1	no	—
158	C₄×C₂₃	—	4	1	no	—
159	C₄×C₂₃	—	4	1	no	—
160	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
161	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
162	Dic₂₃	—	2	1	no	—
163	Dic₂₃	—	2	1	no	—
164	Dic₂₃	—	2	1	no	—
165	Dic₂₃	—	2	1	no	—
166	Dic₂₃	—	4	1	no	—
167	Dic₂₃	—	4	1	no	—
168	Dic₂₃	—	4	1	no	—
169	Dic₂₃	—	4	1	no	—
170	#541	—	23	1	no	—
171	#518	—	23	1	no	—
172	#485	—	23	1	no	—
173	#485	—	23	1	no	—
174	#478	—	23	1	no	—
175	#478	—	23	1	no	—
176	#461	—	23	1	no	—
177	#461	—	23	1	no	—
178	#450	—	23	1	no	—
179	#450	D₂₃	1	1	yes	characteristic
180	#440	—	23	1	no	—
181	#440	—	23	1	no	—
182	C₄×SD₁₆	—	23	1	no	—
183	C₄×SD₁₆	—	23	1	no	—
184	#433	—	23	1	no	—
185	C₂×C₂₃	—	4	1	no	—
186	C₂×C₂₃	—	4	2	no	—
187	C₂×C₂₃	—	4	2	no	—
188	C₂×C₂₃	#525	1	1	yes	characteristic
189	C₂×C₂₃	#577	1	1	yes	characteristic
190	C₂×C₂₃	#577	1	1	yes	characteristic
191	C₂²×D₄	C₂×D₂₃	1	1	yes	characteristic
192	C₂×SD₁₆	—	23	1	no	—
193	C₂×SD₁₆	—	23	1	no	—
194	C₂×SD₁₆	C₂×D₂₃	1	1	yes	characteristic
195	C₂×SD₁₆	C₂×D₂₃	1	1	yes	characteristic
196	C₄⋊Q₈	—	23	1	no	—
197	#126	—	46	1	no	—
198	#126	—	46	1	no	—
199	#125	—	23	1	no	—
200	#125	—	46	1	no	—
201	#125	—	46	1	no	—
202	C₂²⋊Q₈	—	23	1	no	—
203	C₂²⋊Q₈	C₂×D₂₃	1	1	yes	characteristic
204	C₄×Q₈	—	23	1	no	—
205	C₄×Q₈	—	23	1	no	—
206	C₄×D₄	—	23	1	no	—
207	C₄×D₄	—	23	1	no	—
208	C₄×D₄	—	23	1	no	—
209	C₄×D₄	—	23	1	no	—
210	C₄×D₄	—	46	1	no	—
211	C₄×D₄	—	46	1	no	—
212	#117	—	23	1	no	—
213	#117	—	23	1	no	—
214	C₂×C₂²⋊C₄	—	46	1	no	—
215	C₂×C₂²⋊C₄	—	46	1	no	—
216	#107	—	23	1	no	—
217	#107	—	23	1	no	—
218	#106	—	23	1	no	—
219	#106	—	23	1	no	—
220	C₄⋊C₈	—	23	1	no	—
221	C₄⋊C₈	—	23	1	no	—
222	#103	—	23	1	no	—
223	#103	—	23	1	no	—
224	#103	—	23	1	no	—
225	#103	—	23	1	no	—
226	#102	—	23	1	no	—
227	#102	—	23	1	no	—
228	#102	C₂×D₂₃	1	1	yes	characteristic
229	#102	C₂×D₂₃	1	1	yes	characteristic
230	C₂²⋊C₈	—	23	1	no	—
231	C₂²⋊C₈	C₂×D₂₃	1	1	yes	characteristic
232	#97	—	23	1	no	—
233	C₄×C₈	—	23	1	no	—
234	C₂₃	#3123	1	1	yes	characteristic, complemented, Sylow
235	C₂⁴	—	2	1	no	—
236	C₂×Q₈	—	23	1	no	—
237	C₂×Q₈	C₂²×D₂₃	1	1	yes	characteristic
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	C₂²×D₂₃	1	1	yes	characteristic
242	C₂×D₄	C₂²×D₂₃	1	1	yes	characteristic
243	C₂²×C₄	—	46	1	no	—
244	C₂²×C₄	—	46	1	no	—
245	C₂²×C₄	—	46	1	no	—
246	C₂²×C₄	—	46	1	no	—
247	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
248	SD₁₆	—	2	2	no	—
249	SD₁₆	—	2	2	no	—
250	SD₁₆	—	46	2	no	—
251	SD₁₆	—	46	2	no	—
252	C₂×C₈	—	23	1	no	—
253	C₂×C₈	—	23	1	no	—
254	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
255	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
256	C₄⋊C₄	—	2	1	no	—
257	C₄⋊C₄	—	23	1	no	—
258	C₄⋊C₄	—	23	1	no	—
259	C₄⋊C₄	—	23	1	no	—
260	C₄⋊C₄	—	23	1	no	—
261	C₄⋊C₄	—	23	1	no	—
262	C₄⋊C₄	—	46	1	no	—
263	C₄⋊C₄	—	46	1	no	—
264	C₄⋊C₄	—	46	1	no	—
265	C₄⋊C₄	—	46	1	no	—
266	C₄⋊C₄	—	46	1	no	—
267	C₄⋊C₄	—	46	1	no	—
268	C₄⋊C₄	—	46	1	no	—
269	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
270	C₂²⋊C₄	—	2	1	no	—
271	C₂²⋊C₄	—	23	1	no	—
272	C₂²⋊C₄	—	23	1	no	—
273	C₂²⋊C₄	—	46	1	no	—
274	C₂²⋊C₄	—	46	1	no	—
275	C₂²⋊C₄	—	46	1	no	—
276	C₂²⋊C₄	—	46	1	no	—
277	C₂²⋊C₄	—	46	1	no	—
278	C₂²⋊C₄	—	46	1	no	—
279	C₄²	—	23	1	no	—
280	C₄²	—	23	1	no	—
281	C₄²	—	46	1	no	—
282	C₄²	—	46	1	no	—
283	C₄²	—	46	1	no	—
284	C₂³	—	2	4	no	—
285	C₂³	—	2	1	no	—
286	C₂³	—	2	1	no	—
287	C₂³	—	8	1	no	—
288	C₂³	—	8	1	no	—
289	C₂³	D₄×D₂₃	1	1	yes	characteristic
290	Q₈	—	2	1	no	—
291	Q₈	—	2	1	no	—
292	Q₈	—	46	1	no	—
293	Q₈	—	46	1	no	—
294	D₄	—	2	4	no	—
295	D₄	—	2	1	no	—
296	D₄	—	2	1	no	—
297	D₄	#67290	1	2	yes	—
298	D₄	#67290	1	2	yes	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	23	1	no	—
304	C₂×C₄	—	23	1	no	—
305	C₂×C₄	—	23	1	no	—
306	C₂×C₄	—	23	1	no	—
307	C₂×C₄	—	46	1	no	—
308	C₂×C₄	—	46	1	no	—
309	C₂×C₄	—	46	1	no	—
310	C₂×C₄	—	46	1	no	—
311	C₂×C₄	—	184	1	no	—
312	C₂×C₄	—	184	1	no	—
313	C₂×C₄	—	184	1	no	—
314	C₂×C₄	—	184	1	no	—
315	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
316	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
317	C₈	—	2	1	no	—
318	C₈	—	2	1	no	—
319	C₈	—	46	1	no	—
320	C₈	—	46	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	4	1	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	—	8	4	no	—
331	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	4	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	46	1	no	—
336	C₄	—	46	1	no	—
337	C₄	—	46	1	no	—
338	C₄	—	46	1	no	—
339	C₄	—	92	1	no	—
340	C₄	—	92	1	no	—
341	C₄	—	92	1	no	—
342	C₄	—	92	1	no	—
343	C₄	C₂×#67290	1	1	yes	characteristic
344	C₄	C₂×#67290	1	1	yes	characteristic
345	C₂	—	4	1	no	—
346	C₂	—	4	2	no	—
347	C₂	—	4	2	no	—
348	C₂	#13092923	1	1	yes	characteristic
349	C₂	#13093223	1	1	yes	characteristic
350	C₂	#13093227	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂₃×#450	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂²×D₄×C₂₃	C₂²
hypercenter	#450	D₂₃
hypocenter	C₂₃	#3123
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #450

Complements

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

N	H ≅ G/N	type
1. #13093027	C₂	semidirect
3. #13092759	C₂	semidirect
4. #13092755	C₂	semidirect
5. #13092719	C₂	semidirect
6. #13092715	C₂	semidirect
7. #13092620	C₂	semidirect
8. #13092620	C₂	semidirect
10. #13092518	C₂	semidirect
11. #13092514	C₂	semidirect
12. #13092508	C₂	semidirect
13. SD₁₆×Dic₂₃	C₂	semidirect
14. #13092475	C₂	semidirect
22. Q₈×Dic₂₃	C₂²	semidirect
23. #10625115	C₂²	semidirect
36. #10625061	C₂²	semidirect
38. #10625058	C₂²	semidirect
44. #10625017	C₂²	semidirect
45. #10625017	C₂²	semidirect
49. #10625000	C₂²	semidirect
50. #10624999	C₂²	semidirect
51. #10624998	C₂²	semidirect
52. #10624998	C₂²	semidirect
53. #10624997	C₂²	semidirect
54. Dic₂₃⋊C₈	C₂²	semidirect
55. Dic₂₃⋊C₈	C₂²	semidirect
56. C₈×Dic₂₃	C₂²	semidirect
57. #10624990	C₂²	semidirect
58. #10624989	C₂²	semidirect
234. C₂₃	#3123	semidirect

Maximal subgroups

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

subgroup	index
#13093027	2
#13092892	2
#13092759	2
#13092755	2
#13092719	2
#13092715	2
#13092620	2
#13092568	2
#13092518	2
#13092514	2
#13092508	2
SD₁₆×Dic₂₃	2
#13092475	2
C₂₃×#450	2
#3123	23