Subgroups — #15683951

Order 2944.

412 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092969	C₂	1	1	yes	maximal, characteristic, complemented
2	#13092962	C₂	1	1	yes	maximal, characteristic
3	#13092814	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092788	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092766	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092762	C₂	1	1	yes	maximal, characteristic, complemented
7	#13092648	C₂	1	1	yes	maximal, characteristic, complemented
8	#13092646	C₂	1	1	yes	maximal, characteristic, complemented
9	#13092644	C₂	1	1	yes	maximal, characteristic, complemented
10	#13092508	C₂	1	1	yes	maximal, characteristic, complemented
11	SD₁₆×Dic₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#13092504	C₂	1	1	yes	maximal, characteristic, complemented
13	#13092502	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×C₈×Dic₂₃	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#465	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#67290	—	2	1	no	—
17	C₂×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
20	C₂₃×#121	C₂²	1	1	yes	characteristic
21	Q₈×Dic₂₃	C₂²	1	1	yes	characteristic
22	#10625115	C₂²	1	1	yes	characteristic
23	C₂×#67274	C₂²	1	1	yes	characteristic
24	#10625111	C₂²	1	1	yes	characteristic
25	#10625110	—	2	1	no	—
26	D₄×Dic₂₃	C₂²	1	1	yes	characteristic
27	C₂×#67273	C₂²	1	1	yes	characteristic
28	#10625101	C₂²	1	1	yes	characteristic
29	#10625095	C₂²	1	1	yes	characteristic
30	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
31	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
32	C₂×#67263	C₂²	1	1	yes	characteristic
33	#10625064	C₂²	1	1	yes	characteristic
34	#10625060	—	2	1	no	—
35	#10625059	C₂²	1	1	yes	characteristic
36	#10625059	C₂²	1	1	yes	characteristic
37	#10625058	—	2	1	no	—
38	#10625058	C₂²	1	1	yes	characteristic
39	#10625056	—	2	1	no	—
40	(C₂²⋊D₂₃)⋊C₄	—	2	1	no	—
41	#10625049	—	2	1	no	—
42	#10625048	—	2	1	no	—
43	C₂₃×#106	C₂²	1	1	yes	characteristic
44	C₂₃×#103	C₂²	1	1	yes	characteristic
45	C₂₃×#102	C₂²	1	1	yes	characteristic
46	#10625017	C₂²	1	1	yes	characteristic
47	#10625014	C₂²	1	1	yes	characteristic
48	#10625003	C₂²	1	1	yes	characteristic
49	#10624999	C₂²	1	1	yes	characteristic
50	#10624998	C₂²	1	1	yes	characteristic
51	C₈×Dic₂₃	C₂²	1	1	yes	characteristic
52	C₈×Dic₂₃	C₂²	1	1	yes	characteristic
53	C₈×Dic₂₃	C₂²	1	1	yes	characteristic
54	C₈×Dic₂₃	C₂²	1	1	yes	characteristic
55	#10624992	C₂²	1	1	yes	characteristic
56	#10624991	C₂²	1	1	yes	characteristic
57	#10624990	C₂²	1	1	yes	characteristic
58	#10624990	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₂₃	—	2	1	no	—
61	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₂₃	—	2	1	no	—
64	C₂×C₂²⋊D₂₃	—	2	1	no	—
65	C₂×C₂²⋊D₂₃	—	2	1	no	—
66	C₂²×Dic₂₃	—	2	1	no	—
67	C₂²×Dic₂₃	D₄	1	1	yes	characteristic
68	C₂²×Dic₂₃	D₄	1	1	yes	characteristic
69	#67290	—	4	4	no	—
70	C₂×D₉₂	C₂³	1	1	yes	characteristic
71	C₂×C₄×D₂₃	—	2	1	no	—
72	C₂×Dic₄₆	—	2	1	no	—
73	C₂×Dic₄₆	—	2	1	no	—
74	C₂×Dic₄₆	C₂³	1	1	yes	characteristic
75	SD₁₆×C₂₃	—	2	2	no	—
76	C₂×C₈×C₂₃	—	2	2	no	—
77	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
78	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
79	C₄⋊C₄×C₂₃	—	2	1	no	—
80	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
81	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
82	C₂²⋊C₄×C₂₃	—	2	1	no	—
83	C₂²⋊C₄×C₂₃	—	2	1	no	—
84	C₂²⋊Dic₂₃	—	2	1	no	—
85	C₂²⋊Dic₂₃	—	2	1	no	—
86	#67274	—	2	2	no	—
87	#67273	—	2	2	no	—
88	(C₂×D₂₃)⋊C₄	—	2	1	no	—
89	(C₂×D₂₃)⋊C₄	—	2	1	no	—
90	C₄⋊Dic₂₃	—	2	1	no	—
91	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
92	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
93	Dic₂₃⋊C₄	—	2	1	no	—
94	Dic₂₃⋊C₄	—	2	1	no	—
95	Dic₂₃⋊C₄	—	2	1	no	—
96	Dic₂₃⋊C₄	—	2	1	no	—
97	Dic₂₃⋊C₄	—	2	1	no	—
98	Dic₂₃⋊C₄	—	2	1	no	—
99	C₄×Dic₂₃	—	2	1	no	—
100	C₄×Dic₂₃	—	2	1	no	—
101	C₄×Dic₂₃	D₄	1	1	yes	characteristic
102	C₄×Dic₂₃	D₄	1	1	yes	characteristic
103	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
104	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
105	C₂×C₂₃⋊C₈	—	2	2	no	—
106	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
107	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
108	#67263	—	2	2	no	—
109	C₂³×C₂₃	—	2	1	no	—
110	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
111	C₂²×D₂₃	—	2	1	no	—
112	Q₈×C₂₃	—	2	1	no	—
113	Q₈×C₂₃	—	2	1	no	—
114	D₄×C₂₃	—	2	1	no	—
115	D₄×C₂₃	—	2	1	no	—
116	D₄×C₂₃	—	4	2	no	—
117	C₂×C₄×C₂₃	—	2	1	no	—
118	C₂×C₄×C₂₃	—	2	1	no	—
119	C₂×C₄×C₂₃	—	2	1	no	—
120	C₂×C₄×C₂₃	—	2	1	no	—
121	C₂×C₄×C₂₃	—	2	1	no	—
122	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
124	C₂²⋊D₂₃	—	4	2	no	—
125	C₂²⋊D₂₃	—	4	2	no	—
126	C₂²⋊D₂₃	—	4	2	no	—
127	C₂×Dic₂₃	—	2	2	no	—
128	C₂×Dic₂₃	—	2	1	no	—
129	C₂×Dic₂₃	—	2	1	no	—
130	C₂×Dic₂₃	—	2	1	no	—
131	C₂×Dic₂₃	—	4	2	no	—
132	C₂×Dic₂₃	SD₁₆	1	4	yes	complemented
133	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
134	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
135	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
136	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
137	D₉₂	—	2	1	no	—
138	D₉₂	—	2	1	no	—
139	C₄×D₂₃	—	4	2	no	—
140	Dic₄₆	—	2	1	no	—
141	Dic₄₆	—	2	1	no	—
142	Dic₄₆	—	4	2	no	—
143	Dic₄₆	—	4	2	no	—
144	C₈×C₂₃	—	2	1	no	—
145	C₈×C₂₃	#41	1	2	yes	complemented
146	C₂₃⋊C₈	—	2	1	no	—
147	C₂₃⋊C₈	#41	1	2	yes	complemented
148	#3167	—	23	1	no	maximal, Sylow
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	C₂×SD₁₆	1	2	yes	—
155	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
156	C₂×D₂₃	—	4	1	no	—
157	C₂×D₂₃	—	4	1	no	—
158	C₂×D₂₃	—	4	1	no	—
159	C₄×C₂₃	—	2	1	no	—
160	C₄×C₂₃	—	4	1	no	—
161	C₄×C₂₃	—	4	1	no	—
162	C₄×C₂₃	—	4	1	no	—
163	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
164	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
165	Dic₂₃	—	2	1	no	—
166	Dic₂₃	—	4	1	no	—
167	Dic₂₃	—	4	1	no	—
168	Dic₂₃	—	4	1	no	—
169	Dic₂₃	C₂×SD₁₆	1	2	yes	—
170	Dic₂₃	C₂×SD₁₆	1	2	yes	—
171	Dic₂₃	#135	1	2	yes	—
172	#533	—	23	1	no	—
173	#532	—	23	1	no	—
174	#498	—	23	1	no	—
175	#492	—	23	1	no	—
176	#487	—	23	1	no	—
177	#486	—	23	1	no	—
178	#465	—	23	1	no	—
179	#465	—	23	1	no	—
180	#465	—	23	1	no	—
181	#465	D₂₃	1	1	yes	characteristic, complemented
182	C₄×SD₁₆	—	23	1	no	—
183	C₄×SD₁₆	—	23	1	no	—
184	C₄×SD₁₆	—	23	1	no	—
185	C₄×SD₁₆	—	23	1	no	—
186	C₂×C₄×C₈	—	23	1	no	—
187	C₂×C₂₃	—	2	2	no	—
188	C₂×C₂₃	—	8	1	no	—
189	C₂×C₂₃	#532	1	1	yes	characteristic
190	C₂×C₂₃	C₂²×SD₁₆	1	1	yes	characteristic
191	C₂×C₂₃	C₂×#135	1	1	yes	characteristic
192	D₂₃	—	8	1	no	—
193	C₂×#41	—	46	1	no	—
194	C₂×SD₁₆	—	23	1	no	—
195	C₂×SD₁₆	—	23	1	no	—
196	C₂×SD₁₆	—	23	1	no	—
197	C₂×SD₁₆	C₂×D₂₃	1	1	yes	characteristic
198	C₂²×C₈	—	23	1	no	—
199	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
200	C₄⋊Q₈	—	23	1	no	—
201	C₄⋊Q₈	—	23	1	no	—
202	C₄⋊Q₈	—	23	1	no	—
203	#127	—	23	1	no	—
204	#125	—	46	1	no	—
205	#124	—	46	1	no	—
206	C₂²⋊Q₈	—	23	1	no	—
207	C₂²⋊Q₈	—	46	1	no	—
208	C₂²⋊Q₈	C₂×D₂₃	1	1	yes	characteristic
209	#121	—	23	1	no	—
210	#121	—	46	1	no	—
211	#121	C₂×D₂₃	1	1	yes	characteristic
212	C₄×Q₈	—	23	1	no	—
213	C₄×Q₈	—	23	1	no	—
214	C₄×Q₈	—	46	1	no	—
215	C₄×D₄	—	23	1	no	—
216	C₄×D₄	—	23	1	no	—
217	C₄×D₄	—	46	1	no	—
218	#117	—	46	1	no	—
219	C₂×C₄²	—	23	1	no	—
220	#106	—	23	1	no	—
221	#106	—	23	1	no	—
222	#106	—	23	1	no	—
223	#106	C₂×D₂₃	1	1	yes	characteristic
224	#103	—	23	1	no	—
225	#103	—	23	1	no	—
226	#103	—	23	1	no	—
227	#103	C₂×D₂₃	1	1	yes	characteristic
228	#102	—	23	1	no	—
229	#102	—	23	1	no	—
230	#102	—	23	1	no	—
231	#102	C₂×D₂₃	1	1	yes	characteristic
232	C₄×C₈	—	23	1	no	—
233	C₄×C₈	—	23	1	no	—
234	C₄×C₈	—	23	1	no	—
235	C₄×C₈	—	23	1	no	—
236	C₂₃	#3167	1	1	yes	characteristic, complemented, Sylow
237	#41	—	92	4	no	—
238	C₂×Q₈	—	23	1	no	—
239	C₂×Q₈	—	46	1	no	—
240	C₂×Q₈	—	46	1	no	—
241	C₂×Q₈	C₂²×D₂₃	1	1	yes	characteristic
242	C₂×D₄	—	2	1	no	—
243	C₂×D₄	—	23	1	no	—
244	C₂×D₄	—	46	1	no	—
245	C₂×D₄	—	46	1	no	—
246	C₂×D₄	—	46	1	no	—
247	C₂×D₄	C₂²×D₂₃	1	1	yes	characteristic
248	C₂²×C₄	—	23	1	no	—
249	C₂²×C₄	—	23	1	no	—
250	C₂²×C₄	—	46	1	no	—
251	C₂²×C₄	—	46	1	no	—
252	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
253	SD₁₆	—	2	2	no	—
254	SD₁₆	—	46	2	no	—
255	SD₁₆	—	46	2	no	—
256	SD₁₆	—	46	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	23	1	no	—
259	C₂×C₈	—	23	1	no	—
260	C₂×C₈	—	46	2	no	—
261	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
262	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	23	1	no	—
265	C₄⋊C₄	—	23	1	no	—
266	C₄⋊C₄	—	46	1	no	—
267	C₄⋊C₄	—	46	1	no	—
268	C₄⋊C₄	—	46	1	no	—
269	C₄⋊C₄	—	46	1	no	—
270	C₄⋊C₄	—	46	1	no	—
271	C₄⋊C₄	—	46	1	no	—
272	C₄⋊C₄	—	46	1	no	—
273	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
274	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
275	C₂²⋊C₄	—	2	1	no	—
276	C₂²⋊C₄	—	2	1	no	—
277	C₂²⋊C₄	—	46	1	no	—
278	C₂²⋊C₄	—	46	1	no	—
279	C₂²⋊C₄	—	46	1	no	—
280	C₂²⋊C₄	—	46	1	no	—
281	C₄²	—	23	1	no	—
282	C₄²	—	23	1	no	—
283	C₄²	—	23	1	no	—
284	C₄²	—	23	1	no	—
285	C₄²	—	46	1	no	—
286	C₄²	—	46	1	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	46	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	Q₈	—	92	2	no	—
295	Q₈	—	92	2	no	—
296	D₄	—	2	1	no	—
297	D₄	—	2	1	no	—
298	D₄	—	4	2	no	—
299	D₄	—	46	1	no	—
300	D₄	—	46	1	no	—
301	D₄	—	92	2	no	—
302	D₄	—	92	2	no	—
303	D₄	—	92	2	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	23	4	no	—
310	C₂×C₄	—	23	1	no	—
311	C₂×C₄	—	23	1	no	—
312	C₂×C₄	—	23	1	no	—
313	C₂×C₄	—	23	1	no	—
314	C₂×C₄	—	46	2	no	—
315	C₂×C₄	—	46	1	no	—
316	C₂×C₄	—	46	1	no	—
317	C₂×C₄	—	46	1	no	—
318	C₂×C₄	—	92	2	no	—
319	C₂×C₄	—	92	2	no	—
320	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
321	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
322	C₈	—	2	1	no	—
323	C₈	—	23	2	no	—
324	C₈	—	46	1	no	—
325	C₈	#67290	1	2	yes	complemented
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	—	4	1	no	—
331	C₂²	—	92	1	no	—
332	C₂²	—	92	1	no	—
333	C₂²	—	92	1	no	—
334	C₂²	SD₁₆×D₂₃	1	2	yes	—
335	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	4	1	no	—
338	C₄	—	4	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	23	2	no	—
341	C₄	—	23	2	no	—
342	C₄	—	23	2	no	—
343	C₄	—	46	1	no	—
344	C₄	—	92	1	no	—
345	C₄	—	92	1	no	—
346	C₄	—	92	1	no	—
347	C₄	C₂×#67290	1	1	yes	characteristic
348	C₄	#41×D₂₃	1	1	yes	characteristic
349	C₂	—	2	2	no	—
350	C₂	—	8	1	no	—
351	C₂	—	184	1	no	—
352	C₂	#13092962	1	1	yes	characteristic
353	C₂	C₂×SD₁₆×D₂₃	1	1	yes	characteristic
354	C₂	#135×D₂₃	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×#465	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	#13092962	C₂
hypercenter	#465	D₂₃
hypocenter	C₂₃	#3167
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #465

Complements

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

N	H ≅ G/N	type
1. #13092969	C₂	semidirect
3. #13092814	C₂	semidirect
4. #13092788	C₂	semidirect
5. #13092766	C₂	semidirect
6. #13092762	C₂	semidirect
7. #13092648	C₂	semidirect
8. #13092646	C₂	semidirect
9. #13092644	C₂	semidirect
10. #13092508	C₂	semidirect
11. SD₁₆×Dic₂₃	C₂	semidirect
12. #13092504	C₂	semidirect
13. #13092502	C₂	semidirect
14. C₂×C₈×Dic₂₃	C₂	semidirect
15. C₂₃×#465	C₂	semidirect
132. C₂×Dic₂₃	SD₁₆	semidirect
145. C₈×C₂₃	#41	semidirect
147. C₂₃⋊C₈	#41	semidirect
181. #465	D₂₃	semidirect
236. C₂₃	#3167	semidirect
325. C₈	#67290	semidirect

Maximal subgroups

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

subgroup	index
#13092969	2
#13092962	2
#13092814	2
#13092788	2
#13092766	2
#13092762	2
#13092648	2
#13092646	2
#13092644	2
#13092508	2
SD₁₆×Dic₂₃	2
#13092504	2
#13092502	2
C₂×C₈×Dic₂₃	2
C₂₃×#465	2
#3167	23