Subgroups — #15686079

Order 2944.

336 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13093144	C₂	1	1	yes	maximal, characteristic, complemented
2	#13093136	C₂	1	1	yes	maximal, characteristic
3	#13092770	C₂	1	1	yes	maximal, characteristic
4	#13092766	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092726	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092721	C₂	1	1	yes	maximal, characteristic, complemented
7	#13092695	C₂	1	1	yes	maximal, characteristic
8	#13092638	C₂	1	1	yes	maximal, characteristic
9	#13092636	C₂	1	1	yes	maximal, characteristic
10	#13092626	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×Dic₄₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×Dic₉₂	C₂	1	1	yes	maximal, characteristic, complemented
13	#13092509	C₂	1	1	yes	maximal, characteristic, complemented
14	#13092508	C₂	1	1	yes	maximal, characteristic
15	C₂₃×#488	C₂	1	1	yes	maximal, characteristic
16	C₂₃×#125	C₂²	1	1	yes	characteristic
17	C₂₃×#124	C₂²	1	1	yes	characteristic
18	Q₈×Dic₂₃	C₂²	1	1	yes	characteristic
19	#10625115	C₂²	1	1	yes	characteristic
20	C₂×#67275	C₂²	1	1	yes	characteristic
21	#10625107	C₂²	1	1	yes	characteristic
22	D₄×Dic₂₃	C₂²	1	1	yes	characteristic
23	C₂×#67273	C₂²	1	1	yes	characteristic
24	C₂×Dic₉₂	C₂²	1	1	yes	characteristic
25	#10625061	C₂²	1	1	yes	characteristic
26	#10625060	—	2	1	no	—
27	#10625060	—	2	1	no	—
28	#10625059	—	2	1	no	—
29	#10625059	C₂²	1	1	yes	characteristic
30	#10625058	—	2	1	no	—
31	#10625058	C₂²	1	1	yes	characteristic
32	#10625058	C₂²	1	1	yes	characteristic
33	#10625050	—	2	1	no	—
34	#10625050	—	2	1	no	—
35	#10625049	—	2	1	no	—
36	#10625048	—	2	1	no	—
37	C₄×Dic₄₆	C₂²	1	1	yes	characteristic
38	C₄×Dic₄₆	C₂²	1	1	yes	characteristic
39	C₄×Dic₄₆	C₂²	1	1	yes	characteristic
40	C₂₃×#103	C₂²	1	1	yes	characteristic
41	C₂₃×#103	C₂²	1	1	yes	characteristic
42	C₂₃×#102	C₂²	1	1	yes	characteristic
43	C₂₃×#102	C₂²	1	1	yes	characteristic
44	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
45	#10625017	C₂²	1	1	yes	characteristic
46	#10625014	C₂²	1	1	yes	characteristic
47	#10625000	C₂²	1	1	yes	characteristic
48	#10624998	C₂²	1	1	yes	characteristic
49	#10624998	C₂²	1	1	yes	characteristic
50	Dic₂₃⋊C₈	C₂²	1	1	yes	characteristic
51	Dic₂₃⋊C₈	C₂²	1	1	yes	characteristic
52	C₈×Dic₂₃	C₂²	1	1	yes	characteristic
53	C₈×Dic₂₃	C₂²	1	1	yes	characteristic
54	#10624992	C₂²	1	1	yes	characteristic
55	#10624992	C₂²	1	1	yes	characteristic
56	#10624990	C₂²	1	1	yes	characteristic
57	#10624989	C₂²	1	1	yes	characteristic
58	C₄⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
61	C₂²×Dic₂₃	—	2	1	no	—
62	C₂×Dic₄₆	—	2	1	no	—
63	C₂×Dic₄₆	D₄	1	1	yes	characteristic
64	C₂×Dic₄₆	C₂³	1	1	yes	characteristic
65	C₂×Dic₄₆	C₂³	1	1	yes	characteristic
66	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
67	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
68	C₄⋊C₄×C₂₃	—	2	1	no	—
69	C₄⋊C₄×C₂₃	—	2	1	no	—
70	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
72	C₂²⋊C₄×C₂₃	—	2	1	no	—
73	C₂²⋊C₄×C₂₃	—	2	1	no	—
74	C₄²×C₂₃	C₂³	1	1	yes	characteristic
75	C₂²⋊Dic₂₃	—	2	1	no	—
76	C₂²⋊Dic₂₃	—	2	1	no	—
77	C₂²⋊Dic₂₃	—	2	1	no	—
78	#67275	—	2	2	no	—
79	#67273	—	2	2	no	—
80	C₄⋊Dic₂₃	—	2	1	no	—
81	C₄⋊Dic₂₃	—	2	1	no	—
82	C₄⋊Dic₂₃	D₄	1	1	yes	characteristic
83	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
84	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
85	Dic₂₃⋊C₄	—	2	1	no	—
86	Dic₂₃⋊C₄	—	2	1	no	—
87	Dic₂₃⋊C₄	—	2	1	no	—
88	Dic₂₃⋊C₄	—	2	1	no	—
89	Dic₂₃⋊C₄	—	2	1	no	—
90	Dic₂₃⋊C₄	—	2	1	no	—
91	Dic₂₃⋊C₄	—	2	1	no	—
92	Dic₂₃⋊C₄	—	2	1	no	—
93	Dic₂₃⋊C₄	D₄	1	1	yes	characteristic
94	Dic₂₃⋊C₄	D₄	1	1	yes	characteristic
95	C₄×Dic₂₃	—	2	1	no	—
96	C₄×Dic₂₃	—	2	1	no	—
97	C₄×Dic₂₃	—	2	1	no	—
98	C₄×Dic₂₃	—	2	1	no	—
99	C₄×Dic₂₃	—	2	1	no	—
100	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
101	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
102	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
103	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
104	Dic₉₂	—	2	2	no	—
105	C₂³×C₂₃	—	2	1	no	—
106	Q₈×C₂₃	—	2	1	no	—
107	Q₈×C₂₃	—	2	1	no	—
108	D₄×C₂₃	—	2	1	no	—
109	D₄×C₂₃	—	2	1	no	—
110	C₂×C₄×C₂₃	—	2	1	no	—
111	C₂×C₄×C₂₃	—	2	1	no	—
112	C₂×C₄×C₂₃	—	2	1	no	—
113	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
116	C₂×Dic₂₃	—	2	1	no	—
117	C₂×Dic₂₃	—	2	1	no	—
118	C₂×Dic₂₃	—	2	1	no	—
119	C₂×Dic₂₃	—	2	1	no	—
120	C₂×Dic₂₃	—	4	2	no	—
121	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
124	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
125	Dic₄₆	—	2	1	no	—
126	Dic₄₆	—	2	2	no	—
127	Dic₄₆	—	2	1	no	—
128	Dic₄₆	—	2	1	no	—
129	Dic₄₆	—	4	2	no	—
130	Dic₄₆	#41	1	2	yes	—
131	C₈×C₂₃	—	2	1	no	—
132	C₈×C₂₃	—	2	1	no	—
133	C₂₃⋊C₈	—	2	1	no	—
134	C₂₃⋊C₈	—	2	1	no	—
135	#3364	—	23	1	no	maximal, Sylow
136	C₂²×C₂₃	—	4	1	no	—
137	C₂²×C₂₃	—	4	1	no	—
138	C₂²×C₂₃	—	4	1	no	—
139	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
140	C₄×C₂₃	—	2	1	no	—
141	C₄×C₂₃	—	2	1	no	—
142	C₄×C₂₃	—	4	1	no	—
143	C₄×C₂₃	—	4	1	no	—
144	C₄×C₂₃	—	4	1	no	—
145	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
146	C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
147	Dic₂₃	—	2	2	no	—
148	Dic₂₃	—	2	1	no	—
149	Dic₂₃	—	2	1	no	—
150	Dic₂₃	—	2	1	no	—
151	Dic₂₃	—	4	1	no	—
152	Dic₂₃	—	4	1	no	—
153	Dic₂₃	—	4	1	no	—
154	Dic₂₃	#135	1	2	yes	—
155	#557	—	23	1	no	—
156	#556	—	23	1	no	—
157	#488	—	23	1	no	—
158	#488	D₂₃	1	1	yes	characteristic
159	#487	—	23	1	no	—
160	#479	—	23	1	no	—
161	#479	—	23	1	no	—
162	#474	—	23	1	no	—
163	#464	—	23	1	no	—
164	#464	—	23	1	no	—
165	#462	—	23	1	no	—
166	C₈×Q₈	—	23	1	no	—
167	C₄×Q₁₆	—	23	1	no	—
168	C₄×Q₁₆	—	23	1	no	—
169	C₄×SD₁₆	—	23	1	no	—
170	C₂×C₂₃	—	8	1	no	—
171	C₂×C₂₃	#548	1	1	yes	characteristic
172	C₂×C₂₃	C₂×#135	1	1	yes	characteristic
173	C₂×C₂₃	#578	1	1	yes	characteristic
174	C₂×Q₁₆	—	23	1	no	—
175	C₂×Q₁₆	—	23	1	no	—
176	C₂×SD₁₆	—	23	1	no	—
177	C₄⋊Q₈	—	23	1	no	—
178	C₄⋊Q₈	—	23	1	no	—
179	C₄⋊Q₈	—	46	1	no	—
180	#126	—	46	1	no	—
181	#126	—	46	1	no	—
182	#125	—	23	1	no	—
183	#125	—	46	1	no	—
184	#125	—	46	1	no	—
185	#125	C₂×D₂₃	1	1	yes	characteristic
186	#124	—	23	1	no	—
187	#124	C₂×D₂₃	1	1	yes	characteristic
188	C₂²⋊Q₈	—	46	1	no	—
189	C₄×Q₈	—	23	1	no	—
190	C₄×Q₈	—	23	1	no	—
191	C₄×Q₈	—	23	1	no	—
192	C₄×Q₈	—	23	1	no	—
193	C₄×Q₈	—	23	1	no	—
194	C₄×Q₈	—	23	1	no	—
195	C₄×Q₈	—	46	1	no	—
196	C₄×D₄	—	23	1	no	—
197	#117	—	46	1	no	—
198	#107	—	23	1	no	—
199	#107	—	23	1	no	—
200	#106	—	23	1	no	—
201	C₄⋊C₈	—	23	1	no	—
202	C₄⋊C₈	—	23	1	no	—
203	C₄⋊C₈	—	23	1	no	—
204	#103	—	23	1	no	—
205	#103	—	23	1	no	—
206	#103	—	23	1	no	—
207	#103	—	23	1	no	—
208	#103	—	23	1	no	—
209	#103	C₂×D₂₃	1	1	yes	characteristic
210	#103	C₂×D₂₃	1	1	yes	characteristic
211	#102	—	23	1	no	—
212	#102	C₂×D₂₃	1	1	yes	characteristic
213	#102	C₂×D₂₃	1	1	yes	characteristic
214	C₄×C₈	—	23	1	no	—
215	C₄×C₈	—	23	1	no	—
216	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
217	C₂₃	#3364	1	1	yes	characteristic, complemented, Sylow
218	C₂×Q₈	—	23	1	no	—
219	C₂×Q₈	—	23	1	no	—
220	C₂×Q₈	—	23	1	no	—
221	C₂×Q₈	—	46	1	no	—
222	C₂×Q₈	C₂²×D₂₃	1	1	yes	characteristic
223	C₂×D₄	C₂²×D₂₃	1	1	yes	characteristic
224	C₂²×C₄	—	46	1	no	—
225	Q₁₆	—	46	2	no	—
226	Q₁₆	—	46	2	no	—
227	SD₁₆	—	46	2	no	—
228	C₂×C₈	—	23	1	no	—
229	C₂×C₈	—	23	1	no	—
230	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
231	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	23	1	no	—
235	C₄⋊C₄	—	23	1	no	—
236	C₄⋊C₄	—	23	1	no	—
237	C₄⋊C₄	—	23	1	no	—
238	C₄⋊C₄	—	23	1	no	—
239	C₄⋊C₄	—	46	1	no	—
240	C₄⋊C₄	—	46	1	no	—
241	C₄⋊C₄	—	46	1	no	—
242	C₄⋊C₄	—	46	1	no	—
243	C₄⋊C₄	—	46	1	no	—
244	C₄⋊C₄	—	46	1	no	—
245	C₄⋊C₄	—	46	1	no	—
246	C₄⋊C₄	—	46	1	no	—
247	C₄⋊C₄	—	46	1	no	—
248	C₄⋊C₄	—	46	1	no	—
249	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
250	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
251	C₂²⋊C₄	—	2	1	no	—
252	C₂²⋊C₄	—	2	1	no	—
253	C₂²⋊C₄	—	46	1	no	—
254	C₂²⋊C₄	—	46	1	no	—
255	C₂²⋊C₄	—	46	1	no	—
256	C₄²	—	23	1	no	—
257	C₄²	—	23	1	no	—
258	C₄²	—	46	1	no	—
259	C₄²	—	46	1	no	—
260	C₄²	—	46	1	no	—
261	C₄²	—	46	1	no	—
262	C₄²	—	46	1	no	—
263	C₄²	C₂²×D₂₃	1	1	yes	characteristic
264	C₂³	—	2	1	no	—
265	Q₈	—	2	1	no	—
266	Q₈	—	2	1	no	—
267	Q₈	—	23	2	no	—
268	Q₈	—	46	2	no	—
269	Q₈	—	46	1	no	—
270	Q₈	—	46	1	no	—
271	Q₈	—	46	1	no	—
272	Q₈	—	92	2	no	—
273	D₄	—	2	1	no	—
274	D₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	23	1	no	—
279	C₂×C₄	—	23	1	no	—
280	C₂×C₄	—	23	1	no	—
281	C₂×C₄	—	23	1	no	—
282	C₂×C₄	—	46	1	no	—
283	C₂×C₄	—	46	1	no	—
284	C₂×C₄	—	46	1	no	—
285	C₂×C₄	—	46	1	no	—
286	C₂×C₄	—	92	2	no	—
287	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
288	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
289	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	—	46	1	no	—
293	C₈	—	46	1	no	—
294	C₂²	—	4	1	no	—
295	C₂²	—	4	1	no	—
296	C₂²	—	4	1	no	—
297	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	4	1	no	—
301	C₄	—	4	1	no	—
302	C₄	—	4	1	no	—
303	C₄	—	23	2	no	—
304	C₄	—	46	1	no	—
305	C₄	—	46	1	no	—
306	C₄	—	46	1	no	—
307	C₄	—	46	2	no	—
308	C₄	—	92	1	no	—
309	C₄	—	92	1	no	—
310	C₄	—	92	1	no	—
311	C₄	#41×D₂₃	1	1	yes	characteristic
312	C₄	#10625161	1	1	yes	characteristic
313	C₂	—	8	1	no	—
314	C₂	#13093075	1	1	yes	characteristic
315	C₂	#135×D₂₃	1	1	yes	characteristic
316	C₂	#13093231	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×#488	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂×D₄×C₂₃	C₂³
hypercenter	#488	D₂₃
hypocenter	C₂₃	#3364
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #488

Complements

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

N	H ≅ G/N	type
1. #13093144	C₂	semidirect
4. #13092766	C₂	semidirect
5. #13092726	C₂	semidirect
6. #13092721	C₂	semidirect
10. #13092626	C₂	semidirect
11. C₈×Dic₄₆	C₂	semidirect
12. C₄×Dic₉₂	C₂	semidirect
13. #13092509	C₂	semidirect
217. C₂₃	#3364	semidirect

Maximal subgroups

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

subgroup	index
#13093144	2
#13093136	2
#13092770	2
#13092766	2
#13092726	2
#13092721	2
#13092695	2
#13092638	2
#13092636	2
#13092626	2
C₈×Dic₄₆	2
C₄×Dic₉₂	2
#13092509	2
#13092508	2
C₂₃×#488	2
#3364	23