Subgroups — #28743739

Order 15104.

542 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	C₂×C₄×Dic₂₃₆	C₂	1	1	yes	maximal, characteristic
2	#21426291	C₂	1	2	yes	maximal
3	#21426235	C₂	1	2	yes	maximal, complemented
4	#21426143	C₂	1	2	yes	maximal, complemented
5	#21426029	C₂	1	2	yes	maximal
6	#21425965	C₂	1	4	yes	maximal, complemented
7	#21425899	C₂	1	1	yes	maximal, characteristic
8	C₅₉×#3170	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₂₃₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₁₁₈	—	2	1	no	—
11	C₂×#14273673	C₂²	1	4	yes	—
12	C₄×Dic₂₃₆	—	2	1	no	—
13	C₄×Dic₂₃₆	—	2	1	no	—
14	C₄×Dic₂₃₆	C₂²	1	2	yes	—
15	C₄×Dic₂₃₆	C₂²	1	2	yes	—
16	C₂×#14273661	C₂²	1	1	yes	characteristic
17	C₂×#14273652	—	2	1	no	—
18	#16837229	C₂²	1	4	yes	—
19	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	2	yes	—
20	#16837206	C₂²	1	4	yes	—
21	#16837205	C₂²	1	4	yes	—
22	C₂²⋊(C₅₉⋊C₁₆)	C₂²	1	2	yes	—
23	#16837178	C₂²	1	2	yes	—
24	C₅₉×#533	—	2	1	no	—
25	C₅₉×#500	C₂²	1	1	yes	characteristic
26	C₅₉×#494	C₂²	1	1	yes	characteristic
27	C₅₉×#487	—	2	1	no	—
28	C₅₉×#467	—	2	1	no	—
29	C₅₉×#467	C₂²	1	2	yes	—
30	C₄×Q₁₆×C₅₉	—	2	1	no	—
31	C₄×Q₁₆×C₅₉	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
33	C₂²×Dic₁₁₈	—	2	1	no	—
34	C₂×Dic₂₃₆	—	2	1	no	—
35	C₂×Dic₂₃₆	—	2	4	no	—
36	C₂×Dic₂₃₆	C₂³	1	2	yes	—
37	C₄×Dic₁₁₈	—	2	2	no	—
38	C₄×Dic₁₁₈	—	2	2	no	—
39	C₄×Dic₁₁₈	—	4	1	no	—
40	C₄×Dic₁₁₈	—	4	1	no	—
41	C₂×C₄⋊Dic₅₉	—	2	1	no	—
42	C₂×Dic₅₉⋊C₄	—	4	1	no	—
43	C₂×C₄×Dic₅₉	—	4	1	no	—
44	#14273673	—	2	8	no	—
45	C₂×C₅₉⋊C₁₆	C₂³	1	4	yes	—
46	#14273661	—	2	1	no	—
47	#14273661	C₂³	1	2	yes	—
48	#14273652	—	2	2	no	—
49	#14273652	—	2	2	no	—
50	C₂×Q₁₆×C₅₉	—	2	1	no	—
51	C₂×Q₁₆×C₅₉	C₂³	1	2	yes	—
52	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
54	C₄⋊Q₈×C₅₉	—	2	1	no	—
55	C₄⋊Q₈×C₅₉	—	2	1	no	—
56	#125×C₅₉	—	4	1	no	—
57	C₂²⋊Q₈×C₅₉	—	2	2	no	—
58	C₂²⋊Q₈×C₅₉	—	4	1	no	—
59	C₄×Q₈×C₅₉	—	2	2	no	—
60	C₄×Q₈×C₅₉	—	4	1	no	—
61	#117×C₅₉	—	4	1	no	—
62	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
63	#107×C₅₉	—	2	1	no	—
64	#107×C₅₉	C₂³	1	2	yes	—
65	#103×C₅₉	—	2	2	no	—
66	#103×C₅₉	—	2	2	no	—
67	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
68	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
69	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
71	C₂×Q₈×C₅₉	—	2	2	no	—
72	C₂×Q₈×C₅₉	—	4	1	no	—
73	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
74	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
75	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
76	C₂²×Dic₅₉	—	4	1	no	—
77	C₂²×Dic₅₉	—	4	1	no	—
78	C₂×Dic₁₁₈	—	2	4	no	—
79	C₂×Dic₁₁₈	—	2	2	no	—
80	C₂×Dic₁₁₈	—	4	2	no	—
81	C₂×Dic₁₁₈	—	4	1	no	—
82	Q₁₆×C₅₉	—	2	4	no	—
83	Q₁₆×C₅₉	—	4	2	no	—
84	C₂×C₈×C₅₉	—	2	2	no	—
85	C₂×C₈×C₅₉	—	4	1	no	—
86	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
90	C₄⋊C₄×C₅₉	—	2	2	no	—
91	C₄⋊C₄×C₅₉	—	4	1	no	—
92	C₄⋊C₄×C₅₉	—	4	1	no	—
93	C₄⋊C₄×C₅₉	—	4	2	no	—
94	C₄⋊C₄×C₅₉	—	4	2	no	—
95	C₄⋊C₄×C₅₉	—	4	1	no	—
96	C₂²⋊C₄×C₅₉	—	4	2	no	—
97	C₄²×C₅₉	—	4	2	no	—
98	C₄²×C₅₉	D₈	1	1	yes	characteristic
99	C₄²×C₅₉	D₈	1	1	yes	characteristic
100	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
102	C₄⋊Dic₅₉	—	2	2	no	—
103	C₄⋊Dic₅₉	—	4	1	no	—
104	Dic₅₉⋊C₄	—	4	2	no	—
105	Dic₅₉⋊C₄	—	4	2	no	—
106	C₄×Dic₅₉	—	4	2	no	—
107	C₄×Dic₅₉	—	4	2	no	—
108	Dic₂₃₆	—	2	4	no	—
109	Dic₂₃₆	—	2	2	no	—
110	Dic₂₃₆	#41	1	4	yes	—
111	C₅₉⋊C₁₆	—	2	4	no	—
112	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
113	Q₈×C₅₉	—	4	2	no	—
114	Q₈×C₅₉	—	4	2	no	—
115	Q₈×C₅₉	—	8	2	no	—
116	C₂×C₄×C₅₉	—	2	2	no	—
117	C₂×C₄×C₅₉	—	2	1	no	—
118	C₂×C₄×C₅₉	—	2	1	no	—
119	C₂×C₄×C₅₉	—	2	2	no	—
120	C₂×C₄×C₅₉	—	4	2	no	—
121	C₂×C₄×C₅₉	—	4	2	no	—
122	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
128	C₂×Dic₅₉	—	4	4	no	—
129	C₂×Dic₅₉	—	4	2	no	—
130	C₂×Dic₅₉	—	4	2	no	—
131	C₂×Dic₅₉	—	8	2	no	—
132	Dic₁₁₈	—	2	4	no	—
133	Dic₁₁₈	—	4	4	no	—
134	Dic₁₁₈	—	4	2	no	—
135	C₈×C₅₉	—	2	1	no	—
136	C₈×C₅₉	—	2	1	no	—
137	C₈×C₅₉	—	2	1	no	—
138	C₈×C₅₉	C₂×#41	1	2	yes	—
139	#33673	—	59	1	no	maximal, Sylow
140	C₂²×C₅₉	—	2	1	no	—
141	C₂²×C₅₉	—	2	1	no	—
142	C₂²×C₅₉	#510	1	2	yes	—
143	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
144	C₄×C₅₉	—	2	1	no	—
145	C₄×C₅₉	—	2	1	no	—
146	C₄×C₅₉	—	2	1	no	—
147	C₄×C₅₉	—	2	1	no	—
148	C₄×C₅₉	—	8	2	no	—
149	C₄×C₅₉	—	8	2	no	—
150	C₄×C₅₉	#510	1	2	yes	—
151	C₄×C₅₉	#525	1	1	yes	characteristic
152	C₄×C₅₉	#575	1	1	yes	characteristic
153	Dic₅₉	—	4	4	no	—
154	Dic₅₉	—	8	2	no	—
155	#3170	D₅₉	1	1	yes	characteristic
156	C₂×C₄×Q₁₆	—	59	1	no	—
157	#2235	—	59	2	no	—
158	#2225	—	59	2	no	—
159	#2209	—	59	2	no	—
160	#2188	—	59	2	no	—
161	#2176	—	59	4	no	—
162	#2163	—	59	1	no	—
163	C₂×C₅₉	—	2	2	no	—
164	C₂×C₅₉	#3100	1	1	yes	characteristic
165	C₂×C₅₉	C₂×#510	1	1	yes	characteristic
166	C₂×C₅₉	C₂×#510	1	1	yes	characteristic
167	C₂²×Q₁₆	—	59	1	no	—
168	#533	—	2	1	no	—
169	C₂×C₄×Q₈	—	118	1	no	—
170	C₂×Q₃₂	—	59	4	no	—
171	#500	C₂×D₅₉	1	1	yes	characteristic
172	#494	C₂×D₅₉	1	1	yes	characteristic
173	#487	—	2	1	no	—
174	#467	—	2	1	no	—
175	#467	C₂×D₅₉	1	2	yes	—
176	C₄×Q₁₆	—	2	1	no	—
177	C₄×Q₁₆	—	59	2	no	—
178	C₄×Q₁₆	—	59	2	no	—
179	C₄×Q₁₆	—	118	1	no	—
180	C₄×Q₁₆	—	118	1	no	—
181	C₄×Q₁₆	C₂×D₅₉	1	2	yes	—
182	C₂×#107	—	59	1	no	—
183	C₂×#103	—	118	1	no	—
184	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
185	#367	—	59	4	no	—
186	C₄⋊C₁₆	—	59	2	no	—
187	#358	—	59	4	no	—
188	#358	—	59	4	no	—
189	C₂²⋊C₁₆	—	59	2	no	—
190	#346	—	59	2	no	—
191	C₅₉	#33673	1	1	yes	characteristic, complemented, Sylow
192	C₂²×Q₈	—	118	1	no	—
193	C₂×Q₁₆	—	2	1	no	—
194	C₂×Q₁₆	—	59	2	no	—
195	C₂×Q₁₆	—	118	4	no	—
196	C₂×Q₁₆	—	118	1	no	—
197	C₂×Q₁₆	C₂²×D₅₉	1	2	yes	—
198	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
199	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
200	C₄⋊Q₈	—	2	1	no	—
201	C₄⋊Q₈	—	2	1	no	—
202	#125	—	4	1	no	—
203	C₂²⋊Q₈	—	2	2	no	—
204	C₂²⋊Q₈	—	4	1	no	—
205	C₄×Q₈	—	2	2	no	—
206	C₄×Q₈	—	4	1	no	—
207	C₄×Q₈	—	118	2	no	—
208	C₄×Q₈	—	118	2	no	—
209	C₄×Q₈	—	236	1	no	—
210	C₄×Q₈	—	236	1	no	—
211	#117	—	4	1	no	—
212	C₂×C₄⋊C₄	—	118	1	no	—
213	C₂×C₄⋊C₄	—	236	1	no	—
214	C₂×C₄²	—	236	1	no	—
215	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
216	Q₃₂	—	118	8	no	—
217	C₂×C₁₆	—	59	4	no	—
218	#107	—	2	1	no	—
219	#107	—	59	2	no	—
220	#107	—	118	1	no	—
221	#107	C₂²×D₅₉	1	2	yes	—
222	#103	—	2	2	no	—
223	#103	—	2	2	no	—
224	#103	—	118	2	no	—
225	#103	—	118	2	no	—
226	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
227	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
228	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
229	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
230	C₂×Q₈	—	2	2	no	—
231	C₂×Q₈	—	4	1	no	—
232	C₂×Q₈	—	118	4	no	—
233	C₂×Q₈	—	118	2	no	—
234	C₂×Q₈	—	236	2	no	—
235	C₂×Q₈	—	236	1	no	—
236	C₂²×C₄	—	236	1	no	—
237	C₂²×C₄	—	236	1	no	—
238	C₂²×C₄	#11745669	1	1	yes	characteristic
239	C₂²×C₄	#11745669	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
241	Q₁₆	—	2	4	no	—
242	Q₁₆	—	4	2	no	—
243	Q₁₆	—	59	4	no	—
244	Q₁₆	—	118	4	no	—
245	Q₁₆	—	118	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	4	1	no	—
248	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
249	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
250	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
251	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	4	1	no	—
254	C₄⋊C₄	—	4	1	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	4	1	no	—
258	C₄⋊C₄	—	118	2	no	—
259	C₄⋊C₄	—	236	1	no	—
260	C₄⋊C₄	—	236	2	no	—
261	C₄⋊C₄	—	236	2	no	—
262	C₂²⋊C₄	—	4	2	no	—
263	C₄²	—	4	2	no	—
264	C₄²	—	236	2	no	—
265	C₄²	—	236	2	no	—
266	C₄²	#11745669	1	1	yes	characteristic
267	C₄²	#11745669	1	1	yes	characteristic
268	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
269	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
270	C₁₆	—	118	4	no	—
271	C₂³	C₂×#11745669	1	1	yes	characteristic
272	Q₈	—	4	2	no	—
273	Q₈	—	4	2	no	—
274	Q₈	—	8	2	no	—
275	Q₈	—	118	4	no	—
276	Q₈	—	236	4	no	—
277	Q₈	—	236	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	472	2	no	—
285	C₂×C₄	—	236	2	no	—
286	C₂×C₄	—	236	4	no	—
287	C₂×C₄	—	236	2	no	—
288	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
289	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
290	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
291	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
292	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
293	C₂×C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
294	C₈	—	2	1	no	—
295	C₈	—	2	1	no	—
296	C₈	—	2	1	no	—
297	C₈	#41×D₅₉	1	2	yes	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	#16837843	1	2	yes	—
301	C₂²	C₂²×#11745669	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	8	2	no	—
307	C₄	—	8	2	no	—
308	C₄	—	472	2	no	—
309	C₄	—	236	4	no	—
310	C₄	#16837842	1	2	yes	—
311	C₄	#16837909	1	1	yes	characteristic
312	C₄	#16838200	1	1	yes	characteristic
313	C₂	—	2	2	no	—
314	C₂	#21434092	1	1	yes	characteristic
315	C₂	C₂×#16837842	1	1	yes	characteristic
316	C₂	C₂×#16837843	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11745669
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₅₉×#3170	C₂
socle	C₂²×C₅₉	C₂²×D₈
prime radical	C₂³×C₅₉	C₂×D₈
hypercenter	#3170	D₅₉
hypocenter	C₅₉	#33673
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3170

Complements

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

N	H ≅ G/N	type
3. #21426235	C₂	semidirect
4. #21426143	C₂	semidirect
6. #21425965	C₂	semidirect
191. C₅₉	#33673	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₂₃₆	2
#21426291	2
#21426235	2
#21426143	2
#21426029	2
#21425965	2
#21425899	2
C₅₉×#3170	2
#33673	59