Subgroups — #28743372

Order 15104.

556 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21432949	C₂	1	1	yes	maximal, characteristic
2	#21426250	C₂	1	2	yes	maximal, complemented
3	#21426250	C₂	1	2	yes	maximal, complemented
4	#21426240	C₂	1	2	yes	maximal, complemented
5	#21426230	C₂	1	2	yes	maximal, complemented
6	#21425882	C₂	1	1	yes	maximal, characteristic, complemented
7	#21425878	C₂	1	1	yes	maximal, characteristic, complemented
8	#21425855	C₂	1	1	yes	maximal, characteristic, complemented
9	#21425811	C₂	1	1	yes	maximal, characteristic
10	C₂×#16837202	C₂	1	1	yes	maximal, characteristic
11	C₅₉×#3164	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₄₇₂	C₂²	1	1	yes	characteristic
13	#16837876	—	2	1	no	—
14	C₂×#14273667	C₂²	1	1	yes	characteristic
15	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
16	#16837515	—	2	2	no	—
17	C₄×D₄₇₂	C₂²	1	2	yes	—
18	C₄×D₄₇₂	C₂²	1	2	yes	—
19	#16837440	C₂²	1	1	yes	characteristic
20	#16837408	—	2	1	no	—
21	#16837229	C₂²	1	1	yes	characteristic, complemented
22	#16837229	C₂²	1	2	yes	complemented
23	#16837229	C₂²	1	1	yes	characteristic, complemented
24	#16837227	C₂²	1	1	yes	characteristic, complemented
25	#16837227	C₂²	1	2	yes	complemented
26	#16837227	C₂²	1	1	yes	characteristic, complemented
27	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	2	yes	complemented
28	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	2	yes	complemented
29	#16837202	C₂²	1	2	yes	—
30	#16837202	C₂²	1	2	yes	—
31	#16837202	C₂²	1	2	yes	—
32	#16837202	C₂²	1	2	yes	—
33	C₅₉×#544	—	2	1	no	—
34	C₅₉×#501	—	2	2	no	—
35	C₅₉×#500	C₂²	1	2	yes	complemented
36	C₅₉×#499	C₂²	1	2	yes	complemented
37	C₅₉×#428	—	2	1	no	—
38	C₅₉×#427	C₂²	1	1	yes	characteristic
39	C₂×#107×C₅₉	C₂²	1	1	yes	characteristic
40	C₅₉×#405	C₂²	1	1	yes	characteristic
41	C₂²×D₂₃₆	—	2	1	no	—
42	C₂×D₄₇₂	—	2	1	no	—
43	C₂×D₄₇₂	—	2	2	no	—
44	C₂×D₄₇₂	Q₈	1	4	yes	complemented
45	C₂×D₄₇₂	C₂³	1	2	yes	—
46	C₄×D₂₃₆	—	2	2	no	—
47	C₄×D₂₃₆	—	2	2	no	—
48	#14273689	—	4	1	no	—
49	#14273689	—	4	1	no	—
50	#14273686	—	2	1	no	—
51	C₂²⋊C₄×D₅₉	—	4	2	no	—
52	#14273667	—	2	2	no	—
53	C₂×C₅₉⋊C₁₆	—	2	2	no	—
54	C₂×C₅₉⋊C₁₆	C₂³	1	1	yes	characteristic
55	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
56	C₂×C₅₉⋊C₁₆	C₂³	1	1	yes	characteristic
57	#14273661	C₂³	1	1	yes	characteristic
58	#14273661	C₂³	1	1	yes	characteristic
59	#14273659	—	2	1	no	—
60	#14273649	—	2	2	no	—
61	#14273649	—	2	2	no	—
62	C₂×M₁₆×C₅₉	D₄	1	1	yes	characteristic
63	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
64	C₄⋊Q₈×C₅₉	—	2	2	no	—
65	C₄⋊Q₈×C₅₉	—	4	1	no	—
66	#125×C₅₉	—	2	2	no	—
67	#125×C₅₉	—	4	1	no	—
68	C₂²⋊Q₈×C₅₉	—	4	2	no	—
69	#117×C₅₉	—	2	1	no	—
70	#117×C₅₉	D₄	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
72	#107×C₅₉	—	2	1	no	—
73	#107×C₅₉	—	2	2	no	—
74	#107×C₅₉	C₂³	1	2	yes	—
75	#107×C₅₉	C₂³	1	1	yes	characteristic
76	#107×C₅₉	C₂³	1	1	yes	characteristic
77	#106×C₅₉	—	2	1	no	—
78	#106×C₅₉	—	2	2	no	—
79	#97×C₅₉	D₄	1	2	yes	—
80	C₄×C₈×C₅₉	C₂³	1	2	yes	—
81	C₂³×D₅₉	—	4	1	no	—
82	C₂×Q₈×C₅₉	—	4	2	no	—
83	C₂²×C₄×C₅₉	—	4	1	no	—
84	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
85	C₂×D₂₃₆	—	2	4	no	—
86	C₂×D₂₃₆	—	2	2	no	—
87	C₂×D₂₃₆	—	4	1	no	—
88	C₂×D₂₃₆	—	4	2	no	—
89	C₂×C₄×D₅₉	—	4	2	no	—
90	C₂×C₄×D₅₉	—	4	2	no	—
91	M₁₆×C₅₉	—	2	2	no	—
92	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
95	C₂×C₈×C₅₉	#41	1	2	yes	—
96	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
97	C₄⋊C₄×C₅₉	—	2	1	no	—
98	C₄⋊C₄×C₅₉	—	2	2	no	—
99	C₄⋊C₄×C₅₉	—	2	1	no	—
100	C₄⋊C₄×C₅₉	—	4	1	no	—
101	C₄⋊C₄×C₅₉	—	4	2	no	—
102	C₄⋊C₄×C₅₉	—	4	2	no	—
103	C₄⋊C₄×C₅₉	—	4	2	no	—
104	C₄⋊C₄×C₅₉	D₈	1	1	yes	characteristic
105	C₄⋊C₄×C₅₉	D₈	1	1	yes	characteristic
106	C₂²⋊C₄×C₅₉	—	4	1	no	—
107	C₂²⋊C₄×C₅₉	D₈	1	2	yes	—
108	C₄²×C₅₉	—	4	1	no	—
109	C₄²×C₅₉	C₂×D₄	1	2	yes	—
110	C₂²⋊Dic₅₉	—	4	1	no	—
111	(C₂×D₅₉)⋊C₄	—	4	2	no	—
112	(C₂×D₅₉)⋊C₄	—	4	2	no	—
113	C₄⋊Dic₅₉	—	2	1	no	—
114	C₄⋊Dic₅₉	—	2	1	no	—
115	C₄×Dic₅₉	—	4	1	no	—
116	D₄₇₂	—	2	4	no	—
117	D₄₇₂	—	2	2	no	—
118	D₄₇₂	C₂×Q₈	1	4	yes	—
119	C₅₉⋊C₁₆	—	2	1	no	—
120	C₅₉⋊C₁₆	—	2	1	no	—
121	C₅₉⋊C₁₆	—	2	2	no	—
122	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
123	C₂²×D₅₉	—	4	2	no	—
124	C₂²×D₅₉	—	4	4	no	—
125	C₂²×D₅₉	—	16	1	no	—
126	C₂²×D₅₉	—	16	1	no	—
127	Q₈×C₅₉	—	16	2	no	—
128	C₂×C₄×C₅₉	—	4	1	no	—
129	C₂×C₄×C₅₉	—	4	1	no	—
130	C₂×C₄×C₅₉	—	4	2	no	—
131	C₂×C₄×C₅₉	—	16	1	no	—
132	C₂×C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
133	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
134	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
135	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₅₉	#137	1	2	yes	—
137	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
138	C₂×Dic₅₉	—	4	1	no	—
139	C₂×Dic₅₉	—	4	1	no	—
140	D₂₃₆	—	2	4	no	—
141	D₂₃₆	—	4	2	no	—
142	D₂₃₆	—	4	4	no	—
143	C₄×D₅₉	—	16	2	no	—
144	C₄×D₅₉	—	16	2	no	—
145	C₈×C₅₉	—	2	2	no	—
146	C₈×C₅₉	C₂²⋊Q₈	1	2	yes	—
147	C₈×C₅₉	C₂²×Q₈	1	1	yes	characteristic
148	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
149	#33631	—	59	1	no	maximal, Sylow
150	C₂²×C₅₉	—	2	1	no	—
151	C₂²×C₅₉	—	2	1	no	—
152	C₂²×C₅₉	#474	1	2	yes	—
153	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
154	C₂×D₅₉	—	4	4	no	—
155	C₂×D₅₉	—	8	2	no	—
156	C₂×D₅₉	—	8	2	no	—
157	C₂×D₅₉	—	16	4	no	—
158	C₄×C₅₉	—	2	2	no	—
159	C₄×C₅₉	—	2	2	no	—
160	C₄×C₅₉	—	8	2	no	—
161	C₄×C₅₉	—	8	1	no	—
162	C₄×C₅₉	—	8	1	no	—
163	C₄×C₅₉	#474	1	2	yes	—
164	C₄×C₅₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
165	C₄×C₅₉	C₂×#137	1	1	yes	characteristic
166	Dic₅₉	—	8	1	no	—
167	Dic₅₉	—	8	1	no	—
168	#3164	D₅₉	1	1	yes	characteristic, complemented
169	#2948	—	59	1	no	—
170	#2228	—	59	2	no	—
171	#2228	—	59	2	no	—
172	#2226	—	59	2	no	—
173	#2224	—	59	2	no	—
174	#2159	—	59	1	no	—
175	#2158	—	59	1	no	—
176	#2151	—	59	1	no	—
177	#2140	—	59	1	no	—
178	C₂×#357	—	59	1	no	—
179	C₂×C₅₉	—	2	2	no	—
180	C₂×C₅₉	C₂×#474	1	1	yes	characteristic
181	C₂×C₅₉	#3415	1	1	yes	characteristic
182	C₂×C₅₉	#3416	1	1	yes	characteristic
183	D₅₉	—	8	4	no	—
184	C₂²×D₈	—	59	1	no	—
185	#544	—	2	1	no	—
186	#518	—	118	1	no	—
187	C₂×M₃₂	—	59	1	no	—
188	C₂²×C₁₆	—	59	1	no	—
189	#501	—	2	2	no	—
190	#500	C₂×D₅₉	1	2	yes	complemented
191	#499	C₂×D₅₉	1	2	yes	complemented
192	#442	—	118	2	no	—
193	C₄×D₈	—	59	2	no	—
194	C₄×D₈	—	59	2	no	—
195	#428	—	2	1	no	—
196	#427	—	59	1	no	—
197	#427	C₂×D₅₉	1	1	yes	characteristic
198	C₂×#107	C₂×D₅₉	1	1	yes	characteristic
199	#418	—	118	1	no	—
200	#405	C₂×D₅₉	1	1	yes	characteristic
201	#367	—	59	1	no	—
202	#367	—	59	1	no	—
203	#367	—	59	2	no	—
204	#366	—	59	1	no	—
205	#366	—	59	1	no	—
206	#366	—	59	2	no	—
207	C₄⋊C₁₆	—	59	2	no	—
208	C₄⋊C₁₆	—	59	2	no	—
209	#357	—	59	2	no	—
210	#357	—	59	2	no	—
211	#357	—	59	2	no	—
212	#357	—	59	2	no	—
213	C₅₉	#33631	1	1	yes	characteristic, complemented, Sylow
214	C₂²×D₄	—	118	1	no	—
215	C₂×D₈	—	59	4	no	—
216	C₂×D₈	—	59	2	no	—
217	C₂×D₈	—	118	2	no	—
218	C₂×D₈	—	118	1	no	—
219	C₂×M₁₆	C₂²⋊D₅₉	1	1	yes	characteristic
220	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
221	C₄⋊Q₈	—	2	2	no	—
222	C₄⋊Q₈	—	4	1	no	—
223	#125	—	2	2	no	—
224	#125	—	4	1	no	—
225	C₂²⋊Q₈	—	4	2	no	—
226	C₄×D₄	—	118	2	no	—
227	C₄×D₄	—	118	2	no	—
228	C₄×D₄	—	236	1	no	—
229	C₄×D₄	—	236	1	no	—
230	#117	—	2	1	no	—
231	#117	—	118	1	no	—
232	#117	C₂²⋊D₅₉	1	1	yes	characteristic
233	C₂×C₄⋊C₄	—	2	1	no	—
234	C₂×C₂²⋊C₄	—	236	2	no	—
235	M₃₂	—	118	2	no	—
236	C₂×C₁₆	—	59	1	no	—
237	C₂×C₁₆	—	59	2	no	—
238	C₂×C₁₆	—	59	1	no	—
239	C₂×C₁₆	—	118	2	no	—
240	#107	—	2	1	no	—
241	#107	—	2	2	no	—
242	#107	—	59	1	no	—
243	#107	—	59	1	no	—
244	#107	C₂²×D₅₉	1	2	yes	—
245	#107	C₂²×D₅₉	1	1	yes	characteristic
246	#107	C₂²×D₅₉	1	1	yes	characteristic
247	#106	—	2	1	no	—
248	#106	—	2	2	no	—
249	#106	—	118	1	no	—
250	#102	—	118	2	no	—
251	#102	—	118	2	no	—
252	#97	C₂²⋊D₅₉	1	2	yes	—
253	C₄×C₈	C₂²×D₅₉	1	2	yes	—
254	C₂⁴	—	236	1	no	—
255	C₂×Q₈	—	4	2	no	—
256	C₂×D₄	—	118	4	no	—
257	C₂×D₄	—	118	2	no	—
258	C₂×D₄	—	236	1	no	—
259	C₂×D₄	—	236	2	no	—
260	C₂²×C₄	—	4	1	no	—
261	C₂²×C₄	—	236	2	no	—
262	C₂²×C₄	—	236	2	no	—
263	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
264	D₈	—	59	4	no	—
265	D₈	—	118	4	no	—
266	D₈	—	118	2	no	—
267	M₁₆	—	2	2	no	—
268	C₂×C₈	Q₈×D₅₉	1	2	yes	—
269	C₂×C₈	#11745689	1	2	yes	—
270	C₂×C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
271	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
272	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	—	4	2	no	—
280	C₄⋊C₄	—	118	1	no	—
281	C₄⋊C₄	—	118	1	no	—
282	C₄⋊C₄	#11745669	1	1	yes	characteristic
283	C₄⋊C₄	#11745669	1	1	yes	characteristic
284	C₂²⋊C₄	—	4	1	no	—
285	C₂²⋊C₄	—	236	1	no	—
286	C₂²⋊C₄	—	236	2	no	—
287	C₂²⋊C₄	—	236	2	no	—
288	C₂²⋊C₄	#11745669	1	2	yes	—
289	C₄²	—	4	1	no	—
290	C₄²	—	236	1	no	—
291	C₄²	C₂×C₂²⋊D₅₉	1	2	yes	—
292	C₁₆	—	118	1	no	—
293	C₁₆	—	118	1	no	—
294	C₁₆	—	118	2	no	—
295	C₂³	—	944	1	no	—
296	C₂³	—	944	1	no	—
297	C₂³	—	236	4	no	—
298	C₂³	—	236	2	no	—
299	C₂³	C₂×#11745669	1	1	yes	characteristic
300	Q₈	—	16	2	no	—
301	D₄	—	118	4	no	—
302	D₄	—	236	4	no	—
303	D₄	—	236	2	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	16	1	no	—
308	C₂×C₄	—	944	2	no	—
309	C₂×C₄	—	944	2	no	—
310	C₂×C₄	—	236	1	no	—
311	C₂×C₄	—	236	1	no	—
312	C₂×C₄	#14273703	1	2	yes	—
313	C₂×C₄	C₂×#11745669	1	2	yes	—
314	C₂×C₄	C₂×#11745669	1	2	yes	—
315	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
316	C₂×C₄	#14273758	1	2	yes	—
317	C₂×C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
318	C₈	—	2	2	no	—
319	C₈	#14273703	1	2	yes	—
320	C₈	C₂×Q₈×D₅₉	1	1	yes	characteristic
321	C₈	C₂×#11745689	1	1	yes	characteristic
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	944	4	no	—
325	C₂²	—	472	2	no	—
326	C₂²	—	472	2	no	—
327	C₂²	—	236	4	no	—
328	C₂²	#16837679	1	2	yes	—
329	C₂²	C₂²×#11745669	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	8	2	no	—
333	C₄	—	8	1	no	—
334	C₄	—	8	1	no	—
335	C₄	—	472	1	no	—
336	C₄	—	472	1	no	—
337	C₄	#16837679	1	2	yes	—
338	C₄	C₂×#14273703	1	1	yes	characteristic
339	C₄	C₂×#14273758	1	1	yes	characteristic
340	C₂	—	2	2	no	—
341	C₂	—	472	4	no	—
342	C₂	C₂×#16837679	1	1	yes	characteristic
343	C₂	#21437235	1	1	yes	characteristic
344	C₂	#21437240	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11745669
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₅₉×#3164	C₂
socle	C₂²×C₅₉	C₂²×D₈
prime radical	C₂²×D₄₇₂	C₂²
hypercenter	#3164	D₅₉
hypocenter	C₅₉	#33631
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #3164

Complements

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

N	H ≅ G/N	type
2. #21426250	C₂	semidirect
3. #21426250	C₂	semidirect
4. #21426240	C₂	semidirect
5. #21426230	C₂	semidirect
6. #21425882	C₂	semidirect
7. #21425878	C₂	semidirect
8. #21425855	C₂	semidirect
11. C₅₉×#3164	C₂	semidirect
21. #16837229	C₂²	semidirect
22. #16837229	C₂²	semidirect
23. #16837229	C₂²	semidirect
24. #16837227	C₂²	semidirect
25. #16837227	C₂²	semidirect
26. #16837227	C₂²	semidirect
27. C₄⋊(C₅₉⋊C₁₆)	C₂²	semidirect
28. C₄⋊(C₅₉⋊C₁₆)	C₂²	semidirect
35. C₅₉×#500	C₂²	semidirect
36. C₅₉×#499	C₂²	semidirect
44. C₂×D₄₇₂	Q₈	semidirect
168. #3164	D₅₉	semidirect
190. #500	C₂×D₅₉	semidirect
191. #499	C₂×D₅₉	semidirect
213. C₅₉	#33631	semidirect

Maximal subgroups

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

subgroup	index
#21432949	2
#21426250	2
#21426240	2
#21426230	2
#21425882	2
#21425878	2
#21425855	2
#21425811	2
C₂×#16837202	2
C₅₉×#3164	2
#33631	59