Subgroups — #28742154

Order 15104.

632 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	C₂×#16837843	C₂	1	1	yes	maximal, characteristic
2	#21433014	C₂	1	1	yes	maximal, characteristic
3	#21425985	C₂	1	8	yes	maximal, complemented
4	#21425868	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16837218	C₂	1	1	yes	maximal, characteristic, complemented
6	#21425831	C₂	1	1	yes	maximal, characteristic
7	#21425701	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₉×#2950	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14273749	C₂²	1	1	yes	characteristic
10	#16838168	—	2	1	no	—
11	#16837843	C₄	1	4	yes	complemented
12	#16837843	C₂²	1	4	yes	—
13	C₂×#14273673	C₄	1	2	yes	complemented
14	C₂×#14273670	C₄	1	2	yes	complemented
15	C₂×#14273667	C₂²	1	1	yes	characteristic
16	C₂×#14273667	C₂²	1	1	yes	characteristic
17	#16837525	C₂²	1	4	yes	—
18	#16837520	—	2	2	no	—
19	C₂×#14273663	C₂²	1	1	yes	characteristic
20	#16837420	—	2	1	no	—
21	#16837224	C₂²	1	4	yes	complemented
22	#16837218	C₂²	1	4	yes	complemented
23	#16837211	C₂²	1	4	yes	—
24	#16837180	C₂²	1	4	yes	complemented
25	C₂²×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
26	C₅₉×#519	—	2	1	no	—
27	C₅₉×#441	C₂²	1	4	yes	complemented
28	C₄×Q₁₆×C₅₉	—	2	2	no	—
29	C₅₉×#427	C₂²	1	1	yes	characteristic
30	C₅₉×#419	—	2	1	no	—
31	C₅₉×#405	C₂²	1	1	yes	characteristic
32	C₂×#11745685	—	2	1	no	—
33	#14273749	—	2	1	no	—
34	#14273749	—	2	1	no	—
35	#14273749	C₂×C₄	1	2	yes	—
36	#14273749	C₂³	1	2	yes	—
37	C₂×Dic₂₃₆	C₂×C₄	1	1	yes	characteristic
38	C₂×#11745660	—	2	1	no	—
39	C₂×D₄₇₂	C₂×C₄	1	1	yes	characteristic
40	#14273735	—	4	2	no	—
41	#14273733	—	2	4	no	—
42	C₂×#11745664	—	2	1	no	—
43	M₁₆×D₅₉	—	4	2	no	—
44	#14273673	C₂×C₄	1	8	yes	—
45	#14273670	C₂×C₄	1	8	yes	—
46	#14273667	—	2	1	no	—
47	#14273667	C₂×C₄	1	2	yes	—
48	#14273667	C₂³	1	2	yes	—
49	#14273667	C₂³	1	2	yes	—
50	C₂×C₅₉⋊C₁₆	—	2	1	no	—
51	C₂×C₅₉⋊C₁₆	C₂×C₄	1	2	yes	—
52	#14273663	—	2	1	no	—
53	#14273663	C₂³	1	2	yes	—
54	#14273655	—	2	4	no	—
55	C₂²×Q₈×C₅₉	—	2	1	no	—
56	C₂×Q₁₆×C₅₉	—	2	1	no	—
57	C₂×Q₁₆×C₅₉	—	2	1	no	—
58	C₂×Q₁₆×C₅₉	—	2	1	no	—
59	C₂×Q₁₆×C₅₉	C₂×C₄	1	2	yes	—
60	C₂×Q₁₆×C₅₉	C₂×C₄	1	2	yes	—
61	C₂×Q₁₆×C₅₉	C₂³	1	2	yes	—
62	C₂×M₁₆×C₅₉	D₄	1	1	yes	characteristic
63	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
64	C₄×Q₈×C₅₉	—	2	4	no	—
65	C₄×Q₈×C₅₉	—	4	2	no	—
66	#117×C₅₉	—	2	1	no	—
67	#117×C₅₉	—	4	2	no	—
68	#117×C₅₉	D₄	1	1	yes	characteristic
69	#107×C₅₉	—	2	1	no	—
70	#106×C₅₉	C₂³	1	2	yes	—
71	#103×C₅₉	—	2	4	no	—
72	#97×C₅₉	C₂³	1	2	yes	—
73	C₄×C₈×C₅₉	D₄	1	2	yes	—
74	C₂×Q₈×C₅₉	—	2	2	no	—
75	C₂×Q₈×C₅₉	—	2	2	no	—
76	C₂×Q₈×C₅₉	—	2	2	no	—
77	C₂×Q₈×C₅₉	—	4	1	no	—
78	C₂×Q₈×C₅₉	—	4	1	no	—
79	C₂×Q₈×C₅₉	—	4	1	no	—
80	C₂²×C₄×C₅₉	—	4	1	no	—
81	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
82	C₂×C₂²⋊D₅₉	—	4	1	no	—
83	#11745685	—	2	2	no	—
84	#11745685	—	2	2	no	—
85	#11745685	—	4	1	no	—
86	#11745685	—	4	1	no	—
87	C₂×D₂₃₆	—	2	1	no	—
88	C₂×C₄×D₅₉	—	4	1	no	—
89	C₂×Dic₁₁₈	—	2	1	no	—
90	Q₁₆×C₅₉	—	2	2	no	—
91	Q₁₆×C₅₉	—	2	2	no	—
92	Q₁₆×C₅₉	—	2	2	no	—
93	Q₁₆×C₅₉	C₂²×C₄	1	4	yes	—
94	M₁₆×C₅₉	—	2	1	no	—
95	M₁₆×C₅₉	—	2	1	no	—
96	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
101	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
102	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
103	C₄⋊C₄×C₅₉	—	2	1	no	—
104	C₄⋊C₄×C₅₉	—	2	2	no	—
105	C₄⋊C₄×C₅₉	—	4	4	no	—
106	C₂²⋊C₄×C₅₉	—	2	1	no	—
107	C₂²⋊C₄×C₅₉	—	4	1	no	—
108	C₄²×C₅₉	—	4	4	no	—
109	C₄²×C₅₉	—	4	1	no	—
110	C₄²×C₅₉	C₂×D₄	1	2	yes	—
111	#11745664	—	2	2	no	—
112	#11745664	—	4	1	no	—
113	C₂×C₅₉⋊C₈	—	4	1	no	—
114	Dic₂₃₆	—	2	1	no	—
115	Dic₂₃₆	C₂²×C₄	1	2	yes	—
116	D₄₇₂	—	2	1	no	—
117	D₄₇₂	C₂²×C₄	1	2	yes	—
118	#11745660	—	2	2	no	—
119	#11745660	—	2	2	no	—
120	#11745659	—	4	4	no	—
121	C₈×D₅₉	—	4	4	no	—
122	C₅₉⋊C₁₆	—	2	2	no	—
123	C₅₉⋊C₁₆	C₂²×C₄	1	4	yes	—
124	C₂³×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂²×D₅₉	—	4	1	no	—
126	Q₈×C₅₉	—	2	4	no	—
127	Q₈×C₅₉	—	4	2	no	—
128	Q₈×C₅₉	—	4	2	no	—
129	Q₈×C₅₉	—	4	2	no	—
130	C₂×C₄×C₅₉	—	2	2	no	—
131	C₂×C₄×C₅₉	—	4	2	no	—
132	C₂×C₄×C₅₉	—	4	2	no	—
133	C₂×C₄×C₅₉	—	4	2	no	—
134	C₂×C₄×C₅₉	—	4	2	no	—
135	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
138	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
139	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
140	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
141	C₂²⋊D₅₉	—	4	2	no	—
142	C₂²⋊D₅₉	—	4	2	no	—
143	C₂×Dic₅₉	—	4	1	no	—
144	D₂₃₆	—	2	2	no	—
145	D₂₃₆	—	4	1	no	—
146	C₄×D₅₉	—	4	2	no	—
147	C₄×D₅₉	—	4	2	no	—
148	Dic₁₁₈	—	2	2	no	—
149	Dic₁₁₈	—	4	1	no	—
150	C₈×C₅₉	—	2	2	no	—
151	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
152	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
153	C₈×C₅₉	C₂³×C₄	1	1	yes	characteristic
154	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
155	C₅₉⋊C₈	—	4	2	no	—
156	#33470	—	59	1	no	maximal, Sylow
157	C₂²×C₅₉	—	4	1	no	—
158	C₂²×C₅₉	#442	1	1	yes	characteristic
159	C₂²×C₅₉	#442	1	1	yes	characteristic
160	C₂²×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
161	C₂×D₅₉	—	4	2	no	—
162	C₂×D₅₉	—	16	1	no	—
163	C₄×C₅₉	—	4	2	no	—
164	C₄×C₅₉	—	4	4	no	—
165	C₄×C₅₉	—	8	2	no	—
166	C₄×C₅₉	#442	1	1	yes	characteristic
167	C₄×C₅₉	#442	1	1	yes	characteristic
168	C₄×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
169	C₄×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
170	Dic₅₉	—	4	2	no	—
171	C₂×#510	—	59	1	no	—
172	#2957	—	59	1	no	—
173	#2950	D₅₉	1	1	yes	characteristic, complemented
174	#2180	—	59	8	no	—
175	#2155	—	59	1	no	—
176	C₂×#362	—	59	1	no	—
177	#2145	—	59	1	no	—
178	#2110	—	59	1	no	—
179	C₂×C₅₉	—	2	1	no	—
180	C₂×C₅₉	—	2	1	no	—
181	C₂×C₅₉	—	2	1	no	—
182	C₂×C₅₉	C₂×#442	1	1	yes	characteristic
183	D₅₉	—	8	2	no	—
184	C₂×#135	—	59	1	no	—
185	C₂²×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
186	#568	—	118	1	no	—
187	#519	—	2	1	no	—
188	#510	—	59	4	no	—
189	#510	—	59	4	no	—
190	C₂×Q₃₂	—	59	2	no	—
191	C₂×SD₃₂	—	59	2	no	—
192	C₂×M₃₂	—	59	1	no	—
193	C₂×M₃₂	—	59	1	no	—
194	#444	—	59	4	no	—
195	#443	—	118	2	no	—
196	#441	C₂×D₅₉	1	4	yes	complemented
197	C₄×Q₁₆	—	2	2	no	—
198	C₂×#108	—	59	1	no	—
199	#427	C₂×D₅₉	1	1	yes	characteristic
200	#421	—	118	1	no	—
201	#419	—	2	1	no	—
202	#405	C₂×D₅₉	1	1	yes	characteristic
203	#365	—	59	4	no	—
204	#362	—	59	4	no	—
205	#360	—	59	4	no	—
206	#347	—	59	4	no	—
207	C₅₉	#33470	1	1	yes	characteristic, complemented, Sylow
208	C₂×#41	—	118	1	no	—
209	C₂²×Q₈	—	2	1	no	—
210	#135	—	59	2	no	—
211	#135	—	59	2	no	—
212	#135	—	118	1	no	—
213	#135	—	118	1	no	—
214	C₂×Q₁₆	—	2	1	no	—
215	C₂×Q₁₆	—	2	1	no	—
216	C₂×Q₁₆	—	2	1	no	—
217	C₂×Q₁₆	—	59	1	no	—
218	C₂×Q₁₆	C₄×D₅₉	1	2	yes	—
219	C₂×Q₁₆	C₄×D₅₉	1	2	yes	—
220	C₂×Q₁₆	C₂²×D₅₉	1	2	yes	—
221	C₂×SD₁₆	—	118	1	no	—
222	C₂×D₈	—	59	1	no	—
223	#131	—	118	4	no	—
224	#131	—	236	2	no	—
225	C₂×M₁₆	—	118	1	no	—
226	C₂×M₁₆	—	236	2	no	—
227	C₂×M₁₆	C₂²⋊D₅₉	1	1	yes	characteristic
228	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
229	C₄×Q₈	—	2	4	no	—
230	C₄×Q₈	—	4	2	no	—
231	#117	—	2	1	no	—
232	#117	—	4	2	no	—
233	#117	C₂²⋊D₅₉	1	1	yes	characteristic
234	Q₃₂	—	59	8	no	—
235	SD₃₂	—	59	8	no	—
236	M₃₂	—	59	2	no	—
237	M₃₂	—	59	2	no	—
238	M₃₂	—	59	2	no	—
239	M₃₂	—	118	1	no	—
240	C₂×C₁₆	—	59	2	no	—
241	C₂×C₁₆	—	118	1	no	—
242	#108	—	59	2	no	—
243	#108	—	118	1	no	—
244	#107	—	2	1	no	—
245	#106	C₂²×D₅₉	1	2	yes	—
246	#104	—	118	4	no	—
247	#103	—	2	4	no	—
248	#97	C₂²×D₅₉	1	2	yes	—
249	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
250	#41	—	118	2	no	—
251	#41	—	118	2	no	—
252	#41	—	236	1	no	—
253	#41	—	236	1	no	—
254	C₂×Q₈	—	2	2	no	—
255	C₂×Q₈	—	2	2	no	—
256	C₂×Q₈	—	2	2	no	—
257	C₂×Q₈	—	4	1	no	—
258	C₂×Q₈	—	4	1	no	—
259	C₂×Q₈	—	4	1	no	—
260	C₂×Q₈	—	118	1	no	—
261	C₂×D₄	—	118	1	no	—
262	C₂×D₄	—	236	1	no	—
263	C₂²×C₄	—	4	1	no	—
264	C₂²×C₄	—	236	1	no	—
265	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
266	Q₁₆	—	2	2	no	—
267	Q₁₆	—	2	2	no	—
268	Q₁₆	—	2	2	no	—
269	Q₁₆	—	59	2	no	—
270	Q₁₆	—	118	1	no	—
271	Q₁₆	C₂×C₄×D₅₉	1	4	yes	—
272	SD₁₆	—	118	2	no	—
273	SD₁₆	—	118	2	no	—
274	D₈	—	59	2	no	—
275	D₈	—	118	1	no	—
276	M₁₆	—	2	1	no	—
277	M₁₆	—	2	1	no	—
278	M₁₆	—	118	2	no	—
279	M₁₆	—	236	4	no	—
280	M₁₆	—	236	1	no	—
281	C₂×C₈	—	236	4	no	—
282	C₂×C₈	—	236	1	no	—
283	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
284	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
285	C₂×C₈	#11745685	1	1	yes	characteristic
286	C₂×C₈	#11745685	1	1	yes	characteristic
287	C₂×C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
288	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
289	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
290	C₄⋊C₄	—	2	1	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	4	4	no	—
293	C₂²⋊C₄	—	2	1	no	—
294	C₂²⋊C₄	—	4	1	no	—
295	C₄²	—	4	4	no	—
296	C₄²	—	4	1	no	—
297	C₄²	C₂×C₂²⋊D₅₉	1	2	yes	—
298	C₁₆	—	59	4	no	—
299	C₁₆	—	118	2	no	—
300	C₂³	—	236	1	no	—
301	C₂³	#14273753	1	1	yes	characteristic
302	Q₈	—	2	4	no	—
303	Q₈	—	4	2	no	—
304	Q₈	—	4	2	no	—
305	Q₈	—	4	2	no	—
306	Q₈	—	118	2	no	—
307	Q₈	—	236	1	no	—
308	D₄	—	118	2	no	—
309	D₄	—	236	2	no	—
310	D₄	—	236	2	no	—
311	D₄	—	236	1	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	236	1	no	—
318	C₂×C₄	—	236	2	no	—
319	C₂×C₄	—	236	2	no	—
320	C₂×C₄	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
321	C₂×C₄	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
322	C₂×C₄	#14273752	1	1	yes	characteristic
323	C₂×C₄	#14273752	1	1	yes	characteristic
324	C₂×C₄	#14273753	1	1	yes	characteristic
325	C₂×C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
326	C₈	—	2	2	no	—
327	C₈	—	236	2	no	—
328	C₈	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
329	C₈	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
330	C₈	C₂²×C₄×D₅₉	1	1	yes	characteristic
331	C₈	C₂×#11745685	1	1	yes	characteristic
332	C₂²	—	4	1	no	—
333	C₂²	—	944	1	no	—
334	C₂²	—	236	2	no	—
335	C₂²	#16837512	1	1	yes	characteristic
336	C₂²	#16837512	1	1	yes	characteristic
337	C₂²	C₂×#14273753	1	1	yes	characteristic
338	C₄	—	4	2	no	—
339	C₄	—	4	4	no	—
340	C₄	—	8	2	no	—
341	C₄	—	236	2	no	—
342	C₄	#16837512	1	1	yes	characteristic
343	C₄	#16837512	1	1	yes	characteristic
344	C₄	C₂×C₄×C₂²⋊D₅₉	1	1	yes	characteristic
345	C₄	C₂×#14273752	1	1	yes	characteristic
346	C₂	—	2	1	no	—
347	C₂	—	2	1	no	—
348	C₂	—	2	1	no	—
349	C₂	—	472	2	no	—
350	C₂	C₂×#16837512	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#16837512
commutator	C₈×C₅₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₅₉×#2950	C₂
socle	C₂×C₅₉	C₂×#442
prime radical	C₂×#14273749	C₂²
cosocle	C₂×C₈×C₅₉	C₂⁴
hypercenter	#2950	D₅₉
hypocenter	C₅₉	#33470
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #2950

Complements

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

N	H ≅ G/N	type
3. #21425985	C₂	semidirect
4. #21425868	C₂	semidirect
5. C₂×#16837218	C₂	semidirect
7. #21425701	C₂	semidirect
8. C₅₉×#2950	C₂	semidirect
11. #16837843	C₄	semidirect
13. C₂×#14273673	C₄	semidirect
14. C₂×#14273670	C₄	semidirect
21. #16837224	C₂²	semidirect
22. #16837218	C₂²	semidirect
24. #16837180	C₂²	semidirect
27. C₅₉×#441	C₂²	semidirect
173. #2950	D₅₉	semidirect
196. #441	C₂×D₅₉	semidirect
207. C₅₉	#33470	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837843	2
#21433014	2
#21425985	2
#21425868	2
C₂×#16837218	2
#21425831	2
#21425701	2
C₅₉×#2950	2
#33470	59