Subgroups — #28505693

Order 15104.

508 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837643	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837217	C₂	1	2	yes	maximal, complemented
3	#21425707	C₂	1	1	yes	maximal, characteristic
4	#21422206	C₂	1	2	yes	maximal, complemented
5	C₂²×#108×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₃₆	C₂²	1	1	yes	characteristic
7	C₂×#14273706	—	2	1	no	—
8	C₂×#14273667	C₂²	1	2	yes	—
9	#16837643	—	2	1	no	—
10	#16837643	—	2	1	no	—
11	#16837643	C₄	1	2	yes	—
12	#16837643	C₄	1	2	yes	—
13	C₂×#14273661	C₂²	1	1	yes	characteristic
14	C₂×#14273652	—	2	1	no	—
15	#16837217	—	2	4	no	—
16	C₂²⋊(C₅₉⋊C₁₆)	—	2	1	no	—
17	C₂²⋊(C₅₉⋊C₁₆)	—	2	1	no	—
18	C₂²×M₁₆×C₅₉	—	2	1	no	—
19	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
20	C₂×#108×C₅₉	—	2	1	no	—
21	C₂×#108×C₅₉	—	2	4	no	—
22	C₂×#108×C₅₉	C₂²	1	2	yes	—
23	C₂²×Dic₁₁₈	—	2	1	no	—
24	C₂×Dic₂₃₆	—	2	2	no	—
25	C₂×Dic₂₃₆	—	2	2	no	—
26	C₂×Dic₂₃₆	—	2	1	no	—
27	C₂×Dic₂₃₆	C₂×C₄	1	2	yes	—
28	#14273706	—	2	2	no	—
29	#14273706	—	2	2	no	—
30	#14273706	—	4	1	no	—
31	#14273706	—	4	1	no	—
32	C₂×C₄⋊Dic₅₉	—	2	1	no	—
33	C₂×Dic₅₉⋊C₄	—	4	1	no	—
34	C₂×C₂²⋊Dic₅₉	—	4	1	no	—
35	#14273667	—	2	2	no	—
36	#14273667	—	2	2	no	—
37	C₂×C₅₉⋊C₁₆	—	2	2	no	—
38	#14273661	—	2	1	no	—
39	#14273661	C₂×C₄	1	2	yes	—
40	#14273652	—	2	2	no	—
41	#14273652	—	2	2	no	—
42	C₂³×C₄×C₅₉	D₄	1	1	yes	characteristic
43	C₂×M₁₆×C₅₉	—	2	4	no	—
44	C₂×M₁₆×C₅₉	—	2	2	no	—
45	C₂×M₁₆×C₅₉	—	4	2	no	—
46	C₂×M₁₆×C₅₉	—	4	1	no	—
47	C₂²×C₈×C₅₉	—	2	2	no	—
48	C₂²×C₈×C₅₉	—	2	2	no	—
49	C₂²×C₈×C₅₉	—	4	1	no	—
50	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
56	#108×C₅₉	—	2	4	no	—
57	#108×C₅₉	—	2	2	no	—
58	#108×C₅₉	D₄	1	4	yes	—
59	C₂⁴×C₅₉	SD₁₆	1	1	yes	characteristic
60	C₂²×C₄×C₅₉	—	2	1	no	—
61	C₂²×C₄×C₅₉	—	2	1	no	—
62	C₂²×C₄×C₅₉	—	2	1	no	—
63	C₂²×C₄×C₅₉	—	2	1	no	—
64	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
67	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
68	C₂²×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
69	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
70	C₂²×Dic₅₉	—	4	1	no	—
71	C₂²×Dic₅₉	—	4	1	no	—
72	C₂×Dic₁₁₈	—	2	2	no	—
73	C₂×Dic₁₁₈	—	4	1	no	—
74	C₂×Dic₁₁₈	—	4	1	no	—
75	C₂×Dic₁₁₈	—	4	1	no	—
76	C₂×Dic₁₁₈	—	4	1	no	—
77	C₂×Dic₁₁₈	—	4	1	no	—
78	M₁₆×C₅₉	—	2	4	no	—
79	M₁₆×C₅₉	—	4	4	no	—
80	M₁₆×C₅₉	—	4	2	no	—
81	C₂×C₈×C₅₉	—	2	1	no	—
82	C₂×C₈×C₅₉	—	2	1	no	—
83	C₂×C₈×C₅₉	—	2	2	no	—
84	C₂×C₈×C₅₉	—	2	2	no	—
85	C₂×C₈×C₅₉	—	2	2	no	—
86	C₂×C₈×C₅₉	—	2	2	no	—
87	C₂×C₈×C₅₉	—	4	4	no	—
88	C₂×C₈×C₅₉	—	4	2	no	—
89	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₅₉	#41	1	2	yes	—
95	C₂²⋊Dic₅₉	—	4	2	no	—
96	C₂²⋊Dic₅₉	—	4	2	no	—
97	C₄⋊Dic₅₉	—	2	2	no	—
98	C₄⋊Dic₅₉	—	4	1	no	—
99	Dic₅₉⋊C₄	—	4	2	no	—
100	Dic₅₉⋊C₄	—	4	2	no	—
101	Dic₂₃₆	—	2	4	no	—
102	Dic₂₃₆	—	4	2	no	—
103	C₅₉⋊C₁₆	—	4	2	no	—
104	C₂³×C₅₉	—	2	2	no	—
105	C₂³×C₅₉	—	2	1	no	—
106	C₂³×C₅₉	—	2	1	no	—
107	C₂³×C₅₉	—	4	1	no	—
108	C₂³×C₅₉	#102	1	1	yes	characteristic
109	C₂³×C₅₉	#102	1	1	yes	characteristic
110	C₂³×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
111	C₂×C₄×C₅₉	—	2	1	no	—
112	C₂×C₄×C₅₉	—	2	1	no	—
113	C₂×C₄×C₅₉	—	2	1	no	—
114	C₂×C₄×C₅₉	—	2	1	no	—
115	C₂×C₄×C₅₉	—	2	1	no	—
116	C₂×C₄×C₅₉	—	2	1	no	—
117	C₂×C₄×C₅₉	—	2	1	no	—
118	C₂×C₄×C₅₉	—	2	1	no	—
119	C₂×C₄×C₅₉	—	2	1	no	—
120	C₂×C₄×C₅₉	—	2	1	no	—
121	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
122	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
123	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₅₉	#120	1	1	yes	characteristic
125	C₂×C₄×C₅₉	#123	1	1	yes	characteristic
126	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
128	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
129	C₂×Dic₅₉	—	4	2	no	—
130	C₂×Dic₅₉	—	4	2	no	—
131	C₂×Dic₅₉	—	8	2	no	—
132	C₂×Dic₅₉	—	8	1	no	—
133	C₂×Dic₅₉	—	8	1	no	—
134	Dic₁₁₈	—	4	2	no	—
135	Dic₁₁₈	—	4	2	no	—
136	Dic₁₁₈	—	8	1	no	—
137	Dic₁₁₈	—	8	1	no	—
138	C₈×C₅₉	—	2	1	no	—
139	C₈×C₅₉	—	2	1	no	—
140	C₈×C₅₉	—	4	4	no	—
141	C₈×C₅₉	C₄×D₄	1	2	yes	—
142	C₈×C₅₉	#121	1	2	yes	—
143	#13003	—	59	1	no	maximal, Sylow
144	C₂²×C₅₉	—	2	1	no	—
145	C₂²×C₅₉	—	2	1	no	—
146	C₂²×C₅₉	—	2	1	no	—
147	C₂²×C₅₉	—	2	1	no	—
148	C₂²×C₅₉	—	2	1	no	—
149	C₂²×C₅₉	—	2	1	no	—
150	C₂²×C₅₉	—	2	1	no	—
151	C₂²×C₅₉	—	4	2	no	—
152	C₂²×C₅₉	—	4	2	no	—
153	C₂²×C₅₉	#362	1	2	yes	complemented
154	C₂²×C₅₉	C₂×#102	1	1	yes	characteristic
155	C₂²×C₅₉	#448	1	1	yes	characteristic
156	C₂²×C₅₉	#481	1	1	yes	characteristic
157	C₄×C₅₉	—	2	1	no	—
158	C₄×C₅₉	—	2	1	no	—
159	C₄×C₅₉	#386	1	1	yes	characteristic
160	C₄×C₅₉	#417	1	1	yes	characteristic
161	C₄×C₅₉	#447	1	1	yes	characteristic
162	C₄×C₅₉	#480	1	1	yes	characteristic
163	Dic₅₉	—	8	2	no	—
164	Dic₅₉	—	8	2	no	—
165	C₂×#467	—	59	1	no	—
166	C₂²×#108	D₅₉	1	1	yes	characteristic
167	C₂×#362	—	59	2	no	—
168	#2112	—	59	1	no	—
169	#1387	—	59	2	no	—
170	C₂×C₅₉	—	2	2	no	—
171	C₂×C₅₉	—	2	1	no	—
172	C₂×C₅₉	—	2	1	no	—
173	C₂×C₅₉	—	4	1	no	—
174	C₂×C₅₉	#1874	1	1	yes	characteristic
175	C₂×C₅₉	C₂×#362	1	1	yes	characteristic
176	C₂×C₅₉	#2148	1	1	yes	characteristic
177	C₂²×Q₁₆	—	59	1	no	—
178	C₂²×M₁₆	—	2	1	no	—
179	C₂³×C₈	C₂×D₅₉	1	1	yes	characteristic
180	C₂×C₂²⋊Q₈	—	118	1	no	—
181	C₂×M₃₂	—	59	2	no	—
182	#467	—	59	2	no	—
183	#467	—	59	2	no	—
184	#467	—	118	1	no	—
185	#467	—	118	1	no	—
186	C₂×#108	—	2	1	no	—
187	C₂×#108	—	2	4	no	—
188	C₂×#108	C₂×D₅₉	1	2	yes	—
189	C₂×#107	—	59	1	no	—
190	C₂×#103	—	118	1	no	—
191	#362	—	118	4	no	—
192	C₂²⋊C₁₆	—	118	1	no	—
193	C₂²⋊C₁₆	—	118	1	no	—
194	C₅₉	#13003	1	1	yes	characteristic, complemented, Sylow
195	C₂²×Q₈	—	118	1	no	—
196	C₂³×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
197	C₂×Q₁₆	—	59	2	no	—
198	C₂×Q₁₆	—	118	1	no	—
199	C₂×Q₁₆	—	118	2	no	—
200	C₂×Q₁₆	—	118	2	no	—
201	C₂×M₁₆	—	2	4	no	—
202	C₂×M₁₆	—	2	2	no	—
203	C₂×M₁₆	—	4	2	no	—
204	C₂×M₁₆	—	4	1	no	—
205	C₂²×C₈	—	2	2	no	—
206	C₂²×C₈	—	2	2	no	—
207	C₂²×C₈	—	4	1	no	—
208	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
209	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
210	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
211	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
212	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
213	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
214	C₂²⋊Q₈	—	118	2	no	—
215	C₂²⋊Q₈	—	118	2	no	—
216	C₂²⋊Q₈	—	236	1	no	—
217	C₂²⋊Q₈	—	236	1	no	—
218	C₂×C₄⋊C₄	—	118	1	no	—
219	C₂×C₄⋊C₄	—	236	1	no	—
220	C₂×C₂²⋊C₄	—	236	1	no	—
221	M₃₂	—	118	2	no	—
222	M₃₂	—	118	2	no	—
223	C₂×C₁₆	—	118	2	no	—
224	#108	—	2	4	no	—
225	#108	—	2	2	no	—
226	#108	C₂²⋊D₅₉	1	4	yes	—
227	#107	—	59	2	no	—
228	#107	—	118	1	no	—
229	#103	—	118	2	no	—
230	#103	—	118	2	no	—
231	C₂⁴	#11745671	1	1	yes	characteristic
232	C₂×Q₈	—	118	2	no	—
233	C₂×Q₈	—	236	1	no	—
234	C₂×Q₈	—	236	1	no	—
235	C₂×Q₈	—	236	1	no	—
236	C₂×Q₈	—	236	1	no	—
237	C₂×Q₈	—	236	1	no	—
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	2	1	no	—
242	C₂²×C₄	—	236	1	no	—
243	C₂²×C₄	—	236	1	no	—
244	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
245	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
246	C₂²×C₄	#11745669	1	1	yes	characteristic
247	C₂²×C₄	#11745669	1	1	yes	characteristic
248	C₂²×C₄	#11745671	1	1	yes	characteristic
249	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
250	Q₁₆	—	118	4	no	—
251	Q₁₆	—	236	2	no	—
252	M₁₆	—	2	4	no	—
253	M₁₆	—	4	4	no	—
254	M₁₆	—	4	2	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	4	4	no	—
262	C₂×C₈	—	4	2	no	—
263	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
264	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
265	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
266	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
267	C₂×C₈	#11745685	1	2	yes	—
268	C₂×C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
269	C₄⋊C₄	—	118	2	no	—
270	C₄⋊C₄	—	236	2	no	—
271	C₄⋊C₄	—	236	2	no	—
272	C₄⋊C₄	—	236	1	no	—
273	C₂²⋊C₄	—	236	2	no	—
274	C₂²⋊C₄	—	236	2	no	—
275	C₁₆	—	236	2	no	—
276	C₂³	—	2	1	no	—
277	C₂³	—	2	2	no	—
278	C₂³	—	2	1	no	—
279	C₂³	—	4	1	no	—
280	C₂³	#14273647	1	1	yes	characteristic
281	C₂³	#14273647	1	1	yes	characteristic
282	C₂³	C₂×#11745671	1	1	yes	characteristic
283	Q₈	—	472	1	no	—
284	Q₈	—	472	1	no	—
285	Q₈	—	236	2	no	—
286	Q₈	—	236	2	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	472	2	no	—
298	C₂×C₄	—	472	1	no	—
299	C₂×C₄	—	472	1	no	—
300	C₂×C₄	—	236	2	no	—
301	C₂×C₄	—	236	2	no	—
302	C₂×C₄	#14273647	1	1	yes	characteristic
303	C₂×C₄	#14273647	1	1	yes	characteristic
304	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
305	C₂×C₄	#14273696	1	1	yes	characteristic
306	C₂×C₄	#14273710	1	1	yes	characteristic
307	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
308	C₂×C₄	#14273753	1	1	yes	characteristic
309	C₂×C₄	#14273758	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	4	4	no	—
313	C₈	C₄×C₂²⋊D₅₉	1	2	yes	—
314	C₈	#14273701	1	2	yes	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	4	2	no	—
323	C₂²	—	4	2	no	—
324	C₂²	#16837217	1	2	yes	complemented
325	C₂²	C₂×#14273647	1	1	yes	characteristic
326	C₂²	#16837541	1	1	yes	characteristic
327	C₂²	#16837720	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	472	2	no	—
331	C₄	—	472	2	no	—
332	C₄	#16837281	1	1	yes	characteristic
333	C₄	#16837401	1	1	yes	characteristic
334	C₄	#16837536	1	1	yes	characteristic
335	C₄	#16837715	1	1	yes	characteristic
336	C₂	—	2	1	no	—
337	C₂	—	2	2	no	—
338	C₂	—	2	1	no	—
339	C₂	—	4	1	no	—
340	C₂	#21424413	1	1	yes	characteristic
341	C₂	C₂×#16837217	1	1	yes	characteristic
342	C₂	#21425843	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14273647
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂²×#108×C₅₉	C₂
socle	C₂²×C₅₉	C₂×#102
prime radical	C₂⁴×C₅₉	SD₁₆
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂²×#108	D₅₉
hypocenter	C₅₉	#13003
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
2. C₂×#16837217	C₂	semidirect
4. #21422206	C₂	semidirect
153. C₂²×C₅₉	#362	semidirect
194. C₅₉	#13003	semidirect
324. C₂²	#16837217	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837643	2
C₂×#16837217	2
#21425707	2
#21422206	2
C₂²×#108×C₅₉	2
#13003	59