Subgroups — #28607218

Order 15104.

808 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21433014	C₂	1	1	yes	maximal, characteristic
2	#21433012	C₂	1	1	yes	maximal, characteristic
3	C₂×#16837523	C₂	1	1	yes	maximal, characteristic
4	#21432706	C₂	1	2	yes	maximal, complemented
5	#21424887	C₂	1	1	yes	maximal, characteristic, complemented
6	#21424339	C₂	1	2	yes	maximal, complemented
7	#21424333	C₂	1	2	yes	maximal, complemented
8	#21424269	C₂	1	1	yes	maximal, characteristic, complemented
9	#21424149	C₂	1	2	yes	maximal, complemented
10	#21423981	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₉×#1953	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14273749	C₂²	1	1	yes	characteristic
13	#16838168	C₂²	1	2	yes	—
14	#16838168	C₂²	1	2	yes	—
15	C₂×#14273735	C₂²	1	2	yes	—
16	#16837525	—	2	2	no	—
17	#16837523	—	2	2	no	—
18	#16837523	C₄	1	4	yes	complemented
19	#16837523	C₄	1	4	yes	complemented
20	#16837520	—	2	2	no	—
21	#16837518	—	2	2	no	—
22	#16837451	C₂²	1	2	yes	—
23	#16837449	—	2	4	no	—
24	C₂×#14273663	C₂²	1	1	yes	characteristic
25	C₂×#14273662	C₂²	1	2	yes	—
26	C₂×#14273662	C₂²	1	1	yes	characteristic
27	#108×D₅₉	—	2	4	no	—
28	#16837438	C₂²	1	1	yes	characteristic
29	#16837425	C₂²	1	2	yes	—
30	#16837420	C₂²	1	2	yes	—
31	#16837418	C₂²	1	2	yes	—
32	C₂×#14273653	C₂²	1	2	yes	—
33	#16837357	C₂²	1	1	yes	characteristic
34	C₂×#14273637	C₂²	1	1	yes	characteristic
35	#16837174	C₂²	1	2	yes	—
36	#16837163	C₂²	1	1	yes	characteristic
37	#16837161	C₂²	1	1	yes	characteristic
38	C₅₉×#430	C₂²	1	2	yes	—
39	C₂×#108×C₅₉	C₂²	1	1	yes	characteristic
40	C₂×#108×C₅₉	C₂²	1	1	yes	characteristic
41	C₅₉×#405	C₂²	1	1	yes	characteristic
42	C₅₉×#344	C₂²	1	2	yes	—
43	C₂×#11745685	C₂³	1	2	yes	—
44	#14273749	C₂×C₄	1	2	yes	—
45	#14273749	C₂×C₄	1	2	yes	—
46	#14273749	D₄	1	2	yes	—
47	#14273749	D₄	1	2	yes	—
48	C₂×Dic₂₃₆	Q₈	1	1	yes	characteristic
49	C₂×#11745660	Q₈	1	2	yes	—
50	C₂×D₄₇₂	Q₈	1	1	yes	characteristic
51	#14273735	—	2	2	no	—
52	#14273735	—	2	2	no	—
53	#14273735	—	2	4	no	—
54	#14273735	—	4	2	no	—
55	#14273735	C₂×C₄	1	4	yes	—
56	#14273735	C₂×C₄	1	4	yes	—
57	#14273733	—	2	4	no	—
58	#14273733	—	4	2	no	—
59	C₂×#11745664	C₂³	1	2	yes	—
60	C₂×#11745664	C₂³	1	1	yes	characteristic
61	C₂×#11745659	—	2	2	no	—
62	M₁₆×D₅₉	—	2	4	no	—
63	M₁₆×D₅₉	—	2	4	no	—
64	C₂²×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
65	C₂×C₈×D₅₉	—	2	2	no	—
66	#14273686	C₂³	1	1	yes	characteristic
67	C₂×C₄×Dic₅₉	C₂³	1	1	yes	characteristic
68	#14273663	—	2	1	no	—
69	#14273663	—	2	1	no	—
70	#14273663	—	2	4	no	—
71	#14273662	—	2	2	no	—
72	#14273662	—	2	2	no	—
73	#14273662	D₄	1	4	yes	—
74	#14273660	—	2	1	no	—
75	#14273658	—	2	1	no	—
76	Dic₅₉⋊C₈	—	4	2	no	—
77	#14273655	—	2	4	no	—
78	#14273653	—	2	4	no	—
79	#14273653	C₂×C₄	1	8	yes	—
80	#14273637	—	2	1	no	—
81	#14273637	C₂×C₄	1	2	yes	—
82	#14273637	C₂×C₄	1	2	yes	—
83	C₈×Dic₅₉	—	2	1	no	—
84	C₂×M₁₆×C₅₉	C₂³	1	2	yes	—
85	C₂×M₁₆×C₅₉	C₂³	1	2	yes	—
86	C₂×M₁₆×C₅₉	C₂³	1	1	yes	characteristic
87	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
88	#117×C₅₉	C₂³	1	1	yes	characteristic
89	#108×C₅₉	—	2	4	no	—
90	#108×C₅₉	—	2	1	no	—
91	#108×C₅₉	—	2	1	no	—
92	#108×C₅₉	C₂×C₄	1	2	yes	—
93	#108×C₅₉	C₂×C₄	1	2	yes	—
94	#97×C₅₉	—	2	1	no	—
95	C₄×C₈×C₅₉	—	2	1	no	—
96	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
97	C₂×C₂²⋊D₅₉	—	2	2	no	—
98	C₂²×Dic₅₉	—	2	1	no	—
99	#11745685	—	2	2	no	—
100	#11745685	—	2	2	no	—
101	#11745685	C₂²×C₄	1	4	yes	—
102	#11745685	C₂×D₄	1	4	yes	—
103	C₂×D₂₃₆	C₂×Q₈	1	2	yes	—
104	C₂×C₄×D₅₉	—	2	2	no	—
105	C₂×Dic₁₁₈	C₂×Q₈	1	2	yes	—
106	M₁₆×C₅₉	—	2	2	no	—
107	M₁₆×C₅₉	—	2	1	no	—
108	M₁₆×C₅₉	—	2	2	no	—
109	M₁₆×C₅₉	—	2	2	no	—
110	M₁₆×C₅₉	C₂²×C₄	1	4	yes	—
111	M₁₆×C₅₉	C₂×D₄	1	2	yes	—
112	C₂×C₈×C₅₉	—	2	1	no	—
113	C₂×C₈×C₅₉	—	2	2	no	—
114	C₂×C₈×C₅₉	—	2	2	no	—
115	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
120	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
121	C₄⋊C₄×C₅₉	—	2	1	no	—
122	C₂²⋊C₄×C₅₉	—	2	1	no	—
123	C₄²×C₅₉	—	2	1	no	—
124	C₂²⋊Dic₅₉	—	2	1	no	—
125	C₄⋊Dic₅₉	—	2	1	no	—
126	C₄×Dic₅₉	—	2	1	no	—
127	C₄×Dic₅₉	—	2	1	no	—
128	C₄×Dic₅₉	C₂²×C₄	1	2	yes	—
129	#11745664	—	2	1	no	—
130	#11745664	—	2	2	no	—
131	#11745664	—	2	2	no	—
132	#11745664	C₂×D₄	1	2	yes	—
133	C₂×C₅₉⋊C₈	—	2	1	no	—
134	C₂×C₅₉⋊C₈	—	2	2	no	—
135	C₂×C₅₉⋊C₈	C₂²×C₄	1	2	yes	—
136	C₂×C₅₉⋊C₈	C₂×D₄	1	2	yes	—
137	C₂×C₅₉⋊C₈	#41	1	1	yes	characteristic
138	C₂×C₅₉⋊C₈	#41	1	1	yes	characteristic
139	Dic₂₃₆	C₄⋊C₄	1	2	yes	—
140	Dic₂₃₆	C₄⋊C₄	1	2	yes	—
141	D₄₇₂	C₄⋊C₄	1	2	yes	—
142	D₄₇₂	C₄⋊C₄	1	2	yes	—
143	#11745660	C₄⋊C₄	1	4	yes	—
144	#11745660	C₄⋊C₄	1	4	yes	—
145	#11745659	—	2	4	no	—
146	#11745659	—	2	4	no	—
147	#11745659	—	4	4	no	—
148	#11745659	—	4	4	no	—
149	C₈×D₅₉	—	2	4	no	—
150	C₈×D₅₉	—	2	4	no	—
151	C₈×D₅₉	—	4	4	no	—
152	C₈×D₅₉	—	4	4	no	—
153	C₂³×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
154	C₂²×D₅₉	—	2	2	no	—
155	C₂×C₄×C₅₉	—	4	1	no	—
156	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
157	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
160	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
161	C₂×C₄×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
162	C₂²⋊D₅₉	—	2	4	no	—
163	C₂²⋊D₅₉	—	2	4	no	—
164	C₂×Dic₅₉	—	2	2	no	—
165	C₂×Dic₅₉	—	2	2	no	—
166	C₂×Dic₅₉	—	4	1	no	—
167	C₂×Dic₅₉	—	8	1	no	—
168	D₂₃₆	—	2	2	no	—
169	D₂₃₆	C₂×C₄⋊C₄	1	4	yes	—
170	C₄×D₅₉	—	2	4	no	—
171	C₄×D₅₉	—	2	4	no	—
172	Dic₁₁₈	—	2	2	no	—
173	Dic₁₁₈	C₂×C₄⋊C₄	1	4	yes	—
174	C₈×C₅₉	—	2	4	no	—
175	C₈×C₅₉	—	2	2	no	—
176	C₈×C₅₉	—	4	2	no	—
177	C₈×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
178	C₈×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
179	C₈×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
180	C₈×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
181	C₅₉⋊C₈	—	2	2	no	—
182	C₅₉⋊C₈	—	4	2	no	—
183	C₅₉⋊C₈	C₄×D₄	1	2	yes	—
184	C₅₉⋊C₈	C₄×D₄	1	2	yes	—
185	#22695	—	59	1	no	maximal, Sylow
186	C₂²×C₅₉	—	4	1	no	—
187	C₂²×C₅₉	#520	1	1	yes	characteristic
188	C₂²×C₅₉	#547	1	1	yes	characteristic
189	C₂²×C₅₉	D₄×Q₈	1	1	yes	characteristic
190	C₂×D₅₉	—	2	4	no	—
191	C₂×D₅₉	—	8	2	no	—
192	C₄×C₅₉	—	8	1	no	—
193	C₄×C₅₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
194	C₄×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
195	C₄×C₅₉	D₄²	1	1	yes	characteristic
196	C₄×C₅₉	#554	1	1	yes	characteristic
197	Dic₅₉	—	2	4	no	—
198	Dic₅₉	—	4	2	no	—
199	Dic₅₉	—	8	1	no	—
200	#2957	—	59	1	no	—
201	#2957	—	59	1	no	—
202	C₂×#444	—	59	1	no	—
203	#2916	—	59	2	no	—
204	#1953	—	59	1	no	—
205	#1953	D₅₉	1	1	yes	characteristic, complemented
206	#1861	—	59	2	no	—
207	#1861	—	59	2	no	—
208	#1849	—	59	1	no	—
209	#1814	—	59	2	no	—
210	#1773	—	59	1	no	—
211	C₂×C₅₉	—	2	1	no	—
212	C₂×C₅₉	—	2	1	no	—
213	C₂×C₅₉	—	2	1	no	—
214	C₂×C₅₉	D₄×C₄⋊C₄	1	1	yes	characteristic
215	D₅₉	—	4	4	no	—
216	C₂×#135	—	59	1	no	—
217	#568	—	59	2	no	—
218	#568	—	59	2	no	—
219	C₂×#131	—	59	2	no	—
220	#444	—	59	4	no	—
221	#444	—	59	4	no	—
222	#444	—	118	2	no	—
223	#444	—	118	2	no	—
224	#443	—	118	2	no	—
225	#443	—	118	2	no	—
226	#430	—	59	2	no	—
227	#430	—	118	4	no	—
228	#430	C₂×D₅₉	1	2	yes	—
229	C₂×#108	—	59	1	no	—
230	C₂×#108	—	59	2	no	—
231	C₂×#108	—	59	1	no	—
232	C₂×#108	—	118	4	no	—
233	C₂×#108	C₂×D₅₉	1	1	yes	characteristic
234	C₂×#108	C₂×D₅₉	1	1	yes	characteristic
235	#427	—	59	1	no	—
236	#423	—	59	2	no	—
237	#421	—	59	2	no	—
238	#421	—	59	2	no	—
239	C₂×#104	—	59	2	no	—
240	#405	—	59	1	no	—
241	#405	C₂×D₅₉	1	1	yes	characteristic
242	C₂×#97	—	59	1	no	—
243	#344	—	59	2	no	—
244	#344	C₂×D₅₉	1	2	yes	—
245	#339	—	59	1	no	—
246	#339	—	59	1	no	—
247	C₅₉	#22695	1	1	yes	characteristic, complemented, Sylow
248	C₂×#41	—	59	2	no	—
249	#135	—	59	2	no	—
250	#135	—	59	2	no	—
251	#135	—	59	2	no	—
252	#135	—	59	2	no	—
253	C₂×Q₁₆	—	59	1	no	—
254	C₂×SD₁₆	—	59	2	no	—
255	C₂×D₈	—	59	1	no	—
256	#131	—	59	4	no	—
257	#131	—	59	4	no	—
258	#131	—	118	4	no	—
259	#131	—	118	2	no	—
260	#131	—	118	2	no	—
261	#131	—	118	4	no	—
262	#131	—	236	2	no	—
263	#131	—	236	2	no	—
264	C₂×M₁₆	—	59	1	no	—
265	C₂×M₁₆	—	59	2	no	—
266	C₂×M₁₆	—	118	2	no	—
267	C₂×M₁₆	—	118	4	no	—
268	C₂×M₁₆	—	118	4	no	—
269	C₂×M₁₆	C₂²×D₅₉	1	2	yes	—
270	C₂×M₁₆	C₂²×D₅₉	1	2	yes	—
271	C₂×M₁₆	C₂²×D₅₉	1	1	yes	characteristic
272	C₂²×C₈	—	59	1	no	—
273	C₂²×C₈	—	118	2	no	—
274	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
275	#117	—	59	1	no	—
276	#117	C₂²×D₅₉	1	1	yes	characteristic
277	C₂×C₄²	—	59	1	no	—
278	#108	—	2	4	no	—
279	#108	—	2	1	no	—
280	#108	—	2	1	no	—
281	#108	—	59	4	no	—
282	#108	—	118	4	no	—
283	#108	—	118	2	no	—
284	#108	—	118	2	no	—
285	#108	—	118	1	no	—
286	#108	—	118	1	no	—
287	#108	C₄×D₅₉	1	2	yes	—
288	#108	C₄×D₅₉	1	2	yes	—
289	#107	—	118	1	no	—
290	#106	—	118	1	no	—
291	C₄⋊C₈	—	236	2	no	—
292	#104	—	59	8	no	—
293	#104	—	118	4	no	—
294	#104	—	118	4	no	—
295	#97	—	2	1	no	—
296	#97	—	59	2	no	—
297	#97	—	59	2	no	—
298	#97	—	118	1	no	—
299	C₄×C₈	—	2	1	no	—
300	C₄×C₈	—	118	1	no	—
301	#41	—	59	4	no	—
302	#41	—	59	4	no	—
303	#41	—	118	2	no	—
304	#41	—	118	2	no	—
305	C₂×Q₈	—	59	2	no	—
306	C₂×D₄	—	59	2	no	—
307	C₂×D₄	—	118	2	no	—
308	C₂²×C₄	—	118	2	no	—
309	C₂²×C₄	—	118	1	no	—
310	C₂²×C₄	C₂³×D₅₉	1	1	yes	characteristic
311	Q₁₆	—	59	2	no	—
312	Q₁₆	—	59	2	no	—
313	SD₁₆	—	59	4	no	—
314	SD₁₆	—	59	4	no	—
315	D₈	—	59	2	no	—
316	D₈	—	59	2	no	—
317	M₁₆	—	2	2	no	—
318	M₁₆	—	2	1	no	—
319	M₁₆	—	2	2	no	—
320	M₁₆	—	2	2	no	—
321	M₁₆	—	59	2	no	—
322	M₁₆	—	118	4	no	—
323	M₁₆	—	118	4	no	—
324	M₁₆	—	118	1	no	—
325	M₁₆	—	118	2	no	—
326	M₁₆	—	118	2	no	—
327	M₁₆	—	236	4	no	—
328	M₁₆	—	236	4	no	—
329	M₁₆	C₂×C₄×D₅₉	1	4	yes	—
330	M₁₆	D₄×D₅₉	1	2	yes	—
331	C₂×C₈	—	2	1	no	—
332	C₂×C₈	—	2	2	no	—
333	C₂×C₈	—	2	2	no	—
334	C₂×C₈	—	59	2	no	—
335	C₂×C₈	—	59	2	no	—
336	C₂×C₈	—	59	1	no	—
337	C₂×C₈	—	59	1	no	—
338	C₂×C₈	—	118	4	no	—
339	C₂×C₈	—	118	4	no	—
340	C₂×C₈	—	118	1	no	—
341	C₂×C₈	—	118	2	no	—
342	C₂×C₈	—	236	4	no	—
343	C₂×C₈	—	236	4	no	—
344	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
345	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
346	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
347	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
348	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
349	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
350	C₄⋊C₄	—	2	1	no	—
351	C₄⋊C₄	—	118	1	no	—
352	C₂²⋊C₄	—	2	1	no	—
353	C₂²⋊C₄	—	118	1	no	—
354	C₄²	—	2	1	no	—
355	C₄²	—	59	2	no	—
356	C₄²	—	118	1	no	—
357	C₄²	—	118	1	no	—
358	C₂³	—	118	2	no	—
359	C₂³	#14273775	1	1	yes	characteristic
360	Q₈	—	59	4	no	—
361	Q₈	—	118	2	no	—
362	D₄	—	59	4	no	—
363	D₄	—	118	4	no	—
364	D₄	—	118	4	no	—
365	D₄	—	118	2	no	—
366	C₂×C₄	—	4	1	no	—
367	C₂×C₄	—	118	2	no	—
368	C₂×C₄	—	118	4	no	—
369	C₂×C₄	—	118	4	no	—
370	C₂×C₄	—	118	2	no	—
371	C₂×C₄	—	472	1	no	—
372	C₂×C₄	—	236	1	no	—
373	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
374	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
375	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
376	C₂×C₄	C₂×Q₈×D₅₉	1	1	yes	characteristic
377	C₂×C₄	#41×D₅₉	1	1	yes	characteristic
378	C₂×C₄	#14273773	1	1	yes	characteristic
379	C₈	—	2	2	no	—
380	C₈	—	2	4	no	—
381	C₈	—	4	2	no	—
382	C₈	—	59	2	no	—
383	C₈	—	59	2	no	—
384	C₈	—	118	2	no	—
385	C₈	—	236	2	no	—
386	C₈	C₄⋊C₄×D₅₉	1	1	yes	characteristic
387	C₈	C₄⋊C₄×D₅₉	1	1	yes	characteristic
388	C₈	C₄⋊C₄×D₅₉	1	1	yes	characteristic
389	C₈	C₄⋊C₄×D₅₉	1	1	yes	characteristic
390	C₂²	—	4	1	no	—
391	C₂²	—	118	4	no	—
392	C₂²	—	472	2	no	—
393	C₂²	#16837884	1	1	yes	characteristic
394	C₂²	#16838054	1	1	yes	characteristic
395	C₂²	#16838067	1	1	yes	characteristic
396	C₄	—	8	1	no	—
397	C₄	—	118	4	no	—
398	C₄	—	472	1	no	—
399	C₄	—	236	2	no	—
400	C₄	C₂×C₄⋊C₄×D₅₉	1	1	yes	characteristic
401	C₄	C₄×D₄×D₅₉	1	1	yes	characteristic
402	C₄	#16838036	1	1	yes	characteristic
403	C₄	#16838108	1	1	yes	characteristic
404	C₂	—	2	1	no	—
405	C₂	—	2	1	no	—
406	C₂	—	2	1	no	—
407	C₂	—	236	4	no	—
408	C₂	#21426997	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16838036
commutator	C₂×C₄×C₅₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₉
Fitting	C₅₉×#1953	C₂
socle	C₂×C₅₉	D₄×C₄⋊C₄
prime radical	C₂×#14273749	C₂²
cosocle	C₂²×C₄×C₅₉	C₂⁴
hypercenter	#1953	D₅₉
hypocenter	C₅₉	#22695
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1953

Complements

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

N	H ≅ G/N	type
4. #21432706	C₂	semidirect
5. #21424887	C₂	semidirect
6. #21424339	C₂	semidirect
7. #21424333	C₂	semidirect
8. #21424269	C₂	semidirect
9. #21424149	C₂	semidirect
10. #21423981	C₂	semidirect
11. C₅₉×#1953	C₂	semidirect
18. #16837523	C₄	semidirect
19. #16837523	C₄	semidirect
205. #1953	D₅₉	semidirect
247. C₅₉	#22695	semidirect

Maximal subgroups

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

subgroup	index
#21433014	2
#21433012	2
C₂×#16837523	2
#21432706	2
#21424887	2
#21424339	2
#21424333	2
#21424269	2
#21424149	2
#21423981	2
C₅₉×#1953	2
#22695	59