Subgroups — #28745487

Order 15104.

646 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21437234	C₂	1	1	yes	maximal, characteristic, complemented
2	#21435882	C₂	1	2	yes	maximal, complemented
3	#21426358	C₂	1	1	yes	maximal, characteristic, complemented
4	#21426218	C₂	1	1	yes	maximal, characteristic, complemented
5	#21426216	C₂	1	1	yes	maximal, characteristic, complemented
6	#21426167	C₂	1	2	yes	maximal, complemented
7	#21426064	C₂	1	2	yes	maximal, complemented
8	#21426060	C₂	1	2	yes	maximal, complemented
9	#21425977	C₂	1	1	yes	maximal, characteristic, complemented
10	#21425931	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₉×#3417	C₂	1	1	yes	maximal, characteristic, complemented
12	#16838210	—	2	2	no	—
13	#16838204	C₂²	1	2	yes	—
14	#16837839	—	2	2	no	—
15	#16837837	C₂²	1	2	yes	—
16	#16837835	—	2	1	no	—
17	C₂×#14273670	C₂²	1	2	yes	complemented
18	C₂×D₉₄₄	C₂²	1	1	yes	characteristic
19	D₁₆×D₅₉	—	2	1	no	—
20	#16837818	C₂²	1	1	yes	characteristic
21	#16837787	C₂²	1	2	yes	complemented
22	#16837674	C₂²	1	2	yes	—
23	#16837672	C₂²	1	2	yes	complemented
24	#16837667	—	2	2	no	—
25	#16837578	—	2	2	no	—
26	#16837575	—	2	2	no	—
27	#16837519	C₂²	1	2	yes	complemented
28	#16837517	C₂²	1	1	yes	characteristic
29	#16837232	C₂²	1	1	yes	characteristic, complemented
30	#16837222	C₂²	1	1	yes	characteristic, complemented
31	#16837218	C₂²	1	2	yes	complemented
32	#16837216	C₂²	1	2	yes	complemented
33	#16837209	C₂²	1	2	yes	complemented
34	#16837187	C₂²	1	1	yes	characteristic
35	#16837186	C₂²	1	1	yes	characteristic, complemented
36	C₁₆×Dic₅₉	C₂²	1	1	yes	characteristic, complemented
37	C₅₉×#578	C₂²	1	2	yes	complemented
38	C₅₉×#510	—	2	2	no	—
39	C₅₉×#510	C₂²	1	2	yes	complemented
40	C₅₉×#508	—	2	1	no	—
41	C₅₉×#508	C₂²	1	1	yes	characteristic
42	C₂×Q₃₂×C₅₉	—	2	1	no	—
43	C₂×Q₃₂×C₅₉	C₂²	1	1	yes	characteristic, complemented
44	C₅₉×#504	C₂²	1	1	yes	characteristic, complemented
45	#14273773	—	2	2	no	—
46	#14273758	—	2	2	no	—
47	#14273758	—	4	2	no	—
48	#14273754	—	4	2	no	—
49	#14273752	—	2	2	no	—
50	#14273752	—	2	2	no	—
51	#14273752	—	4	2	no	—
52	#14273750	—	2	2	no	—
53	#14273747	—	4	2	no	—
54	#14273747	D₄	1	1	yes	characteristic
55	C₂×#11745671	—	2	2	no	—
56	C₂×#11745671	—	4	2	no	—
57	C₂×D₄₇₂	C₂³	1	2	yes	—
58	D₈×D₅₉	—	2	2	no	—
59	#14273734	—	2	2	no	—
60	#14273733	C₂³	1	1	yes	characteristic
61	#14273716	—	2	2	no	—
62	#14273670	—	2	2	no	—
63	#14273670	—	4	2	no	—
64	D₉₄₄	—	2	1	no	—
65	D₉₄₄	—	2	1	no	—
66	#14273668	—	2	2	no	—
67	#14273667	C₂³	1	1	yes	characteristic
68	#14273666	—	2	1	no	—
69	C₂×C₅₉⋊C₁₆	C₂³	1	1	yes	characteristic
70	C₁₆×D₅₉	—	2	1	no	—
71	#14273663	C₂³	1	2	yes	—
72	#14273662	D₄	1	1	yes	characteristic
73	#14273654	—	2	2	no	—
74	#14273654	D₄	1	1	yes	characteristic
75	#14273653	D₄	1	1	yes	characteristic
76	#14273646	—	2	2	no	—
77	#14273643	—	2	2	no	—
78	C₈×Dic₅₉	C₂³	1	1	yes	characteristic
79	2₋¹⁺⁴×C₅₉	—	2	2	no	—
80	#137×C₅₉	—	2	2	no	—
81	#137×C₅₉	—	4	2	no	—
82	#135×C₅₉	—	2	2	no	—
83	#135×C₅₉	C₂³	1	2	yes	—
84	C₂×Q₁₆×C₅₉	—	2	2	no	—
85	C₂×Q₁₆×C₅₉	C₂³	1	2	yes	—
86	#131×C₅₉	C₂³	1	1	yes	characteristic
87	Q₃₂×C₅₉	—	2	2	no	—
88	Q₃₂×C₅₉	—	2	1	no	—
89	Q₃₂×C₅₉	—	2	1	no	—
90	Q₃₂×C₅₉	D₄	1	1	yes	characteristic
91	SD₃₂×C₅₉	—	2	2	no	—
92	SD₃₂×C₅₉	D₄	1	2	yes	—
93	D₁₆×C₅₉	D₄	1	1	yes	characteristic
94	M₃₂×C₅₉	—	2	1	no	—
95	M₃₂×C₅₉	C₂³	1	1	yes	characteristic
96	C₂×C₁₆×C₅₉	—	2	1	no	—
97	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
98	#41×C₅₉	—	2	2	no	—
99	#41×C₅₉	—	4	2	no	—
100	#41×C₅₉	—	4	2	no	—
101	#41×C₅₉	—	8	2	no	—
102	#41×C₅₉	C₂×D₄	1	1	yes	characteristic
103	C₂×Q₈×C₅₉	—	2	2	no	—
104	C₂×Q₈×C₅₉	—	4	2	no	—
105	C₂×Q₈×C₅₉	—	4	2	no	—
106	#11745689	—	4	2	no	—
107	D₄×D₅₉	—	4	2	no	—
108	D₄×D₅₉	—	8	2	no	—
109	#11745685	—	4	2	no	—
110	#11745685	C₂×D₄	1	1	yes	characteristic
111	C₂×D₂₃₆	—	2	2	no	—
112	C₂×D₂₃₆	—	4	2	no	—
113	Q₁₆×C₅₉	—	2	2	no	—
114	Q₁₆×C₅₉	—	2	2	no	—
115	Q₁₆×C₅₉	—	4	2	no	—
116	Q₁₆×C₅₉	C₂×D₄	1	2	yes	—
117	SD₁₆×C₅₉	—	2	2	no	—
118	SD₁₆×C₅₉	—	4	2	no	—
119	D₈×C₅₉	C₂×D₄	1	2	yes	—
120	M₁₆×C₅₉	—	2	1	no	—
121	M₁₆×C₅₉	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈×C₅₉	—	2	1	no	—
123	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
124	#11745672	—	4	2	no	—
125	#11745672	D₈	1	1	yes	characteristic
126	#11745671	—	4	2	no	—
127	#11745671	—	4	2	no	—
128	#11745671	—	4	2	no	—
129	#11745671	—	4	2	no	—
130	#11745671	—	4	2	no	—
131	#11745671	D₈	1	1	yes	characteristic
132	#11745670	D₈	1	1	yes	characteristic
133	#11745669	—	4	2	no	—
134	#11745669	—	4	2	no	—
135	#11745669	D₈	1	1	yes	characteristic
136	(C₂×D₅₉)⋊C₄	—	8	2	no	—
137	C₄×Dic₅₉	C₂×D₄	1	1	yes	characteristic
138	#11745664	—	2	2	no	—
139	#11745664	—	4	2	no	—
140	#11745664	C₂×D₄	1	1	yes	characteristic
141	C₂×C₅₉⋊C₈	—	4	2	no	—
142	C₂×C₅₉⋊C₈	C₂×D₄	1	1	yes	characteristic
143	D₄₇₂	—	2	2	no	—
144	D₄₇₂	—	2	2	no	—
145	#11745659	—	2	1	no	—
146	C₈×D₅₉	—	2	1	no	—
147	C₁₆×C₅₉	—	2	1	no	—
148	C₁₆×C₅₉	C₂×D₄	1	1	yes	characteristic
149	C₁₆×C₅₉	C₂×D₄	1	1	yes	characteristic
150	C₅₉⋊C₁₆	—	2	1	no	—
151	C₅₉⋊C₁₆	—	2	1	no	—
152	C₂²×D₅₉	—	4	2	no	—
153	C₂²×D₅₉	—	8	2	no	—
154	Q₈×C₅₉	—	2	2	no	—
155	Q₈×C₅₉	—	4	2	no	—
156	Q₈×C₅₉	—	4	2	no	—
157	Q₈×C₅₉	—	8	2	no	—
158	Q₈×C₅₉	C₂×D₈	1	1	yes	characteristic
159	D₄×C₅₉	—	2	1	no	—
160	D₄×C₅₉	—	2	2	no	—
161	D₄×C₅₉	—	4	2	no	—
162	D₄×C₅₉	—	8	2	no	—
163	D₄×C₅₉	C₂×D₈	1	1	yes	characteristic
164	C₂×C₄×C₅₉	—	2	1	no	—
165	C₂×C₄×C₅₉	—	4	2	no	—
166	C₂×C₄×C₅₉	—	4	2	no	—
167	C₂×C₄×C₅₉	—	8	2	no	—
168	C₂×C₄×C₅₉	—	8	2	no	—
169	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
170	C₂²⋊D₅₉	—	2	1	no	—
171	C₂²⋊D₅₉	—	8	2	no	—
172	C₂×Dic₅₉	—	2	1	no	—
173	D₂₃₆	—	4	2	no	—
174	D₂₃₆	—	4	2	no	—
175	D₂₃₆	—	8	2	no	—
176	D₂₃₆	C₂×D₈	1	1	yes	characteristic
177	C₄×D₅₉	—	2	1	no	—
178	C₄×D₅₉	—	8	2	no	—
179	Dic₁₁₈	C₂×D₈	1	1	yes	characteristic
180	C₈×C₅₉	—	2	1	no	—
181	C₈×C₅₉	C₂²×D₄	1	1	yes	characteristic
182	C₈×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
183	C₅₉⋊C₈	—	2	1	no	—
184	C₅₉⋊C₈	—	4	2	no	—
185	C₅₉⋊C₈	C₂×D₈	1	1	yes	characteristic
186	C₅₉⋊C₈	C₂×D₈	1	1	yes	characteristic
187	#33859	—	59	1	no	maximal, Sylow
188	C₂²×C₅₉	—	2	1	no	—
189	C₂²×C₅₉	—	4	2	no	—
190	C₂²×C₅₉	#576	1	1	yes	characteristic
191	C₂×D₅₉	—	2	1	no	—
192	C₂×D₅₉	—	8	2	no	—
193	C₂×D₅₉	—	16	2	no	—
194	C₂×D₅₉	—	32	2	no	—
195	C₄×C₅₉	—	2	1	no	—
196	C₄×C₅₉	—	4	2	no	—
197	C₄×C₅₉	—	8	2	no	—
198	C₄×C₅₉	D₄²	1	1	yes	characteristic
199	C₄×C₅₉	C₂²×D₈	1	1	yes	characteristic
200	Dic₅₉	—	2	1	no	—
201	Dic₅₉	—	4	1	no	—
202	#3417	D₅₉	1	1	yes	characteristic, complemented
203	#3415	—	59	1	no	—
204	#3290	—	59	2	no	—
205	#2249	—	59	1	no	—
206	#2222	—	59	1	no	—
207	#2221	—	59	1	no	—
208	#2212	—	59	2	no	—
209	#2194	—	59	2	no	—
210	#2193	—	59	2	no	—
211	#2178	—	59	1	no	—
212	#2170	—	59	1	no	—
213	C₂×C₅₉	—	2	1	no	—
214	C₂×C₅₉	—	4	1	no	—
215	C₂×C₅₉	—	8	2	no	—
216	C₂×C₅₉	D₄×D₈	1	1	yes	characteristic
217	D₅₉	—	4	1	no	—
218	D₅₉	—	16	2	no	—
219	#578	C₂×D₅₉	1	2	yes	complemented
220	#577	—	118	2	no	—
221	#576	—	59	2	no	—
222	#510	—	2	2	no	—
223	#510	C₂×D₅₉	1	2	yes	complemented
224	#509	—	59	2	no	—
225	#509	—	118	2	no	—
226	#508	—	2	1	no	—
227	#508	—	118	1	no	—
228	#508	C₂×D₅₉	1	1	yes	characteristic
229	C₂×Q₃₂	—	2	1	no	—
230	C₂×Q₃₂	C₂×D₅₉	1	1	yes	characteristic, complemented
231	C₂×SD₃₂	—	59	2	no	—
232	C₂×D₁₆	—	59	1	no	—
233	C₂×D₁₆	—	118	1	no	—
234	#504	—	59	1	no	—
235	#504	C₂×D₅₉	1	1	yes	characteristic, complemented
236	#495	—	59	2	no	—
237	#473	—	59	2	no	—
238	#472	—	59	2	no	—
239	#471	—	118	2	no	—
240	#455	—	118	2	no	—
241	#454	—	118	2	no	—
242	#443	—	59	1	no	—
243	#443	—	59	2	no	—
244	#368	—	59	1	no	—
245	#364	—	59	1	no	—
246	#362	—	59	2	no	—
247	#361	—	59	2	no	—
248	#359	—	59	2	no	—
249	#350	—	59	1	no	—
250	#350	—	59	1	no	—
251	C₄×C₁₆	—	59	1	no	—
252	C₅₉	#33859	1	1	yes	characteristic, complemented, Sylow
253	2₋¹⁺⁴	—	2	2	no	—
254	H(5,𝔽₂)	—	118	2	no	—
255	#137	—	2	2	no	—
256	#137	—	4	2	no	—
257	#137	—	118	2	no	—
258	#137	—	236	2	no	—
259	AGL(1,ℤ/8)	—	118	2	no	—
260	AGL(1,ℤ/8)	—	118	2	no	—
261	AGL(1,ℤ/8)	—	236	2	no	—
262	AGL(1,ℤ/8)	—	236	2	no	—
263	#135	—	2	2	no	—
264	#135	—	59	1	no	—
265	#135	—	118	2	no	—
266	#135	—	236	2	no	—
267	#135	C₂²×D₅₉	1	2	yes	—
268	C₂×Q₁₆	—	2	2	no	—
269	C₂×Q₁₆	C₂²×D₅₉	1	2	yes	—
270	C₂×SD₁₆	—	118	2	no	—
271	C₂×SD₁₆	—	236	2	no	—
272	C₂×D₈	—	59	2	no	—
273	C₂×D₈	—	118	2	no	—
274	#131	—	59	1	no	—
275	#131	—	118	2	no	—
276	#131	C₂²×D₅₉	1	1	yes	characteristic
277	#124	—	118	2	no	—
278	Q₃₂	—	2	2	no	—
279	Q₃₂	—	2	1	no	—
280	Q₃₂	—	2	1	no	—
281	Q₃₂	C₂²⋊D₅₉	1	1	yes	characteristic
282	SD₃₂	—	2	2	no	—
283	SD₃₂	—	118	2	no	—
284	SD₃₂	—	236	2	no	—
285	SD₃₂	C₂²⋊D₅₉	1	2	yes	—
286	D₁₆	—	118	2	no	—
287	D₁₆	—	118	1	no	—
288	D₁₆	—	118	1	no	—
289	D₁₆	C₂²⋊D₅₉	1	1	yes	characteristic
290	M₃₂	—	2	1	no	—
291	M₃₂	—	59	1	no	—
292	M₃₂	—	118	1	no	—
293	M₃₂	C₂²×D₅₉	1	1	yes	characteristic
294	C₂×C₁₆	—	2	1	no	—
295	C₂×C₁₆	—	59	1	no	—
296	C₂×C₁₆	—	118	1	no	—
297	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
298	#108	—	59	1	no	—
299	#108	—	59	2	no	—
300	#104	—	59	1	no	—
301	#104	—	59	1	no	—
302	#104	—	118	2	no	—
303	#101	—	118	2	no	—
304	#100	—	118	2	no	—
305	C₄×C₈	—	59	1	no	—
306	#41	—	2	2	no	—
307	#41	—	4	2	no	—
308	#41	—	4	2	no	—
309	#41	—	8	2	no	—
310	#41	—	59	1	no	—
311	#41	—	236	2	no	—
312	#41	—	236	2	no	—
313	#41	D₄×D₅₉	1	1	yes	characteristic
314	C₂×Q₈	—	2	2	no	—
315	C₂×Q₈	—	4	2	no	—
316	C₂×Q₈	—	4	2	no	—
317	C₂×D₄	—	118	2	no	—
318	C₂×D₄	—	472	2	no	—
319	C₂×D₄	—	236	2	no	—
320	C₂×D₄	—	236	2	no	—
321	Q₁₆	—	2	2	no	—
322	Q₁₆	—	2	2	no	—
323	Q₁₆	—	4	2	no	—
324	Q₁₆	—	59	1	no	—
325	Q₁₆	—	236	2	no	—
326	Q₁₆	C₂×C₂²⋊D₅₉	1	2	yes	—
327	SD₁₆	—	2	2	no	—
328	SD₁₆	—	4	2	no	—
329	SD₁₆	—	59	1	no	—
330	SD₁₆	—	59	1	no	—
331	SD₁₆	—	236	2	no	—
332	SD₁₆	—	236	2	no	—
333	SD₁₆	—	236	2	no	—
334	SD₁₆	—	236	2	no	—
335	SD₁₆	—	236	2	no	—
336	D₈	—	59	1	no	—
337	D₈	—	118	2	no	—
338	D₈	—	118	2	no	—
339	D₈	—	236	2	no	—
340	D₈	—	236	2	no	—
341	D₈	C₂×C₂²⋊D₅₉	1	2	yes	—
342	M₁₆	—	2	1	no	—
343	M₁₆	—	59	1	no	—
344	M₁₆	—	118	1	no	—
345	M₁₆	—	118	2	no	—
346	M₁₆	—	236	2	no	—
347	M₁₆	D₄×D₅₉	1	1	yes	characteristic
348	C₂×C₈	—	2	1	no	—
349	C₂×C₈	—	59	1	no	—
350	C₂×C₈	—	118	1	no	—
351	C₂×C₈	—	236	2	no	—
352	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
353	C₂²⋊C₄	—	472	2	no	—
354	C₄²	—	59	1	no	—
355	C₁₆	—	2	1	no	—
356	C₁₆	—	118	1	no	—
357	C₁₆	—	118	1	no	—
358	C₁₆	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
359	C₁₆	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
360	C₂³	—	472	2	no	—
361	C₂³	—	236	2	no	—
362	Q₈	—	2	2	no	—
363	Q₈	—	4	2	no	—
364	Q₈	—	4	2	no	—
365	Q₈	—	8	2	no	—
366	Q₈	—	59	1	no	—
367	Q₈	D₈×D₅₉	1	1	yes	characteristic
368	D₄	—	2	1	no	—
369	D₄	—	2	2	no	—
370	D₄	—	4	2	no	—
371	D₄	—	8	2	no	—
372	D₄	—	59	1	no	—
373	D₄	—	118	1	no	—
374	D₄	—	472	2	no	—
375	D₄	—	472	2	no	—
376	D₄	—	236	2	no	—
377	D₄	—	236	2	no	—
378	D₄	D₈×D₅₉	1	1	yes	characteristic
379	C₂×C₄	—	2	1	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	4	2	no	—
382	C₂×C₄	—	8	2	no	—
383	C₂×C₄	—	8	2	no	—
384	C₂×C₄	—	118	1	no	—
385	C₂×C₄	—	118	1	no	—
386	C₂×C₄	—	472	2	no	—
387	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
388	C₈	—	2	1	no	—
389	C₈	—	59	1	no	—
390	C₈	—	59	1	no	—
391	C₈	—	118	1	no	—
392	C₈	—	236	2	no	—
393	C₈	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
394	C₈	#14273772	1	1	yes	characteristic
395	C₂²	—	2	1	no	—
396	C₂²	—	4	2	no	—
397	C₂²	—	1888	2	no	—
398	C₂²	—	118	1	no	—
399	C₂²	—	944	2	no	—
400	C₂²	—	472	2	no	—
401	C₂²	#16838203	1	1	yes	characteristic
402	C₄	—	2	1	no	—
403	C₄	—	4	2	no	—
404	C₄	—	8	2	no	—
405	C₄	—	118	1	no	—
406	C₄	—	236	1	no	—
407	C₄	D₄×C₂²⋊D₅₉	1	1	yes	characteristic
408	C₄	C₂×D₈×D₅₉	1	1	yes	characteristic
409	C₂	—	2	1	no	—
410	C₂	—	4	1	no	—
411	C₂	—	8	2	no	—
412	C₂	—	944	2	no	—
413	C₂	—	236	1	no	—
414	C₂	D₈×C₂²⋊D₅₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×C₂²⋊D₅₉
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₅₉×#3417	C₂
socle	C₂×C₅₉	D₄×D₈
hypercenter	#3417	D₅₉
hypocenter	C₅₉	#33859
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131 ◁ #3417

Complements

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

N	H ≅ G/N	type
1. #21437234	C₂	semidirect
2. #21435882	C₂	semidirect
3. #21426358	C₂	semidirect
4. #21426218	C₂	semidirect
5. #21426216	C₂	semidirect
6. #21426167	C₂	semidirect
7. #21426064	C₂	semidirect
8. #21426060	C₂	semidirect
9. #21425977	C₂	semidirect
10. #21425931	C₂	semidirect
11. C₅₉×#3417	C₂	semidirect
17. C₂×#14273670	C₂²	semidirect
21. #16837787	C₂²	semidirect
23. #16837672	C₂²	semidirect
27. #16837519	C₂²	semidirect
29. #16837232	C₂²	semidirect
30. #16837222	C₂²	semidirect
31. #16837218	C₂²	semidirect
32. #16837216	C₂²	semidirect
33. #16837209	C₂²	semidirect
35. #16837186	C₂²	semidirect
36. C₁₆×Dic₅₉	C₂²	semidirect
37. C₅₉×#578	C₂²	semidirect
39. C₅₉×#510	C₂²	semidirect
43. C₂×Q₃₂×C₅₉	C₂²	semidirect
44. C₅₉×#504	C₂²	semidirect
202. #3417	D₅₉	semidirect
219. #578	C₂×D₅₉	semidirect
223. #510	C₂×D₅₉	semidirect
230. C₂×Q₃₂	C₂×D₅₉	semidirect
235. #504	C₂×D₅₉	semidirect
252. C₅₉	#33859	semidirect

Maximal subgroups

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

subgroup	index
#21437234	2
#21435882	2
#21426358	2
#21426218	2
#21426216	2
#21426167	2
#21426064	2
#21426060	2
#21425977	2
#21425931	2
C₅₉×#3417	2
#33859	59