Subgroups — #28743429

Order 15104.

524 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21434608	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16837519	C₂	1	1	yes	maximal, characteristic
3	#21426270	C₂	1	4	yes	maximal, complemented
4	#21426265	C₂	1	4	yes	maximal, complemented
5	#21425869	C₂	1	1	yes	maximal, characteristic, complemented
6	#21425868	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#16837222	C₂	1	1	yes	maximal, characteristic, complemented
8	#21425832	C₂	1	1	yes	maximal, characteristic
9	C₅₉×#2145	C₂	1	1	yes	maximal, characteristic
10	C₂×#14273734	—	2	1	no	—
11	#16837955	—	2	1	no	—
12	C₂×#14273667	C₂²	1	1	yes	characteristic
13	#16837806	—	2	1	no	—
14	#16837803	—	2	1	no	—
15	#16837802	C₂²	1	2	yes	complemented
16	#16837799	C₂²	1	2	yes	complemented
17	#16837519	—	2	1	no	—
18	#16837519	—	2	1	no	—
19	#16837519	C₂²	1	2	yes	—
20	#16837519	C₂²	1	2	yes	—
21	C₂×#14273663	C₂²	1	1	yes	characteristic
22	#16837440	C₂²	1	1	yes	characteristic
23	#16837438	—	2	1	no	—
24	C₂×#14273654	—	2	1	no	—
25	C₂×C₈×Dic₅₉	C₂²	1	1	yes	characteristic
26	#16837225	C₂²	1	4	yes	complemented
27	#16837224	C₂²	1	4	yes	complemented
28	#16837222	C₂²	1	2	yes	complemented
29	#16837222	C₂²	1	2	yes	complemented
30	#16837212	C₂²	1	4	yes	—
31	C₂×#135×C₅₉	C₂²	1	1	yes	characteristic
32	C₂×M₃₂×C₅₉	C₂²	1	1	yes	characteristic
33	C₅₉×#427	C₂²	1	1	yes	characteristic
34	C₅₉×#360	C₂²	1	4	yes	—
35	#14273734	—	2	2	no	—
36	#14273734	—	2	2	no	—
37	#14273734	—	4	1	no	—
38	#14273734	—	4	1	no	—
39	C₂×#11745664	—	2	1	no	—
40	C₂×#11745664	—	4	1	no	—
41	C₂²×C₅₉⋊C₈	—	4	1	no	—
42	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
43	#14273726	—	2	2	no	—
44	#14273718	—	2	2	no	—
45	#14273704	—	4	2	no	—
46	C₄×Dic₁₁₈	—	4	2	no	—
47	#14273686	—	2	1	no	—
48	C₂×C₄×Dic₅₉	D₄	1	1	yes	characteristic
49	#14273667	—	2	1	no	—
50	#14273667	C₂³	1	2	yes	—
51	C₂×C₅₉⋊C₁₆	—	2	1	no	—
52	#14273663	—	2	1	no	—
53	#14273663	C₂³	1	2	yes	—
54	#14273661	—	2	1	no	—
55	#14273660	—	2	2	no	—
56	#14273659	C₂³	1	2	yes	—
57	#14273658	—	2	2	no	—
58	#14273654	—	2	2	no	—
59	#14273654	—	2	2	no	—
60	C₈×Dic₅₉	D₄	1	1	yes	characteristic
61	C₈×Dic₅₉	D₄	1	1	yes	characteristic
62	C₈×Dic₅₉	C₂³	1	1	yes	characteristic
63	C₈×Dic₅₉	C₂³	1	1	yes	characteristic
64	C₂×#41×C₅₉	—	2	1	no	—
65	#135×C₅₉	—	2	1	no	—
66	#135×C₅₉	—	2	1	no	—
67	#135×C₅₉	Q₈	1	2	yes	complemented
68	#135×C₅₉	C₂³	1	2	yes	—
69	C₂×Q₁₆×C₅₉	Q₈	1	1	yes	characteristic, complemented
70	C₂×SD₁₆×C₅₉	—	2	1	no	—
71	C₂×D₈×C₅₉	Q₈	1	1	yes	characteristic, complemented
72	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
73	#117×C₅₉	—	2	1	no	—
74	M₃₂×C₅₉	—	2	1	no	—
75	M₃₂×C₅₉	C₂³	1	2	yes	—
76	C₂×C₁₆×C₅₉	—	2	1	no	—
77	#107×C₅₉	—	2	1	no	—
78	#106×C₅₉	C₂³	1	2	yes	—
79	#41×C₅₉	—	2	2	no	—
80	#41×C₅₉	—	2	2	no	—
81	#41×C₅₉	—	4	1	no	—
82	#41×C₅₉	—	4	1	no	—
83	C₂×Q₈×C₅₉	—	2	1	no	—
84	C₂×D₄×C₅₉	—	2	1	no	—
85	C₂×D₄×C₅₉	—	4	1	no	—
86	C₂²×C₄×C₅₉	—	4	1	no	—
87	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
88	C₂²×Dic₅₉	—	2	1	no	—
89	C₂×Dic₁₁₈	—	4	2	no	—
90	Q₁₆×C₅₉	—	2	1	no	—
91	Q₁₆×C₅₉	C₂×Q₈	1	2	yes	—
92	SD₁₆×C₅₉	—	2	2	no	—
93	SD₁₆×C₅₉	—	2	2	no	—
94	D₈×C₅₉	—	2	1	no	—
95	D₈×C₅₉	C₂×Q₈	1	2	yes	—
96	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
98	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
99	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
100	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
101	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
102	C₄⋊C₄×C₅₉	—	2	2	no	—
103	C₂²⋊C₄×C₅₉	—	4	1	no	—
104	C₄²×C₅₉	—	4	1	no	—
105	C₂²⋊Dic₅₉	—	4	1	no	—
106	C₄⋊Dic₅₉	—	2	2	no	—
107	C₄⋊Dic₅₉	—	4	2	no	—
108	Dic₅₉⋊C₄	—	4	2	no	—
109	Dic₅₉⋊C₄	—	4	2	no	—
110	C₄×Dic₅₉	—	2	1	no	—
111	C₄×Dic₅₉	—	4	1	no	—
112	C₄×Dic₅₉	C₂×D₄	1	1	yes	characteristic
113	C₄×Dic₅₉	C₂×D₄	1	1	yes	characteristic
114	#11745664	—	2	2	no	—
115	#11745664	—	4	1	no	—
116	#11745664	—	4	2	no	—
117	#11745664	—	4	2	no	—
118	C₂×C₅₉⋊C₈	—	4	1	no	—
119	C₂×C₅₉⋊C₈	—	4	2	no	—
120	C₂×C₅₉⋊C₈	—	4	2	no	—
121	C₂×C₅₉⋊C₈	SD₁₆	1	1	yes	characteristic
122	C₂×C₅₉⋊C₈	SD₁₆	1	1	yes	characteristic
123	C₂×C₅₉⋊C₈	SD₁₆	1	1	yes	characteristic
124	C₂×C₅₉⋊C₈	SD₁₆	1	1	yes	characteristic
125	C₂×C₅₉⋊C₈	C₂×D₄	1	1	yes	characteristic
126	C₂×C₅₉⋊C₈	C₂×D₄	1	1	yes	characteristic
127	C₁₆×C₅₉	—	2	2	no	—
128	C₅₉⋊C₁₆	—	2	2	no	—
129	C₂³×C₅₉	—	4	1	no	—
130	C₂³×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
131	Q₈×C₅₉	—	2	2	no	—
132	Q₈×C₅₉	—	4	1	no	—
133	D₄×C₅₉	—	2	2	no	—
134	D₄×C₅₉	—	4	2	no	—
135	D₄×C₅₉	—	4	2	no	—
136	D₄×C₅₉	—	4	1	no	—
137	C₂×C₄×C₅₉	—	4	1	no	—
138	C₂×C₄×C₅₉	—	4	2	no	—
139	C₂×C₄×C₅₉	—	4	2	no	—
140	C₂×C₄×C₅₉	—	4	2	no	—
141	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
142	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
143	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
144	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
145	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
146	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
147	C₂×Dic₅₉	—	2	1	no	—
148	C₂×Dic₅₉	—	2	1	no	—
149	C₂×Dic₅₉	—	4	2	no	—
150	C₂×Dic₅₉	—	4	2	no	—
151	Dic₁₁₈	—	8	4	no	—
152	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
153	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
154	C₈×C₅₉	C₂²×Q₈	1	1	yes	characteristic
155	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
156	C₅₉⋊C₈	—	4	2	no	—
157	C₅₉⋊C₈	C₂×SD₁₆	1	1	yes	characteristic
158	C₅₉⋊C₈	C₂×SD₁₆	1	1	yes	characteristic
159	C₅₉⋊C₈	C₂×SD₁₆	1	1	yes	characteristic
160	C₅₉⋊C₈	C₂×SD₁₆	1	1	yes	characteristic
161	#33637	—	59	1	no	maximal, Sylow
162	C₂²×C₅₉	—	4	1	no	—
163	C₂²×C₅₉	—	4	2	no	—
164	C₂²×C₅₉	—	8	2	no	—
165	C₂²×C₅₉	#476	1	1	yes	characteristic
166	C₂²×C₅₉	#476	1	1	yes	characteristic
167	C₂²×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
168	C₄×C₅₉	—	4	2	no	—
169	C₄×C₅₉	—	8	2	no	—
170	C₄×C₅₉	#476	1	1	yes	characteristic
171	C₄×C₅₉	#476	1	1	yes	characteristic
172	C₄×C₅₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
173	C₄×C₅₉	C₂²×SD₁₆	1	1	yes	characteristic
174	Dic₅₉	—	2	2	no	—
175	Dic₅₉	—	4	1	no	—
176	Dic₅₉	—	8	2	no	—
177	#3160	—	59	1	no	—
178	C₂×#443	—	59	1	no	—
179	#2231	—	59	4	no	—
180	#2230	—	59	4	no	—
181	#2155	—	59	1	no	—
182	#2155	—	59	1	no	—
183	C₂×#364	—	59	1	no	—
184	#2145	—	59	1	no	—
185	#2145	D₅₉	1	1	yes	characteristic
186	C₂×C₅₉	—	2	1	no	—
187	C₂×C₅₉	—	2	1	no	—
188	C₂×C₅₉	—	2	1	no	—
189	C₂×C₅₉	—	8	2	no	—
190	C₂×C₅₉	C₂×#476	1	1	yes	characteristic
191	C₂×#135	C₂×D₅₉	1	1	yes	characteristic
192	C₂×#131	—	118	1	no	—
193	#533	—	118	1	no	—
194	C₂×M₃₂	—	59	1	no	—
195	C₂×M₃₂	C₂×D₅₉	1	1	yes	characteristic
196	#500	—	118	1	no	—
197	#499	—	59	2	no	—
198	#499	—	118	1	no	—
199	#498	—	59	2	no	—
200	#443	—	59	2	no	—
201	#443	—	59	2	no	—
202	#443	—	118	1	no	—
203	#443	—	118	1	no	—
204	C₂×#108	—	59	1	no	—
205	#427	—	59	1	no	—
206	#427	—	118	1	no	—
207	#427	C₂×D₅₉	1	1	yes	characteristic
208	C₂×#104	—	118	1	no	—
209	C₂×C₄×C₈	—	59	1	no	—
210	#365	—	59	4	no	—
211	#365	—	59	4	no	—
212	#364	—	59	2	no	—
213	#364	—	59	2	no	—
214	#360	—	59	4	no	—
215	#360	C₂×D₅₉	1	4	yes	—
216	C₅₉	#33637	1	1	yes	characteristic, complemented, Sylow
217	C₂×#41	—	2	1	no	—
218	#135	—	2	1	no	—
219	#135	—	2	1	no	—
220	#135	Dic₁₁₈	1	2	yes	complemented
221	#135	C₂²×D₅₉	1	2	yes	—
222	C₂×Q₁₆	Dic₁₁₈	1	1	yes	characteristic, complemented
223	C₂×SD₁₆	—	2	1	no	—
224	C₂×D₈	Dic₁₁₈	1	1	yes	characteristic, complemented
225	#131	—	118	2	no	—
226	#131	—	118	2	no	—
227	#131	—	236	1	no	—
228	#131	—	236	1	no	—
229	C₂×M₁₆	—	118	1	no	—
230	C₂×M₁₆	—	236	1	no	—
231	C₂²×C₈	—	59	1	no	—
232	C₂²×C₈	—	236	1	no	—
233	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
234	C₄⋊Q₈	—	118	2	no	—
235	#125	—	118	2	no	—
236	C₂²⋊Q₈	—	236	2	no	—
237	C₄×Q₈	—	236	2	no	—
238	#117	—	2	1	no	—
239	#117	—	118	1	no	—
240	C₂×C₄²	—	59	1	no	—
241	M₃₂	—	2	1	no	—
242	M₃₂	—	59	2	no	—
243	M₃₂	—	118	1	no	—
244	M₃₂	C₂²×D₅₉	1	2	yes	—
245	C₂×C₁₆	—	2	1	no	—
246	C₂×C₁₆	—	118	1	no	—
247	#108	—	59	2	no	—
248	#108	—	118	1	no	—
249	#107	—	2	1	no	—
250	#107	—	118	2	no	—
251	#107	—	118	1	no	—
252	#106	—	59	2	no	—
253	#106	—	118	2	no	—
254	#106	C₂²×D₅₉	1	2	yes	—
255	#104	—	118	2	no	—
256	#104	—	118	2	no	—
257	C₄×C₈	—	59	1	no	—
258	C₄×C₈	—	59	1	no	—
259	C₄×C₈	—	59	1	no	—
260	C₄×C₈	—	59	1	no	—
261	#41	—	2	2	no	—
262	#41	—	2	2	no	—
263	#41	—	4	1	no	—
264	#41	—	4	1	no	—
265	C₂×Q₈	—	2	1	no	—
266	C₂×Q₈	—	236	2	no	—
267	C₂×D₄	—	2	1	no	—
268	C₂×D₄	—	4	1	no	—
269	C₂²×C₄	—	4	1	no	—
270	C₂²×C₄	—	118	1	no	—
271	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
272	Q₁₆	—	2	1	no	—
273	Q₁₆	C₂×Dic₁₁₈	1	2	yes	—
274	SD₁₆	—	2	2	no	—
275	SD₁₆	—	2	2	no	—
276	D₈	—	2	1	no	—
277	D₈	C₂×Dic₁₁₈	1	2	yes	—
278	M₁₆	—	118	2	no	—
279	M₁₆	—	236	2	no	—
280	M₁₆	—	236	2	no	—
281	M₁₆	—	236	1	no	—
282	C₂×C₈	—	59	1	no	—
283	C₂×C₈	—	59	1	no	—
284	C₂×C₈	—	59	1	no	—
285	C₂×C₈	—	59	1	no	—
286	C₂×C₈	—	59	1	no	—
287	C₂×C₈	—	59	1	no	—
288	C₂×C₈	—	236	2	no	—
289	C₂×C₈	—	236	2	no	—
290	C₂×C₈	—	236	1	no	—
291	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
292	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
293	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
294	C₂×C₈	#11745687	1	1	yes	characteristic
295	C₂×C₈	#11745687	1	1	yes	characteristic
296	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	118	2	no	—
299	C₄⋊C₄	—	236	2	no	—
300	C₄⋊C₄	—	236	2	no	—
301	C₄⋊C₄	—	236	2	no	—
302	C₂²⋊C₄	—	4	1	no	—
303	C₂²⋊C₄	—	236	1	no	—
304	C₄²	—	4	1	no	—
305	C₄²	—	59	1	no	—
306	C₄²	—	59	1	no	—
307	C₄²	—	118	1	no	—
308	C₄²	—	236	1	no	—
309	C₁₆	—	2	2	no	—
310	C₁₆	—	118	2	no	—
311	C₂³	—	4	1	no	—
312	C₂³	#14273751	1	1	yes	characteristic
313	Q₈	—	2	2	no	—
314	Q₈	—	4	1	no	—
315	Q₈	—	472	4	no	—
316	D₄	—	2	2	no	—
317	D₄	—	4	2	no	—
318	D₄	—	4	2	no	—
319	D₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	118	1	no	—
325	C₂×C₄	—	118	1	no	—
326	C₂×C₄	—	236	2	no	—
327	C₂×C₄	—	236	2	no	—
328	C₂×C₄	#14273704	1	1	yes	characteristic
329	C₂×C₄	#14273704	1	1	yes	characteristic
330	C₂×C₄	SD₁₆×D₅₉	1	1	yes	characteristic
331	C₂×C₄	SD₁₆×D₅₉	1	1	yes	characteristic
332	C₂×C₄	#14273751	1	1	yes	characteristic
333	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
334	C₈	—	59	1	no	—
335	C₈	—	59	1	no	—
336	C₈	—	59	1	no	—
337	C₈	—	59	1	no	—
338	C₈	—	236	2	no	—
339	C₈	#14273704	1	1	yes	characteristic
340	C₈	#14273704	1	1	yes	characteristic
341	C₈	C₂²×Dic₁₁₈	1	1	yes	characteristic
342	C₈	C₂×#11745687	1	1	yes	characteristic
343	C₂²	—	4	1	no	—
344	C₂²	—	4	2	no	—
345	C₂²	—	8	2	no	—
346	C₂²	#16837690	1	1	yes	characteristic
347	C₂²	#16837690	1	1	yes	characteristic
348	C₂²	C₂×#14273751	1	1	yes	characteristic
349	C₄	—	4	2	no	—
350	C₄	—	8	2	no	—
351	C₄	—	118	2	no	—
352	C₄	—	472	2	no	—
353	C₄	—	236	1	no	—
354	C₄	#16837690	1	1	yes	characteristic
355	C₄	#16837690	1	1	yes	characteristic
356	C₄	C₂×#14273704	1	1	yes	characteristic
357	C₄	C₂×SD₁₆×D₅₉	1	1	yes	characteristic
358	C₂	—	2	1	no	—
359	C₂	—	2	1	no	—
360	C₂	—	2	1	no	—
361	C₂	—	8	2	no	—
362	C₂	C₂×#16837690	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16837690
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₅₉×#2145	C₂
socle	C₂×C₅₉	C₂×#476
prime radical	C₂×#135×C₅₉	C₂²
hypercenter	#2145	D₅₉
hypocenter	C₅₉	#33637
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2145

Complements

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

N	H ≅ G/N	type
1. #21434608	C₂	semidirect
3. #21426270	C₂	semidirect
4. #21426265	C₂	semidirect
5. #21425869	C₂	semidirect
6. #21425868	C₂	semidirect
7. C₂×#16837222	C₂	semidirect
15. #16837802	C₂²	semidirect
16. #16837799	C₂²	semidirect
26. #16837225	C₂²	semidirect
27. #16837224	C₂²	semidirect
28. #16837222	C₂²	semidirect
29. #16837222	C₂²	semidirect
67. #135×C₅₉	Q₈	semidirect
69. C₂×Q₁₆×C₅₉	Q₈	semidirect
71. C₂×D₈×C₅₉	Q₈	semidirect
216. C₅₉	#33637	semidirect
220. #135	Dic₁₁₈	semidirect
222. C₂×Q₁₆	Dic₁₁₈	semidirect
224. C₂×D₈	Dic₁₁₈	semidirect

Maximal subgroups

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

subgroup	index
#21434608	2
C₂×#16837519	2
#21426270	2
#21426265	2
#21425869	2
#21425868	2
C₂×#16837222	2
#21425832	2
C₅₉×#2145	2
#33637	59