Subgroups — #28507188

Order 15104.

472 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837223	C₂	1	1	yes	maximal, characteristic, complemented
2	#21425832	C₂	1	2	yes	maximal, complemented
3	#21425824	C₂	1	2	yes	maximal, complemented
4	#21423975	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₉×#3147	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273667	C₂²	1	2	yes	—
7	C₂×#14273663	C₂²	1	1	yes	characteristic
8	#16837440	C₂²	1	1	yes	characteristic
9	#16837357	—	2	1	no	—
10	#16837223	—	2	1	no	—
11	#16837223	—	2	1	no	—
12	#16837212	—	2	4	no	—
13	#16837209	—	2	4	no	—
14	#16837162	—	2	1	no	—
15	C₂×#135×C₅₉	—	2	1	no	—
16	C₂×#135×C₅₉	C₂²	1	2	yes	—
17	C₅₉×#532	—	2	1	no	—
18	C₅₉×#495	—	2	1	no	—
19	C₅₉×#495	C₄	1	2	yes	complemented
20	C₅₉×#494	C₄	1	1	yes	characteristic, complemented
21	C₅₉×#493	C₄	1	1	yes	characteristic, complemented
22	C₅₉×#492	—	2	1	no	—
23	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
24	C₂×#11745664	—	2	1	no	—
25	#14273686	—	2	1	no	—
26	#14273667	—	2	2	no	—
27	#14273667	—	2	2	no	—
28	C₂×C₅₉⋊C₁₆	—	2	2	no	—
29	#14273663	—	2	1	no	—
30	#14273663	D₄	1	2	yes	complemented
31	#14273661	—	2	1	no	—
32	#14273659	D₄	1	2	yes	complemented
33	#14273637	—	4	1	no	—
34	C₈×Dic₅₉	—	4	1	no	—
35	C₂×#41×C₅₉	—	2	2	no	—
36	#135×C₅₉	—	2	2	no	—
37	#135×C₅₉	—	2	2	no	—
38	#135×C₅₉	—	2	2	no	—
39	#135×C₅₉	—	2	2	no	—
40	#135×C₅₉	—	4	1	no	—
41	#135×C₅₉	—	4	1	no	—
42	#135×C₅₉	—	4	1	no	—
43	#135×C₅₉	—	4	1	no	—
44	C₂×Q₁₆×C₅₉	—	2	1	no	—
45	C₂×Q₁₆×C₅₉	C₂×C₄	1	2	yes	—
46	C₂×SD₁₆×C₅₉	—	2	2	no	—
47	C₂×SD₁₆×C₅₉	—	2	2	no	—
48	C₂×D₈×C₅₉	—	2	1	no	—
49	C₂×D₈×C₅₉	C₂×C₄	1	2	yes	—
50	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
52	C₄⋊Q₈×C₅₉	—	2	1	no	—
53	#127×C₅₉	—	2	1	no	—
54	#124×C₅₉	—	2	2	no	—
55	#121×C₅₉	—	4	2	no	—
56	C₄×D₄×C₅₉	—	4	2	no	—
57	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
58	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
61	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
62	#41×C₅₉	—	4	2	no	—
63	#41×C₅₉	—	4	2	no	—
64	#41×C₅₉	—	4	2	no	—
65	#41×C₅₉	—	4	2	no	—
66	C₂×Q₈×C₅₉	—	2	2	no	—
67	C₂×D₄×C₅₉	—	2	2	no	—
68	C₂×D₄×C₅₉	—	4	2	no	—
69	C₂×D₄×C₅₉	—	4	2	no	—
70	C₂²×C₄×C₅₉	—	2	1	no	—
71	C₂²×C₄×C₅₉	—	4	2	no	—
72	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
73	Q₁₆×C₅₉	—	2	2	no	—
74	Q₁₆×C₅₉	—	2	2	no	—
75	Q₁₆×C₅₉	—	4	1	no	—
76	Q₁₆×C₅₉	—	4	1	no	—
77	SD₁₆×C₅₉	—	4	2	no	—
78	SD₁₆×C₅₉	—	4	2	no	—
79	SD₁₆×C₅₉	—	4	2	no	—
80	SD₁₆×C₅₉	—	4	2	no	—
81	D₈×C₅₉	—	2	2	no	—
82	D₈×C₅₉	—	2	2	no	—
83	D₈×C₅₉	—	4	1	no	—
84	D₈×C₅₉	—	4	1	no	—
85	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₈×C₅₉	D₈	1	1	yes	characteristic
88	C₂×C₈×C₅₉	D₈	1	1	yes	characteristic
89	C₂×C₈×C₅₉	SD₁₆	1	1	yes	characteristic
90	C₂×C₈×C₅₉	SD₁₆	1	1	yes	characteristic
91	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
96	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
97	C₄⋊C₄×C₅₉	—	4	2	no	—
98	C₂²⋊C₄×C₅₉	—	4	4	no	—
99	C₄²×C₅₉	—	2	1	no	—
100	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
101	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
102	C₂²⋊Dic₅₉	—	4	1	no	—
103	C₄⋊Dic₅₉	—	2	2	no	—
104	C₄×Dic₅₉	—	4	1	no	—
105	#11745664	—	2	2	no	—
106	#11745664	—	4	1	no	—
107	C₂×C₅₉⋊C₈	—	4	1	no	—
108	C₅₉⋊C₁₆	—	4	2	no	—
109	C₂³×C₅₉	—	4	2	no	—
110	C₂³×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
111	Q₈×C₅₉	—	4	2	no	—
112	Q₈×C₅₉	—	4	2	no	—
113	D₄×C₅₉	—	4	2	no	—
114	D₄×C₅₉	—	4	2	no	—
115	D₄×C₅₉	—	8	4	no	—
116	D₄×C₅₉	—	8	2	no	—
117	D₄×C₅₉	—	8	2	no	—
118	C₂×C₄×C₅₉	—	2	1	no	—
119	C₂×C₄×C₅₉	—	2	1	no	—
120	C₂×C₄×C₅₉	—	4	2	no	—
121	C₂×C₄×C₅₉	—	4	2	no	—
122	C₂×C₄×C₅₉	—	8	2	no	—
123	C₂×C₄×C₅₉	—	8	2	no	—
124	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
126	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
127	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
129	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
130	C₂×Dic₅₉	—	4	2	no	—
131	C₈×C₅₉	#102	1	1	yes	characteristic
132	C₈×C₅₉	#102	1	1	yes	characteristic
133	C₈×C₅₉	#102	1	1	yes	characteristic
134	C₈×C₅₉	#102	1	1	yes	characteristic
135	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
136	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
137	C₈×C₅₉	#124	1	1	yes	characteristic
138	C₈×C₅₉	#127	1	1	yes	characteristic
139	C₅₉⋊C₈	—	4	2	no	—
140	#13259	—	59	1	no	maximal, Sylow
141	C₂²×C₅₉	—	4	1	no	—
142	C₂²×C₅₉	—	8	4	no	—
143	C₂²×C₅₉	—	8	2	no	—
144	C₂²×C₅₉	#418	1	1	yes	characteristic
145	C₂²×C₅₉	#459	1	1	yes	characteristic
146	C₂²×C₅₉	#460	1	1	yes	characteristic
147	C₄×C₅₉	—	2	2	no	—
148	C₄×C₅₉	—	4	1	no	—
149	C₄×C₅₉	—	8	2	no	—
150	C₄×C₅₉	#390	1	1	yes	characteristic
151	C₄×C₅₉	C₂×#102	1	1	yes	characteristic
152	C₄×C₅₉	#459	1	1	yes	characteristic
153	C₄×C₅₉	#460	1	1	yes	characteristic
154	Dic₅₉	—	8	2	no	—
155	#3147	D₅₉	1	1	yes	characteristic
156	C₂×#364	—	59	1	no	—
157	#2145	—	59	2	no	—
158	#2143	—	59	2	no	—
159	#1772	—	59	1	no	—
160	C₂×C₅₉	—	2	1	no	—
161	C₂×C₅₉	—	2	1	no	—
162	C₂×C₅₉	—	2	1	no	—
163	C₂×C₅₉	—	16	2	no	—
164	C₂×C₅₉	#1982	1	1	yes	characteristic
165	C₂×#135	—	2	1	no	—
166	C₂×#135	C₂×D₅₉	1	2	yes	—
167	#532	—	2	1	no	—
168	C₂×M₃₂	—	59	2	no	—
169	#495	—	2	1	no	—
170	#495	Dic₅₉	1	2	yes	complemented
171	#494	Dic₅₉	1	1	yes	characteristic, complemented
172	#493	Dic₅₉	1	1	yes	characteristic, complemented
173	#492	—	2	1	no	—
174	C₂×#108	—	59	1	no	—
175	#427	—	59	1	no	—
176	#405	—	118	1	no	—
177	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
178	#364	—	118	1	no	—
179	#364	—	118	1	no	—
180	#360	—	118	4	no	—
181	#359	—	118	4	no	—
182	#339	—	118	1	no	—
183	C₅₉	#13259	1	1	yes	characteristic, complemented, Sylow
184	C₂×#41	—	2	2	no	—
185	#135	—	2	2	no	—
186	#135	—	2	2	no	—
187	#135	—	2	2	no	—
188	#135	—	2	2	no	—
189	#135	—	4	1	no	—
190	#135	—	4	1	no	—
191	#135	—	4	1	no	—
192	#135	—	4	1	no	—
193	C₂×Q₁₆	—	2	1	no	—
194	C₂×Q₁₆	C₂×Dic₅₉	1	2	yes	—
195	C₂×SD₁₆	—	2	2	no	—
196	C₂×SD₁₆	—	2	2	no	—
197	C₂×D₈	—	2	1	no	—
198	C₂×D₈	C₂×Dic₅₉	1	2	yes	—
199	C₂×M₁₆	—	118	1	no	—
200	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
201	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
202	C₄⋊Q₈	—	2	1	no	—
203	#127	—	2	1	no	—
204	#124	—	2	2	no	—
205	#121	—	4	2	no	—
206	C₄×D₄	—	4	2	no	—
207	#117	—	118	1	no	—
208	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
209	M₃₂	—	118	2	no	—
210	M₃₂	—	118	2	no	—
211	C₂×C₁₆	—	118	2	no	—
212	#108	—	59	2	no	—
213	#108	—	118	1	no	—
214	#107	—	118	1	no	—
215	#106	—	59	2	no	—
216	#97	—	236	1	no	—
217	C₄×C₈	—	236	1	no	—
218	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
219	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
220	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
221	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
222	#41	—	4	2	no	—
223	#41	—	4	2	no	—
224	#41	—	4	2	no	—
225	#41	—	4	2	no	—
226	C₂×Q₈	—	2	2	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	4	2	no	—
229	C₂×D₄	—	4	2	no	—
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	4	2	no	—
232	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
233	Q₁₆	—	2	2	no	—
234	Q₁₆	—	2	2	no	—
235	Q₁₆	—	4	1	no	—
236	Q₁₆	—	4	1	no	—
237	SD₁₆	—	4	2	no	—
238	SD₁₆	—	4	2	no	—
239	SD₁₆	—	4	2	no	—
240	SD₁₆	—	4	2	no	—
241	D₈	—	2	2	no	—
242	D₈	—	2	2	no	—
243	D₈	—	4	1	no	—
244	D₈	—	4	1	no	—
245	M₁₆	—	118	2	no	—
246	M₁₆	—	236	1	no	—
247	C₂×C₈	—	236	1	no	—
248	C₂×C₈	#11745669	1	1	yes	characteristic
249	C₂×C₈	#11745669	1	1	yes	characteristic
250	C₂×C₈	#11745670	1	1	yes	characteristic
251	C₂×C₈	#11745670	1	1	yes	characteristic
252	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
253	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
254	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
255	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
256	C₂×C₈	#11745687	1	1	yes	characteristic
257	C₂×C₈	#11745687	1	1	yes	characteristic
258	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
259	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
260	C₄⋊C₄	—	4	2	no	—
261	C₄⋊C₄	—	118	2	no	—
262	C₂²⋊C₄	—	4	4	no	—
263	C₂²⋊C₄	—	236	1	no	—
264	C₄²	—	2	1	no	—
265	C₄²	—	236	1	no	—
266	C₄²	C₂²⋊Dic₅₉	1	1	yes	characteristic
267	C₄²	C₂²⋊Dic₅₉	1	1	yes	characteristic
268	C₁₆	—	236	2	no	—
269	C₂³	—	4	2	no	—
270	C₂³	#14273753	1	1	yes	characteristic
271	Q₈	—	4	2	no	—
272	Q₈	—	4	2	no	—
273	D₄	—	4	2	no	—
274	D₄	—	4	2	no	—
275	D₄	—	8	4	no	—
276	D₄	—	8	2	no	—
277	D₄	—	8	2	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	8	2	no	—
283	C₂×C₄	—	8	2	no	—
284	C₂×C₄	—	236	2	no	—
285	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
286	C₂×C₄	#14273698	1	1	yes	characteristic
287	C₂×C₄	#14273698	1	1	yes	characteristic
288	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
289	C₂×C₄	C₂×#11745670	1	1	yes	characteristic
290	C₂×C₄	#14273753	1	1	yes	characteristic
291	C₈	—	236	2	no	—
292	C₈	#14273648	1	1	yes	characteristic
293	C₈	#14273648	1	1	yes	characteristic
294	C₈	#14273648	1	1	yes	characteristic
295	C₈	#14273648	1	1	yes	characteristic
296	C₈	D₄×Dic₅₉	1	1	yes	characteristic
297	C₈	D₄×Dic₅₉	1	1	yes	characteristic
298	C₈	#14273714	1	1	yes	characteristic
299	C₈	#14273723	1	1	yes	characteristic
300	C₂²	—	4	1	no	—
301	C₂²	—	8	4	no	—
302	C₂²	—	8	2	no	—
303	C₂²	#16837407	1	1	yes	characteristic
304	C₂²	#16837594	1	1	yes	characteristic
305	C₂²	#16837599	1	1	yes	characteristic
306	C₄	—	2	2	no	—
307	C₄	—	4	1	no	—
308	C₄	—	8	2	no	—
309	C₄	—	472	2	no	—
310	C₄	#16837305	1	1	yes	characteristic
311	C₄	C₂×#14273648	1	1	yes	characteristic
312	C₄	#16837594	1	1	yes	characteristic
313	C₄	#16837599	1	1	yes	characteristic
314	C₂	—	2	1	no	—
315	C₂	—	2	1	no	—
316	C₂	—	2	1	no	—
317	C₂	—	16	2	no	—
318	C₂	#21425024	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16837594
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting, prime radical	C₅₉×#3147	C₂
socle	C₂×C₅₉	#1982
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#3147	D₅₉
hypocenter	C₅₉	#13259
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #3147

Complements

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

N	H ≅ G/N	type
1. C₂×#16837223	C₂	semidirect
2. #21425832	C₂	semidirect
3. #21425824	C₂	semidirect
4. #21423975	C₂	semidirect
19. C₅₉×#495	C₄	semidirect
20. C₅₉×#494	C₄	semidirect
21. C₅₉×#493	C₄	semidirect
30. #14273663	D₄	semidirect
32. #14273659	D₄	semidirect
170. #495	Dic₅₉	semidirect
171. #494	Dic₅₉	semidirect
172. #493	Dic₅₉	semidirect
183. C₅₉	#13259	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837223	2
#21425832	2
#21425824	2
#21423975	2
C₅₉×#3147	2
#13259	59