Subgroups — #28507210

Order 15104.

386 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₁₁₈	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×#16837177	C₂	1	1	yes	maximal, characteristic
4	#21422174	C₂	1	1	yes	maximal, characteristic
5	#21422173	C₂	1	1	yes	maximal, characteristic
6	#21422173	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#1374	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Dic₁₁₈	C₄	1	1	yes	characteristic
9	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₈×Dic₁₁₈	—	2	2	no	—
12	C₈×Dic₁₁₈	—	2	2	no	—
13	C₂×C₄⋊(C₅₉⋊C₈)	C₄	1	1	yes	characteristic
14	C₂×Dic₅₉⋊C₈	C₄	1	1	yes	characteristic
15	C₂×Dic₅₉⋊C₈	C₄	1	1	yes	characteristic
16	C₂×C₈×Dic₅₉	C₂²	1	1	yes	characteristic
17	C₂×C₈×Dic₅₉	C₂²	1	1	yes	characteristic
18	Dic₅₉⋊C₁₆	—	2	2	no	—
19	#16837177	—	2	2	no	—
20	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
23	C₂²×Dic₁₁₈	C₈	1	1	yes	characteristic
24	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×Dic₁₁₈	—	2	2	no	—
27	C₄×Dic₁₁₈	—	2	2	no	—
28	C₂×C₄⋊Dic₅₉	C₈	1	1	yes	characteristic
29	C₂×Dic₅₉⋊C₄	C₈	1	1	yes	characteristic
30	C₂×Dic₅₉⋊C₄	C₈	1	1	yes	characteristic
31	C₂×C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₅₉⋊C₁₆	—	2	1	no	—
34	C₂×C₅₉⋊C₁₆	—	2	1	no	—
35	C₂×C₅₉⋊C₁₆	—	2	1	no	—
36	C₂×C₅₉⋊C₁₆	—	2	1	no	—
37	C₂×C₅₉⋊C₁₆	—	2	1	no	—
38	C₂×C₅₉⋊C₁₆	Q₈	1	2	yes	complemented
39	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
40	Dic₅₉⋊C₈	—	2	2	no	—
41	Dic₅₉⋊C₈	—	2	2	no	—
42	C₈×Dic₅₉	—	2	1	no	—
43	C₈×Dic₅₉	—	2	1	no	—
44	C₈×Dic₅₉	D₄	1	2	yes	—
45	C₈×Dic₅₉	D₄	1	2	yes	—
46	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
49	C₂×C₁₆×C₅₉	—	2	1	no	—
50	C₂×C₁₆×C₅₉	—	2	1	no	—
51	C₂×C₁₆×C₅₉	—	2	1	no	—
52	C₂×C₁₆×C₅₉	—	2	1	no	—
53	C₂×C₁₆×C₅₉	—	2	1	no	—
54	C₂×C₁₆×C₅₉	Q₈	1	2	yes	complemented
55	C₄×C₈×C₅₉	—	2	1	no	—
56	C₄×C₈×C₅₉	—	2	1	no	—
57	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
58	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
60	C₂²×Dic₅₉	C₂×C₈	1	1	yes	characteristic
61	C₂²×Dic₅₉	C₂×C₈	1	1	yes	characteristic
62	C₂²×Dic₅₉	C₂×C₈	1	1	yes	characteristic
63	C₂²×Dic₅₉	C₂×C₈	1	1	yes	characteristic
64	C₂×Dic₁₁₈	—	2	2	no	—
65	C₂×Dic₁₁₈	—	2	2	no	—
66	C₂×Dic₁₁₈	—	2	2	no	—
67	C₂×C₈×C₅₉	—	2	1	no	—
68	C₂×C₈×C₅₉	—	2	1	no	—
69	C₂×C₈×C₅₉	—	2	1	no	—
70	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
73	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
74	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
75	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
76	C₄²×C₅₉	—	2	1	no	—
77	C₄²×C₅₉	—	2	1	no	—
78	C₄⋊Dic₅₉	—	2	2	no	—
79	Dic₅₉⋊C₄	—	2	2	no	—
80	Dic₅₉⋊C₄	—	2	2	no	—
81	C₄×Dic₅₉	—	2	1	no	—
82	C₄×Dic₅₉	—	2	1	no	—
83	C₄×Dic₅₉	C₂²⋊C₄	1	2	yes	—
84	C₄×Dic₅₉	M₁₆	1	2	yes	—
85	C₂×C₅₉⋊C₈	—	2	1	no	—
86	C₂×C₅₉⋊C₈	—	2	1	no	—
87	C₂×C₅₉⋊C₈	—	2	1	no	—
88	C₂×C₅₉⋊C₈	—	2	1	no	—
89	C₂×C₅₉⋊C₈	—	2	1	no	—
90	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	2	yes	—
91	C₁₆×C₅₉	—	2	2	no	—
92	C₁₆×C₅₉	—	4	1	no	—
93	C₅₉⋊C₁₆	—	2	2	no	—
94	C₅₉⋊C₁₆	—	4	1	no	—
95	C₂³×C₅₉	C₂²×C₈	1	1	yes	characteristic
96	C₂×C₄×C₅₉	—	2	1	no	—
97	C₂×C₄×C₅₉	—	2	1	no	—
98	C₂×C₄×C₅₉	—	2	1	no	—
99	C₂×C₄×C₅₉	—	2	1	no	—
100	C₂×C₄×C₅₉	—	2	1	no	—
101	C₂×C₄×C₅₉	M₃₂	1	2	yes	—
102	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₅₉	C₄×Q₈	1	1	yes	characteristic
104	C₂×C₄×C₅₉	C₄×Q₈	1	1	yes	characteristic
105	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
106	C₂×C₄×C₅₉	#131	1	1	yes	characteristic
107	C₂×C₄×C₅₉	#131	1	1	yes	characteristic
108	C₂×Dic₅₉	—	2	1	no	—
109	C₂×Dic₅₉	—	2	1	no	—
110	C₂×Dic₅₉	—	2	1	no	—
111	C₂×Dic₅₉	—	2	1	no	—
112	C₂×Dic₅₉	—	2	1	no	—
113	C₂×Dic₅₉	—	2	1	no	—
114	C₂×Dic₅₉	—	2	1	no	—
115	C₂×Dic₅₉	—	2	1	no	—
116	C₂×Dic₅₉	—	2	1	no	—
117	C₂×Dic₅₉	C₂²⋊C₈	1	2	yes	—
118	C₂×Dic₅₉	C₂²⋊C₈	1	2	yes	—
119	C₂×Dic₅₉	M₃₂	1	2	yes	—
120	Dic₁₁₈	—	2	8	no	—
121	C₈×C₅₉	—	4	1	no	—
122	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
123	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
124	C₈×C₅₉	#124	1	1	yes	characteristic
125	C₈×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
126	C₅₉⋊C₈	—	2	2	no	—
127	C₅₉⋊C₈	—	4	1	no	—
128	#13264	—	59	1	no	maximal, Sylow
129	C₂²×C₅₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
130	C₂²×C₅₉	C₈×Q₈	1	1	yes	characteristic
131	C₂²×C₅₉	C₈×Q₈	1	1	yes	characteristic
132	C₂²×C₅₉	C₂×M₃₂	1	1	yes	characteristic
133	C₂²×C₅₉	C₂×M₃₂	1	1	yes	characteristic
134	C₂²×C₅₉	#504	1	1	yes	characteristic
135	C₂²×C₅₉	#504	1	1	yes	characteristic
136	C₄×C₅₉	—	2	2	no	—
137	C₄×C₅₉	—	4	1	no	—
138	C₄×C₅₉	#391	1	1	yes	characteristic
139	C₄×C₅₉	#408	1	1	yes	characteristic
140	C₄×C₅₉	#446	1	1	yes	characteristic
141	C₄×C₅₉	#446	1	1	yes	characteristic
142	Dic₅₉	—	2	2	no	—
143	Dic₅₉	—	2	2	no	—
144	Dic₅₉	—	2	2	no	—
145	Dic₅₉	—	4	1	no	—
146	C₂×C₈×Q₈	—	59	1	no	—
147	C₂×C₄⋊C₁₆	—	59	1	no	—
148	C₂×#346	—	59	1	no	—
149	#1374	—	59	1	no	—
150	#1374	—	59	1	no	—
151	#1374	—	59	1	no	—
152	#1374	D₅₉	1	1	yes	characteristic
153	C₂×C₅₉	#1984	1	1	yes	characteristic
154	C₂×C₅₉	#2112	1	1	yes	characteristic
155	C₂×C₅₉	#2113	1	1	yes	characteristic
156	C₂×C₅₉	#2183	1	1	yes	characteristic
157	C₂×C₅₉	#2183	1	1	yes	characteristic
158	C₂×C₅₉	#2183	1	1	yes	characteristic
159	C₂×C₅₉	#2183	1	1	yes	characteristic
160	C₂×C₄×Q₈	—	59	1	no	—
161	C₂²×C₁₆	—	59	1	no	—
162	C₂²×C₁₆	—	59	1	no	—
163	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
164	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
165	C₈×Q₈	—	118	2	no	—
166	C₈×Q₈	—	118	2	no	—
167	C₂×C₄⋊C₈	—	59	1	no	—
168	C₂×C₄⋊C₈	—	59	1	no	—
169	C₂×C₄⋊C₈	—	59	1	no	—
170	C₂×C₄×C₈	—	59	1	no	—
171	C₂×C₄×C₈	—	59	1	no	—
172	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
173	C₄⋊C₁₆	—	118	2	no	—
174	#346	—	118	2	no	—
175	C₅₉	#13264	1	1	yes	characteristic, complemented, Sylow
176	C₂²×Q₈	—	59	1	no	—
177	C₂²×C₈	—	59	1	no	—
178	C₂²×C₈	—	59	1	no	—
179	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
180	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
181	C₄×Q₈	—	118	2	no	—
182	C₄×Q₈	—	118	2	no	—
183	C₂×C₄⋊C₄	—	59	1	no	—
184	C₂×C₄⋊C₄	—	59	1	no	—
185	C₂×C₄⋊C₄	—	59	1	no	—
186	C₂×C₄²	—	59	1	no	—
187	C₂×C₄²	—	59	1	no	—
188	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
189	C₂×C₁₆	—	2	1	no	—
190	C₂×C₁₆	—	2	1	no	—
191	C₂×C₁₆	—	2	1	no	—
192	C₂×C₁₆	—	2	1	no	—
193	C₂×C₁₆	—	2	1	no	—
194	C₂×C₁₆	—	59	2	no	—
195	C₂×C₁₆	—	118	1	no	—
196	C₂×C₁₆	—	118	1	no	—
197	C₂×C₁₆	—	118	1	no	—
198	C₂×C₁₆	—	118	1	no	—
199	C₂×C₁₆	—	118	1	no	—
200	C₂×C₁₆	Dic₁₁₈	1	2	yes	complemented
201	C₄⋊C₈	—	118	2	no	—
202	C₄⋊C₈	—	118	2	no	—
203	C₄⋊C₈	—	118	2	no	—
204	C₄×C₈	—	2	1	no	—
205	C₄×C₈	—	2	1	no	—
206	C₄×C₈	—	59	2	no	—
207	C₄×C₈	—	59	2	no	—
208	C₄×C₈	—	118	1	no	—
209	C₄×C₈	—	118	1	no	—
210	C₂×Q₈	—	118	2	no	—
211	C₂×Q₈	—	118	2	no	—
212	C₂×Q₈	—	118	2	no	—
213	C₂²×C₄	—	59	1	no	—
214	C₂²×C₄	—	59	1	no	—
215	C₂²×C₄	—	59	1	no	—
216	C₂²×C₄	—	59	1	no	—
217	C₂²×C₄	C₈×D₅₉	1	1	yes	characteristic
218	C₂²×C₄	C₈×D₅₉	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	59	2	no	—
224	C₂×C₈	—	118	1	no	—
225	C₂×C₈	—	118	1	no	—
226	C₂×C₈	—	118	1	no	—
227	C₂×C₈	—	118	1	no	—
228	C₂×C₈	—	118	1	no	—
229	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
230	C₂×C₈	#11745685	1	1	yes	characteristic
231	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
232	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
233	C₂×C₈	#11745687	1	1	yes	characteristic
234	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
235	C₄⋊C₄	—	118	2	no	—
236	C₄⋊C₄	—	118	2	no	—
237	C₄⋊C₄	—	118	2	no	—
238	C₄²	—	2	1	no	—
239	C₄²	—	2	1	no	—
240	C₄²	—	59	2	no	—
241	C₄²	—	59	2	no	—
242	C₄²	—	118	1	no	—
243	C₄²	—	118	1	no	—
244	C₁₆	—	2	2	no	—
245	C₁₆	—	4	1	no	—
246	C₁₆	—	118	2	no	—
247	C₁₆	—	236	1	no	—
248	C₂³	C₂×C₈×D₅₉	1	1	yes	characteristic
249	Q₈	—	118	8	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	59	2	no	—
256	C₂×C₄	—	59	2	no	—
257	C₂×C₄	—	59	2	no	—
258	C₂×C₄	—	118	1	no	—
259	C₂×C₄	—	118	1	no	—
260	C₂×C₄	—	118	1	no	—
261	C₂×C₄	—	118	1	no	—
262	C₂×C₄	—	118	1	no	—
263	C₂×C₄	—	118	1	no	—
264	C₂×C₄	—	118	1	no	—
265	C₂×C₄	—	118	1	no	—
266	C₂×C₄	—	118	1	no	—
267	C₂×C₄	#14273666	1	2	yes	—
268	C₂×C₄	C₂²⋊C₄×D₅₉	1	1	yes	characteristic
269	C₂×C₄	#14273692	1	1	yes	characteristic
270	C₂×C₄	C₄×Dic₁₁₈	1	1	yes	characteristic
271	C₂×C₄	M₁₆×D₅₉	1	1	yes	characteristic
272	C₂×C₄	#14273733	1	1	yes	characteristic
273	C₂×C₄	#14273735	1	1	yes	characteristic
274	C₈	—	4	1	no	—
275	C₈	—	118	2	no	—
276	C₈	—	236	1	no	—
277	C₈	#14273704	1	1	yes	characteristic
278	C₈	#14273705	1	1	yes	characteristic
279	C₈	#14273713	1	1	yes	characteristic
280	C₈	#14273726	1	1	yes	characteristic
281	C₂²	C₂²⋊C₈×D₅₉	1	1	yes	characteristic
282	C₂²	#16837526	1	1	yes	characteristic
283	C₂²	C₈×Dic₁₁₈	1	1	yes	characteristic
284	C₂²	M₃₂×D₅₉	1	1	yes	characteristic
285	C₂²	C₂×#14273666	1	1	yes	characteristic
286	C₂²	#16837818	1	1	yes	characteristic
287	C₂²	#16837820	1	1	yes	characteristic
288	C₄	—	2	2	no	—
289	C₄	—	4	1	no	—
290	C₄	—	118	2	no	—
291	C₄	—	118	2	no	—
292	C₄	—	118	2	no	—
293	C₄	—	236	1	no	—
294	C₄	#16837306	1	1	yes	characteristic
295	C₄	#16837367	1	1	yes	characteristic
296	C₄	#16837530	1	1	yes	characteristic
297	C₄	#16837531	1	1	yes	characteristic
298	C₂	#21425030	1	1	yes	characteristic
299	C₂	#21425705	1	1	yes	characteristic
300	C₂	#21425708	1	1	yes	characteristic
301	C₂	#21426001	1	1	yes	characteristic
302	C₂	#21426002	1	1	yes	characteristic
303	C₂	#21426003	1	1	yes	characteristic
304	C₂	#21426004	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₉
commutator, socle, prime radical	C₂³×C₅₉	C₂²×C₈
Fitting	C₅₉×#1374	C₂
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#1374	D₅₉
hypocenter	C₅₉	#13264
LCS	G ▷ C₂³×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
38. C₂×C₅₉⋊C₁₆	Q₈	semidirect
54. C₂×C₁₆×C₅₉	Q₈	semidirect
175. C₅₉	#13264	semidirect
200. C₂×C₁₆	Dic₁₁₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₁₁₈	2
C₂×Dic₅₉⋊C₁₆	2
C₂×#16837177	2
#21422174	2
#21422173	2
C₅₉×#1374	2
#13264	59