Subgroups — #28742200

Order 15104.

530 conjugacy classes of subgroups.

The subgroup lattice has 300 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	#21426003	C₂	1	4	yes	maximal, complemented
3	#21426001	C₂	1	2	yes	maximal, complemented
4	#21425998	C₂	1	2	yes	maximal, complemented
5	#21425851	C₂	1	2	yes	maximal
6	C₄×#14273667	C₂	1	1	yes	maximal, characteristic
7	M₃₂×Dic₅₉	C₂	1	2	yes	maximal
8	C₅₉×#2150	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₁₁₈	C₄	1	1	yes	characteristic
10	C₂×#14273667	C₂²	1	1	yes	characteristic
11	C₂×#14273667	C₂²	1	1	yes	characteristic
12	C₈×Dic₁₁₈	C₄	1	2	yes	—
13	C₈×Dic₁₁₈	C₄	1	1	yes	characteristic
14	C₈×Dic₁₁₈	C₄	1	1	yes	characteristic
15	C₈×Dic₁₁₈	C₂²	1	1	yes	characteristic
16	C₈×Dic₁₁₈	C₂²	1	1	yes	characteristic
17	C₈×Dic₁₁₈	C₂²	1	2	yes	—
18	C₂×C₄⋊(C₅₉⋊C₈)	C₄	1	1	yes	characteristic
19	C₂×Dic₅₉⋊C₈	C₄	1	2	yes	—
20	C₂×C₈×Dic₅₉	C₂²	1	2	yes	—
21	Dic₅₉⋊C₁₆	C₂²	1	4	yes	—
22	Dic₅₉⋊C₁₆	C₂²	1	4	yes	—
23	#16837178	C₂²	1	2	yes	—
24	#16837177	C₂²	1	4	yes	—
25	C₄×C₅₉⋊C₁₆	C₂²	1	2	yes	—
26	C₁₆×Dic₅₉	C₂²	1	4	yes	—
27	C₂×M₃₂×C₅₉	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
29	C₅₉×C₄⋊C₁₆	C₂²	1	2	yes	—
30	C₅₉×C₄⋊C₁₆	C₂²	1	2	yes	—
31	C₂²×Dic₁₁₈	C₈	1	1	yes	characteristic
32	C₂²×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
33	C₄×Dic₁₁₈	C₈	1	1	yes	characteristic
34	C₄×Dic₁₁₈	C₈	1	2	yes	—
35	C₄×Dic₁₁₈	C₈	1	1	yes	characteristic
36	C₄×Dic₁₁₈	C₂×C₄	1	1	yes	characteristic
37	C₄×Dic₁₁₈	C₂×C₄	1	2	yes	—
38	C₄×Dic₁₁₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄⋊Dic₅₉	C₈	1	1	yes	characteristic
40	C₂×Dic₅₉⋊C₄	C₈	1	2	yes	—
41	C₂×C₄×Dic₅₉	C₂×C₄	1	2	yes	—
42	#14273667	—	2	2	no	—
43	#14273667	Q₈	1	4	yes	complemented
44	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
45	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
46	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	1	yes	characteristic
47	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	1	yes	characteristic
48	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	2	yes	—
49	Dic₅₉⋊C₈	C₂×C₄	1	2	yes	—
50	Dic₅₉⋊C₈	C₂×C₄	1	2	yes	—
51	Dic₅₉⋊C₈	C₂×C₄	1	2	yes	—
52	Dic₅₉⋊C₈	C₂×C₄	1	2	yes	—
53	C₈×Dic₅₉	C₂×C₄	1	2	yes	—
54	C₈×Dic₅₉	C₂×C₄	1	2	yes	—
55	C₈×Dic₅₉	C₂³	1	2	yes	—
56	C₈×Dic₅₉	C₂³	1	2	yes	—
57	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
60	M₃₂×C₅₉	—	2	4	no	—
61	C₂×C₁₆×C₅₉	C₂³	1	4	yes	—
62	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
65	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
66	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
67	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
68	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
69	C₂²×Dic₅₉	C₂×C₈	1	2	yes	—
70	C₂²×Dic₅₉	C₂×C₈	1	2	yes	—
71	C₂×Dic₁₁₈	—	2	4	no	—
72	C₂×Dic₁₁₈	C₂×C₈	1	1	yes	characteristic
73	C₂×Dic₁₁₈	C₂×C₈	1	2	yes	—
74	C₂×Dic₁₁₈	C₂×C₈	1	1	yes	characteristic
75	C₂×C₈×C₅₉	—	2	2	no	—
76	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
80	C₂×C₈×C₅₉	#41	1	2	yes	—
81	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
82	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
83	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
84	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
86	C₄⋊Dic₅₉	C₂×C₈	1	2	yes	—
87	C₄⋊Dic₅₉	C₂×C₈	1	1	yes	characteristic
88	C₄⋊Dic₅₉	C₂×C₈	1	1	yes	characteristic
89	Dic₅₉⋊C₄	C₂×C₈	1	2	yes	—
90	Dic₅₉⋊C₄	C₂×C₈	1	2	yes	—
91	Dic₅₉⋊C₄	C₂×C₈	1	2	yes	—
92	Dic₅₉⋊C₄	C₂×C₈	1	2	yes	—
93	C₄×Dic₅₉	C₂×C₈	1	2	yes	—
94	C₄×Dic₅₉	C₂×C₈	1	2	yes	—
95	C₄×Dic₅₉	C₂²×C₄	1	2	yes	—
96	C₄×Dic₅₉	C₂²×C₄	1	2	yes	—
97	C₂×C₅₉⋊C₈	—	2	4	no	—
98	C₂×C₅₉⋊C₈	C₂²×C₄	1	2	yes	—
99	C₂×C₅₉⋊C₈	C₂²×C₄	1	2	yes	—
100	C₁₆×C₅₉	—	2	4	no	—
101	C₅₉⋊C₁₆	—	2	2	no	—
102	C₅₉⋊C₁₆	C₂×Q₈	1	4	yes	complemented
103	C₂³×C₅₉	C₂²×C₈	1	1	yes	characteristic
104	C₂×C₄×C₅₉	—	2	2	no	—
105	C₂×C₄×C₅₉	—	4	1	no	—
106	C₂×C₄×C₅₉	C₄×Q₈	1	2	yes	—
107	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₄×C₅₉	#131	1	2	yes	—
113	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
114	C₂×Dic₅₉	—	2	4	no	—
115	C₂×Dic₅₉	—	4	2	no	—
116	C₂×Dic₅₉	C₂²×C₈	1	2	yes	—
117	C₂×Dic₅₉	C₂²×C₈	1	2	yes	—
118	C₂×Dic₅₉	C₂²×C₈	1	2	yes	—
119	C₂×Dic₅₉	C₂²×C₈	1	2	yes	—
120	Dic₁₁₈	—	2	2	no	—
121	Dic₁₁₈	—	2	4	no	—
122	Dic₁₁₈	M₃₂	1	4	yes	complemented
123	C₈×C₅₉	—	2	1	no	—
124	C₈×C₅₉	—	2	1	no	—
125	C₈×C₅₉	C₄×Q₈	1	2	yes	—
126	C₈×C₅₉	C₂²×Q₈	1	1	yes	characteristic
127	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
128	C₅₉⋊C₈	—	2	2	no	—
129	C₅₉⋊C₈	—	2	2	no	—
130	Q₈×M₃₂	—	59	1	no	maximal, Sylow
131	C₂²×C₅₉	—	2	1	no	—
132	C₂²×C₅₉	—	2	1	no	—
133	C₂²×C₅₉	C₈×Q₈	1	2	yes	—
134	C₂²×C₅₉	C₂³×C₈	1	1	yes	characteristic
135	C₄×C₅₉	—	2	1	no	—
136	C₄×C₅₉	—	2	1	no	—
137	C₄×C₅₉	—	2	1	no	—
138	C₄×C₅₉	C₈×Q₈	1	2	yes	—
139	C₄×C₅₉	C₂×M₃₂	1	2	yes	—
140	C₄×C₅₉	C₂×C₄×Q₈	1	1	yes	characteristic
141	C₄×C₅₉	C₂×#131	1	1	yes	characteristic
142	Dic₅₉	—	2	2	no	—
143	Dic₅₉	—	2	2	no	—
144	Dic₅₉	—	2	2	no	—
145	Dic₅₉	C₂×M₃₂	1	4	yes	—
146	C₂×C₈×Q₈	—	59	1	no	—
147	#2183	—	59	2	no	—
148	#2183	—	59	4	no	—
149	Q₈×C₁₆	—	59	2	no	—
150	#2150	—	59	2	no	—
151	#2150	D₅₉	1	1	yes	characteristic
152	C₄×M₃₂	—	59	2	no	—
153	C₄×M₃₂	—	59	1	no	—
154	C₂×C₅₉	—	2	2	no	—
155	C₂×C₅₉	C₂×C₈×Q₈	1	1	yes	characteristic
156	C₂×C₅₉	C₂²×M₃₂	1	1	yes	characteristic
157	C₂×C₅₉	#3407	1	1	yes	characteristic
158	C₂×C₄×Q₈	—	59	1	no	—
159	C₂×M₃₂	—	59	1	no	—
160	C₂×M₃₂	—	59	1	no	—
161	C₂×M₃₂	C₂×D₅₉	1	2	yes	—
162	C₈×Q₈	—	59	2	no	—
163	C₈×Q₈	—	59	1	no	—
164	C₈×Q₈	—	59	1	no	—
165	C₈×Q₈	—	59	1	no	—
166	C₈×Q₈	—	59	2	no	—
167	C₈×Q₈	—	59	1	no	—
168	C₂×C₄⋊C₈	—	59	2	no	—
169	C₂×C₄⋊C₈	—	59	1	no	—
170	C₂×C₄×C₈	—	59	2	no	—
171	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
172	C₄⋊C₁₆	—	59	4	no	—
173	C₄⋊C₁₆	—	59	4	no	—
174	C₄⋊C₁₆	C₂×D₅₉	1	2	yes	—
175	C₄⋊C₁₆	C₂×D₅₉	1	2	yes	—
176	#346	—	59	4	no	—
177	#346	—	59	2	no	—
178	C₄×C₁₆	—	59	4	no	—
179	C₄×C₁₆	—	59	2	no	—
180	C₅₉	Q₈×M₃₂	1	1	yes	characteristic, complemented, Sylow
181	C₂²×Q₈	—	59	1	no	—
182	C₂²×C₈	—	59	2	no	—
183	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
184	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
185	C₄×Q₈	—	59	1	no	—
186	C₄×Q₈	—	59	2	no	—
187	C₄×Q₈	—	59	1	no	—
188	C₄×Q₈	—	59	1	no	—
189	C₄×Q₈	—	59	2	no	—
190	C₄×Q₈	—	59	1	no	—
191	C₂×C₄⋊C₄	—	59	2	no	—
192	C₂×C₄⋊C₄	—	59	1	no	—
193	C₂×C₄²	—	59	2	no	—
194	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
195	M₃₂	—	2	4	no	—
196	M₃₂	—	59	4	no	—
197	M₃₂	—	118	2	no	—
198	C₂×C₁₆	—	59	2	no	—
199	C₂×C₁₆	—	59	2	no	—
200	C₂×C₁₆	C₂²×D₅₉	1	4	yes	—
201	C₄⋊C₈	—	59	2	no	—
202	C₄⋊C₈	—	59	2	no	—
203	C₄⋊C₈	—	59	2	no	—
204	C₄⋊C₈	—	59	2	no	—
205	C₄⋊C₈	—	59	1	no	—
206	C₄⋊C₈	—	59	2	no	—
207	C₄⋊C₈	—	59	1	no	—
208	C₄×C₈	—	59	2	no	—
209	C₄×C₈	—	59	2	no	—
210	C₄×C₈	—	59	2	no	—
211	C₄×C₈	—	59	2	no	—
212	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
213	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
214	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
215	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
216	C₂×Q₈	—	59	1	no	—
217	C₂×Q₈	—	59	2	no	—
218	C₂×Q₈	—	59	1	no	—
219	C₂×Q₈	—	118	4	no	—
220	C₂²×C₄	—	59	2	no	—
221	C₂²×C₄	—	59	2	no	—
222	C₂²×C₄	C₈×D₅₉	1	1	yes	characteristic
223	C₂²×C₄	C₈×D₅₉	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	59	2	no	—
227	C₂×C₈	—	59	2	no	—
228	C₂×C₈	—	118	4	no	—
229	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
230	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
231	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
232	C₂×C₈	#11745687	1	2	yes	—
233	C₂×C₈	Q₈×D₅₉	1	2	yes	—
234	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
235	C₄⋊C₄	—	59	2	no	—
236	C₄⋊C₄	—	59	2	no	—
237	C₄⋊C₄	—	59	2	no	—
238	C₄⋊C₄	—	59	2	no	—
239	C₄⋊C₄	—	59	2	no	—
240	C₄⋊C₄	—	59	1	no	—
241	C₄⋊C₄	—	59	1	no	—
242	C₄²	—	59	2	no	—
243	C₄²	—	59	2	no	—
244	C₄²	—	59	2	no	—
245	C₄²	—	59	2	no	—
246	C₄²	C₈×D₅₉	1	1	yes	characteristic
247	C₄²	C₈×D₅₉	1	1	yes	characteristic
248	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
249	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
250	C₁₆	—	2	4	no	—
251	C₁₆	—	59	4	no	—
252	C₁₆	—	118	2	no	—
253	C₂³	C₂×C₈×D₅₉	1	1	yes	characteristic
254	Q₈	—	59	4	no	—
255	Q₈	—	118	4	no	—
256	Q₈	—	118	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	59	2	no	—
260	C₂×C₄	—	59	2	no	—
261	C₂×C₄	—	59	2	no	—
262	C₂×C₄	—	59	2	no	—
263	C₂×C₄	—	118	4	no	—
264	C₂×C₄	—	236	2	no	—
265	C₂×C₄	#14273692	1	2	yes	—
266	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
267	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
268	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
269	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
270	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
271	C₂×C₄	#14273735	1	2	yes	—
272	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
273	C₈	—	2	1	no	—
274	C₈	—	2	1	no	—
275	C₈	—	118	2	no	—
276	C₈	—	118	2	no	—
277	C₈	#14273692	1	2	yes	—
278	C₈	C₂×Q₈×D₅₉	1	1	yes	characteristic
279	C₈	C₂×#11745687	1	1	yes	characteristic
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	#16837526	1	2	yes	—
283	C₂²	C₂²×C₈×D₅₉	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	59	4	no	—
288	C₄	—	118	2	no	—
289	C₄	—	118	2	no	—
290	C₄	—	118	2	no	—
291	C₄	#16837526	1	2	yes	—
292	C₄	M₃₂×D₅₉	1	2	yes	—
293	C₄	C₂×#14273692	1	1	yes	characteristic
294	C₄	C₂×#14273735	1	1	yes	characteristic
295	C₂	—	2	2	no	—
296	C₂	C₂×#16837526	1	1	yes	characteristic
297	C₂	C₂×M₃₂×D₅₉	1	1	yes	characteristic
298	C₂	#21437198	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₉
commutator, socle	C₂²×C₅₉	C₂³×C₈
Fitting	C₅₉×#2150	C₂
prime radical	C₂³×C₅₉	C₂²×C₈
cosocle	C₂×C₈×C₅₉	C₂⁴
hypercenter	#2150	D₅₉
hypocenter	C₅₉	Q₈×M₃₂
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
2. #21426003	C₂	semidirect
3. #21426001	C₂	semidirect
4. #21425998	C₂	semidirect
43. #14273667	Q₈	semidirect
102. C₅₉⋊C₁₆	C₂×Q₈	semidirect
122. Dic₁₁₈	M₃₂	semidirect
180. C₅₉	Q₈×M₃₂	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
#21426003	2
#21426001	2
#21425998	2
#21425851	2
C₄×#14273667	2
M₃₂×Dic₅₉	2
C₅₉×#2150	2
Q₈×M₃₂	59