Subgroups — #28741336

Order 15104.

818 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837818	C₂	1	2	yes	maximal
2	C₄×#14273733	C₂	1	1	yes	maximal, characteristic
3	#21425694	C₂	1	4	yes	maximal, complemented
4	#21425692	C₂	1	2	yes	maximal, complemented
5	C₄×#14273667	C₂	1	1	yes	maximal, characteristic, complemented
6	#21425689	C₂	1	2	yes	maximal, complemented
7	#346×D₅₉	C₂	1	2	yes	maximal, complemented
8	C₂×C₅₉×#346	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14273733	C₄	1	1	yes	characteristic
10	C₂×#14273733	C₂²	1	1	yes	characteristic
11	C₄×#11745685	C₄	1	1	yes	characteristic
12	#16837818	C₄	1	16	yes	complemented
13	C₂×#14273667	C₂²	1	2	yes	—
14	C₂×#14273666	C₂²	1	4	yes	—
15	C₂×C₁₆×D₅₉	C₂²	1	4	yes	—
16	#16837357	C₄	1	1	yes	characteristic
17	#16837357	C₂²	1	1	yes	characteristic
18	#16837356	C₄	1	2	yes	—
19	#16837356	C₂²	1	2	yes	—
20	C₄×#11745664	C₄	1	1	yes	characteristic
21	C₄×#11745659	C₄	1	2	yes	—
22	C₄×C₈×D₅₉	C₂²	1	2	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	C₂²×C₁₆×C₅₉	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
29	C₅₉×#346	C₂²	1	1	yes	characteristic
30	C₅₉×#346	C₂²	1	1	yes	characteristic
31	C₅₉×#346	C₂²	1	2	yes	—
32	C₂×#11745685	C₂×C₄	1	1	yes	characteristic
33	#14273733	C₈	1	8	yes	—
34	#14273733	C₂×C₄	1	8	yes	—
35	C₂×#11745664	C₂×C₄	1	1	yes	characteristic
36	C₂×#11745664	C₂×C₄	1	1	yes	characteristic
37	C₂×#11745659	C₂×C₄	1	2	yes	—
38	C₂×#11745659	C₂×C₄	1	2	yes	—
39	C₂×C₈×D₅₉	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₅₉	C₂³	1	2	yes	—
41	C₄×Dic₁₁₈	C₈	1	2	yes	—
42	C₄×D₂₃₆	C₈	1	2	yes	—
43	C₄×C₂²⋊D₅₉	C₈	1	4	yes	—
44	#14273686	C₂×C₄	1	1	yes	characteristic
45	#14273684	C₂×C₄	1	2	yes	—
46	C₄²×D₅₉	C₂×C₄	1	2	yes	—
47	#14273667	C₂×C₄	1	8	yes	complemented
48	#14273666	C₂×C₄	1	16	yes	complemented
49	C₂×C₅₉⋊C₁₆	C₂³	1	4	yes	—
50	C₁₆×D₅₉	C₂×C₄	1	16	yes	complemented
51	#14273638	C₂×C₄	1	2	yes	—
52	#14273637	C₂×C₄	1	2	yes	—
53	#14273637	C₂×C₄	1	2	yes	—
54	C₄×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
55	C₈×Dic₅₉	C₂×C₄	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	C₂×C₁₆×C₅₉	C₂×C₄	1	8	yes	complemented
61	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
62	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
63	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
67	C₂²×C₄×C₅₉	C₂×C₈	1	2	yes	—
68	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₂²⋊D₅₉	C₂×C₈	1	2	yes	—
70	#11745685	C₄²	1	4	yes	—
71	#11745685	C₂×C₈	1	4	yes	—
72	C₂×D₂₃₆	C₂×C₈	1	1	yes	characteristic
73	C₂×C₄×D₅₉	C₂×C₈	1	4	yes	—
74	C₂×C₄×D₅₉	C₂²×C₄	1	2	yes	—
75	C₂×Dic₁₁₈	C₂×C₈	1	1	yes	characteristic
76	C₂×C₈×C₅₉	C₂×C₈	1	4	yes	—
77	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₅₉	C₂²×C₄	1	4	yes	—
79	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
82	C₄²×C₅₉	C₂×C₈	1	2	yes	—
83	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
85	C₂²⋊Dic₅₉	C₂×C₈	1	2	yes	—
86	(C₂×D₅₉)⋊C₄	C₂×C₈	1	4	yes	—
87	C₄⋊Dic₅₉	C₂×C₈	1	2	yes	—
88	Dic₅₉⋊C₄	C₂×C₈	1	4	yes	—
89	C₄×Dic₅₉	C₂×C₈	1	4	yes	—
90	C₄×Dic₅₉	C₂²×C₄	1	2	yes	—
91	#11745664	C₄²	1	4	yes	—
92	#11745664	C₂×C₈	1	4	yes	—
93	C₂×C₅₉⋊C₈	C₂²×C₄	1	2	yes	—
94	C₂×C₅₉⋊C₈	C₂²×C₄	1	2	yes	—
95	#11745659	C₄²	1	8	yes	—
96	#11745659	C₂×C₈	1	8	yes	—
97	C₈×D₅₉	C₂×C₈	1	8	yes	—
98	C₈×D₅₉	C₂²×C₄	1	8	yes	—
99	C₁₆×C₅₉	C₂²×C₄	1	4	yes	—
100	C₁₆×C₅₉	C₂²×C₄	1	4	yes	—
101	C₅₉⋊C₁₆	C₂²×C₄	1	8	yes	—
102	C₂³×C₅₉	C₂²×C₈	1	1	yes	characteristic
103	C₂²×D₅₉	C₂²×C₈	1	2	yes	—
104	C₂×C₄×C₅₉	—	4	2	no	—
105	C₂×C₄×C₅₉	C₂×C₄²	1	2	yes	—
106	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
107	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
108	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
109	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
111	C₂²⋊D₅₉	C₄×C₈	1	8	yes	—
112	C₂×Dic₅₉	C₂²×C₈	1	4	yes	—
113	C₂×Dic₅₉	C₂²×C₈	1	2	yes	—
114	D₂₃₆	C₄×C₈	1	4	yes	—
115	C₄×D₅₉	—	4	4	no	—
116	C₄×D₅₉	C₂×C₄²	1	4	yes	—
117	C₄×D₅₉	C₂²×C₈	1	4	yes	—
118	Dic₁₁₈	C₄×C₈	1	4	yes	—
119	C₈×C₅₉	C₂×C₄²	1	2	yes	—
120	C₈×C₅₉	C₂²×C₈	1	2	yes	—
121	C₈×C₅₉	C₂²×C₈	1	2	yes	—
122	C₈×C₅₉	C₂³×C₄	1	2	yes	—
123	C₅₉⋊C₈	C₂×C₄²	1	4	yes	—
124	C₅₉⋊C₈	C₂²×C₈	1	4	yes	—
125	#33326	—	59	1	no	maximal, Sylow
126	C₂²×C₅₉	—	2	2	no	—
127	C₂²×C₅₉	C₂×C₄×C₈	1	2	yes	—
128	C₂²×C₅₉	C₂³×C₈	1	1	yes	characteristic
129	C₂×D₅₉	—	2	4	no	—
130	C₂×D₅₉	C₂×C₄×C₈	1	4	yes	—
131	C₄×C₅₉	—	2	2	no	—
132	C₄×C₅₉	—	2	2	no	—
133	C₄×C₅₉	C₂×C₄×C₈	1	2	yes	—
134	C₄×C₅₉	C₂²×C₄²	1	1	yes	characteristic
135	C₄×C₅₉	C₂³×C₈	1	1	yes	characteristic
136	Dic₅₉	—	2	4	no	—
137	Dic₅₉	C₂×C₄×C₈	1	4	yes	—
138	C₂×#504	—	59	2	no	—
139	C₄×#131	—	59	1	no	—
140	#2108	—	59	2	no	—
141	#2108	—	59	4	no	—
142	C₄×M₃₂	—	59	1	no	—
143	C₄×M₃₂	—	59	2	no	—
144	C₂×#346	—	59	2	no	—
145	C₂×#346	D₅₉	1	1	yes	characteristic, complemented
146	C₂×C₅₉	—	2	2	no	—
147	C₂×C₅₉	C₂²×C₄×C₈	1	1	yes	characteristic
148	C₂×C₅₉	#3407	1	2	yes	—
149	D₅₉	—	2	4	no	—
150	C₂×#131	—	59	1	no	—
151	C₂×#131	—	59	1	no	—
152	C₄×#41	—	59	1	no	—
153	#504	—	59	16	no	—
154	C₂×M₃₂	—	59	2	no	—
155	C₂×M₃₂	—	59	4	no	—
156	C₂²×C₁₆	—	59	4	no	—
157	C₂²×C₁₆	C₂×D₅₉	1	2	yes	—
158	#405	—	59	1	no	—
159	#405	—	59	1	no	—
160	#405	—	59	2	no	—
161	#405	—	59	2	no	—
162	C₄×M₁₆	—	59	1	no	—
163	C₄×M₁₆	—	59	2	no	—
164	C₂×C₄×C₈	—	59	2	no	—
165	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
166	#346	—	59	4	no	—
167	#346	—	59	2	no	—
168	#346	C₂×D₅₉	1	1	yes	characteristic
169	#346	C₂×D₅₉	1	1	yes	characteristic
170	#346	C₂×D₅₉	1	2	yes	—
171	C₄×C₁₆	—	59	4	no	—
172	C₄×C₁₆	—	59	2	no	—
173	C₅₉	#33326	1	1	yes	characteristic, complemented, Sylow
174	C₂×#41	—	59	1	no	—
175	#131	—	59	8	no	—
176	#131	—	59	8	no	—
177	C₂×M₁₆	—	59	1	no	—
178	C₂×M₁₆	—	59	1	no	—
179	C₂×M₁₆	—	59	2	no	—
180	C₂×M₁₆	—	59	2	no	—
181	C₂²×C₈	—	59	2	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	2	no	—
186	C₄×D₄	—	59	4	no	—
187	C₄×D₄	—	59	2	no	—
188	#117	—	59	1	no	—
189	#117	—	59	2	no	—
190	C₂×C₄²	—	59	2	no	—
191	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
192	M₃₂	—	59	8	no	—
193	M₃₂	—	59	16	no	—
194	C₂×C₁₆	—	59	4	no	—
195	C₂×C₁₆	—	59	16	no	—
196	C₂×C₁₆	C₄×D₅₉	1	8	yes	complemented
197	C₂×C₁₆	C₂²×D₅₉	1	2	yes	—
198	C₂×C₁₆	C₂²×D₅₉	1	2	yes	—
199	#97	—	59	2	no	—
200	#97	—	59	2	no	—
201	#97	—	59	2	no	—
202	C₄×C₈	—	59	2	no	—
203	C₄×C₈	—	59	2	no	—
204	C₄×C₈	—	59	2	no	—
205	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
206	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
207	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
208	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
209	#41	—	59	4	no	—
210	#41	—	59	4	no	—
211	C₂×Q₈	—	59	1	no	—
212	C₂×D₄	—	59	2	no	—
213	C₂×D₄	—	59	1	no	—
214	C₂²×C₄	—	59	2	no	—
215	C₂²×C₄	—	59	4	no	—
216	C₂²×C₄	C₈×D₅₉	1	2	yes	—
217	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
218	M₁₆	—	59	4	no	—
219	M₁₆	—	59	4	no	—
220	M₁₆	—	59	8	no	—
221	M₁₆	—	59	8	no	—
222	C₂×C₈	—	59	2	no	—
223	C₂×C₈	—	59	2	no	—
224	C₂×C₈	—	59	8	no	—
225	C₂×C₈	—	59	8	no	—
226	C₂×C₈	C₈×D₅₉	1	4	yes	—
227	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
228	C₂×C₈	C₂×C₄×D₅₉	1	4	yes	—
229	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
230	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
231	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
232	C₄⋊C₄	—	59	4	no	—
233	C₄⋊C₄	—	59	2	no	—
234	C₂²⋊C₄	—	59	2	no	—
235	C₂²⋊C₄	—	59	4	no	—
236	C₄²	—	59	4	no	—
237	C₄²	—	59	2	no	—
238	C₄²	C₈×D₅₉	1	2	yes	—
239	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
240	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
241	C₁₆	—	59	8	no	—
242	C₁₆	C₂×C₄×D₅₉	1	4	yes	—
243	C₁₆	C₂×C₄×D₅₉	1	4	yes	—
244	C₂³	—	59	2	no	—
245	C₂³	C₂×C₈×D₅₉	1	1	yes	characteristic
246	Q₈	—	59	4	no	—
247	D₄	—	59	8	no	—
248	D₄	—	59	4	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	59	2	no	—
251	C₂×C₄	—	59	4	no	—
252	C₂×C₄	—	59	4	no	—
253	C₂×C₄	—	59	4	no	—
254	C₂×C₄	—	236	4	no	—
255	C₂×C₄	C₄²×D₅₉	1	2	yes	—
256	C₂×C₄	C₂×C₈×D₅₉	1	2	yes	—
257	C₂×C₄	C₂×C₈×D₅₉	1	2	yes	—
258	C₂×C₄	C₂×C₈×D₅₉	1	2	yes	—
259	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
260	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
261	C₈	—	59	4	no	—
262	C₈	—	59	4	no	—
263	C₈	C₄²×D₅₉	1	2	yes	—
264	C₈	C₂×C₈×D₅₉	1	2	yes	—
265	C₈	C₂×C₈×D₅₉	1	2	yes	—
266	C₈	C₂²×C₄×D₅₉	1	2	yes	—
267	C₂²	—	2	2	no	—
268	C₂²	—	59	4	no	—
269	C₂²	—	118	4	no	—
270	C₂²	C₄×C₈×D₅₉	1	2	yes	—
271	C₂²	C₂²×C₈×D₅₉	1	1	yes	characteristic
272	C₄	—	2	2	no	—
273	C₄	—	2	2	no	—
274	C₄	—	59	4	no	—
275	C₄	—	118	4	no	—
276	C₄	C₄×C₈×D₅₉	1	2	yes	—
277	C₄	C₂×C₄²×D₅₉	1	1	yes	characteristic
278	C₄	C₂²×C₈×D₅₉	1	1	yes	characteristic
279	C₂	—	2	2	no	—
280	C₂	—	118	4	no	—
281	C₂	C₂×C₄×C₈×D₅₉	1	1	yes	characteristic
282	C₂	#21437198	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₉
commutator	C₂×C₅₉	C₂²×C₄×C₈
Fitting	C₂×C₅₉×#346	C₂
socle	C₂²×C₅₉	C₂³×C₈
prime radical	C₂×#11745685	C₂×C₄
cosocle	C₂×C₈×C₅₉	C₂⁴
hypercenter	C₂×#346	D₅₉
hypocenter	C₅₉	#33326
LCS	G ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
3. #21425694	C₂	semidirect
4. #21425692	C₂	semidirect
5. C₄×#14273667	C₂	semidirect
6. #21425689	C₂	semidirect
7. #346×D₅₉	C₂	semidirect
8. C₂×C₅₉×#346	C₂	semidirect
12. #16837818	C₄	semidirect
47. #14273667	C₂×C₄	semidirect
48. #14273666	C₂×C₄	semidirect
50. C₁₆×D₅₉	C₂×C₄	semidirect
60. C₂×C₁₆×C₅₉	C₂×C₄	semidirect
145. C₂×#346	D₅₉	semidirect
173. C₅₉	#33326	semidirect
196. C₂×C₁₆	C₄×D₅₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837818	2
C₄×#14273733	2
#21425694	2
#21425692	2
C₄×#14273667	2
#21425689	2
#346×D₅₉	2
C₂×C₅₉×#346	2
#33326	59