Subgroups — #28559221

Order 15104.

614 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21433031	C₂	1	2	yes	maximal
2	#21422993	C₂	1	4	yes	maximal, complemented
3	Q₁₆×C₅₉⋊C₈	C₂	1	4	yes	maximal, complemented
4	#21422855	C₂	1	1	yes	maximal, characteristic
5	#21422551	C₂	1	2	yes	maximal
6	C₂×C₈×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#3170	C₂	1	1	yes	maximal, characteristic
8	#16837533	—	2	2	no	—
9	#16837533	C₂²	1	4	yes	—
10	Q₈×C₅₉⋊C₈	—	2	2	no	—
11	Q₈×C₅₉⋊C₈	C₂²	1	4	yes	—
12	#16837429	—	2	2	no	—
13	#16837426	C₂²	1	2	yes	—
14	#16837358	—	2	2	no	—
15	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
17	#16837155	C₂²	1	4	yes	—
18	Q₈⋊(C₅₉⋊C₈)	C₂²	1	8	yes	—
19	C₈×C₅₉⋊C₈	C₂²	1	2	yes	—
20	C₈×C₅₉⋊C₈	C₂²	1	2	yes	—
21	C₅₉×#533	C₂²	1	2	yes	—
22	C₅₉×#500	C₄	1	1	yes	characteristic
23	C₅₉×#494	C₄	1	1	yes	characteristic
24	C₅₉×#487	C₄	1	2	yes	—
25	C₅₉×#467	C₄	1	4	yes	—
26	C₄×Q₁₆×C₅₉	C₂²	1	4	yes	—
27	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
28	C₂×#11745664	—	2	2	no	—
29	C₂²×C₅₉⋊C₈	—	2	1	no	—
30	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
31	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
32	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
33	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
34	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
35	C₄⋊(C₅₉⋊C₈)	C₂³	1	4	yes	—
36	#14273638	—	2	4	no	—
37	C₄×C₅₉⋊C₈	—	2	1	no	—
38	C₄×C₅₉⋊C₈	—	2	4	no	—
39	C₄×C₅₉⋊C₈	D₄	1	1	yes	characteristic
40	C₄×C₅₉⋊C₈	D₄	1	1	yes	characteristic
41	C₄×C₅₉⋊C₈	C₂³	1	2	yes	—
42	C₄×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
43	C₄×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
44	C₂×Q₁₆×C₅₉	C₂×C₄	1	4	yes	—
45	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
46	C₄⋊Q₈×C₅₉	C₂×C₄	1	2	yes	—
47	C₄⋊Q₈×C₅₉	C₂×C₄	1	2	yes	—
48	#125×C₅₉	—	2	2	no	—
49	C₂²⋊Q₈×C₅₉	—	2	2	no	—
50	C₂²⋊Q₈×C₅₉	C₂×C₄	1	4	yes	—
51	C₄×Q₈×C₅₉	—	2	2	no	—
52	C₄×Q₈×C₅₉	C₂³	1	4	yes	—
53	#117×C₅₉	—	2	2	no	—
54	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
55	#107×C₅₉	C₂×C₄	1	4	yes	—
56	#103×C₅₉	C₂×C₄	1	8	yes	—
57	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₅₉	C₂³	1	2	yes	—
60	C₂×Q₈×C₅₉	—	2	2	no	—
61	C₂×Q₈×C₅₉	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
65	Q₁₆×C₅₉	—	2	8	no	—
66	C₂×C₈×C₅₉	—	2	4	no	—
67	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
69	C₄⋊C₄×C₅₉	—	2	2	no	—
70	C₄⋊C₄×C₅₉	—	2	2	no	—
71	C₄⋊C₄×C₅₉	—	2	2	no	—
72	C₄⋊C₄×C₅₉	—	2	4	no	—
73	C₄⋊C₄×C₅₉	—	2	4	no	—
74	C₄⋊C₄×C₅₉	C₂²×C₄	1	4	yes	—
75	C₂²⋊C₄×C₅₉	—	2	4	no	—
76	C₄²×C₅₉	—	2	4	no	—
77	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₅₉	#41	1	1	yes	characteristic
80	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
81	#11745664	—	4	4	no	—
82	C₂×C₅₉⋊C₈	—	2	4	no	—
83	C₂×C₅₉⋊C₈	—	2	2	no	—
84	C₂×C₅₉⋊C₈	—	2	2	no	—
85	C₂×C₅₉⋊C₈	—	2	4	no	—
86	C₂×C₅₉⋊C₈	Q₁₆	1	4	yes	complemented
87	C₂×C₅₉⋊C₈	C₂×D₄	1	1	yes	characteristic
88	C₂×C₅₉⋊C₈	C₂×D₄	1	1	yes	characteristic
89	C₂×C₅₉⋊C₈	C₂×D₄	1	1	yes	characteristic
90	C₂×C₅₉⋊C₈	C₂×D₄	1	1	yes	characteristic
91	C₂³×C₅₉	C₄×D₄	1	1	yes	characteristic
92	Q₈×C₅₉	—	2	8	no	—
93	Q₈×C₅₉	—	4	4	no	—
94	C₂×C₄×C₅₉	—	2	2	no	—
95	C₂×C₄×C₅₉	—	2	2	no	—
96	C₂×C₄×C₅₉	—	2	4	no	—
97	C₂×C₄×C₅₉	—	2	4	no	—
98	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
99	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
100	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
101	C₂×C₄×C₅₉	C₂×Q₁₆	1	2	yes	—
102	C₂×C₄×C₅₉	#135	1	2	yes	—
103	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
106	C₈×C₅₉	—	2	2	no	—
107	C₈×C₅₉	#131	1	4	yes	—
108	C₅₉⋊C₈	—	2	4	no	—
109	C₅₉⋊C₈	—	2	1	no	—
110	C₅₉⋊C₈	—	2	1	no	—
111	C₅₉⋊C₈	—	4	4	no	—
112	C₅₉⋊C₈	C₂×Q₁₆	1	2	yes	—
113	C₅₉⋊C₈	C₂×Q₁₆	1	2	yes	—
114	#17958	—	59	1	no	maximal, Sylow
115	C₂²×C₅₉	—	2	1	no	—
116	C₂²×C₅₉	—	2	1	no	—
117	C₂²×C₅₉	C₄×Q₁₆	1	2	yes	—
118	C₂²×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
119	C₄×C₅₉	—	2	1	no	—
120	C₄×C₅₉	—	2	1	no	—
121	C₄×C₅₉	—	4	4	no	—
122	C₄×C₅₉	—	4	4	no	—
123	C₄×C₅₉	C₄×Q₁₆	1	2	yes	—
124	C₄×C₅₉	#443	1	2	yes	—
125	C₄×C₅₉	C₂×#131	1	2	yes	—
126	C₄×C₅₉	C₂²×Q₁₆	1	1	yes	characteristic
127	C₄×C₅₉	C₂×#135	1	1	yes	characteristic
128	#3170	D₅₉	1	1	yes	characteristic
129	#2960	—	59	2	no	—
130	#1591	—	59	4	no	—
131	C₈×Q₁₆	—	59	4	no	—
132	#1567	—	59	1	no	—
133	#1493	—	59	2	no	—
134	C₂×C₈²	—	59	1	no	—
135	C₂×C₅₉	—	2	2	no	—
136	C₂×C₅₉	#2931	1	1	yes	characteristic
137	C₂×C₅₉	C₂×C₄×Q₁₆	1	1	yes	characteristic
138	C₂×C₅₉	C₂×#443	1	1	yes	characteristic
139	#533	C₂×D₅₉	1	2	yes	—
140	#500	Dic₅₉	1	1	yes	characteristic
141	#494	Dic₅₉	1	1	yes	characteristic
142	#487	Dic₅₉	1	2	yes	—
143	#467	Dic₅₉	1	4	yes	—
144	#446	—	59	4	no	—
145	#446	—	118	2	no	—
146	C₈×Q₈	—	59	4	no	—
147	C₈×Q₈	—	118	2	no	—
148	C₄×Q₁₆	C₂×D₅₉	1	4	yes	—
149	#424	—	118	2	no	—
150	#423	—	59	2	no	—
151	#405	—	118	2	no	—
152	C₂×C₄×C₈	—	59	1	no	—
153	C₂×C₄×C₈	—	59	1	no	—
154	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
155	#335	—	59	4	no	—
156	Q₈⋊C₈	—	59	8	no	—
157	C₈²	—	59	2	no	—
158	C₈²	—	59	2	no	—
159	C₅₉	#17958	1	1	yes	characteristic, complemented, Sylow
160	C₂×Q₁₆	C₂×Dic₅₉	1	4	yes	—
161	C₂×M₁₆	—	118	2	no	—
162	C₂²×C₈	—	59	1	no	—
163	C₂²×C₈	—	59	1	no	—
164	C₂²×C₈	—	118	1	no	—
165	C₂²×C₈	C₂×Dic₅₉	1	2	yes	—
166	C₄⋊Q₈	C₂×Dic₅₉	1	2	yes	—
167	C₄⋊Q₈	C₂×Dic₅₉	1	2	yes	—
168	#125	—	2	2	no	—
169	C₂²⋊Q₈	—	2	2	no	—
170	C₂²⋊Q₈	C₂×Dic₅₉	1	4	yes	—
171	C₄×Q₈	—	2	2	no	—
172	C₄×Q₈	C₂²×D₅₉	1	4	yes	—
173	#117	—	2	2	no	—
174	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
175	#107	C₂×Dic₅₉	1	4	yes	—
176	C₄⋊C₈	—	59	4	no	—
177	C₄⋊C₈	—	118	4	no	—
178	C₄⋊C₈	—	118	4	no	—
179	C₄⋊C₈	—	118	2	no	—
180	#103	C₂×Dic₅₉	1	8	yes	—
181	#97	—	118	4	no	—
182	C₄×C₈	—	59	1	no	—
183	C₄×C₈	—	59	1	no	—
184	C₄×C₈	—	59	1	no	—
185	C₄×C₈	—	59	1	no	—
186	C₄×C₈	—	59	2	no	—
187	C₄×C₈	—	118	4	no	—
188	C₄×C₈	—	118	1	no	—
189	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
190	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
191	C₄×C₈	C₂²×D₅₉	1	2	yes	—
192	C₂×Q₈	—	2	2	no	—
193	C₂×Q₈	C₂²×Dic₅₉	1	4	yes	—
194	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
195	C₂²×C₄	#11745687	1	1	yes	characteristic
196	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
197	Q₁₆	—	2	8	no	—
198	M₁₆	—	236	4	no	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	59	4	no	—
201	C₂×C₈	—	59	1	no	—
202	C₂×C₈	—	59	1	no	—
203	C₂×C₈	—	59	1	no	—
204	C₂×C₈	—	59	1	no	—
205	C₂×C₈	—	118	4	no	—
206	C₂×C₈	—	118	2	no	—
207	C₂×C₈	—	118	2	no	—
208	C₂×C₈	—	118	4	no	—
209	C₂×C₈	C₂²×Dic₅₉	1	2	yes	—
210	C₂×C₈	C₂²×Dic₅₉	1	2	yes	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	—	2	4	no	—
215	C₄⋊C₄	—	2	4	no	—
216	C₄⋊C₄	C₂²×Dic₅₉	1	4	yes	—
217	C₂²⋊C₄	—	2	4	no	—
218	C₄²	—	2	4	no	—
219	C₄²	D₄×D₅₉	1	1	yes	characteristic
220	C₄²	#11745687	1	1	yes	characteristic
221	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
222	C₄²	C₂³×D₅₉	1	1	yes	characteristic
223	C₂³	D₄×Dic₅₉	1	1	yes	characteristic
224	Q₈	—	2	8	no	—
225	Q₈	—	4	4	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
231	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
232	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
233	C₂×C₄	Q₁₆×D₅₉	1	2	yes	—
234	C₂×C₄	#14273750	1	2	yes	—
235	C₂×C₄	C₂³×Dic₅₉	1	1	yes	characteristic
236	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
237	C₂×C₄	C₂×#11745687	1	1	yes	characteristic
238	C₈	—	2	2	no	—
239	C₈	—	59	2	no	—
240	C₈	—	59	2	no	—
241	C₈	—	118	4	no	—
242	C₈	—	118	1	no	—
243	C₈	—	118	1	no	—
244	C₈	—	236	4	no	—
245	C₈	#14273734	1	4	yes	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	Q₁₆×Dic₅₉	1	2	yes	—
249	C₂²	C₂×D₄×Dic₅₉	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	4	4	no	—
253	C₄	—	4	4	no	—
254	C₄	Q₁₆×Dic₅₉	1	2	yes	—
255	C₄	#16837519	1	2	yes	—
256	C₄	C₂×#14273734	1	2	yes	—
257	C₄	C₂×Q₁₆×D₅₉	1	1	yes	characteristic
258	C₄	C₂×#14273750	1	1	yes	characteristic
259	C₂	—	2	2	no	—
260	C₂	#21432808	1	1	yes	characteristic
261	C₂	C₂×Q₁₆×Dic₅₉	1	1	yes	characteristic
262	C₂	C₂×#16837519	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂³×C₄
Frattini	C₄²	C₂³×D₅₉
Fitting	C₅₉×#3170	C₂
socle	C₂²×C₅₉	C₂×C₄×D₄
prime radical	C₂³×C₅₉	C₄×D₄
cosocle	C₄²×C₅₉	C₂⁴
hypercenter	#3170	D₅₉
hypocenter	C₅₉	#17958
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3170

Complements

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

N	H ≅ G/N	type
2. #21422993	C₂	semidirect
3. Q₁₆×C₅₉⋊C₈	C₂	semidirect
86. C₂×C₅₉⋊C₈	Q₁₆	semidirect
159. C₅₉	#17958	semidirect

Maximal subgroups

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

subgroup	index
#21433031	2
#21422993	2
Q₁₆×C₅₉⋊C₈	2
#21422855	2
#21422551	2
C₂×C₈×C₅₉⋊C₈	2
C₅₉×#3170	2
#17958	59