Subgroups — #28499631

Order 15104.

378 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21425895	C₂	1	1	yes	maximal, characteristic, complemented
2	#21424224	C₂	1	1	yes	maximal, characteristic, complemented
3	#21422105	C₂	1	4	yes	maximal, complemented
4	C₅₉×#3169	C₂	1	1	yes	maximal, characteristic
5	#16837272	—	2	1	no	—
6	C₄⋊(C₅₉⋊C₁₆)	—	2	1	no	—
7	C₂²⋊(C₅₉⋊C₁₆)	C₂²	1	2	yes	—
8	C₄×C₅₉⋊C₁₆	—	2	1	no	—
9	#16837165	—	2	1	no	—
10	#16837165	C₂²	1	2	yes	—
11	C₅₉×#532	—	2	1	no	—
12	C₅₉×#500	C₄	1	1	yes	characteristic, complemented
13	C₅₉×#493	C₄	1	1	yes	characteristic, complemented
14	C₅₉×#486	—	2	1	no	—
15	C₅₉×#466	—	2	1	no	—
16	C₅₉×#466	C₂²	1	2	yes	—
17	C₄×D₈×C₅₉	—	2	1	no	—
18	C₄×D₈×C₅₉	C₄	1	2	yes	complemented
19	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₅₉	—	2	2	no	—
21	C₂×C₄⋊Dic₅₉	—	4	1	no	—
22	C₂×C₄⋊Dic₅₉	—	4	1	no	—
23	C₂×C₅₉⋊C₁₆	—	2	2	no	—
24	C₂×#41×C₅₉	—	4	1	no	—
25	C₂×D₈×C₅₉	—	2	1	no	—
26	C₂×D₈×C₅₉	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
29	C₄⋊Q₈×C₅₉	—	2	1	no	—
30	#127×C₅₉	—	2	1	no	—
31	#124×C₅₉	—	4	1	no	—
32	#121×C₅₉	—	2	2	no	—
33	#121×C₅₉	—	4	1	no	—
34	C₄×D₄×C₅₉	—	2	2	no	—
35	C₄×D₄×C₅₉	—	4	1	no	—
36	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
37	#107×C₅₉	—	2	1	no	—
38	#107×C₅₉	C₂×C₄	1	2	yes	—
39	#102×C₅₉	—	2	2	no	—
40	#102×C₅₉	—	2	2	no	—
41	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
44	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
45	#41×C₅₉	—	8	2	no	—
46	#41×C₅₉	—	8	2	no	—
47	C₂×Q₈×C₅₉	—	4	1	no	—
48	C₂×D₄×C₅₉	—	2	2	no	—
49	C₂×D₄×C₅₉	—	4	2	no	—
50	C₂×D₄×C₅₉	—	4	1	no	—
51	C₂×D₄×C₅₉	—	4	1	no	—
52	C₂²×C₄×C₅₉	—	4	2	no	—
53	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
56	C₂²×Dic₅₉	—	4	2	no	—
57	D₈×C₅₉	—	4	2	no	—
58	D₈×C₅₉	—	4	2	no	—
59	C₂×C₈×C₅₉	—	4	1	no	—
60	C₂×C₈×C₅₉	—	4	1	no	—
61	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
65	C₄⋊C₄×C₅₉	—	2	2	no	—
66	C₄⋊C₄×C₅₉	—	4	1	no	—
67	C₂²⋊C₄×C₅₉	—	4	2	no	—
68	C₂²⋊C₄×C₅₉	—	4	2	no	—
69	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
70	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
71	C₄²×C₅₉	SD₁₆	1	1	yes	characteristic
72	C₄²×C₅₉	SD₁₆	1	1	yes	characteristic
73	C₄⋊Dic₅₉	—	4	2	no	—
74	C₄⋊Dic₅₉	—	4	2	no	—
75	C₄⋊Dic₅₉	—	8	2	no	—
76	C₄⋊Dic₅₉	—	8	2	no	—
77	C₅₉⋊C₁₆	—	4	2	no	—
78	C₂³×C₅₉	—	4	2	no	—
79	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
80	Q₈×C₅₉	—	8	2	no	—
81	D₄×C₅₉	—	4	2	no	—
82	D₄×C₅₉	—	4	2	no	—
83	D₄×C₅₉	—	8	2	no	—
84	D₄×C₅₉	—	8	2	no	—
85	D₄×C₅₉	—	8	2	no	—
86	D₄×C₅₉	—	8	2	no	—
87	C₂×C₄×C₅₉	—	2	2	no	—
88	C₂×C₄×C₅₉	—	2	1	no	—
89	C₂×C₄×C₅₉	—	2	1	no	—
90	C₂×C₄×C₅₉	—	4	2	no	—
91	C₂×C₄×C₅₉	—	8	4	no	—
92	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
93	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
94	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
95	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
96	C₂×C₄×C₅₉	D₁₆	1	2	yes	—
97	C₂×C₄×C₅₉	SD₃₂	1	2	yes	—
98	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
100	C₂×Dic₅₉	—	8	2	no	—
101	C₂×Dic₅₉	—	8	2	no	—
102	C₂×Dic₅₉	—	8	2	no	—
103	C₈×C₅₉	—	2	1	no	—
104	C₈×C₅₉	—	2	1	no	—
105	C₈×C₅₉	—	2	1	no	—
106	C₈×C₅₉	—	2	1	no	—
107	#11856	—	59	1	no	maximal, Sylow
108	C₂²×C₅₉	—	2	1	no	—
109	C₂²×C₅₉	—	2	1	no	—
110	C₂²×C₅₉	—	8	2	no	—
111	C₂²×C₅₉	—	8	2	no	—
112	C₂²×C₅₉	—	8	2	no	—
113	C₂²×C₅₉	C₂×#102	1	1	yes	characteristic
114	C₂²×C₅₉	C₂×D₁₆	1	1	yes	characteristic
115	C₂²×C₅₉	C₂×SD₃₂	1	1	yes	characteristic
116	C₄×C₅₉	—	2	1	no	—
117	C₄×C₅₉	—	2	1	no	—
118	C₄×C₅₉	—	8	2	no	—
119	C₄×C₅₉	#357	1	2	yes	—
120	C₄×C₅₉	#357	1	2	yes	—
121	C₄×C₅₉	#360	1	2	yes	—
122	C₄×C₅₉	#410	1	1	yes	characteristic
123	C₄×C₅₉	#421	1	1	yes	characteristic
124	Dic₅₉	—	16	2	no	—
125	#3169	D₅₉	1	1	yes	characteristic
126	#2162	—	59	1	no	—
127	#1834	—	59	1	no	—
128	#1347	—	59	4	no	—
129	C₂×C₅₉	—	2	1	no	—
130	C₂×C₅₉	—	2	1	no	—
131	C₂×C₅₉	—	16	2	no	—
132	C₂×C₅₉	#1517	1	1	yes	characteristic
133	C₂×C₅₉	C₂×#357	1	1	yes	characteristic
134	C₂×C₅₉	C₂×#360	1	1	yes	characteristic
135	#532	—	2	1	no	—
136	#500	Dic₅₉	1	1	yes	characteristic, complemented
137	#493	Dic₅₉	1	1	yes	characteristic, complemented
138	#486	—	2	1	no	—
139	#466	—	2	1	no	—
140	#466	C₂×D₅₉	1	2	yes	—
141	C₄×D₈	—	2	1	no	—
142	C₄×D₈	Dic₅₉	1	2	yes	complemented
143	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
144	#384	—	118	1	no	—
145	C₄⋊C₁₆	—	118	1	no	—
146	C₂²⋊C₁₆	—	59	2	no	—
147	C₄×C₁₆	—	118	1	no	—
148	#340	—	59	2	no	—
149	#340	—	118	1	no	—
150	C₅₉	#11856	1	1	yes	characteristic, complemented, Sylow
151	C₂×#41	—	4	1	no	—
152	C₂×D₈	—	2	1	no	—
153	C₂×D₈	C₂×Dic₅₉	1	2	yes	—
154	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
155	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
156	C₄⋊Q₈	—	2	1	no	—
157	#127	—	2	1	no	—
158	#124	—	4	1	no	—
159	#121	—	2	2	no	—
160	#121	—	4	1	no	—
161	C₄×D₄	—	2	2	no	—
162	C₄×D₄	—	4	1	no	—
163	C₂×C₄⋊C₄	—	118	2	no	—
164	C₂×C₄⋊C₄	—	236	1	no	—
165	C₂×C₄⋊C₄	—	236	1	no	—
166	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
167	C₂×C₁₆	—	118	2	no	—
168	#107	—	2	1	no	—
169	#107	C₂×Dic₅₉	1	2	yes	—
170	#102	—	2	2	no	—
171	#102	—	2	2	no	—
172	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
173	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
174	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
175	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
176	#41	—	8	2	no	—
177	#41	—	8	2	no	—
178	C₂×Q₈	—	4	1	no	—
179	C₂×D₄	—	2	2	no	—
180	C₂×D₄	—	4	2	no	—
181	C₂×D₄	—	4	1	no	—
182	C₂×D₄	—	4	1	no	—
183	C₂²×C₄	—	4	2	no	—
184	C₂²×C₄	—	236	2	no	—
185	C₂²×C₄	#11745669	1	1	yes	characteristic
186	C₂²×C₄	#11745669	1	1	yes	characteristic
187	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
188	D₈	—	4	2	no	—
189	D₈	—	4	2	no	—
190	C₂×C₈	—	4	1	no	—
191	C₂×C₈	—	4	1	no	—
192	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
193	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
194	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
195	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
196	C₄⋊C₄	—	2	2	no	—
197	C₄⋊C₄	—	4	1	no	—
198	C₄⋊C₄	—	472	2	no	—
199	C₄⋊C₄	—	472	2	no	—
200	C₄⋊C₄	—	236	2	no	—
201	C₄⋊C₄	—	236	2	no	—
202	C₂²⋊C₄	—	4	2	no	—
203	C₂²⋊C₄	—	4	2	no	—
204	C₄²	#11745670	1	1	yes	characteristic
205	C₄²	#11745670	1	1	yes	characteristic
206	C₄²	C₂²⋊Dic₅₉	1	1	yes	characteristic
207	C₄²	C₂²⋊Dic₅₉	1	1	yes	characteristic
208	C₁₆	—	236	2	no	—
209	C₂³	—	4	2	no	—
210	C₂³	C₂×#11745669	1	1	yes	characteristic
211	Q₈	—	8	2	no	—
212	D₄	—	4	2	no	—
213	D₄	—	4	2	no	—
214	D₄	—	8	2	no	—
215	D₄	—	8	2	no	—
216	D₄	—	8	2	no	—
217	D₄	—	8	2	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	8	4	no	—
223	C₂×C₄	—	472	2	no	—
224	C₂×C₄	—	472	2	no	—
225	C₂×C₄	—	472	2	no	—
226	C₂×C₄	#14273648	1	1	yes	characteristic
227	C₂×C₄	#14273648	1	1	yes	characteristic
228	C₂×C₄	#14273648	1	1	yes	characteristic
229	C₂×C₄	#14273648	1	1	yes	characteristic
230	C₂×C₄	#14273668	1	2	yes	—
231	C₂×C₄	#14273671	1	2	yes	—
232	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
233	C₂×C₄	C₂×#11745670	1	1	yes	characteristic
234	C₈	—	2	1	no	—
235	C₈	—	2	1	no	—
236	C₈	—	2	1	no	—
237	C₈	—	2	1	no	—
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	8	2	no	—
241	C₂²	—	8	2	no	—
242	C₂²	—	8	2	no	—
243	C₂²	C₂×#14273648	1	1	yes	characteristic
244	C₂²	C₂×#14273668	1	1	yes	characteristic
245	C₂²	C₂×#14273671	1	1	yes	characteristic
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	8	2	no	—
249	C₄	—	944	2	no	—
250	C₄	#16837203	1	2	yes	—
251	C₄	#16837203	1	2	yes	—
252	C₄	#16837212	1	2	yes	—
253	C₄	#16837376	1	1	yes	characteristic
254	C₄	#16837419	1	1	yes	characteristic
255	C₂	—	2	1	no	—
256	C₂	—	2	1	no	—
257	C₂	—	16	2	no	—
258	C₂	#21422656	1	1	yes	characteristic
259	C₂	C₂×#16837203	1	1	yes	characteristic
260	C₂	C₂×#16837212	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14273648
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting, prime radical	C₅₉×#3169	C₂
socle	C₂²×C₅₉	C₂×#102
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#3169	D₅₉
hypocenter	C₅₉	#11856
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3169

Complements

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

N	H ≅ G/N	type
1. #21425895	C₂	semidirect
2. #21424224	C₂	semidirect
3. #21422105	C₂	semidirect
12. C₅₉×#500	C₄	semidirect
13. C₅₉×#493	C₄	semidirect
18. C₄×D₈×C₅₉	C₄	semidirect
136. #500	Dic₅₉	semidirect
137. #493	Dic₅₉	semidirect
142. C₄×D₈	Dic₅₉	semidirect
150. C₅₉	#11856	semidirect

Maximal subgroups

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

subgroup	index
#21425895	2
#21424224	2
#21422105	2
C₅₉×#3169	2
#11856	59