Subgroups — #28481795

Order 15104.

418 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837532	C₂	1	1	yes	maximal, characteristic
2	#21424098	C₂	1	2	yes	maximal
3	C₂×#16837155	C₂	1	1	yes	maximal, characteristic
4	#21421979	C₂	1	2	yes	maximal
5	C₅₉×#1834	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
7	#16837532	—	2	2	no	—
8	#16837532	C₄	1	4	yes	complemented
9	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
10	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂×Dic₅₉⋊C₈	—	2	1	no	—
12	C₂×#14273650	—	2	2	no	—
13	C₂×#14273637	—	2	1	no	—
14	#16837164	—	2	2	no	—
15	#16837155	—	2	2	no	—
16	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₅₉×#384	C₂²	1	2	yes	—
18	C₅₉×#340	—	2	1	no	—
19	C₅₉×#340	—	2	1	no	—
20	C₂²×Dic₁₁₈	D₄	1	1	yes	characteristic
21	C₂²×C₅₉⋊C₈	—	2	1	no	—
22	C₂²×C₅₉⋊C₈	—	2	2	no	—
23	C₄×Dic₁₁₈	—	2	1	no	—
24	C₄×Dic₁₁₈	—	2	1	no	—
25	C₄×Dic₁₁₈	C₂×C₄	1	2	yes	—
26	C₄×Dic₁₁₈	Q₈	1	2	yes	—
27	C₂×C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
28	C₂×Dic₅₉⋊C₄	—	2	1	no	—
29	C₂×C₄×Dic₅₉	—	2	1	no	—
30	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
31	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
32	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
33	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	2	yes	—
34	Dic₅₉⋊C₈	—	2	4	no	—
35	#14273650	—	4	4	no	—
36	#14273637	—	2	4	no	—
37	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
40	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
41	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
42	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
43	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₅₉	D₄	1	2	yes	—
45	C₂²×C₄×C₅₉	—	4	2	no	—
46	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
49	C₂²×Dic₅₉	—	2	1	no	—
50	C₂²×Dic₅₉	—	2	1	no	—
51	C₂×Dic₁₁₈	—	2	2	no	—
52	C₂×Dic₁₁₈	—	2	2	no	—
53	C₂×Dic₁₁₈	—	2	1	no	—
54	C₂×Dic₁₁₈	C₄⋊C₄	1	2	yes	—
55	C₂×C₈×C₅₉	—	2	2	no	—
56	C₂×C₈×C₅₉	—	2	2	no	—
57	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
58	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
59	C₄⋊C₄×C₅₉	—	4	2	no	—
60	C₄⋊C₄×C₅₉	—	4	2	no	—
61	C₄⋊C₄×C₅₉	—	4	2	no	—
62	C₄⋊C₄×C₅₉	—	4	2	no	—
63	C₄⋊C₄×C₅₉	—	8	2	no	—
64	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
65	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
67	C₄²×C₅₉	#41	1	1	yes	characteristic
68	C₄⋊Dic₅₉	—	2	1	no	—
69	C₄⋊Dic₅₉	C₄⋊C₄	1	2	yes	—
70	Dic₅₉⋊C₄	—	2	2	no	—
71	Dic₅₉⋊C₄	—	2	2	no	—
72	C₄×Dic₅₉	—	2	2	no	—
73	C₄×Dic₅₉	—	2	2	no	—
74	C₂×C₅₉⋊C₈	—	2	2	no	—
75	C₂×C₅₉⋊C₈	—	4	2	no	—
76	C₂×C₅₉⋊C₈	—	4	2	no	—
77	C₂×C₅₉⋊C₈	—	4	2	no	—
78	C₂×C₅₉⋊C₈	—	4	2	no	—
79	C₂³×C₅₉	#121	1	1	yes	characteristic
80	C₂×C₄×C₅₉	—	2	1	no	—
81	C₂×C₄×C₅₉	—	2	1	no	—
82	C₂×C₄×C₅₉	—	8	2	no	—
83	C₂×C₄×C₅₉	—	8	2	no	—
84	C₂×C₄×C₅₉	—	8	2	no	—
85	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
88	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
89	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
90	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
91	C₂×C₄×C₅₉	#135	1	2	yes	—
92	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
93	C₂×C₄×C₅₉	#137	1	2	yes	—
94	C₂×C₄×C₅₉	#137	1	2	yes	—
95	C₂×Dic₅₉	—	2	2	no	—
96	C₂×Dic₅₉	—	2	2	no	—
97	C₂×Dic₅₉	—	4	2	no	—
98	C₂×Dic₅₉	—	4	1	no	—
99	C₂×Dic₅₉	—	4	1	no	—
100	Dic₁₁₈	—	2	4	no	—
101	Dic₁₁₈	—	4	1	no	—
102	Dic₁₁₈	—	4	1	no	—
103	C₈×C₅₉	—	2	2	no	—
104	C₈×C₅₉	—	2	2	no	—
105	C₅₉⋊C₈	—	4	2	no	—
106	C₅₉⋊C₈	—	8	2	no	—
107	#8655	—	59	1	no	maximal, Sylow
108	C₂²×C₅₉	#380	1	1	yes	characteristic
109	C₂²×C₅₉	#464	1	2	yes	—
110	C₂²×C₅₉	#468	1	2	yes	—
111	C₂²×C₅₉	#472	1	1	yes	characteristic
112	C₂²×C₅₉	#473	1	1	yes	characteristic
113	C₄×C₅₉	—	2	1	no	—
114	C₄×C₅₉	—	2	1	no	—
115	C₄×C₅₉	—	16	2	no	—
116	C₄×C₅₉	#416	1	1	yes	characteristic
117	C₄×C₅₉	#419	1	1	yes	characteristic
118	C₄×C₅₉	#428	1	2	yes	—
119	C₄×C₅₉	#469	1	1	yes	characteristic
120	C₄×C₅₉	#470	1	1	yes	characteristic
121	C₄×C₅₉	#479	1	2	yes	—
122	Dic₅₉	—	4	2	no	—
123	Dic₅₉	—	4	2	no	—
124	C₂×#446	—	59	1	no	—
125	#1834	D₅₉	1	1	yes	characteristic
126	#1803	—	59	2	no	—
127	C₂×#335	—	59	1	no	—
128	#1296	—	59	2	no	—
129	C₂×C₅₉	#1698	1	2	yes	—
130	C₂×C₅₉	#1700	1	2	yes	—
131	C₂×C₅₉	#1805	1	1	yes	characteristic
132	C₂×C₅₉	#1809	1	1	yes	characteristic
133	C₂×C₅₉	#1816	1	1	yes	characteristic
134	C₂×C₄×Q₈	—	59	1	no	—
135	#446	—	59	4	no	—
136	#446	—	118	2	no	—
137	C₂×C₄⋊C₈	—	59	1	no	—
138	C₂×C₄⋊C₈	—	59	2	no	—
139	C₂×C₄⋊C₈	—	118	1	no	—
140	C₂×#103	—	118	2	no	—
141	C₂×#97	—	118	1	no	—
142	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
143	#384	C₂×D₅₉	1	2	yes	—
144	#340	—	2	1	no	—
145	#340	—	2	1	no	—
146	#340	—	118	2	no	—
147	#335	—	118	2	no	—
148	C₅₉	#8655	1	1	yes	characteristic, complemented, Sylow
149	C₂²×Q₈	—	59	1	no	—
150	C₂²×C₈	—	118	1	no	—
151	C₂²×C₈	—	118	2	no	—
152	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
153	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
154	C₄×Q₈	—	59	2	no	—
155	C₄×Q₈	—	59	2	no	—
156	C₄×Q₈	—	118	1	no	—
157	C₄×Q₈	—	118	1	no	—
158	C₂×C₄⋊C₄	—	2	2	no	—
159	C₂×C₄⋊C₄	—	2	2	no	—
160	C₂×C₄⋊C₄	—	2	2	no	—
161	C₂×C₄⋊C₄	—	59	1	no	—
162	C₂×C₄⋊C₄	—	118	1	no	—
163	C₂×C₄²	—	118	1	no	—
164	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
165	C₄⋊C₈	—	59	2	no	—
166	C₄⋊C₈	—	118	4	no	—
167	C₄⋊C₈	—	118	1	no	—
168	C₄⋊C₈	—	118	2	no	—
169	C₄⋊C₈	—	118	2	no	—
170	#103	—	236	4	no	—
171	#97	—	118	4	no	—
172	C₄×C₈	C₄×D₅₉	1	2	yes	—
173	C₄×C₈	D₂₃₆	1	2	yes	—
174	C₂×Q₈	—	59	2	no	—
175	C₂×Q₈	—	118	1	no	—
176	C₂×Q₈	—	118	2	no	—
177	C₂×Q₈	—	118	2	no	—
178	C₂²×C₄	—	4	2	no	—
179	C₂²×C₄	—	118	1	no	—
180	C₂²×C₄	—	118	1	no	—
181	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
182	C₂²×C₄	#11745687	1	1	yes	characteristic
183	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	118	2	no	—
187	C₂×C₈	—	236	2	no	—
188	C₂×C₈	—	236	2	no	—
189	C₂×C₈	—	236	2	no	—
190	C₂×C₈	—	236	2	no	—
191	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
192	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
193	C₄⋊C₄	—	4	2	no	—
194	C₄⋊C₄	—	4	2	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄⋊C₄	—	8	2	no	—
198	C₄⋊C₄	—	59	2	no	—
199	C₄⋊C₄	—	118	2	no	—
200	C₄⋊C₄	—	118	1	no	—
201	C₄⋊C₄	—	118	2	no	—
202	C₄²	—	118	2	no	—
203	C₄²	—	118	2	no	—
204	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
205	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
206	C₄²	Q₈×D₅₉	1	1	yes	characteristic
207	C₄²	#11745689	1	1	yes	characteristic
208	C₂³	#14273698	1	1	yes	characteristic
209	Q₈	—	118	4	no	—
210	Q₈	—	236	1	no	—
211	Q₈	—	236	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	8	2	no	—
215	C₂×C₄	—	8	2	no	—
216	C₂×C₄	—	8	2	no	—
217	C₂×C₄	—	118	2	no	—
218	C₂×C₄	—	118	2	no	—
219	C₂×C₄	—	236	2	no	—
220	C₂×C₄	—	236	1	no	—
221	C₂×C₄	—	236	1	no	—
222	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
223	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
224	C₂×C₄	#14273685	1	1	yes	characteristic
225	C₂×C₄	#14273702	1	1	yes	characteristic
226	C₂×C₄	#14273703	1	1	yes	characteristic
227	C₂×C₄	#14273707	1	1	yes	characteristic
228	C₂×C₄	#14273747	1	2	yes	—
229	C₂×C₄	#14273754	1	2	yes	—
230	C₂×C₄	#14273756	1	2	yes	—
231	C₂×C₄	#14273757	1	2	yes	—
232	C₈	—	2	2	no	—
233	C₈	—	2	2	no	—
234	C₈	—	472	2	no	—
235	C₈	—	236	2	no	—
236	C₂²	#16837259	1	1	yes	characteristic
237	C₂²	#16837621	1	2	yes	—
238	C₂²	#16837645	1	2	yes	—
239	C₂²	#16837669	1	1	yes	characteristic
240	C₂²	#16837674	1	1	yes	characteristic
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	16	2	no	—
244	C₄	—	236	2	no	—
245	C₄	—	236	2	no	—
246	C₄	#16837396	1	1	yes	characteristic
247	C₄	#16837410	1	1	yes	characteristic
248	C₄	#16837441	1	2	yes	—
249	C₄	#16837656	1	1	yes	characteristic
250	C₄	#16837661	1	1	yes	characteristic
251	C₄	#16837706	1	2	yes	—
252	C₂	#21423633	1	2	yes	—
253	C₂	#21423647	1	2	yes	—
254	C₂	#21424108	1	1	yes	characteristic
255	C₂	#21424126	1	1	yes	characteristic
256	C₂	#21424161	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14273698
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1834	C₂
socle, prime radical	C₂³×C₅₉	#121
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1834	D₅₉
hypocenter	C₅₉	#8655
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

Complements

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

N	H ≅ G/N	type
8. #16837532	C₄	semidirect
148. C₅₉	#8655	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837532	2
#21424098	2
C₂×#16837155	2
#21421979	2
C₅₉×#1834	2
#8655	59