Subgroups — #28481769

Order 15104.

438 conjugacy classes of subgroups.

The subgroup lattice has 268 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	#21424224	C₂	1	1	yes	maximal, characteristic
3	#21424102	C₂	1	2	yes	maximal
4	#21421980	C₂	1	2	yes	maximal
5	C₂×C₅₉×#335	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
7	C₈×Dic₁₁₈	—	2	2	no	—
8	C₈×Dic₁₁₈	C₄	1	4	yes	complemented
9	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×Dic₅₉⋊C₈	—	2	1	no	—
11	C₂×#14273652	—	2	2	no	—
12	C₂×C₈×Dic₅₉	—	2	1	no	—
13	#16837272	C₂²	1	2	yes	—
14	#16837165	—	2	2	no	—
15	#16837165	—	2	1	no	—
16	#16837165	—	2	1	no	—
17	C₂×C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
19	C₅₉×#335	—	2	2	no	—
20	C₂²×Dic₁₁₈	D₄	1	1	yes	characteristic
21	C₂²×C₅₉⋊C₈	—	2	1	no	—
22	C₄×Dic₁₁₈	—	2	1	no	—
23	C₄×Dic₁₁₈	—	2	1	no	—
24	C₄×Dic₁₁₈	C₂×C₄	1	2	yes	—
25	C₄×Dic₁₁₈	Q₈	1	2	yes	—
26	C₂×C₄⋊Dic₅₉	—	2	2	no	—
27	C₂×C₄⋊Dic₅₉	—	2	2	no	—
28	C₂×C₄⋊Dic₅₉	—	2	2	no	—
29	C₂×C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
30	C₂×Dic₅₉⋊C₄	—	2	1	no	—
31	C₂×C₄×Dic₅₉	—	2	1	no	—
32	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
33	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	2	yes	—
34	Dic₅₉⋊C₈	—	2	4	no	—
35	#14273652	—	4	4	no	—
36	C₈×Dic₅₉	—	2	4	no	—
37	C₂²×C₈×C₅₉	—	2	2	no	—
38	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₅₉	—	2	2	no	—
42	C₄⋊C₈×C₅₉	—	2	2	no	—
43	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₅₉	D₄	1	2	yes	—
45	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
48	C₂²×Dic₅₉	—	2	1	no	—
49	C₂²×Dic₅₉	—	2	1	no	—
50	C₂²×Dic₅₉	—	4	2	no	—
51	C₂×Dic₁₁₈	—	2	1	no	—
52	C₂×Dic₁₁₈	—	2	1	no	—
53	C₂×Dic₁₁₈	—	2	1	no	—
54	C₂×Dic₁₁₈	C₄⋊C₄	1	2	yes	—
55	C₂×Dic₁₁₈	D₈	1	2	yes	—
56	C₂×Dic₁₁₈	Q₁₆	1	2	yes	—
57	C₂×C₈×C₅₉	—	2	2	no	—
58	C₂×C₈×C₅₉	—	2	2	no	—
59	C₂×C₈×C₅₉	—	4	2	no	—
60	C₂×C₈×C₅₉	—	4	2	no	—
61	C₂×C₈×C₅₉	—	4	2	no	—
62	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
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	—	4	2	no	—
70	C₄⋊Dic₅₉	—	4	2	no	—
71	C₄⋊Dic₅₉	—	4	2	no	—
72	C₄⋊Dic₅₉	—	4	2	no	—
73	C₄⋊Dic₅₉	—	8	2	no	—
74	C₄⋊Dic₅₉	C₄⋊C₄	1	2	yes	—
75	Dic₅₉⋊C₄	—	2	2	no	—
76	Dic₅₉⋊C₄	—	2	2	no	—
77	C₄×Dic₅₉	—	2	2	no	—
78	C₄×Dic₅₉	—	2	2	no	—
79	C₂×C₅₉⋊C₈	—	2	2	no	—
80	C₂×C₅₉⋊C₈	—	4	2	no	—
81	C₂³×C₅₉	#121	1	1	yes	characteristic
82	C₂×C₄×C₅₉	—	2	2	no	—
83	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
86	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
87	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
88	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
89	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
91	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
92	C₂×C₄×C₅₉	#135	1	2	yes	—
93	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
94	C₂×C₄×C₅₉	#137	1	2	yes	—
95	C₂×Dic₅₉	—	2	2	no	—
96	C₂×Dic₅₉	—	2	2	no	—
97	C₂×Dic₅₉	—	2	2	no	—
98	C₂×Dic₅₉	—	2	2	no	—
99	C₂×Dic₅₉	—	4	2	no	—
100	C₂×Dic₅₉	—	8	2	no	—
101	C₂×Dic₅₉	—	8	2	no	—
102	C₂×Dic₅₉	—	8	2	no	—
103	Dic₁₁₈	—	2	2	no	—
104	Dic₁₁₈	—	2	2	no	—
105	Dic₁₁₈	—	2	2	no	—
106	Dic₁₁₈	#107	1	4	yes	—
107	C₈×C₅₉	—	2	2	no	—
108	C₈×C₅₉	—	2	2	no	—
109	C₈×C₅₉	—	8	2	no	—
110	C₅₉⋊C₈	—	4	2	no	—
111	#8649	—	59	1	no	maximal, Sylow
112	C₂²×C₅₉	#380	1	1	yes	characteristic
113	C₂²×C₅₉	#464	1	2	yes	—
114	C₂²×C₅₉	#466	1	1	yes	characteristic
115	C₂²×C₅₉	#467	1	1	yes	characteristic
116	C₂²×C₅₉	#471	1	2	yes	—
117	C₄×C₅₉	—	2	1	no	—
118	C₄×C₅₉	—	2	1	no	—
119	C₄×C₅₉	#416	1	1	yes	characteristic
120	C₄×C₅₉	#419	1	1	yes	characteristic
121	C₄×C₅₉	C₂×#107	1	1	yes	characteristic
122	C₄×C₅₉	#427	1	1	yes	characteristic
123	C₄×C₅₉	#466	1	1	yes	characteristic
124	C₄×C₅₉	#467	1	1	yes	characteristic
125	C₄×C₅₉	#479	1	2	yes	—
126	Dic₅₉	—	2	4	no	—
127	Dic₅₉	—	4	2	no	—
128	Dic₅₉	—	16	2	no	—
129	C₂×C₈×Q₈	—	59	1	no	—
130	#1834	—	59	1	no	—
131	#1803	—	59	2	no	—
132	C₂×#335	D₅₉	1	1	yes	characteristic
133	#1296	—	59	2	no	—
134	C₂×C₅₉	#1682	1	2	yes	—
135	C₂×C₅₉	#1684	1	2	yes	—
136	C₂×C₅₉	#1805	1	1	yes	characteristic
137	C₂×C₅₉	#1808	1	1	yes	characteristic
138	C₂×C₅₉	#1815	1	1	yes	characteristic
139	C₂×C₄×Q₈	—	59	1	no	—
140	C₈×Q₈	—	59	4	no	—
141	C₈×Q₈	—	118	2	no	—
142	C₂×C₄⋊C₈	—	59	1	no	—
143	C₂×C₄⋊C₈	—	118	1	no	—
144	C₂×C₄⋊C₈	C₂×D₅₉	1	2	yes	—
145	C₂×#103	—	118	2	no	—
146	C₂×C₄×C₈	—	118	1	no	—
147	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
148	#384	—	59	2	no	—
149	#340	—	118	2	no	—
150	#340	—	118	1	no	—
151	#340	—	118	1	no	—
152	#335	—	2	2	no	—
153	C₅₉	#8649	1	1	yes	characteristic, complemented, Sylow
154	C₂²×Q₈	—	59	1	no	—
155	C₂²×C₈	—	2	2	no	—
156	C₂²×C₈	—	118	1	no	—
157	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
158	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
159	C₄×Q₈	—	59	2	no	—
160	C₄×Q₈	—	59	2	no	—
161	C₄×Q₈	—	118	1	no	—
162	C₄×Q₈	—	118	1	no	—
163	C₂×C₄⋊C₄	—	59	1	no	—
164	C₂×C₄⋊C₄	—	118	1	no	—
165	C₂×C₄⋊C₄	—	118	2	no	—
166	C₂×C₄⋊C₄	—	118	2	no	—
167	C₂×C₄⋊C₄	—	118	2	no	—
168	C₂×C₄²	—	118	1	no	—
169	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
170	C₄⋊C₈	—	2	2	no	—
171	C₄⋊C₈	—	2	2	no	—
172	C₄⋊C₈	—	59	2	no	—
173	C₄⋊C₈	—	118	4	no	—
174	C₄⋊C₈	—	118	1	no	—
175	#103	—	236	4	no	—
176	C₄×C₈	—	118	4	no	—
177	C₄×C₈	C₄×D₅₉	1	2	yes	—
178	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
179	C₂×Q₈	—	59	2	no	—
180	C₂×Q₈	—	59	2	no	—
181	C₂×Q₈	—	59	2	no	—
182	C₂×Q₈	—	118	1	no	—
183	C₂×Q₈	—	118	1	no	—
184	C₂×Q₈	—	118	1	no	—
185	C₂²×C₄	—	118	1	no	—
186	C₂²×C₄	—	118	1	no	—
187	C₂²×C₄	—	236	2	no	—
188	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
189	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
190	C₂²×C₄	#11745689	1	1	yes	characteristic
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	4	2	no	—
194	C₂×C₈	—	4	2	no	—
195	C₂×C₈	—	4	2	no	—
196	C₂×C₈	—	118	2	no	—
197	C₂×C₈	—	236	2	no	—
198	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
199	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
200	C₄⋊C₄	—	59	2	no	—
201	C₄⋊C₄	—	118	2	no	—
202	C₄⋊C₄	—	118	1	no	—
203	C₄⋊C₄	—	118	2	no	—
204	C₄⋊C₄	—	472	2	no	—
205	C₄⋊C₄	—	236	2	no	—
206	C₄⋊C₄	—	236	2	no	—
207	C₄⋊C₄	—	236	2	no	—
208	C₄⋊C₄	—	236	2	no	—
209	C₄²	—	118	2	no	—
210	C₄²	—	118	2	no	—
211	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
212	C₄²	#11745687	1	1	yes	characteristic
213	C₄²	Q₈×D₅₉	1	1	yes	characteristic
214	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
215	C₂³	#14273702	1	1	yes	characteristic
216	Q₈	—	59	4	no	—
217	Q₈	—	118	2	no	—
218	Q₈	—	118	2	no	—
219	Q₈	—	118	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	118	2	no	—
222	C₂×C₄	—	118	2	no	—
223	C₂×C₄	—	118	2	no	—
224	C₂×C₄	—	118	2	no	—
225	C₂×C₄	—	472	2	no	—
226	C₂×C₄	—	472	2	no	—
227	C₂×C₄	—	472	2	no	—
228	C₂×C₄	—	236	2	no	—
229	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
230	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
231	C₂×C₄	#14273685	1	1	yes	characteristic
232	C₂×C₄	#14273698	1	1	yes	characteristic
233	C₂×C₄	#14273703	1	1	yes	characteristic
234	C₂×C₄	#14273707	1	1	yes	characteristic
235	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
236	C₂×C₄	Q₁₆×D₅₉	1	1	yes	characteristic
237	C₂×C₄	#14273746	1	1	yes	characteristic
238	C₂×C₄	#14273749	1	2	yes	—
239	C₂×C₄	#14273750	1	1	yes	characteristic
240	C₂×C₄	#14273760	1	2	yes	—
241	C₈	—	2	2	no	—
242	C₈	—	2	2	no	—
243	C₈	—	8	2	no	—
244	C₈	—	236	2	no	—
245	C₂²	#16837259	1	1	yes	characteristic
246	C₂²	#16837625	1	2	yes	—
247	C₂²	#16837639	1	1	yes	characteristic
248	C₂²	#16837644	1	1	yes	characteristic
249	C₂²	#16837666	1	2	yes	—
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	118	4	no	—
253	C₄	—	944	2	no	—
254	C₄	—	236	2	no	—
255	C₄	#16837399	1	1	yes	characteristic
256	C₄	#16837412	1	1	yes	characteristic
257	C₄	#107×D₅₉	1	1	yes	characteristic
258	C₄	#16837439	1	1	yes	characteristic
259	C₄	#16837635	1	1	yes	characteristic
260	C₄	#16837640	1	1	yes	characteristic
261	C₄	#16837710	1	2	yes	—
262	C₂	#21423541	1	2	yes	—
263	C₂	#21423555	1	2	yes	—
264	C₂	#21424112	1	1	yes	characteristic
265	C₂	#21424122	1	1	yes	characteristic
266	C₂	#21424152	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14273702
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₂×C₅₉×#335	C₂
socle, prime radical	C₂³×C₅₉	#121
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₂×#335	D₅₉
hypocenter	C₅₉	#8649
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
8. C₈×Dic₁₁₈	C₄	semidirect
153. C₅₉	#8649	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
#21424224	2
#21424102	2
#21421980	2
C₂×C₅₉×#335	2
#8649	59