Subgroups — #28481819

Order 15104.

394 conjugacy classes of subgroups.

The subgroup lattice has 266 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	#21424268	C₂	1	1	yes	maximal, characteristic
3	#21422526	C₂	1	2	yes	maximal
4	#21421992	C₂	1	2	yes	maximal
5	C₅₉×#1564	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
7	C₈×Dic₁₁₈	—	2	1	no	—
8	C₈×Dic₁₁₈	—	2	1	no	—
9	C₈×Dic₁₁₈	C₄	1	2	yes	—
10	C₈×Dic₁₁₈	C₄	1	2	yes	—
11	C₂×#14273663	—	2	1	no	—
12	C₂×#14273663	—	2	1	no	—
13	#16837426	C₂²	1	2	yes	—
14	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
15	C₂×Dic₅₉⋊C₈	—	2	1	no	—
16	C₂×C₈×Dic₅₉	—	2	1	no	—
17	#16837134	—	2	4	no	—
18	C₅₉×#431	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
20	C₅₉×#335	—	2	1	no	—
21	C₅₉×#334	—	2	1	no	—
22	C₂²×Dic₁₁₈	Q₈	1	1	yes	characteristic
23	C₂×#11745664	—	2	2	no	—
24	C₂²×C₅₉⋊C₈	—	2	1	no	—
25	C₄×Dic₁₁₈	—	2	1	no	—
26	C₄×Dic₁₁₈	—	2	1	no	—
27	C₄×Dic₁₁₈	C₂×C₄	1	2	yes	—
28	C₄×Dic₁₁₈	D₄	1	2	yes	—
29	C₂×C₄⋊Dic₅₉	Q₈	1	1	yes	characteristic
30	C₂×Dic₅₉⋊C₄	—	2	1	no	—
31	C₂×C₄×Dic₅₉	—	2	1	no	—
32	#14273663	—	4	2	no	—
33	#14273663	—	4	2	no	—
34	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
35	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
36	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
37	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	2	yes	—
38	Dic₅₉⋊C₈	—	2	2	no	—
39	Dic₅₉⋊C₈	—	2	2	no	—
40	C₈×Dic₅₉	—	2	2	no	—
41	C₈×Dic₅₉	—	2	2	no	—
42	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
45	C₄⋊C₈×C₅₉	—	2	2	no	—
46	C₂²⋊C₈×C₅₉	—	2	2	no	—
47	C₄×C₈×C₅₉	—	2	2	no	—
48	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
51	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
55	C₂²×Dic₅₉	—	2	1	no	—
56	C₂²×Dic₅₉	—	2	1	no	—
57	C₂×Dic₁₁₈	—	2	1	no	—
58	C₂×Dic₁₁₈	—	2	4	no	—
59	C₂×Dic₁₁₈	C₄⋊C₄	1	2	yes	—
60	C₂×C₈×C₅₉	—	2	1	no	—
61	C₂×C₈×C₅₉	—	2	1	no	—
62	C₂×C₈×C₅₉	—	4	2	no	—
63	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
68	C₂×C₈×C₅₉	Q₁₆	1	2	yes	—
69	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
70	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₅₉	#41	1	1	yes	characteristic
73	C₄⋊Dic₅₉	—	2	1	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	#11745664	—	4	2	no	—
80	#11745664	—	4	2	no	—
81	C₂×C₅₉⋊C₈	—	2	2	no	—
82	C₂×C₅₉⋊C₈	—	4	1	no	—
83	C₂×C₅₉⋊C₈	—	4	1	no	—
84	C₂×C₅₉⋊C₈	—	4	2	no	—
85	C₂³×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₅₉	—	2	2	no	—
87	C₂×C₄×C₅₉	—	4	1	no	—
88	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
91	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
92	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
93	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
94	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
95	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
96	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
97	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
98	C₂×Dic₅₉	—	2	4	no	—
99	C₂×Dic₅₉	—	2	2	no	—
100	C₂×Dic₅₉	—	2	2	no	—
101	C₂×Dic₅₉	—	4	2	no	—
102	Dic₁₁₈	—	2	4	no	—
103	Dic₁₁₈	—	2	2	no	—
104	Dic₁₁₈	#108	1	4	yes	complemented
105	C₈×C₅₉	—	2	1	no	—
106	C₈×C₅₉	—	2	1	no	—
107	C₈×C₅₉	—	8	2	no	—
108	C₈×C₅₉	#103	1	2	yes	—
109	C₈×C₅₉	#103	1	2	yes	—
110	C₅₉⋊C₈	—	4	2	no	—
111	C₅₉⋊C₈	—	8	2	no	—
112	#8659	—	59	1	no	maximal, Sylow
113	C₂²×C₅₉	—	2	1	no	—
114	C₂²×C₅₉	—	2	1	no	—
115	C₂²×C₅₉	#380	1	1	yes	characteristic
116	C₂²×C₅₉	#475	1	1	yes	characteristic
117	C₂²×C₅₉	#477	1	1	yes	characteristic
118	C₄×C₅₉	—	2	1	no	—
119	C₄×C₅₉	—	2	1	no	—
120	C₄×C₅₉	—	2	1	no	—
121	C₄×C₅₉	C₂×#103	1	1	yes	characteristic
122	C₄×C₅₉	#417	1	1	yes	characteristic
123	C₄×C₅₉	C₂×#108	1	2	yes	—
124	C₄×C₅₉	#460	1	1	yes	characteristic
125	C₄×C₅₉	#462	1	1	yes	characteristic
126	Dic₅₉	—	2	4	no	—
127	Dic₅₉	—	4	2	no	—
128	C₂×C₈×Q₈	—	59	1	no	—
129	#1848	—	59	1	no	—
130	#1564	D₅₉	1	1	yes	characteristic
131	#1487	—	59	2	no	—
132	#1301	—	59	2	no	—
133	C₂×C₅₉	—	2	1	no	—
134	C₂×C₅₉	—	2	1	no	—
135	C₂×C₅₉	#1803	1	1	yes	characteristic
136	C₂×C₅₉	#1810	1	1	yes	characteristic
137	C₂×C₅₉	#1814	1	1	yes	characteristic
138	C₂×C₄×Q₈	—	59	1	no	—
139	C₈×Q₈	—	59	2	no	—
140	C₈×Q₈	—	59	2	no	—
141	C₈×Q₈	—	118	1	no	—
142	C₈×Q₈	—	118	1	no	—
143	#431	C₂×D₅₉	1	2	yes	—
144	C₂×#108	—	118	1	no	—
145	C₂×#108	—	118	1	no	—
146	#423	—	59	2	no	—
147	C₂×C₄⋊C₈	—	59	1	no	—
148	C₂×C₄⋊C₈	—	118	1	no	—
149	C₂×C₄×C₈	—	118	1	no	—
150	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
151	#335	—	2	1	no	—
152	#334	—	2	1	no	—
153	Q₈⋊C₈	—	118	4	no	—
154	C₅₉	#8659	1	1	yes	characteristic, complemented, Sylow
155	C₂²×Q₈	—	59	1	no	—
156	C₂×M₁₆	—	118	2	no	—
157	C₂²×C₈	—	118	1	no	—
158	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
159	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
160	C₄×Q₈	—	59	2	no	—
161	C₄×Q₈	—	59	2	no	—
162	C₄×Q₈	—	118	1	no	—
163	C₄×Q₈	—	118	1	no	—
164	C₂×C₄⋊C₄	—	59	1	no	—
165	C₂×C₄⋊C₄	—	118	1	no	—
166	C₂×C₄²	—	118	1	no	—
167	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
168	#108	—	236	2	no	—
169	#108	—	236	2	no	—
170	C₄⋊C₈	—	2	2	no	—
171	C₄⋊C₈	—	59	2	no	—
172	C₄⋊C₈	—	118	2	no	—
173	C₄⋊C₈	—	118	2	no	—
174	C₄⋊C₈	—	118	2	no	—
175	C₄⋊C₈	—	118	2	no	—
176	C₄⋊C₈	—	118	1	no	—
177	C₂²⋊C₈	—	2	2	no	—
178	C₄×C₈	—	2	2	no	—
179	C₄×C₈	—	118	2	no	—
180	C₄×C₈	—	118	2	no	—
181	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
182	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
183	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
184	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
185	C₂×Q₈	—	59	2	no	—
186	C₂×Q₈	—	118	4	no	—
187	C₂×Q₈	—	118	1	no	—
188	C₂²×C₄	—	118	1	no	—
189	C₂²×C₄	—	118	1	no	—
190	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
191	C₂²×C₄	#11745687	1	1	yes	characteristic
192	C₂²×C₄	Q₈×D₅₉	1	1	yes	characteristic
193	M₁₆	—	236	2	no	—
194	M₁₆	—	236	2	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	—	118	2	no	—
199	C₂×C₈	—	236	2	no	—
200	C₂×C₈	—	236	1	no	—
201	C₂×C₈	—	236	1	no	—
202	C₂×C₈	#11745660	1	2	yes	—
203	C₂×C₈	Dic₂₃₆	1	2	yes	—
204	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
205	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
206	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
207	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
208	C₄⋊C₄	—	59	2	no	—
209	C₄⋊C₄	—	118	2	no	—
210	C₄⋊C₄	—	118	2	no	—
211	C₄⋊C₄	—	118	1	no	—
212	C₄²	—	118	2	no	—
213	C₄²	—	118	2	no	—
214	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
215	C₄²	D₄×D₅₉	1	1	yes	characteristic
216	C₄²	#11745689	1	1	yes	characteristic
217	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
218	C₂³	#14273707	1	1	yes	characteristic
219	Q₈	—	59	4	no	—
220	Q₈	—	118	4	no	—
221	Q₈	—	118	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	4	1	no	—
224	C₂×C₄	—	118	4	no	—
225	C₂×C₄	—	118	2	no	—
226	C₂×C₄	—	118	2	no	—
227	C₂×C₄	—	236	2	no	—
228	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
229	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
230	C₂×C₄	#14273685	1	1	yes	characteristic
231	C₂×C₄	#14273698	1	1	yes	characteristic
232	C₂×C₄	#14273702	1	1	yes	characteristic
233	C₂×C₄	#14273703	1	1	yes	characteristic
234	C₂×C₄	C₂×#11745660	1	1	yes	characteristic
235	C₂×C₄	C₂×Dic₂₃₆	1	1	yes	characteristic
236	C₂×C₄	#14273755	1	1	yes	characteristic
237	C₂×C₄	#14273760	1	1	yes	characteristic
238	C₈	—	2	1	no	—
239	C₈	—	2	1	no	—
240	C₈	—	8	2	no	—
241	C₈	—	472	2	no	—
242	C₈	—	236	2	no	—
243	C₈	#14273652	1	2	yes	—
244	C₈	#14273652	1	2	yes	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	#16837259	1	1	yes	characteristic
248	C₂²	#16837688	1	1	yes	characteristic
249	C₂²	#16837698	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	118	4	no	—
254	C₄	—	236	2	no	—
255	C₄	C₂×#14273652	1	1	yes	characteristic
256	C₄	#16837404	1	1	yes	characteristic
257	C₄	#108×D₅₉	1	2	yes	—
258	C₄	#16837603	1	1	yes	characteristic
259	C₄	#16837613	1	1	yes	characteristic
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	#21424102	1	1	yes	characteristic
263	C₂	#21424131	1	1	yes	characteristic
264	C₂	#21424147	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14273702
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1564	C₂
socle	C₂²×C₅₉	#380
prime radical	C₂³×C₅₉	C₂²⋊Q₈
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1564	D₅₉
hypocenter	C₅₉	#8659
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
104. Dic₁₁₈	#108	semidirect
154. C₅₉	#8659	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
#21424268	2
#21422526	2
#21421992	2
C₅₉×#1564	2
#8659	59