Subgroups — #28479341

Order 15104.

530 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊(C₅₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₄×#14273638	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊(C₅₉⋊C₈)	C₂	1	2	yes	maximal
4	#21421937	C₂	1	2	yes	maximal
5	C₅₉×#2295	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
7	C₂×#14273638	—	2	2	no	—
8	C₂×C₄×C₅₉⋊C₈	—	2	1	no	—
9	C₂×C₄×C₅₉⋊C₈	C₂²	1	2	yes	—
10	#16837157	—	2	1	no	—
11	Q₈⋊(C₅₉⋊C₈)	C₄	1	8	yes	complemented
12	C₂×C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₅₉×#391	C₄	1	2	yes	—
14	C₅₉×#389	C₄	1	2	yes	—
15	C₅₉×#387	—	2	2	no	—
16	C₄×C₄⋊C₄×C₅₉	—	2	1	no	—
17	C₄×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
18	C₅₉×#376	—	2	1	no	—
19	C₄³×C₅₉	C₂²	1	1	yes	characteristic
20	C₂²×C₅₉⋊C₈	—	2	2	no	—
21	C₂²×C₅₉⋊C₈	—	2	2	no	—
22	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
23	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	4	yes	—
24	#14273638	—	2	8	no	—
25	C₄×C₅₉⋊C₈	—	2	2	no	—
26	C₄×C₅₉⋊C₈	—	4	2	no	—
27	C₄×C₅₉⋊C₈	C₂×C₄	1	4	yes	—
28	C₂²×Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
29	C₄×Q₈×C₅₉	—	2	2	no	—
30	C₄×Q₈×C₅₉	C₂×C₄	1	4	yes	—
31	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
32	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
33	C₂×C₄⋊C₄×C₅₉	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₅₉	—	2	1	no	—
36	C₂×C₄²×C₅₉	—	2	1	no	—
37	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₉	D₄	1	2	yes	—
41	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
42	#95×C₅₉	—	2	2	no	—
43	#95×C₅₉	—	2	2	no	—
44	#95×C₅₉	—	2	2	no	—
45	#95×C₅₉	—	2	1	no	—
46	C₂×Q₈×C₅₉	—	2	1	no	—
47	C₂×Q₈×C₅₉	—	2	4	no	—
48	C₂×Q₈×C₅₉	C₄²	1	2	yes	—
49	C₂²×C₄×C₅₉	—	2	1	no	—
50	C₂²×C₄×C₅₉	—	2	1	no	—
51	C₂²×C₄×C₅₉	—	2	2	no	—
52	C₂²×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
57	C₄⋊C₄×C₅₉	—	2	1	no	—
58	C₄⋊C₄×C₅₉	—	2	4	no	—
59	C₄⋊C₄×C₅₉	—	2	4	no	—
60	C₄⋊C₄×C₅₉	—	4	4	no	—
61	C₄⋊C₄×C₅₉	C₄²	1	2	yes	—
62	C₄²×C₅₉	—	2	4	no	—
63	C₄²×C₅₉	—	2	4	no	—
64	C₄²×C₅₉	—	2	4	no	—
65	C₄²×C₅₉	—	2	1	no	—
66	C₄²×C₅₉	—	2	1	no	—
67	C₄²×C₅₉	—	2	2	no	—
68	C₄²×C₅₉	—	4	1	no	—
69	C₄²×C₅₉	—	4	1	no	—
70	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
71	C₄²×C₅₉	#41	1	2	yes	—
72	C₂×C₅₉⋊C₈	—	2	8	no	—
73	C₂×C₅₉⋊C₈	—	2	4	no	—
74	C₂×C₅₉⋊C₈	—	2	4	no	—
75	C₂×C₅₉⋊C₈	—	4	2	no	—
76	C₂×C₅₉⋊C₈	—	4	2	no	—
77	C₂³×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	Q₈×C₅₉	—	2	4	no	—
79	Q₈×C₅₉	—	4	2	no	—
80	C₂×C₄×C₅₉	—	2	2	no	—
81	C₂×C₄×C₅₉	—	2	2	no	—
82	C₂×C₄×C₅₉	—	2	2	no	—
83	C₂×C₄×C₅₉	—	2	2	no	—
84	C₂×C₄×C₅₉	—	2	2	no	—
85	C₂×C₄×C₅₉	—	2	2	no	—
86	C₂×C₄×C₅₉	—	2	2	no	—
87	C₂×C₄×C₅₉	—	2	2	no	—
88	C₂×C₄×C₅₉	—	4	2	no	—
89	C₂×C₄×C₅₉	—	4	2	no	—
90	C₂×C₄×C₅₉	—	4	2	no	—
91	C₂×C₄×C₅₉	—	4	2	no	—
92	C₂×C₄×C₅₉	—	4	2	no	—
93	C₂×C₄×C₅₉	#104	1	4	yes	—
94	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
95	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
96	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
97	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
98	C₂×C₄×C₅₉	#131	1	4	yes	—
99	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
100	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
101	C₂×C₄×C₅₉	#137	1	2	yes	—
102	C₅₉⋊C₈	—	2	8	no	—
103	C₅₉⋊C₈	—	4	4	no	—
104	#8180	—	59	1	no	maximal, Sylow
105	C₂²×C₅₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
106	C₂²×C₅₉	#408	1	2	yes	—
107	C₂²×C₅₉	#417	1	1	yes	characteristic
108	C₂²×C₅₉	#419	1	1	yes	characteristic
109	C₂²×C₅₉	C₂×#104	1	2	yes	—
110	C₄×C₅₉	—	2	4	no	—
111	C₄×C₅₉	—	2	2	no	—
112	C₄×C₅₉	—	4	2	no	—
113	C₄×C₅₉	—	4	2	no	—
114	C₄×C₅₉	—	4	2	no	—
115	C₄×C₅₉	—	8	2	no	—
116	C₄×C₅₉	#405	1	2	yes	—
117	C₄×C₅₉	#433	1	2	yes	—
118	C₄×C₅₉	#440	1	2	yes	—
119	C₄×C₅₉	#441	1	2	yes	—
120	#2295	D₅₉	1	1	yes	characteristic
121	C₄×C₄⋊C₈	—	59	1	no	—
122	C₄×#97	—	59	1	no	—
123	C₂×Q₈⋊C₈	—	59	2	no	—
124	#1279	—	59	2	no	—
125	C₂×C₅₉	#1483	1	2	yes	—
126	C₂×C₅₉	#1485	1	2	yes	—
127	C₂×C₅₉	#1750	1	1	yes	characteristic
128	C₂×C₅₉	C₄×#104	1	1	yes	characteristic
129	C₂×C₅₉	#1763	1	1	yes	characteristic
130	C₂×C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
131	C₂×C₄⋊C₈	—	59	2	no	—
132	C₂×#97	—	118	2	no	—
133	C₂×C₄×C₈	—	59	2	no	—
134	C₂×C₄×C₈	—	118	1	no	—
135	#391	Dic₅₉	1	2	yes	—
136	#389	Dic₅₉	1	2	yes	—
137	#387	—	2	2	no	—
138	C₄×C₄⋊C₄	—	2	1	no	—
139	C₄×C₄⋊C₄	C₂×D₅₉	1	1	yes	characteristic
140	#376	—	2	1	no	—
141	C₄³	C₂×D₅₉	1	1	yes	characteristic
142	#336	—	118	1	no	—
143	Q₈⋊C₈	—	59	8	no	—
144	C₅₉	#8180	1	1	yes	characteristic, complemented, Sylow
145	C₂²×Q₈	C₂×Dic₅₉	1	1	yes	characteristic
146	C₂²×C₈	—	118	2	no	—
147	C₂²×C₈	—	118	2	no	—
148	C₄×Q₈	—	2	2	no	—
149	C₄×Q₈	C₄×D₅₉	1	4	yes	—
150	C₂×C₄⋊C₄	—	2	1	no	—
151	C₂×C₄⋊C₄	—	2	2	no	—
152	C₂×C₄⋊C₄	C₂×Dic₅₉	1	2	yes	—
153	C₂×C₄⋊C₄	C₂×Dic₅₉	1	1	yes	characteristic
154	C₂×C₄²	—	2	1	no	—
155	C₂×C₄²	—	2	1	no	—
156	C₂×C₄²	C₂×Dic₅₉	1	2	yes	—
157	C₂×C₄²	C₂²⋊D₅₉	1	2	yes	—
158	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
159	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
160	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
161	C₄⋊C₈	—	59	4	no	—
162	C₄⋊C₈	—	118	2	no	—
163	#97	—	118	8	no	—
164	C₄×C₈	—	59	4	no	—
165	C₄×C₈	—	118	2	no	—
166	C₄×C₈	—	236	2	no	—
167	#95	—	2	2	no	—
168	#95	—	2	2	no	—
169	#95	—	2	2	no	—
170	#95	—	2	1	no	—
171	C₂×Q₈	—	2	1	no	—
172	C₂×Q₈	—	2	4	no	—
173	C₂×Q₈	C₄×Dic₅₉	1	2	yes	—
174	C₂²×C₄	—	2	1	no	—
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	C₂²⋊Dic₅₉	1	2	yes	—
178	C₂²×C₄	C₂²⋊Dic₅₉	1	2	yes	—
179	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
180	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
181	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
182	C₂×C₈	—	118	8	no	—
183	C₂×C₈	—	118	4	no	—
184	C₂×C₈	—	118	4	no	—
185	C₂×C₈	—	236	2	no	—
186	C₂×C₈	—	236	2	no	—
187	C₄⋊C₄	—	2	1	no	—
188	C₄⋊C₄	—	2	4	no	—
189	C₄⋊C₄	—	2	4	no	—
190	C₄⋊C₄	—	4	4	no	—
191	C₄⋊C₄	C₄×Dic₅₉	1	2	yes	—
192	C₄²	—	2	4	no	—
193	C₄²	—	2	4	no	—
194	C₄²	—	2	4	no	—
195	C₄²	—	2	1	no	—
196	C₄²	—	2	1	no	—
197	C₄²	—	2	2	no	—
198	C₄²	—	4	1	no	—
199	C₄²	—	4	1	no	—
200	C₄²	C₂×C₄×D₅₉	1	2	yes	—
201	C₄²	#11745685	1	2	yes	—
202	C₂³	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
203	Q₈	—	2	4	no	—
204	Q₈	—	4	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	#14273654	1	4	yes	—
219	C₂×C₄	C₂×C₄×Dic₅₉	1	1	yes	characteristic
220	C₂×C₄	#14273686	1	1	yes	characteristic
221	C₂×C₄	C₄×C₂²⋊D₅₉	1	2	yes	—
222	C₂×C₄	C₄×C₂²⋊D₅₉	1	2	yes	—
223	C₂×C₄	#14273734	1	4	yes	—
224	C₂×C₄	#14273752	1	1	yes	characteristic
225	C₂×C₄	#14273757	1	1	yes	characteristic
226	C₂×C₄	#14273758	1	2	yes	—
227	C₈	—	118	8	no	—
228	C₈	—	236	4	no	—
229	C₂²	C₄×C₂²⋊Dic₅₉	1	1	yes	characteristic
230	C₂²	#16837369	1	2	yes	—
231	C₂²	#16837403	1	1	yes	characteristic
232	C₂²	#16837411	1	1	yes	characteristic
233	C₂²	C₂×#14273654	1	2	yes	—
234	C₄	—	2	2	no	—
235	C₄	—	2	4	no	—
236	C₄	—	4	2	no	—
237	C₄	—	4	2	no	—
238	C₄	—	4	2	no	—
239	C₄	—	8	2	no	—
240	C₄	#16837358	1	2	yes	—
241	C₄	#16837463	1	2	yes	—
242	C₄	#16837500	1	2	yes	—
243	C₄	#16837507	1	2	yes	—
244	C₂	#21422509	1	2	yes	—
245	C₂	#21422517	1	2	yes	—
246	C₂	#21423862	1	1	yes	characteristic
247	C₂	C₄×#14273654	1	1	yes	characteristic
248	C₂	#21423926	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#2295	C₂
socle, prime radical	C₂³×C₅₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#2295	D₅₉
hypocenter	C₅₉	#8180
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #2295

Complements

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

N	H ≅ G/N	type
11. Q₈⋊(C₅₉⋊C₈)	C₄	semidirect
144. C₅₉	#8180	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊(C₅₉⋊C₈)	2
C₄×#14273638	2
C₂×Q₈⋊(C₅₉⋊C₈)	2
#21421937	2
C₅₉×#2295	2
#8180	59