Subgroups — #28500176

Order 15104.

314 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₅₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	#21422855	C₂	1	1	yes	maximal, characteristic
3	#21422154	C₂	1	2	yes	maximal, complemented
4	#21422145	C₂	1	2	yes	maximal, complemented
5	C₅₉×#3170	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅₉⋊C₁₆	—	2	1	no	—
7	#16837426	—	2	1	no	—
8	C₄⋊(C₅₉⋊C₁₆)	—	2	1	no	—
9	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	2	yes	—
10	#16837155	—	2	1	no	—
11	#16837155	C₂²	1	2	yes	—
12	C₅₉×#533	—	2	1	no	—
13	C₅₉×#500	C₂²	1	1	yes	characteristic
14	C₅₉×#494	C₂²	1	1	yes	characteristic
15	C₅₉×#487	—	2	1	no	—
16	C₅₉×#467	—	2	1	no	—
17	C₅₉×#467	C₄	1	2	yes	—
18	C₄×Q₁₆×C₅₉	—	2	1	no	—
19	C₄×Q₁₆×C₅₉	C₄	1	2	yes	—
20	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
21	C₂×#11745664	—	4	1	no	—
22	C₂×C₅₉⋊C₁₆	—	2	2	no	—
23	C₂×C₅₉⋊C₁₆	—	4	2	no	—
24	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
25	C₄⋊(C₅₉⋊C₈)	—	4	1	no	—
26	C₂×Q₁₆×C₅₉	—	2	1	no	—
27	C₂×Q₁₆×C₅₉	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
30	C₄⋊Q₈×C₅₉	—	2	1	no	—
31	C₄⋊Q₈×C₅₉	—	2	1	no	—
32	#125×C₅₉	—	4	1	no	—
33	C₂²⋊Q₈×C₅₉	—	2	2	no	—
34	C₂²⋊Q₈×C₅₉	—	4	1	no	—
35	C₄×Q₈×C₅₉	—	2	2	no	—
36	C₄×Q₈×C₅₉	—	4	1	no	—
37	#117×C₅₉	—	4	1	no	—
38	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
39	#107×C₅₉	—	2	1	no	—
40	#107×C₅₉	C₂×C₄	1	2	yes	—
41	#103×C₅₉	—	2	2	no	—
42	#103×C₅₉	—	2	2	no	—
43	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
45	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
46	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
47	C₂×Q₈×C₅₉	—	2	2	no	—
48	C₂×Q₈×C₅₉	—	4	1	no	—
49	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
51	C₂²×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
52	Q₁₆×C₅₉	—	4	2	no	—
53	Q₁₆×C₅₉	—	4	1	no	—
54	Q₁₆×C₅₉	—	4	1	no	—
55	C₂×C₈×C₅₉	—	2	1	no	—
56	C₂×C₈×C₅₉	—	2	1	no	—
57	C₂×C₈×C₅₉	—	4	1	no	—
58	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
62	C₄⋊C₄×C₅₉	—	2	2	no	—
63	C₄⋊C₄×C₅₉	—	4	2	no	—
64	C₄⋊C₄×C₅₉	—	4	2	no	—
65	C₄⋊C₄×C₅₉	—	4	1	no	—
66	C₄⋊C₄×C₅₉	—	4	1	no	—
67	C₄⋊C₄×C₅₉	—	4	1	no	—
68	C₂²⋊C₄×C₅₉	—	4	2	no	—
69	C₄²×C₅₉	—	4	2	no	—
70	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
71	C₄²×C₅₉	D₈	1	1	yes	characteristic
72	C₄²×C₅₉	SD₁₆	1	1	yes	characteristic
73	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
74	#11745664	—	8	2	no	—
75	C₂×C₅₉⋊C₈	—	4	2	no	—
76	C₅₉⋊C₁₆	—	4	2	no	—
77	C₂³×C₅₉	#102	1	1	yes	characteristic
78	Q₈×C₅₉	—	4	2	no	—
79	Q₈×C₅₉	—	4	2	no	—
80	Q₈×C₅₉	—	8	2	no	—
81	C₂×C₄×C₅₉	—	2	1	no	—
82	C₂×C₄×C₅₉	—	2	1	no	—
83	C₂×C₄×C₅₉	—	4	1	no	—
84	C₂×C₄×C₅₉	—	4	2	no	—
85	C₂×C₄×C₅₉	—	4	2	no	—
86	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
87	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
88	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
89	C₂×C₄×C₅₉	SD₃₂	1	2	yes	—
90	C₂×C₄×C₅₉	Q₃₂	1	2	yes	—
91	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
94	C₈×C₅₉	—	2	1	no	—
95	C₈×C₅₉	—	2	1	no	—
96	C₈×C₅₉	—	2	1	no	—
97	C₈×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
98	C₈×C₅₉	#137	1	1	yes	characteristic
99	C₅₉⋊C₈	—	8	2	no	—
100	#11983	—	59	1	no	maximal, Sylow
101	C₂²×C₅₉	—	2	1	no	—
102	C₂²×C₅₉	—	2	1	no	—
103	C₂²×C₅₉	#358	1	2	yes	—
104	C₂²×C₅₉	C₂×#102	1	1	yes	characteristic
105	C₄×C₅₉	—	2	1	no	—
106	C₄×C₅₉	—	2	1	no	—
107	C₄×C₅₉	—	2	1	no	—
108	C₄×C₅₉	—	8	2	no	—
109	C₄×C₅₉	—	8	2	no	—
110	C₄×C₅₉	#358	1	2	yes	—
111	C₄×C₅₉	#413	1	1	yes	characteristic
112	C₄×C₅₉	#417	1	1	yes	characteristic
113	C₄×C₅₉	C₂×SD₃₂	1	1	yes	characteristic
114	C₄×C₅₉	C₂×Q₃₂	1	1	yes	characteristic
115	#3170	D₅₉	1	1	yes	characteristic
116	C₂×C₄⋊C₁₆	—	59	1	no	—
117	#1567	—	59	1	no	—
118	#1365	—	59	2	no	—
119	#1362	—	59	2	no	—
120	C₂×C₅₉	—	2	2	no	—
121	C₂×C₅₉	#1547	1	1	yes	characteristic
122	C₂×C₅₉	C₂×#358	1	1	yes	characteristic
123	C₂×C₅₉	#2148	1	1	yes	characteristic
124	#533	—	2	1	no	—
125	C₂²×C₁₆	—	118	1	no	—
126	#500	C₂×D₅₉	1	1	yes	characteristic
127	#494	C₂×D₅₉	1	1	yes	characteristic
128	#487	—	2	1	no	—
129	#467	—	2	1	no	—
130	#467	Dic₅₉	1	2	yes	—
131	C₄×Q₁₆	—	2	1	no	—
132	C₄×Q₁₆	Dic₅₉	1	2	yes	—
133	#423	—	118	1	no	—
134	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
135	C₄⋊C₁₆	—	59	2	no	—
136	C₄⋊C₁₆	—	118	1	no	—
137	#335	—	59	2	no	—
138	#335	—	118	1	no	—
139	C₅₉	#11983	1	1	yes	characteristic, complemented, Sylow
140	C₂×Q₁₆	—	2	1	no	—
141	C₂×Q₁₆	C₂×Dic₅₉	1	2	yes	—
142	C₂×M₁₆	—	236	1	no	—
143	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
144	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
145	C₄⋊Q₈	—	2	1	no	—
146	C₄⋊Q₈	—	2	1	no	—
147	#125	—	4	1	no	—
148	C₂²⋊Q₈	—	2	2	no	—
149	C₂²⋊Q₈	—	4	1	no	—
150	C₄×Q₈	—	2	2	no	—
151	C₄×Q₈	—	4	1	no	—
152	#117	—	4	1	no	—
153	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
154	C₂×C₁₆	—	118	2	no	—
155	C₂×C₁₆	—	236	2	no	—
156	#107	—	2	1	no	—
157	#107	C₂×Dic₅₉	1	2	yes	—
158	C₄⋊C₈	—	118	2	no	—
159	C₄⋊C₈	—	236	1	no	—
160	#103	—	2	2	no	—
161	#103	—	2	2	no	—
162	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
163	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
164	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
165	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
166	C₂×Q₈	—	2	2	no	—
167	C₂×Q₈	—	4	1	no	—
168	C₂²×C₄	#11745669	1	1	yes	characteristic
169	C₂²×C₄	#11745670	1	1	yes	characteristic
170	C₂²×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
171	Q₁₆	—	4	2	no	—
172	Q₁₆	—	4	1	no	—
173	Q₁₆	—	4	1	no	—
174	M₁₆	—	472	2	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	4	1	no	—
178	C₂×C₈	—	236	2	no	—
179	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
180	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
181	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
182	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	4	2	no	—
185	C₄⋊C₄	—	4	2	no	—
186	C₄⋊C₄	—	4	1	no	—
187	C₄⋊C₄	—	4	1	no	—
188	C₄⋊C₄	—	4	1	no	—
189	C₂²⋊C₄	—	4	2	no	—
190	C₄²	—	4	2	no	—
191	C₄²	#11745669	1	1	yes	characteristic
192	C₄²	#11745670	1	1	yes	characteristic
193	C₄²	C₂²⋊Dic₅₉	1	1	yes	characteristic
194	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
195	C₁₆	—	236	2	no	—
196	C₂³	#14273648	1	1	yes	characteristic
197	Q₈	—	4	2	no	—
198	Q₈	—	4	2	no	—
199	Q₈	—	8	2	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	4	1	no	—
203	C₂×C₄	—	4	2	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	#14273648	1	1	yes	characteristic
206	C₂×C₄	#14273648	1	1	yes	characteristic
207	C₂×C₄	#14273648	1	1	yes	characteristic
208	C₂×C₄	#14273670	1	2	yes	—
209	C₂×C₄	#14273673	1	2	yes	—
210	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
211	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
212	C₂×C₄	C₂×#11745670	1	1	yes	characteristic
213	C₈	—	2	1	no	—
214	C₈	—	2	1	no	—
215	C₈	—	2	1	no	—
216	C₈	—	472	2	no	—
217	C₈	#14273752	1	1	yes	characteristic
218	C₈	#14273757	1	1	yes	characteristic
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	#16837206	1	2	yes	—
222	C₂²	C₂×#14273648	1	1	yes	characteristic
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	2	1	no	—
226	C₄	—	8	2	no	—
227	C₄	—	8	2	no	—
228	C₄	#16837206	1	2	yes	—
229	C₄	#16837386	1	1	yes	characteristic
230	C₄	#16837403	1	1	yes	characteristic
231	C₄	C₂×#14273670	1	1	yes	characteristic
232	C₄	C₂×#14273673	1	1	yes	characteristic
233	C₂	—	2	2	no	—
234	C₂	#21422773	1	1	yes	characteristic
235	C₂	C₂×#16837206	1	1	yes	characteristic
236	C₂	#21425845	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14273648
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₅₉×#3170	C₂
socle	C₂²×C₅₉	C₂×#102
prime radical	C₂³×C₅₉	#102
cosocle	C₄×C₈×C₅₉	C₂³
hypercenter	#3170	D₅₉
hypocenter	C₅₉	#11983
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3170

Complements

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

N	H ≅ G/N	type
3. #21422154	C₂	semidirect
4. #21422145	C₂	semidirect
139. C₅₉	#11983	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₅₉⋊C₁₆)	2
#21422855	2
#21422154	2
#21422145	2
C₅₉×#3170	2
#11983	59