Subgroups — #28744177

Order 15104.

490 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21434610	C₂	1	1	yes	maximal, characteristic
2	#21426376	C₂	1	2	yes	maximal, complemented
3	#21426372	C₂	1	4	yes	maximal, complemented
4	#21426369	C₂	1	2	yes	maximal, complemented
5	#21425878	C₂	1	4	yes	maximal
6	C₂×C₄×C₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#3160	C₂	1	1	yes	maximal, characteristic
8	#16837959	—	2	1	no	—
9	C₂²×C₅₉⋊C₁₆	C₂²	1	2	yes	—
10	#16837807	C₂²	1	2	yes	—
11	#16837804	—	2	1	no	—
12	#16837804	C₂²	1	2	yes	—
13	#16837800	—	2	1	no	—
14	#16837440	—	2	1	no	—
15	#16837440	C₂²	1	2	yes	—
16	#16837229	C₂²	1	8	yes	—
17	#16837227	C₂²	1	8	yes	—
18	C₄×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
19	C₄×C₅₉⋊C₁₆	C₂²	1	2	yes	—
20	C₄×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
21	C₅₉×#533	—	2	1	no	—
22	C₅₉×#500	C₂²	1	2	yes	—
23	C₅₉×#499	—	2	1	no	—
24	C₅₉×#499	C₂²	1	2	yes	—
25	C₅₉×#498	—	2	1	no	—
26	C₅₉×#427	—	2	1	no	—
27	C₅₉×#427	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
29	#14273727	—	2	2	no	—
30	#14273719	—	2	2	no	—
31	#14273706	—	4	2	no	—
32	C₄×Dic₁₁₈	—	4	2	no	—
33	#14273686	—	2	2	no	—
34	C₂×C₅₉⋊C₁₆	Q₈	1	8	yes	complemented
35	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
36	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
37	#14273661	—	2	2	no	—
38	#14273661	C₂³	1	4	yes	—
39	#14273659	—	2	2	no	—
40	#14273659	—	2	2	no	—
41	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
43	C₄⋊Q₈×C₅₉	—	2	2	no	—
44	#125×C₅₉	—	2	2	no	—
45	C₂²⋊Q₈×C₅₉	—	4	2	no	—
46	C₄×Q₈×C₅₉	—	4	2	no	—
47	#117×C₅₉	—	2	2	no	—
48	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
49	#107×C₅₉	—	2	2	no	—
50	#107×C₅₉	C₂³	1	4	yes	—
51	#106×C₅₉	—	2	2	no	—
52	#106×C₅₉	—	2	2	no	—
53	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
54	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
55	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
56	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
57	C₂×Q₈×C₅₉	—	4	2	no	—
58	C₂²×C₄×C₅₉	D₈	1	2	yes	—
59	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
60	C₂×Dic₁₁₈	—	4	2	no	—
61	C₂×C₈×C₅₉	Q₁₆	1	4	yes	—
62	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₅₉	C₂×Q₈	1	4	yes	—
66	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
67	C₄⋊C₄×C₅₉	—	2	4	no	—
68	C₄⋊C₄×C₅₉	—	4	4	no	—
69	C₄⋊C₄×C₅₉	—	4	2	no	—
70	C₂²⋊C₄×C₅₉	—	4	2	no	—
71	C₄²×C₅₉	—	4	2	no	—
72	C₄²×C₅₉	D₈	1	2	yes	—
73	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
75	C₂²⋊Dic₅₉	—	4	2	no	—
76	C₄⋊Dic₅₉	—	2	4	no	—
77	C₄⋊Dic₅₉	—	4	2	no	—
78	Dic₅₉⋊C₄	—	4	4	no	—
79	C₄×Dic₅₉	—	4	2	no	—
80	C₅₉⋊C₁₆	C₂×Q₈	1	4	yes	—
81	C₅₉⋊C₁₆	C₂×Q₈	1	4	yes	—
82	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
83	Q₈×C₅₉	—	8	4	no	—
84	C₂×C₄×C₅₉	—	2	4	no	—
85	C₂×C₄×C₅₉	—	4	4	no	—
86	C₂×C₄×C₅₉	C₄⋊Q₈	1	2	yes	—
87	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
88	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
89	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₅₉	C₂×Q₁₆	1	2	yes	—
91	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
92	C₂×Dic₅₉	—	4	4	no	—
93	Dic₁₁₈	—	8	4	no	—
94	C₈×C₅₉	C₄⋊Q₈	1	2	yes	—
95	C₈×C₅₉	C₂×Q₁₆	1	2	yes	—
96	C₈×C₅₉	C₂×Q₁₆	1	2	yes	—
97	C₈×C₅₉	C₂²×Q₈	1	2	yes	—
98	#33726	—	59	1	no	maximal, Sylow
99	C₂²×C₅₉	—	2	2	no	—
100	C₂²×C₅₉	#500	1	2	yes	—
101	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
102	C₄×C₅₉	—	2	2	no	—
103	C₄×C₅₉	—	8	4	no	—
104	C₄×C₅₉	#500	1	2	yes	—
105	C₄×C₅₉	#508	1	4	yes	—
106	C₄×C₅₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
107	C₄×C₅₉	C₂²×Q₁₆	1	1	yes	characteristic
108	Dic₅₉	—	8	4	no	—
109	#3160	—	59	1	no	—
110	#3160	D₅₉	1	1	yes	characteristic
111	#2254	—	59	2	no	—
112	#2253	—	59	4	no	—
113	#2252	—	59	2	no	—
114	#2158	—	59	4	no	—
115	C₂×C₄×C₁₆	—	59	1	no	—
116	C₂×C₅₉	—	2	2	no	—
117	C₂×C₅₉	C₂×#500	1	1	yes	characteristic
118	C₂×C₅₉	C₂×#508	1	2	yes	—
119	#533	—	2	1	no	—
120	#533	—	118	1	no	—
121	C₂²×C₁₆	—	59	2	no	—
122	#500	—	59	2	no	—
123	#500	C₂×D₅₉	1	2	yes	—
124	#499	—	2	1	no	—
125	#499	—	59	2	no	—
126	#499	—	118	1	no	—
127	#499	C₂×D₅₉	1	2	yes	—
128	#498	—	2	1	no	—
129	#498	—	118	1	no	—
130	#427	—	2	1	no	—
131	#427	—	59	2	no	—
132	#427	—	118	1	no	—
133	#427	C₂×D₅₉	1	2	yes	—
134	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
135	#367	—	59	8	no	—
136	#366	—	59	8	no	—
137	C₄×C₁₆	—	59	1	no	—
138	C₄×C₁₆	—	59	2	no	—
139	C₄×C₁₆	—	59	1	no	—
140	C₅₉	#33726	1	1	yes	characteristic, complemented, Sylow
141	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
142	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
143	C₄⋊Q₈	—	2	2	no	—
144	C₄⋊Q₈	—	118	2	no	—
145	#125	—	2	2	no	—
146	#125	—	118	2	no	—
147	C₂²⋊Q₈	—	4	2	no	—
148	C₂²⋊Q₈	—	236	2	no	—
149	C₄×Q₈	—	4	2	no	—
150	C₄×Q₈	—	236	2	no	—
151	#117	—	2	2	no	—
152	#117	—	118	2	no	—
153	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
154	C₂×C₁₆	—	59	8	no	—
155	C₂×C₁₆	—	59	2	no	—
156	C₂×C₁₆	—	59	2	no	—
157	#107	—	2	2	no	—
158	#107	—	59	4	no	—
159	#107	—	118	2	no	—
160	#107	C₂²×D₅₉	1	4	yes	—
161	#106	—	2	2	no	—
162	#106	—	2	2	no	—
163	#106	—	118	2	no	—
164	#106	—	118	2	no	—
165	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
166	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
167	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
168	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
169	C₂×Q₈	—	4	2	no	—
170	C₂×Q₈	—	236	2	no	—
171	C₂²×C₄	#11745669	1	2	yes	—
172	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
173	C₂×C₈	#11745672	1	4	yes	—
174	C₂×C₈	Q₈×D₅₉	1	4	yes	—
175	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
176	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
177	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
178	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
179	C₄⋊C₄	—	2	4	no	—
180	C₄⋊C₄	—	4	4	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	118	4	no	—
183	C₄⋊C₄	—	236	4	no	—
184	C₄⋊C₄	—	236	2	no	—
185	C₂²⋊C₄	—	4	2	no	—
186	C₂²⋊C₄	—	236	2	no	—
187	C₄²	—	4	2	no	—
188	C₄²	—	236	2	no	—
189	C₄²	#11745669	1	2	yes	—
190	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
191	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
192	C₁₆	—	59	4	no	—
193	C₁₆	—	59	4	no	—
194	C₂³	C₂×#11745669	1	1	yes	characteristic
195	Q₈	—	8	4	no	—
196	Q₈	—	472	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	4	4	no	—
199	C₂×C₄	—	236	4	no	—
200	C₂×C₄	#14273725	1	2	yes	—
201	C₂×C₄	C₂×#11745669	1	2	yes	—
202	C₂×C₄	C₂×#11745669	1	2	yes	—
203	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
204	C₂×C₄	C₂×#11745672	1	2	yes	—
205	C₂×C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
206	C₈	#14273725	1	2	yes	—
207	C₈	C₂×#11745672	1	2	yes	—
208	C₈	C₂×#11745672	1	2	yes	—
209	C₈	C₂×Q₈×D₅₉	1	2	yes	—
210	C₂²	—	2	2	no	—
211	C₂²	#16837805	1	2	yes	—
212	C₂²	C₂²×#11745669	1	1	yes	characteristic
213	C₄	—	2	2	no	—
214	C₄	—	8	4	no	—
215	C₄	—	472	4	no	—
216	C₄	#16837805	1	2	yes	—
217	C₄	#16837832	1	4	yes	—
218	C₄	C₂×#14273725	1	1	yes	characteristic
219	C₄	C₂²×#11745672	1	1	yes	characteristic
220	C₂	—	2	2	no	—
221	C₂	C₂×#16837805	1	1	yes	characteristic
222	C₂	C₂×#16837832	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#11745669
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₅₉×#3160	C₂
socle	C₂²×C₅₉	C₂²×D₈
prime radical	C₂³×C₅₉	C₂×D₈
hypercenter	#3160	D₅₉
hypocenter	C₅₉	#33726
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
2. #21426376	C₂	semidirect
3. #21426372	C₂	semidirect
4. #21426369	C₂	semidirect
34. C₂×C₅₉⋊C₁₆	Q₈	semidirect
140. C₅₉	#33726	semidirect

Maximal subgroups

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

subgroup	index
#21434610	2
#21426376	2
#21426372	2
#21426369	2
#21425878	2
C₂×C₄×C₅₉⋊C₁₆	2
C₅₉×#3160	2
#33726	59