Subgroups — #28500024

Order 15104.

310 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21425895	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837155	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	C₂×C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
7	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	2	yes	—
8	C₂²⋊(C₅₉⋊C₁₆)	—	2	1	no	—
9	C₄×C₅₉⋊C₁₆	—	2	1	no	—
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	C₂²×C₅₉⋊C₈	—	4	1	no	—
22	C₂×C₅₉⋊C₁₆	—	2	2	no	—
23	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
24	C₄⋊(C₅₉⋊C₈)	—	4	1	no	—
25	C₂×Q₁₆×C₅₉	—	2	1	no	—
26	C₂×Q₁₆×C₅₉	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
29	C₄⋊Q₈×C₅₉	—	2	1	no	—
30	C₄⋊Q₈×C₅₉	—	2	1	no	—
31	#125×C₅₉	—	4	1	no	—
32	C₂²⋊Q₈×C₅₉	—	2	2	no	—
33	C₂²⋊Q₈×C₅₉	—	4	1	no	—
34	C₄×Q₈×C₅₉	—	2	2	no	—
35	C₄×Q₈×C₅₉	—	4	1	no	—
36	#117×C₅₉	—	4	1	no	—
37	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
38	#107×C₅₉	—	2	1	no	—
39	#107×C₅₉	C₂×C₄	1	2	yes	—
40	#103×C₅₉	—	2	2	no	—
41	#103×C₅₉	—	2	2	no	—
42	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
44	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
45	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
46	C₂×Q₈×C₅₉	—	2	2	no	—
47	C₂×Q₈×C₅₉	—	4	1	no	—
48	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
51	Q₁₆×C₅₉	—	4	2	no	—
52	Q₁₆×C₅₉	—	4	2	no	—
53	C₂×C₈×C₅₉	—	2	2	no	—
54	C₂×C₈×C₅₉	—	4	1	no	—
55	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
59	C₄⋊C₄×C₅₉	—	2	2	no	—
60	C₄⋊C₄×C₅₉	—	4	2	no	—
61	C₄⋊C₄×C₅₉	—	4	2	no	—
62	C₄⋊C₄×C₅₉	—	4	1	no	—
63	C₄⋊C₄×C₅₉	—	4	1	no	—
64	C₄⋊C₄×C₅₉	—	4	1	no	—
65	C₂²⋊C₄×C₅₉	—	4	2	no	—
66	C₄²×C₅₉	—	4	2	no	—
67	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₄²×C₅₉	D₈	1	1	yes	characteristic
69	C₄²×C₅₉	SD₁₆	1	1	yes	characteristic
70	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
71	C₂×C₅₉⋊C₈	—	4	2	no	—
72	C₂×C₅₉⋊C₈	—	8	2	no	—
73	C₅₉⋊C₁₆	—	4	2	no	—
74	C₂³×C₅₉	#102	1	1	yes	characteristic
75	Q₈×C₅₉	—	4	2	no	—
76	Q₈×C₅₉	—	4	2	no	—
77	Q₈×C₅₉	—	8	2	no	—
78	C₂×C₄×C₅₉	—	2	1	no	—
79	C₂×C₄×C₅₉	—	2	1	no	—
80	C₂×C₄×C₅₉	—	4	1	no	—
81	C₂×C₄×C₅₉	—	4	2	no	—
82	C₂×C₄×C₅₉	—	4	2	no	—
83	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
84	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
85	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
86	C₂×C₄×C₅₉	SD₃₂	1	2	yes	—
87	C₂×C₄×C₅₉	Q₃₂	1	2	yes	—
88	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
91	C₈×C₅₉	—	2	1	no	—
92	C₈×C₅₉	—	2	1	no	—
93	C₈×C₅₉	—	2	1	no	—
94	C₈×C₅₉	#135	1	2	yes	—
95	C₅₉⋊C₈	—	8	2	no	—
96	#11949	—	59	1	no	maximal, Sylow
97	C₂²×C₅₉	—	2	1	no	—
98	C₂²×C₅₉	—	2	1	no	—
99	C₂²×C₅₉	#358	1	2	yes	—
100	C₂²×C₅₉	C₂×#102	1	1	yes	characteristic
101	C₄×C₅₉	—	2	1	no	—
102	C₄×C₅₉	—	2	1	no	—
103	C₄×C₅₉	—	2	1	no	—
104	C₄×C₅₉	—	8	2	no	—
105	C₄×C₅₉	—	8	2	no	—
106	C₄×C₅₉	#358	1	2	yes	—
107	C₄×C₅₉	#413	1	1	yes	characteristic
108	C₄×C₅₉	#416	1	1	yes	characteristic
109	C₄×C₅₉	C₂×SD₃₂	1	1	yes	characteristic
110	C₄×C₅₉	C₂×Q₃₂	1	1	yes	characteristic
111	#3170	D₅₉	1	1	yes	characteristic
112	#2162	—	59	1	no	—
113	C₂×#335	—	59	1	no	—
114	#1365	—	59	2	no	—
115	#1362	—	59	2	no	—
116	C₂×C₅₉	—	2	2	no	—
117	C₂×C₅₉	#1540	1	1	yes	characteristic
118	C₂×C₅₉	C₂×#358	1	1	yes	characteristic
119	C₂×C₅₉	#2148	1	1	yes	characteristic
120	#533	—	2	1	no	—
121	#500	C₂×D₅₉	1	1	yes	characteristic
122	#494	C₂×D₅₉	1	1	yes	characteristic
123	#487	—	2	1	no	—
124	#467	—	2	1	no	—
125	#467	Dic₅₉	1	2	yes	—
126	C₄×Q₁₆	—	2	1	no	—
127	C₄×Q₁₆	Dic₅₉	1	2	yes	—
128	C₂×C₄⋊C₈	—	118	1	no	—
129	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
130	C₄⋊C₁₆	—	59	2	no	—
131	C₂²⋊C₁₆	—	118	1	no	—
132	C₄×C₁₆	—	118	1	no	—
133	#335	—	59	2	no	—
134	#335	—	118	1	no	—
135	C₅₉	#11949	1	1	yes	characteristic, complemented, Sylow
136	C₂×Q₁₆	—	2	1	no	—
137	C₂×Q₁₆	C₂×Dic₅₉	1	2	yes	—
138	C₂²×C₈	—	236	1	no	—
139	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
140	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
141	C₄⋊Q₈	—	2	1	no	—
142	C₄⋊Q₈	—	2	1	no	—
143	#125	—	4	1	no	—
144	C₂²⋊Q₈	—	2	2	no	—
145	C₂²⋊Q₈	—	4	1	no	—
146	C₄×Q₈	—	2	2	no	—
147	C₄×Q₈	—	4	1	no	—
148	#117	—	4	1	no	—
149	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
150	C₂×C₁₆	—	118	2	no	—
151	#107	—	2	1	no	—
152	#107	C₂×Dic₅₉	1	2	yes	—
153	C₄⋊C₈	—	118	2	no	—
154	C₄⋊C₈	—	236	1	no	—
155	#103	—	2	2	no	—
156	#103	—	2	2	no	—
157	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
158	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
159	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
160	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
161	C₂×Q₈	—	2	2	no	—
162	C₂×Q₈	—	4	1	no	—
163	C₂²×C₄	#11745669	1	1	yes	characteristic
164	C₂²×C₄	#11745670	1	1	yes	characteristic
165	C₂²×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
166	Q₁₆	—	4	2	no	—
167	Q₁₆	—	4	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	4	1	no	—
170	C₂×C₈	—	472	2	no	—
171	C₂×C₈	—	236	2	no	—
172	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
173	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
174	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
175	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄⋊C₄	—	4	2	no	—
179	C₄⋊C₄	—	4	1	no	—
180	C₄⋊C₄	—	4	1	no	—
181	C₄⋊C₄	—	4	1	no	—
182	C₂²⋊C₄	—	4	2	no	—
183	C₄²	—	4	2	no	—
184	C₄²	#11745669	1	1	yes	characteristic
185	C₄²	#11745670	1	1	yes	characteristic
186	C₄²	C₂²⋊Dic₅₉	1	1	yes	characteristic
187	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
188	C₁₆	—	236	2	no	—
189	C₂³	#14273648	1	1	yes	characteristic
190	Q₈	—	4	2	no	—
191	Q₈	—	4	2	no	—
192	Q₈	—	8	2	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	#14273648	1	1	yes	characteristic
199	C₂×C₄	#14273648	1	1	yes	characteristic
200	C₂×C₄	#14273648	1	1	yes	characteristic
201	C₂×C₄	#14273670	1	2	yes	—
202	C₂×C₄	#14273673	1	2	yes	—
203	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
204	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
205	C₂×C₄	C₂×#11745670	1	1	yes	characteristic
206	C₈	—	2	1	no	—
207	C₈	—	2	1	no	—
208	C₈	—	2	1	no	—
209	C₈	—	472	2	no	—
210	C₈	#14273747	1	2	yes	—
211	C₂²	—	2	1	no	—
212	C₂²	—	2	1	no	—
213	C₂²	#16837206	1	2	yes	—
214	C₂²	C₂×#14273648	1	1	yes	characteristic
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	8	2	no	—
219	C₄	—	8	2	no	—
220	C₄	#16837206	1	2	yes	—
221	C₄	#16837386	1	1	yes	characteristic
222	C₄	#16837398	1	1	yes	characteristic
223	C₄	C₂×#14273670	1	1	yes	characteristic
224	C₄	C₂×#14273673	1	1	yes	characteristic
225	C₂	—	2	2	no	—
226	C₂	#21422745	1	1	yes	characteristic
227	C₂	C₂×#16837206	1	1	yes	characteristic
228	C₂	#21425845	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#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	#11949
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ 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
135. C₅₉	#11949	semidirect

Maximal subgroups

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

subgroup	index
#21425895	2
C₂×#16837155	2
#21422154	2
#21422145	2
C₅₉×#3170	2
#11949	59