Subgroups — #28507262

Order 15104.

398 conjugacy classes of subgroups.

The subgroup lattice has 196 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	C₄×#14273661	C₂	1	1	yes	maximal, characteristic
3	#21422187	C₂	1	4	yes	maximal
4	C₂×C₅₉×#500	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₅₉⋊C₁₆	C₂²	1	2	yes	—
6	C₂×#14273661	C₂²	1	2	yes	—
7	C₄×C₄⋊Dic₅₉	—	2	1	no	—
8	C₄⋊(C₅₉⋊C₁₆)	—	2	2	no	—
9	#16837165	—	2	1	no	—
10	C₂×C₄⋊Q₈×C₅₉	—	2	1	no	—
11	C₅₉×#500	—	2	2	no	—
12	C₅₉×#500	C₄	1	4	yes	complemented
13	C₂×#107×C₅₉	—	2	1	no	—
14	C₂×#107×C₅₉	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₅₉	—	2	2	no	—
17	C₂×C₄×Dic₅₉	—	4	1	no	—
18	C₂×C₅₉⋊C₁₆	—	2	2	no	—
19	C₂×C₅₉⋊C₁₆	—	2	4	no	—
20	#14273661	—	2	2	no	—
21	#14273661	Q₈	1	4	yes	complemented
22	C₅₉⋊#95	—	4	1	no	—
23	C₂²×Q₈×C₅₉	—	4	1	no	—
24	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
26	C₄⋊Q₈×C₅₉	—	2	4	no	—
27	C₄⋊Q₈×C₅₉	—	4	2	no	—
28	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
29	C₂×C₄⋊C₄×C₅₉	—	4	1	no	—
30	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
31	#107×C₅₉	—	2	4	no	—
32	#107×C₅₉	—	2	2	no	—
33	#107×C₅₉	C₂×C₄	1	4	yes	—
34	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₅₉	D₄	1	2	yes	—
36	C₂×Q₈×C₅₉	—	4	4	no	—
37	C₂×Q₈×C₅₉	—	8	2	no	—
38	C₂×Q₈×C₅₉	—	8	2	no	—
39	C₂²×C₄×C₅₉	—	4	2	no	—
40	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
41	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
42	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₅₉	—	4	2	no	—
44	C₂×C₈×C₅₉	—	2	2	no	—
45	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
46	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
49	C₂×C₈×C₅₉	#41	1	2	yes	—
50	C₄⋊C₄×C₅₉	—	2	4	no	—
51	C₄⋊C₄×C₅₉	—	4	4	no	—
52	C₄⋊C₄×C₅₉	—	4	2	no	—
53	C₄²×C₅₉	C₂²⋊C₄	1	2	yes	—
54	C₄²×C₅₉	SD₁₆	1	2	yes	—
55	C₄⋊Dic₅₉	—	2	4	no	—
56	C₄⋊Dic₅₉	—	4	2	no	—
57	C₄×Dic₅₉	—	8	2	no	—
58	C₅₉⋊C₁₆	—	4	2	no	—
59	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
60	Q₈×C₅₉	—	8	8	no	—
61	C₂×C₄×C₅₉	—	2	2	no	—
62	C₂×C₄×C₅₉	—	4	4	no	—
63	C₂×C₄×C₅₉	—	8	2	no	—
64	C₂×C₄×C₅₉	—	8	2	no	—
65	C₂×C₄×C₅₉	#102	1	2	yes	—
66	C₂×C₄×C₅₉	#102	1	2	yes	—
67	C₂×C₄×C₅₉	D₁₆	1	2	yes	—
68	C₂×C₄×C₅₉	SD₃₂	1	2	yes	—
69	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
71	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
72	C₂×C₄×C₅₉	#137	1	2	yes	—
73	C₂×Dic₅₉	—	4	4	no	—
74	C₂×Dic₅₉	—	8	2	no	—
75	C₂×Dic₅₉	—	8	2	no	—
76	C₈×C₅₉	—	2	2	no	—
77	C₈×C₅₉	C₄×Q₈	1	2	yes	—
78	C₈×C₅₉	#124	1	1	yes	characteristic
79	C₈×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
80	#13275	—	59	1	no	maximal, Sylow
81	C₂²×C₅₉	C₂×#102	1	1	yes	characteristic
82	C₂²×C₅₉	#474	1	2	yes	—
83	C₂²×C₅₉	C₂×D₁₆	1	1	yes	characteristic
84	C₂²×C₅₉	C₂×SD₃₂	1	1	yes	characteristic
85	C₂²×C₅₉	#510	1	2	yes	—
86	C₄×C₅₉	—	2	2	no	—
87	C₄×C₅₉	—	8	4	no	—
88	C₄×C₅₉	#357	1	4	yes	—
89	C₄×C₅₉	#391	1	1	yes	characteristic
90	C₄×C₅₉	#419	1	1	yes	characteristic
91	C₄×C₅₉	#475	1	2	yes	—
92	Dic₅₉	—	8	4	no	—
93	C₂×#500	D₅₉	1	1	yes	characteristic
94	C₂×C₄⋊C₁₆	—	59	1	no	—
95	C₄×#107	—	59	1	no	—
96	#1380	—	59	4	no	—
97	C₂×C₅₉	#1988	1	1	yes	characteristic
98	C₂×C₅₉	C₂×#357	1	1	yes	characteristic
99	C₂×C₅₉	#2141	1	1	yes	characteristic
100	C₂×C₅₉	#2224	1	2	yes	—
101	C₂×C₅₉	#2225	1	2	yes	—
102	C₂×C₄⋊Q₈	—	2	1	no	—
103	C₂²×C₁₆	—	59	2	no	—
104	#500	—	2	2	no	—
105	#500	Dic₅₉	1	4	yes	complemented
106	C₂×#107	—	2	1	no	—
107	C₂×#107	—	59	2	no	—
108	C₂×#107	C₂×D₅₉	1	2	yes	—
109	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
110	C₄×C₄⋊C₄	—	118	1	no	—
111	C₄⋊C₁₆	—	118	2	no	—
112	#340	—	118	1	no	—
113	C₅₉	#13275	1	1	yes	characteristic, complemented, Sylow
114	C₂²×Q₈	—	4	1	no	—
115	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
116	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
117	C₄⋊Q₈	—	2	4	no	—
118	C₄⋊Q₈	—	4	2	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	4	1	no	—
121	C₂×C₄⋊C₄	—	118	2	no	—
122	C₂×C₄²	—	236	1	no	—
123	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
124	C₂×C₁₆	—	118	4	no	—
125	C₂×C₁₆	—	118	2	no	—
126	#107	—	2	4	no	—
127	#107	—	2	2	no	—
128	#107	—	59	4	no	—
129	#107	—	118	2	no	—
130	#107	C₂×Dic₅₉	1	4	yes	—
131	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
132	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
133	#95	—	236	1	no	—
134	C₂×Q₈	—	4	4	no	—
135	C₂×Q₈	—	8	2	no	—
136	C₂×Q₈	—	8	2	no	—
137	C₂²×C₄	—	4	2	no	—
138	C₂²×C₄	—	236	2	no	—
139	C₂²×C₄	#11745669	1	1	yes	characteristic
140	C₂²×C₄	#11745669	1	1	yes	characteristic
141	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	C₂²⋊Dic₅₉	1	2	yes	—
144	C₂×C₈	Q₈×D₅₉	1	2	yes	—
145	C₂×C₈	#11745689	1	2	yes	—
146	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
147	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
148	C₄⋊C₄	—	2	4	no	—
149	C₄⋊C₄	—	4	4	no	—
150	C₄⋊C₄	—	4	2	no	—
151	C₄⋊C₄	—	118	4	no	—
152	C₄⋊C₄	—	236	2	no	—
153	C₄²	—	472	2	no	—
154	C₄²	#11745670	1	2	yes	—
155	C₄²	C₂²⋊Dic₅₉	1	2	yes	—
156	C₁₆	—	236	2	no	—
157	C₂³	C₂×#11745669	1	1	yes	characteristic
158	Q₈	—	8	8	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	4	4	no	—
161	C₂×C₄	—	8	2	no	—
162	C₂×C₄	—	8	2	no	—
163	C₂×C₄	—	472	2	no	—
164	C₂×C₄	—	472	2	no	—
165	C₂×C₄	—	236	4	no	—
166	C₂×C₄	#14273648	1	2	yes	—
167	C₂×C₄	#14273648	1	2	yes	—
168	C₂×C₄	#14273668	1	2	yes	—
169	C₂×C₄	#14273671	1	2	yes	—
170	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
171	C₂×C₄	#14273703	1	2	yes	—
172	C₂×C₄	C₂×#11745670	1	1	yes	characteristic
173	C₂×C₄	#14273758	1	2	yes	—
174	C₈	—	2	2	no	—
175	C₈	Q₈×Dic₅₉	1	2	yes	—
176	C₈	#14273716	1	1	yes	characteristic
177	C₈	#14273725	1	1	yes	characteristic
178	C₂²	C₂×#14273648	1	1	yes	characteristic
179	C₂²	#16837679	1	2	yes	—
180	C₂²	C₂×#14273668	1	1	yes	characteristic
181	C₂²	C₂×#14273671	1	1	yes	characteristic
182	C₂²	#16837843	1	2	yes	—
183	C₄	—	2	2	no	—
184	C₄	—	8	4	no	—
185	C₄	—	472	4	no	—
186	C₄	#16837203	1	4	yes	—
187	C₄	#16837309	1	1	yes	characteristic
188	C₄	#16837411	1	1	yes	characteristic
189	C₄	#16837684	1	2	yes	—
190	C₂	#21425049	1	1	yes	characteristic
191	C₂	C₂×#16837203	1	1	yes	characteristic
192	C₂	#21425816	1	1	yes	characteristic
193	C₂	#21426230	1	2	yes	—
194	C₂	#21426235	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11745669
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂×C₅₉×#500	C₂
socle, prime radical	C₂³×C₅₉	C₂×D₈
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂×#500	D₅₉
hypocenter	C₅₉	#13275
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
12. C₅₉×#500	C₄	semidirect
21. #14273661	Q₈	semidirect
105. #500	Dic₅₉	semidirect
113. C₅₉	#13275	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
C₄×#14273661	2
#21422187	2
C₂×C₅₉×#500	2
#13275	59