Subgroups — #28507271

Order 15104.

366 conjugacy classes of subgroups.

The subgroup lattice has 212 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	2	yes	maximal
4	#21422187	C₂	1	2	yes	maximal
5	C₂×C₅₉×#499	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅₉⋊C₁₆	C₂²	1	2	yes	—
7	C₂×#14273661	C₂²	1	2	yes	—
8	C₄×C₄⋊Dic₅₉	—	2	1	no	—
9	C₄⋊(C₅₉⋊C₁₆)	—	2	2	no	—
10	#16837165	—	2	1	no	—
11	C₂×#125×C₅₉	—	2	1	no	—
12	C₅₉×#499	—	2	2	no	—
13	C₅₉×#499	C₄	1	4	yes	complemented
14	C₂×#107×C₅₉	C₂²	1	1	yes	characteristic
15	C₂×#107×C₅₉	C₂²	1	1	yes	characteristic
16	C₂×#106×C₅₉	—	2	1	no	—
17	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₅₉	—	2	2	no	—
19	C₂×C₄×Dic₅₉	—	4	1	no	—
20	C₂×C₅₉⋊C₁₆	—	2	2	no	—
21	C₂×C₅₉⋊C₁₆	—	2	4	no	—
22	#14273661	—	2	2	no	—
23	#14273661	Q₈	1	4	yes	—
24	C₅₉⋊#95	—	4	1	no	—
25	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
27	#125×C₅₉	—	2	4	no	—
28	#125×C₅₉	—	4	2	no	—
29	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
30	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
31	C₂×C₄⋊C₄×C₅₉	—	4	1	no	—
32	C₂×C₄⋊C₄×C₅₉	—	4	1	no	—
33	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
34	#107×C₅₉	—	2	1	no	—
35	#107×C₅₉	—	2	1	no	—
36	#107×C₅₉	C₂×C₄	1	2	yes	—
37	#107×C₅₉	C₂×C₄	1	2	yes	—
38	#106×C₅₉	—	2	4	no	—
39	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₅₉	D₄	1	2	yes	—
41	C₂²×C₄×C₅₉	—	4	1	no	—
42	C₂²×C₄×C₅₉	—	4	1	no	—
43	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
44	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
45	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
46	C₂²×Dic₅₉	—	4	2	no	—
47	C₂×C₈×C₅₉	—	2	2	no	—
48	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
49	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
52	C₂×C₈×C₅₉	#41	1	2	yes	—
53	C₄⋊C₄×C₅₉	—	2	2	no	—
54	C₄⋊C₄×C₅₉	—	2	2	no	—
55	C₄⋊C₄×C₅₉	—	4	4	no	—
56	C₄⋊C₄×C₅₉	—	4	4	no	—
57	C₄⋊C₄×C₅₉	—	4	1	no	—
58	C₄⋊C₄×C₅₉	—	4	1	no	—
59	C₄²×C₅₉	C₂²⋊C₄	1	2	yes	—
60	C₄²×C₅₉	SD₁₆	1	2	yes	—
61	C₄⋊Dic₅₉	—	2	4	no	—
62	C₄⋊Dic₅₉	—	4	2	no	—
63	C₄×Dic₅₉	—	8	2	no	—
64	C₅₉⋊C₁₆	—	4	2	no	—
65	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄×C₅₉	—	2	2	no	—
67	C₂×C₄×C₅₉	—	2	1	no	—
68	C₂×C₄×C₅₉	—	2	1	no	—
69	C₂×C₄×C₅₉	—	4	2	no	—
70	C₂×C₄×C₅₉	—	4	2	no	—
71	C₂×C₄×C₅₉	—	8	2	no	—
72	C₂×C₄×C₅₉	—	8	1	no	—
73	C₂×C₄×C₅₉	—	8	1	no	—
74	C₂×C₄×C₅₉	#102	1	2	yes	—
75	C₂×C₄×C₅₉	#102	1	2	yes	—
76	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
78	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₅₉	#137	1	2	yes	—
80	C₂×Dic₅₉	—	4	4	no	—
81	C₂×Dic₅₉	—	8	2	no	—
82	C₂×Dic₅₉	—	8	2	no	—
83	C₈×C₅₉	—	2	2	no	—
84	C₈×C₅₉	C₄×Q₈	1	2	yes	—
85	C₈×C₅₉	#124	1	1	yes	characteristic
86	C₈×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
87	#13277	—	59	1	no	maximal, Sylow
88	C₂²×C₅₉	C₂×#102	1	1	yes	characteristic
89	C₂²×C₅₉	#474	1	2	yes	—
90	C₂²×C₅₉	#508	1	2	yes	—
91	C₂²×C₅₉	#509	1	1	yes	characteristic
92	C₂²×C₅₉	#510	1	1	yes	characteristic
93	C₄×C₅₉	—	2	2	no	—
94	C₄×C₅₉	—	2	2	no	—
95	C₄×C₅₉	—	8	2	no	—
96	C₄×C₅₉	—	8	2	no	—
97	C₄×C₅₉	#391	1	1	yes	characteristic
98	C₄×C₅₉	#419	1	1	yes	characteristic
99	C₄×C₅₉	#475	1	2	yes	—
100	Dic₅₉	—	8	4	no	—
101	C₂×#499	D₅₉	1	1	yes	characteristic
102	C₂×C₄⋊C₁₆	—	59	1	no	—
103	C₄×#107	—	59	1	no	—
104	#1380	—	59	2	no	—
105	#1380	—	59	2	no	—
106	C₂×C₅₉	#1988	1	1	yes	characteristic
107	C₂×C₅₉	#2138	1	1	yes	characteristic
108	C₂×C₅₉	#2139	1	1	yes	characteristic
109	C₂×C₅₉	#2228	1	2	yes	—
110	C₂×C₅₉	#2229	1	2	yes	—
111	C₂×#125	—	2	1	no	—
112	C₂²×C₁₆	—	59	2	no	—
113	#499	—	2	2	no	—
114	#499	Dic₅₉	1	4	yes	complemented
115	C₂×#107	—	59	2	no	—
116	C₂×#107	C₂×D₅₉	1	1	yes	characteristic
117	C₂×#107	C₂×D₅₉	1	1	yes	characteristic
118	C₂×#106	—	2	1	no	—
119	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
120	C₄×C₄⋊C₄	—	118	1	no	—
121	C₄⋊C₁₆	—	118	2	no	—
122	#340	—	118	1	no	—
123	C₅₉	#13277	1	1	yes	characteristic, complemented, Sylow
124	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
125	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
126	#125	—	2	4	no	—
127	#125	—	4	2	no	—
128	C₂×C₄⋊C₄	—	2	1	no	—
129	C₂×C₄⋊C₄	—	2	1	no	—
130	C₂×C₄⋊C₄	—	4	1	no	—
131	C₂×C₄⋊C₄	—	4	1	no	—
132	C₂×C₄⋊C₄	—	118	2	no	—
133	C₂×C₄²	—	236	1	no	—
134	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
135	C₂×C₁₆	—	118	4	no	—
136	C₂×C₁₆	—	118	2	no	—
137	#107	—	2	1	no	—
138	#107	—	2	1	no	—
139	#107	—	59	4	no	—
140	#107	—	118	2	no	—
141	#107	C₂×Dic₅₉	1	2	yes	—
142	#107	C₂×Dic₅₉	1	2	yes	—
143	#106	—	2	4	no	—
144	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
145	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
146	#95	—	236	1	no	—
147	C₂²×C₄	—	4	1	no	—
148	C₂²×C₄	—	4	1	no	—
149	C₂²×C₄	—	236	2	no	—
150	C₂²×C₄	#11745669	1	1	yes	characteristic
151	C₂²×C₄	#11745669	1	1	yes	characteristic
152	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	C₂²⋊Dic₅₉	1	2	yes	—
155	C₂×C₈	Q₈×D₅₉	1	2	yes	—
156	C₂×C₈	#11745689	1	2	yes	—
157	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
158	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
159	C₄⋊C₄	—	2	2	no	—
160	C₄⋊C₄	—	2	2	no	—
161	C₄⋊C₄	—	4	4	no	—
162	C₄⋊C₄	—	4	4	no	—
163	C₄⋊C₄	—	4	1	no	—
164	C₄⋊C₄	—	4	1	no	—
165	C₄⋊C₄	—	118	4	no	—
166	C₄⋊C₄	—	236	2	no	—
167	C₄²	—	472	2	no	—
168	C₄²	#11745670	1	2	yes	—
169	C₄²	C₂²⋊Dic₅₉	1	2	yes	—
170	C₁₆	—	236	2	no	—
171	C₂³	C₂×#11745669	1	1	yes	characteristic
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	8	2	no	—
178	C₂×C₄	—	8	1	no	—
179	C₂×C₄	—	8	1	no	—
180	C₂×C₄	—	472	2	no	—
181	C₂×C₄	—	472	2	no	—
182	C₂×C₄	—	236	4	no	—
183	C₂×C₄	#14273648	1	2	yes	—
184	C₂×C₄	#14273648	1	2	yes	—
185	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
186	C₂×C₄	#14273703	1	2	yes	—
187	C₂×C₄	C₂×#11745670	1	1	yes	characteristic
188	C₂×C₄	#14273758	1	2	yes	—
189	C₈	—	2	2	no	—
190	C₈	Q₈×Dic₅₉	1	2	yes	—
191	C₈	#14273716	1	1	yes	characteristic
192	C₈	#14273725	1	1	yes	characteristic
193	C₂²	C₂×#14273648	1	1	yes	characteristic
194	C₂²	#16837679	1	2	yes	—
195	C₂²	#16837832	1	2	yes	—
196	C₂²	#16837837	1	1	yes	characteristic
197	C₂²	#16837842	1	1	yes	characteristic
198	C₄	—	2	2	no	—
199	C₄	—	2	2	no	—
200	C₄	—	8	2	no	—
201	C₄	—	8	2	no	—
202	C₄	—	472	4	no	—
203	C₄	#16837309	1	1	yes	characteristic
204	C₄	#16837411	1	1	yes	characteristic
205	C₄	#16837684	1	2	yes	—
206	C₂	#21425049	1	1	yes	characteristic
207	C₂	#21425803	1	1	yes	characteristic
208	C₂	#21425808	1	1	yes	characteristic
209	C₂	#21426250	1	2	yes	—
210	C₂	#21426257	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11745669
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂×C₅₉×#499	C₂
socle, prime radical	C₂³×C₅₉	C₂×D₈
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂×#499	D₅₉
hypocenter	C₅₉	#13277
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

Complements

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

N	H ≅ G/N	type
13. C₅₉×#499	C₄	semidirect
114. #499	Dic₅₉	semidirect
123. C₅₉	#13277	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×#499	2
#13277	59