Subgroups — #28507161

Order 15104.

494 conjugacy classes of subgroups.

The subgroup lattice has 214 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	C₂×#16837206	C₂	1	4	yes	maximal
3	C₄×#14273661	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₉×#494	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	#16837206	—	2	8	no	—
10	#16837165	—	2	1	no	—
11	C₂²×Q₁₆×C₅₉	—	2	1	no	—
12	C₂²×Q₁₆×C₅₉	C₂²	1	2	yes	—
13	C₂×C₄⋊Q₈×C₅₉	—	2	1	no	—
14	C₅₉×#494	—	2	2	no	—
15	C₅₉×#494	C₄	1	4	yes	complemented
16	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₅₉	—	2	2	no	—
18	C₂×C₄×Dic₅₉	—	4	1	no	—
19	C₂×C₅₉⋊C₁₆	—	2	4	no	—
20	C₂×C₅₉⋊C₁₆	—	2	2	no	—
21	#14273661	—	2	2	no	—
22	#14273661	D₄	1	4	yes	—
23	C₅₉⋊#95	—	4	1	no	—
24	C₂²×Q₈×C₅₉	—	2	2	no	—
25	C₂×Q₁₆×C₅₉	—	2	4	no	—
26	C₂×Q₁₆×C₅₉	—	2	4	no	—
27	C₂×Q₁₆×C₅₉	—	2	2	no	—
28	C₂×Q₁₆×C₅₉	—	2	4	no	—
29	C₂×Q₁₆×C₅₉	—	4	4	no	—
30	C₂×Q₁₆×C₅₉	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
33	C₄⋊Q₈×C₅₉	—	2	4	no	—
34	C₄⋊Q₈×C₅₉	—	4	2	no	—
35	C₂×C₄⋊C₄×C₅₉	—	4	1	no	—
36	C₂×C₄⋊C₄×C₅₉	—	4	1	no	—
37	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
38	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₅₉	D₄	1	2	yes	—
40	C₂×Q₈×C₅₉	—	2	4	no	—
41	C₂×Q₈×C₅₉	—	4	2	no	—
42	C₂×Q₈×C₅₉	—	4	4	no	—
43	C₂×Q₈×C₅₉	—	4	4	no	—
44	C₂²×C₄×C₅₉	—	4	2	no	—
45	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
46	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
47	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
48	C₂²×Dic₅₉	—	4	2	no	—
49	Q₁₆×C₅₉	—	2	8	no	—
50	Q₁₆×C₅₉	—	4	8	no	—
51	Q₁₆×C₅₉	—	4	2	no	—
52	Q₁₆×C₅₉	—	4	2	no	—
53	C₂×C₈×C₅₉	—	2	2	no	—
54	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
55	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
57	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₅₉	#41	1	2	yes	—
59	C₄⋊C₄×C₅₉	—	4	4	no	—
60	C₄⋊C₄×C₅₉	—	4	4	no	—
61	C₄²×C₅₉	C₂²⋊C₄	1	2	yes	—
62	C₄²×C₅₉	SD₁₆	1	2	yes	—
63	C₄⋊Dic₅₉	—	2	4	no	—
64	C₄⋊Dic₅₉	—	4	2	no	—
65	C₄×Dic₅₉	—	8	2	no	—
66	C₅₉⋊C₁₆	—	4	2	no	—
67	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
68	Q₈×C₅₉	—	4	4	no	—
69	Q₈×C₅₉	—	4	4	no	—
70	Q₈×C₅₉	—	8	4	no	—
71	C₂×C₄×C₅₉	—	2	2	no	—
72	C₂×C₄×C₅₉	—	4	4	no	—
73	C₂×C₄×C₅₉	—	8	4	no	—
74	C₂×C₄×C₅₉	#102	1	2	yes	—
75	C₂×C₄×C₅₉	#102	1	2	yes	—
76	C₂×C₄×C₅₉	SD₃₂	1	2	yes	—
77	C₂×C₄×C₅₉	Q₃₂	1	2	yes	—
78	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₅₉	#121	1	2	yes	—
80	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
81	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
82	C₂×Dic₅₉	—	4	4	no	—
83	C₂×Dic₅₉	—	8	2	no	—
84	C₂×Dic₅₉	—	8	2	no	—
85	C₈×C₅₉	—	2	2	no	—
86	C₈×C₅₉	C₄×D₄	1	2	yes	—
87	C₈×C₅₉	#124	1	1	yes	characteristic
88	C₈×C₅₉	#127	1	1	yes	characteristic
89	#13253	—	59	1	no	maximal, Sylow
90	C₂²×C₅₉	C₂×#102	1	1	yes	characteristic
91	C₂²×C₅₉	#459	1	2	yes	—
92	C₂²×C₅₉	C₂×SD₃₂	1	1	yes	characteristic
93	C₂²×C₅₉	C₂×Q₃₂	1	1	yes	characteristic
94	C₂²×C₅₉	#510	1	2	yes	—
95	C₄×C₅₉	—	2	2	no	—
96	C₄×C₅₉	—	8	4	no	—
97	C₄×C₅₉	#358	1	4	yes	complemented
98	C₄×C₅₉	#390	1	1	yes	characteristic
99	C₄×C₅₉	#418	1	1	yes	characteristic
100	C₄×C₅₉	#460	1	2	yes	—
101	Dic₅₉	—	8	4	no	—
102	C₂×#494	D₅₉	1	1	yes	characteristic
103	C₂×C₄⋊C₁₆	—	59	1	no	—
104	C₂×#358	—	59	4	no	—
105	C₄×#107	—	59	1	no	—
106	C₂×C₅₉	#1982	1	1	yes	characteristic
107	C₂×C₅₉	C₂×#358	1	1	yes	characteristic
108	C₂×C₅₉	#2141	1	1	yes	characteristic
109	C₂×C₅₉	#2208	1	2	yes	—
110	C₂×C₅₉	#2209	1	2	yes	—
111	C₂²×Q₁₆	—	2	1	no	—
112	C₂²×Q₁₆	C₂×D₅₉	1	2	yes	—
113	C₂×C₄⋊Q₈	—	2	1	no	—
114	C₂²×C₁₆	—	59	2	no	—
115	#494	—	2	2	no	—
116	#494	Dic₅₉	1	4	yes	complemented
117	C₂×#107	—	59	2	no	—
118	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
119	C₄×C₄⋊C₄	—	118	1	no	—
120	C₄⋊C₁₆	—	118	2	no	—
121	#358	—	118	8	no	—
122	#340	—	118	1	no	—
123	C₅₉	#13253	1	1	yes	characteristic, complemented, Sylow
124	C₂²×Q₈	—	2	2	no	—
125	C₂×Q₁₆	—	2	4	no	—
126	C₂×Q₁₆	—	2	4	no	—
127	C₂×Q₁₆	—	2	4	no	—
128	C₂×Q₁₆	—	2	2	no	—
129	C₂×Q₁₆	—	4	4	no	—
130	C₂×Q₁₆	C₂×Dic₅₉	1	4	yes	—
131	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
132	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
133	C₄⋊Q₈	—	2	4	no	—
134	C₄⋊Q₈	—	4	2	no	—
135	C₂×C₄⋊C₄	—	4	1	no	—
136	C₂×C₄⋊C₄	—	4	1	no	—
137	C₂×C₄⋊C₄	—	118	2	no	—
138	C₂×C₄²	—	236	1	no	—
139	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
140	C₂×C₁₆	—	118	4	no	—
141	C₂×C₁₆	—	118	2	no	—
142	#107	—	59	4	no	—
143	#107	—	118	2	no	—
144	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
145	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
146	#95	—	236	1	no	—
147	C₂×Q₈	—	2	4	no	—
148	C₂×Q₈	—	4	2	no	—
149	C₂×Q₈	—	4	4	no	—
150	C₂×Q₈	—	4	4	no	—
151	C₂²×C₄	—	4	2	no	—
152	C₂²×C₄	—	236	2	no	—
153	C₂²×C₄	#11745669	1	1	yes	characteristic
154	C₂²×C₄	#11745669	1	1	yes	characteristic
155	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
156	Q₁₆	—	2	8	no	—
157	Q₁₆	—	4	8	no	—
158	Q₁₆	—	4	2	no	—
159	Q₁₆	—	4	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	C₂²⋊Dic₅₉	1	2	yes	—
162	C₂×C₈	D₄×D₅₉	1	2	yes	—
163	C₂×C₈	#11745687	1	2	yes	—
164	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
165	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
166	C₄⋊C₄	—	4	4	no	—
167	C₄⋊C₄	—	4	4	no	—
168	C₄⋊C₄	—	118	4	no	—
169	C₄⋊C₄	—	236	2	no	—
170	C₄²	—	472	2	no	—
171	C₄²	#11745670	1	2	yes	—
172	C₄²	C₂²⋊Dic₅₉	1	2	yes	—
173	C₁₆	—	236	2	no	—
174	C₂³	C₂×#11745669	1	1	yes	characteristic
175	Q₈	—	4	4	no	—
176	Q₈	—	4	4	no	—
177	Q₈	—	8	4	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	—	8	4	no	—
181	C₂×C₄	—	472	2	no	—
182	C₂×C₄	—	472	2	no	—
183	C₂×C₄	—	236	4	no	—
184	C₂×C₄	#14273648	1	2	yes	—
185	C₂×C₄	#14273648	1	2	yes	—
186	C₂×C₄	#14273670	1	2	yes	—
187	C₂×C₄	#14273673	1	2	yes	—
188	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
189	C₂×C₄	#14273698	1	2	yes	—
190	C₂×C₄	C₂×#11745670	1	1	yes	characteristic
191	C₂×C₄	#14273753	1	2	yes	—
192	C₈	—	2	2	no	—
193	C₈	D₄×Dic₅₉	1	2	yes	—
194	C₈	#14273714	1	1	yes	characteristic
195	C₈	#14273723	1	1	yes	characteristic
196	C₂²	C₂×#14273648	1	1	yes	characteristic
197	C₂²	#16837594	1	2	yes	—
198	C₂²	C₂×#14273670	1	1	yes	characteristic
199	C₂²	C₂×#14273673	1	1	yes	characteristic
200	C₂²	#16837843	1	2	yes	—
201	C₄	—	2	2	no	—
202	C₄	—	8	4	no	—
203	C₄	—	472	4	no	—
204	C₄	#16837206	1	4	yes	complemented
205	C₄	#16837305	1	1	yes	characteristic
206	C₄	#16837407	1	1	yes	characteristic
207	C₄	#16837599	1	2	yes	—
208	C₂	#21425024	1	1	yes	characteristic
209	C₂	C₂×#16837206	1	1	yes	characteristic
210	C₂	#21425816	1	1	yes	characteristic
211	C₂	#21426138	1	2	yes	—
212	C₂	#21426143	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11745669
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂×C₅₉×#494	C₂
socle, prime radical	C₂³×C₅₉	C₂×D₈
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂×#494	D₅₉
hypocenter	C₅₉	#13253
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
15. C₅₉×#494	C₄	semidirect
97. C₄×C₅₉	#358	semidirect
116. #494	Dic₅₉	semidirect
123. C₅₉	#13253	semidirect
204. C₄	#16837206	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
C₂×#16837206	2
C₄×#14273661	2
C₂×C₅₉×#494	2
#13253	59