Subgroups — #28744179

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	#21434608	C₂	1	2	yes	maximal
2	#21426377	C₂	1	2	yes	maximal, complemented
3	#21426374	C₂	1	2	yes	maximal, complemented
4	#21426373	C₂	1	2	yes	maximal, complemented
5	#21426370	C₂	1	2	yes	maximal, complemented
6	#21425880	C₂	1	1	yes	maximal, characteristic
7	#21425878	C₂	1	2	yes	maximal
8	C₂×C₁₆×Dic₅₉	C₂	1	1	yes	maximal, characteristic
9	C₅₉×#2158	C₂	1	1	yes	maximal, characteristic
10	#16837955	—	2	2	no	—
11	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
12	#16837806	C₂²	1	4	yes	—
13	#16837803	C₂²	1	4	yes	—
14	#16837802	—	2	2	no	—
15	#16837799	—	2	2	no	—
16	#16837440	C₂²	1	2	yes	—
17	#16837438	—	2	2	no	—
18	C₂×C₈×Dic₅₉	C₂²	1	1	yes	characteristic
19	#16837230	C₂²	1	2	yes	—
20	#16837229	C₂²	1	4	yes	—
21	#16837228	C₂²	1	2	yes	—
22	#16837227	C₂²	1	4	yes	—
23	C₁₆×Dic₅₉	C₂²	1	1	yes	characteristic
24	C₁₆×Dic₅₉	C₂²	1	1	yes	characteristic
25	C₁₆×Dic₅₉	C₂²	1	1	yes	characteristic
26	C₁₆×Dic₅₉	C₂²	1	1	yes	characteristic
27	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
28	C₅₉×#427	C₂²	1	2	yes	—
29	C₅₉×#367	C₂²	1	2	yes	—
30	C₅₉×#366	C₂²	1	2	yes	—
31	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
32	#14273726	—	2	4	no	—
33	#14273718	—	2	4	no	—
34	#14273704	—	4	4	no	—
35	C₄×Dic₁₁₈	—	4	4	no	—
36	#14273686	—	2	2	no	—
37	C₂×C₄×Dic₅₉	D₄	1	1	yes	characteristic
38	C₂×C₅₉⋊C₁₆	Q₈	1	4	yes	complemented
39	C₂×C₅₉⋊C₁₆	C₂³	1	1	yes	characteristic
40	C₂×C₅₉⋊C₁₆	C₂³	1	1	yes	characteristic
41	#14273661	C₂³	1	4	yes	—
42	#14273660	—	2	4	no	—
43	#14273659	—	2	2	no	—
44	#14273658	—	2	4	no	—
45	C₈×Dic₅₉	D₄	1	1	yes	characteristic
46	C₈×Dic₅₉	D₄	1	1	yes	characteristic
47	C₈×Dic₅₉	C₂³	1	1	yes	characteristic
48	C₈×Dic₅₉	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
50	#117×C₅₉	—	2	2	no	—
51	C₂×C₁₆×C₅₉	Q₈	1	4	yes	complemented
52	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
53	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
54	#107×C₅₉	C₂³	1	4	yes	—
55	#106×C₅₉	—	2	2	no	—
56	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
57	C₂²×Dic₅₉	D₈	1	2	yes	—
58	C₂×Dic₁₁₈	—	4	4	no	—
59	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
61	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
62	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
63	C₄⋊C₄×C₅₉	—	2	4	no	—
64	C₂²⋊C₄×C₅₉	—	4	2	no	—
65	C₄²×C₅₉	—	4	2	no	—
66	C₂²⋊Dic₅₉	—	4	2	no	—
67	C₄⋊Dic₅₉	—	2	4	no	—
68	C₄⋊Dic₅₉	—	4	4	no	—
69	Dic₅₉⋊C₄	—	4	4	no	—
70	Dic₅₉⋊C₄	—	4	4	no	—
71	C₄×Dic₅₉	—	4	2	no	—
72	C₄×Dic₅₉	D₈	1	2	yes	—
73	C₄×Dic₅₉	C₂×D₄	1	1	yes	characteristic
74	C₄×Dic₅₉	C₂×D₄	1	1	yes	characteristic
75	C₂×C₅₉⋊C₈	Q₁₆	1	4	yes	—
76	C₂×C₅₉⋊C₈	C₂×D₄	1	1	yes	characteristic
77	C₂×C₅₉⋊C₈	C₂×D₄	1	1	yes	characteristic
78	C₁₆×C₅₉	C₂×Q₈	1	2	yes	—
79	C₁₆×C₅₉	C₂×Q₈	1	2	yes	—
80	C₅₉⋊C₁₆	C₂×Q₈	1	2	yes	—
81	C₅₉⋊C₁₆	C₂×Q₈	1	2	yes	—
82	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₅₉	—	4	4	no	—
84	C₂×C₄×C₅₉	C₄⋊Q₈	1	2	yes	—
85	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₅₉	C₂×Q₁₆	1	2	yes	—
87	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
88	C₂×Dic₅₉	—	2	4	no	—
89	C₂×Dic₅₉	—	4	4	no	—
90	C₂×Dic₅₉	C₂×D₈	1	2	yes	—
91	C₂×Dic₅₉	C₂×D₈	1	2	yes	—
92	Dic₁₁₈	—	8	8	no	—
93	C₈×C₅₉	C₄⋊Q₈	1	2	yes	—
94	C₈×C₅₉	C₂²×Q₈	1	1	yes	characteristic
95	C₈×C₅₉	C₂²×Q₈	1	1	yes	characteristic
96	C₅₉⋊C₈	C₂×Q₁₆	1	2	yes	—
97	C₅₉⋊C₈	C₂×Q₁₆	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	—	8	4	no	—
103	C₄×C₅₉	#500	1	2	yes	—
104	C₄×C₅₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
105	C₄×C₅₉	C₂²×Q₁₆	1	1	yes	characteristic
106	Dic₅₉	—	2	2	no	—
107	Dic₅₉	—	8	4	no	—
108	Dic₅₉	#508	1	4	yes	—
109	#3160	—	59	2	no	—
110	#2254	—	59	2	no	—
111	#2253	—	59	2	no	—
112	#2253	—	59	2	no	—
113	#2252	—	59	2	no	—
114	#2158	—	59	1	no	—
115	#2158	—	59	2	no	—
116	#2158	D₅₉	1	1	yes	characteristic
117	C₂×C₄×C₁₆	—	59	1	no	—
118	C₂×C₅₉	—	2	2	no	—
119	C₂×C₅₉	C₂×#500	1	1	yes	characteristic
120	C₂×C₅₉	C₂×#508	1	2	yes	—
121	#533	—	118	2	no	—
122	C₂²×C₁₆	—	59	1	no	—
123	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
124	#500	—	59	4	no	—
125	#499	—	59	4	no	—
126	#499	—	118	2	no	—
127	#498	—	118	2	no	—
128	#427	—	59	2	no	—
129	#427	—	118	2	no	—
130	#427	C₂×D₅₉	1	2	yes	—
131	C₂×C₄×C₈	—	59	1	no	—
132	#367	—	59	2	no	—
133	#367	—	59	4	no	—
134	#367	C₂×D₅₉	1	2	yes	—
135	#366	—	59	2	no	—
136	#366	—	59	4	no	—
137	#366	C₂×D₅₉	1	2	yes	—
138	C₄×C₁₆	—	59	1	no	—
139	C₄×C₁₆	—	59	1	no	—
140	C₄×C₁₆	—	59	1	no	—
141	C₄×C₁₆	—	59	1	no	—
142	C₅₉	#33726	1	1	yes	characteristic, complemented, Sylow
143	C₂²×C₈	—	59	1	no	—
144	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
145	C₄⋊Q₈	—	118	4	no	—
146	#125	—	118	4	no	—
147	C₂²⋊Q₈	—	236	4	no	—
148	C₄×Q₈	—	236	4	no	—
149	#117	—	2	2	no	—
150	#117	—	118	2	no	—
151	C₂×C₄²	—	59	1	no	—
152	C₂×C₁₆	—	59	4	no	—
153	C₂×C₁₆	—	59	1	no	—
154	C₂×C₁₆	—	59	1	no	—
155	C₂×C₁₆	Dic₁₁₈	1	4	yes	complemented
156	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
157	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
158	#107	—	59	4	no	—
159	#107	—	118	4	no	—
160	#107	C₂²×D₅₉	1	4	yes	—
161	#106	—	2	2	no	—
162	#106	—	118	4	no	—
163	#106	—	118	2	no	—
164	C₄×C₈	—	59	1	no	—
165	C₄×C₈	—	59	1	no	—
166	C₄×C₈	—	59	1	no	—
167	C₄×C₈	—	59	1	no	—
168	C₂×Q₈	—	236	4	no	—
169	C₂²×C₄	—	59	2	no	—
170	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
171	C₂×C₈	—	59	4	no	—
172	C₂×C₈	—	59	1	no	—
173	C₂×C₈	—	59	1	no	—
174	C₂×C₈	C₂×Dic₁₁₈	1	2	yes	—
175	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
176	C₂×C₈	Q₈×D₅₉	1	2	yes	—
177	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
178	C₄⋊C₄	—	2	4	no	—
179	C₄⋊C₄	—	118	4	no	—
180	C₄⋊C₄	—	236	4	no	—
181	C₄⋊C₄	—	236	4	no	—
182	C₄⋊C₄	—	236	4	no	—
183	C₂²⋊C₄	—	4	2	no	—
184	C₂²⋊C₄	—	236	2	no	—
185	C₄²	—	4	2	no	—
186	C₄²	—	59	2	no	—
187	C₄²	—	59	1	no	—
188	C₄²	—	59	1	no	—
189	C₄²	—	236	2	no	—
190	C₁₆	—	59	2	no	—
191	C₁₆	—	59	2	no	—
192	C₁₆	C₂×Dic₁₁₈	1	2	yes	—
193	C₁₆	C₂×Dic₁₁₈	1	2	yes	—
194	C₂³	D₈×D₅₉	1	1	yes	characteristic
195	Q₈	—	472	8	no	—
196	C₂×C₄	—	4	4	no	—
197	C₂×C₄	—	59	2	no	—
198	C₂×C₄	—	59	2	no	—
199	C₂×C₄	—	118	4	no	—
200	C₂×C₄	—	236	4	no	—
201	C₂×C₄	#14273726	1	2	yes	—
202	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
203	C₂×C₄	Q₁₆×D₅₉	1	2	yes	—
204	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
205	C₈	—	59	2	no	—
206	C₈	—	59	2	no	—
207	C₈	#14273726	1	2	yes	—
208	C₈	C₂×Q₈×D₅₉	1	1	yes	characteristic
209	C₈	C₂²×Dic₁₁₈	1	1	yes	characteristic
210	C₂²	—	2	2	no	—
211	C₂²	#16837806	1	2	yes	—
212	C₂²	C₂×D₈×D₅₉	1	1	yes	characteristic
213	C₄	—	8	4	no	—
214	C₄	—	59	4	no	—
215	C₄	—	118	2	no	—
216	C₄	—	472	4	no	—
217	C₄	#16837806	1	2	yes	—
218	C₄	C₂×#14273726	1	1	yes	characteristic
219	C₄	C₂×Q₁₆×D₅₉	1	1	yes	characteristic
220	C₂	—	2	2	no	—
221	C₂	C₂×#16837806	1	1	yes	characteristic
222	C₂	#508×D₅₉	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₅₉
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₅₉×#2158	C₂
socle	C₂²×C₅₉	C₂²×D₈
prime radical	C₂³×C₅₉	C₂×D₈
hypercenter	#2158	D₅₉
hypocenter	C₅₉	#33726
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

Complements

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

N	H ≅ G/N	type
2. #21426377	C₂	semidirect
3. #21426374	C₂	semidirect
4. #21426373	C₂	semidirect
5. #21426370	C₂	semidirect
38. C₂×C₅₉⋊C₁₆	Q₈	semidirect
51. C₂×C₁₆×C₅₉	Q₈	semidirect
142. C₅₉	#33726	semidirect
155. C₂×C₁₆	Dic₁₁₈	semidirect

Maximal subgroups

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

subgroup	index
#21434608	2
#21426377	2
#21426374	2
#21426373	2
#21426370	2
#21425880	2
#21425878	2
C₂×C₁₆×Dic₅₉	2
C₅₉×#2158	2
#33726	59