Subgroups — #28498648

Order 15104.

292 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21425899	C₂	1	1	yes	maximal, characteristic
2	#21425895	C₂	1	1	yes	maximal, characteristic
3	#21422078	C₂	1	4	yes	maximal, complemented
4	C₅₉×#2960	C₂	1	1	yes	maximal, characteristic
5	C₄⋊(C₅₉⋊C₁₆)	—	2	1	no	—
6	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	2	yes	—
7	C₂²⋊(C₅₉⋊C₁₆)	—	2	1	no	—
8	C₂²⋊(C₅₉⋊C₁₆)	—	2	1	no	—
9	#16837178	C₂²	1	2	yes	—
10	C₄×C₅₉⋊C₁₆	—	2	1	no	—
11	C₅₉×#533	C₄	1	1	yes	characteristic
12	C₅₉×#446	—	2	1	no	—
13	C₅₉×#446	C₂²	1	2	yes	—
14	C₈×Q₈×C₅₉	—	2	1	no	—
15	C₈×Q₈×C₅₉	C₄	1	2	yes	—
16	C₅₉×#424	—	2	1	no	—
17	C₅₉×#423	C₄	1	1	yes	characteristic
18	C₅₉×#405	—	2	1	no	—
19	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
20	C₂×C₅₉⋊C₁₆	—	2	2	no	—
21	C₂×C₅₉⋊C₁₆	—	2	2	no	—
22	C₂×M₁₆×C₅₉	—	2	1	no	—
23	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
25	C₄⋊Q₈×C₅₉	C₈	1	1	yes	characteristic
26	C₄⋊Q₈×C₅₉	C₈	1	1	yes	characteristic
27	#125×C₅₉	—	2	1	no	—
28	C₂²⋊Q₈×C₅₉	—	2	1	no	—
29	C₂²⋊Q₈×C₅₉	C₈	1	2	yes	—
30	C₄×Q₈×C₅₉	—	2	1	no	—
31	C₄×Q₈×C₅₉	C₂×C₄	1	2	yes	—
32	#117×C₅₉	—	2	1	no	—
33	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
34	C₄⋊C₈×C₅₉	—	2	2	no	—
35	C₄⋊C₈×C₅₉	—	2	2	no	—
36	C₄⋊C₈×C₅₉	—	2	1	no	—
37	C₄⋊C₈×C₅₉	C₂×C₄	1	2	yes	—
38	#97×C₅₉	—	2	2	no	—
39	C₄×C₈×C₅₉	—	2	2	no	—
40	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
42	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
43	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
44	C₂×Q₈×C₅₉	—	2	1	no	—
45	C₂×Q₈×C₅₉	C₂×C₈	1	2	yes	—
46	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₅₉	M₁₆	1	1	yes	characteristic
49	M₁₆×C₅₉	—	4	2	no	—
50	C₂×C₈×C₅₉	—	2	2	no	—
51	C₂×C₈×C₅₉	—	2	2	no	—
52	C₂×C₈×C₅₉	—	4	1	no	—
53	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
57	C₄⋊C₄×C₅₉	—	2	1	no	—
58	C₄⋊C₄×C₅₉	—	2	1	no	—
59	C₄⋊C₄×C₅₉	—	2	1	no	—
60	C₄⋊C₄×C₅₉	—	2	2	no	—
61	C₄⋊C₄×C₅₉	—	2	2	no	—
62	C₄⋊C₄×C₅₉	C₂×C₈	1	2	yes	—
63	C₂²⋊C₄×C₅₉	—	2	2	no	—
64	C₄²×C₅₉	—	2	2	no	—
65	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
67	C₄²×C₅₉	M₁₆	1	1	yes	characteristic
68	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
69	C₅₉⋊C₁₆	—	4	2	no	—
70	C₅₉⋊C₁₆	—	4	2	no	—
71	C₂³×C₅₉	C₂²⋊C₈	1	1	yes	characteristic
72	Q₈×C₅₉	—	4	2	no	—
73	Q₈×C₅₉	—	4	2	no	—
74	C₂×C₄×C₅₉	—	2	2	no	—
75	C₂×C₄×C₅₉	—	2	1	no	—
76	C₂×C₄×C₅₉	—	2	1	no	—
77	C₂×C₄×C₅₉	—	2	2	no	—
78	C₂×C₄×C₅₉	—	2	2	no	—
79	C₂×C₄×C₅₉	—	4	1	no	—
80	C₂×C₄×C₅₉	C₂²⋊C₈	1	1	yes	characteristic
81	C₂×C₄×C₅₉	C₂²⋊C₈	1	1	yes	characteristic
82	C₂×C₄×C₅₉	C₂²⋊C₈	1	1	yes	characteristic
83	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
86	C₈×C₅₉	—	2	1	no	—
87	C₈×C₅₉	—	2	1	no	—
88	C₈×C₅₉	—	2	1	no	—
89	C₈×C₅₉	—	4	2	no	—
90	C₈×C₅₉	#137	1	2	yes	—
91	#11641	—	59	1	no	maximal, Sylow
92	C₂²×C₅₉	—	2	1	no	—
93	C₂²×C₅₉	—	2	1	no	—
94	C₂²×C₅₉	—	2	1	no	—
95	C₂²×C₅₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
96	C₄×C₅₉	—	2	1	no	—
97	C₄×C₅₉	—	2	1	no	—
98	C₄×C₅₉	—	2	1	no	—
99	C₄×C₅₉	—	2	1	no	—
100	C₄×C₅₉	—	4	2	no	—
101	C₄×C₅₉	—	4	2	no	—
102	C₄×C₅₉	#350	1	2	yes	—
103	C₄×C₅₉	#419	1	1	yes	characteristic
104	C₄×C₅₉	#421	1	1	yes	characteristic
105	#2960	D₅₉	1	1	yes	characteristic
106	#2163	—	59	1	no	—
107	#2162	—	59	1	no	—
108	#1338	—	59	4	no	—
109	C₂×C₅₉	—	4	1	no	—
110	C₂×C₅₉	#1479	1	1	yes	characteristic
111	C₂×C₅₉	#2115	1	1	yes	characteristic
112	C₂×C₅₉	C₂×#350	1	1	yes	characteristic
113	#533	Dic₅₉	1	1	yes	characteristic
114	#446	—	2	1	no	—
115	#446	C₂×D₅₉	1	2	yes	—
116	C₈×Q₈	—	2	1	no	—
117	C₈×Q₈	Dic₅₉	1	2	yes	—
118	#424	—	2	1	no	—
119	#423	Dic₅₉	1	1	yes	characteristic
120	#405	—	2	1	no	—
121	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
122	C₄⋊C₁₆	—	59	2	no	—
123	C₄⋊C₁₆	—	118	1	no	—
124	C₂²⋊C₁₆	—	118	1	no	—
125	C₂²⋊C₁₆	—	118	1	no	—
126	#346	—	59	2	no	—
127	C₄×C₁₆	—	118	1	no	—
128	C₅₉	#11641	1	1	yes	characteristic, complemented, Sylow
129	C₂×M₁₆	—	2	1	no	—
130	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
131	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
132	C₄⋊Q₈	C₅₉⋊C₈	1	1	yes	characteristic
133	C₄⋊Q₈	C₅₉⋊C₈	1	1	yes	characteristic
134	#125	—	2	1	no	—
135	C₂²⋊Q₈	—	2	1	no	—
136	C₂²⋊Q₈	C₅₉⋊C₈	1	2	yes	—
137	C₄×Q₈	—	2	1	no	—
138	C₄×Q₈	C₂×Dic₅₉	1	2	yes	—
139	#117	—	2	1	no	—
140	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
141	C₂×C₁₆	—	118	2	no	—
142	C₂×C₁₆	—	118	2	no	—
143	C₄⋊C₈	—	2	2	no	—
144	C₄⋊C₈	—	2	2	no	—
145	C₄⋊C₈	—	2	1	no	—
146	C₄⋊C₈	C₂×Dic₅₉	1	2	yes	—
147	#97	—	2	2	no	—
148	C₄×C₈	—	2	2	no	—
149	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
150	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
151	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
152	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
153	C₂×Q₈	—	2	1	no	—
154	C₂×Q₈	C₂×C₅₉⋊C₈	1	2	yes	—
155	C₂²×C₄	C₂×C₅₉⋊C₈	1	1	yes	characteristic
156	C₂²×C₄	#11745664	1	1	yes	characteristic
157	C₂²×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
158	M₁₆	—	4	2	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	4	1	no	—
162	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
163	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
164	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
165	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
166	C₄⋊C₄	—	2	1	no	—
167	C₄⋊C₄	—	2	1	no	—
168	C₄⋊C₄	—	2	1	no	—
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	C₂×C₅₉⋊C₈	1	2	yes	—
172	C₂²⋊C₄	—	2	2	no	—
173	C₄²	—	2	2	no	—
174	C₄²	C₂×C₅₉⋊C₈	1	1	yes	characteristic
175	C₄²	#11745664	1	1	yes	characteristic
176	C₄²	C₂²⋊Dic₅₉	1	1	yes	characteristic
177	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
178	C₁₆	—	236	2	no	—
179	C₁₆	—	236	2	no	—
180	C₂³	C₂²⋊(C₅₉⋊C₈)	1	1	yes	characteristic
181	Q₈	—	4	2	no	—
182	Q₈	—	4	2	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	4	1	no	—
189	C₂×C₄	C₂²⋊(C₅₉⋊C₈)	1	1	yes	characteristic
190	C₂×C₄	C₂²⋊(C₅₉⋊C₈)	1	1	yes	characteristic
191	C₂×C₄	C₂²⋊(C₅₉⋊C₈)	1	1	yes	characteristic
192	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
193	C₂×C₄	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
194	C₂×C₄	C₂×#11745664	1	1	yes	characteristic
195	C₈	—	2	1	no	—
196	C₈	—	2	1	no	—
197	C₈	—	2	1	no	—
198	C₈	—	4	2	no	—
199	C₈	#14273758	1	2	yes	—
200	C₂²	—	2	1	no	—
201	C₂²	—	2	1	no	—
202	C₂²	—	2	1	no	—
203	C₂²	C₂×C₂²⋊(C₅₉⋊C₈)	1	1	yes	characteristic
204	C₄	—	2	1	no	—
205	C₄	—	2	1	no	—
206	C₄	—	2	1	no	—
207	C₄	—	2	1	no	—
208	C₄	—	4	2	no	—
209	C₄	—	4	2	no	—
210	C₄	#16837186	1	2	yes	—
211	C₄	#16837411	1	1	yes	characteristic
212	C₄	#16837419	1	1	yes	characteristic
213	C₂	—	4	1	no	—
214	C₂	#21422491	1	1	yes	characteristic
215	C₂	#21425717	1	1	yes	characteristic
216	C₂	C₂×#16837186	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₅₉×#2960	C₂
socle	C₂²×C₅₉	C₂×C₂²⋊C₈
prime radical	C₂³×C₅₉	C₂²⋊C₈
cosocle	C₄×C₈×C₅₉	C₂³
hypercenter	#2960	D₅₉
hypocenter	C₅₉	#11641
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2960

Complements

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

N	H ≅ G/N	type
3. #21422078	C₂	semidirect
128. C₅₉	#11641	semidirect

Maximal subgroups

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

subgroup	index
#21425899	2
#21425895	2
#21422078	2
C₅₉×#2960	2
#11641	59