Subgroups — #28504021

Order 15104.

350 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	#21425852	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837179	C₂	1	2	yes	maximal
3	C₄×#14273637	C₂	1	1	yes	maximal, characteristic
4	#21422175	C₂	1	2	yes	maximal
5	C₅₉×#2150	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273667	C₂²	1	2	yes	—
7	C₂×#14273637	C₄	1	1	yes	characteristic
8	C₂×#14273637	C₄	1	1	yes	characteristic
9	C₂×#14273637	C₂²	1	1	yes	characteristic
10	C₂×#14273637	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₅₉⋊C₈	C₄	1	1	yes	characteristic
12	C₄²×Dic₅₉	C₄	1	1	yes	characteristic
13	C₄⋊(C₅₉⋊C₁₆)	C₄	1	4	yes	complemented
14	#16837179	—	2	4	no	—
15	C₂×M₃₂×C₅₉	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₅₉×C₄⋊C₁₆	C₄	1	4	yes	complemented
18	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₅₉	—	2	2	no	—
21	C₂×C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
23	#14273667	—	2	2	no	—
24	#14273667	—	2	2	no	—
25	C₂×C₅₉⋊C₁₆	C₂×C₄	1	4	yes	—
26	#14273637	—	2	1	no	—
27	#14273637	—	2	1	no	—
28	#14273637	—	2	2	no	—
29	#14273637	—	2	2	no	—
30	#14273637	D₄	1	2	yes	—
31	#14273637	Q₈	1	2	yes	—
32	C₄×C₅₉⋊C₈	—	2	1	no	—
33	C₄×C₅₉⋊C₈	—	2	1	no	—
34	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
37	M₃₂×C₅₉	—	2	2	no	—
38	M₃₂×C₅₉	—	2	2	no	—
39	C₂×C₁₆×C₅₉	C₂×C₄	1	4	yes	—
40	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
42	C₂²×C₄×C₅₉	M₁₆	1	2	yes	—
43	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
44	C₂²×Dic₅₉	—	2	1	no	—
45	C₂²×Dic₅₉	—	2	1	no	—
46	C₂×C₈×C₅₉	—	2	1	no	—
47	C₂×C₈×C₅₉	—	2	1	no	—
48	C₂×C₈×C₅₉	C₄²	1	2	yes	—
49	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
54	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
55	C₄²×C₅₉	C₄²	1	2	yes	—
56	C₄²×C₅₉	M₁₆	1	1	yes	characteristic
57	C₄²×C₅₉	M₁₆	1	1	yes	characteristic
58	C₄×Dic₅₉	—	2	1	no	—
59	C₄×Dic₅₉	—	2	1	no	—
60	C₄×Dic₅₉	—	4	4	no	—
61	C₄×Dic₅₉	C₄⋊C₄	1	2	yes	—
62	C₄×Dic₅₉	M₁₆	1	2	yes	—
63	C₂×C₅₉⋊C₈	—	2	1	no	—
64	C₂×C₅₉⋊C₈	—	2	1	no	—
65	C₂×C₅₉⋊C₈	—	2	1	no	—
66	C₂×C₅₉⋊C₈	—	2	1	no	—
67	C₂×C₅₉⋊C₈	C₄⋊C₄	1	2	yes	—
68	C₂×C₅₉⋊C₈	M₁₆	1	2	yes	—
69	C₁₆×C₅₉	—	2	4	no	—
70	C₅₉⋊C₁₆	—	2	4	no	—
71	C₂³×C₅₉	C₂×M₁₆	1	1	yes	characteristic
72	C₂×C₄×C₅₉	—	2	4	no	—
73	C₂×C₄×C₅₉	#97	1	2	yes	—
74	C₂×C₄×C₅₉	#97	1	2	yes	—
75	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
76	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
78	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
79	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
80	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
81	C₂×Dic₅₉	—	2	2	no	—
82	C₂×Dic₅₉	—	2	2	no	—
83	C₂×Dic₅₉	—	4	2	no	—
84	C₂×Dic₅₉	—	4	2	no	—
85	C₈×C₅₉	—	2	2	no	—
86	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
87	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
88	C₈×C₅₉	C₄×Q₈	1	1	yes	characteristic
89	C₈×C₅₉	C₄×Q₈	1	1	yes	characteristic
90	C₅₉⋊C₈	—	2	2	no	—
91	C₅₉⋊C₈	—	2	2	no	—
92	#12658	—	59	1	no	maximal, Sylow
93	C₂²×C₅₉	—	2	2	no	—
94	C₂²×C₅₉	C₂×#97	1	1	yes	characteristic
95	C₂²×C₅₉	#423	1	1	yes	characteristic
96	C₂²×C₅₉	#432	1	1	yes	characteristic
97	C₄×C₅₉	—	2	2	no	—
98	C₄×C₅₉	#347	1	4	yes	complemented
99	C₄×C₅₉	C₄×C₄⋊C₄	1	1	yes	characteristic
100	C₄×C₅₉	C₄×M₁₆	1	1	yes	characteristic
101	C₄×C₅₉	#423	1	1	yes	characteristic
102	C₄×C₅₉	#432	1	1	yes	characteristic
103	Dic₅₉	—	4	2	no	—
104	Dic₅₉	—	4	2	no	—
105	#2150	—	59	1	no	—
106	#2150	D₅₉	1	1	yes	characteristic
107	C₂×#347	—	59	2	no	—
108	C₄×#97	—	59	1	no	—
109	#1375	—	59	2	no	—
110	C₂×C₅₉	—	2	1	no	—
111	C₂×C₅₉	—	2	1	no	—
112	C₂×C₅₉	#1767	1	1	yes	characteristic
113	C₂×C₅₉	C₂×#347	1	2	yes	—
114	C₂×M₃₂	—	59	2	no	—
115	C₂×M₃₂	C₂×D₅₉	1	2	yes	—
116	C₂×#97	—	59	1	no	—
117	C₂×#97	—	59	1	no	—
118	C₂×#97	—	59	1	no	—
119	C₂×#97	—	59	1	no	—
120	C₂×C₄×C₈	—	59	1	no	—
121	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
122	C₄³	—	59	1	no	—
123	C₄⋊C₁₆	—	59	4	no	—
124	C₄⋊C₁₆	Dic₅₉	1	4	yes	complemented
125	#347	—	118	4	no	—
126	C₅₉	#12658	1	1	yes	characteristic, complemented, Sylow
127	C₂²×C₈	—	59	1	no	—
128	C₂²×C₈	—	59	1	no	—
129	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
130	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
131	C₂×C₄²	—	59	1	no	—
132	C₂×C₄²	—	59	1	no	—
133	C₂×C₄²	—	118	2	no	—
134	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
135	M₃₂	—	2	2	no	—
136	M₃₂	—	2	2	no	—
137	M₃₂	—	118	2	no	—
138	M₃₂	—	118	2	no	—
139	C₂×C₁₆	—	59	4	no	—
140	C₂×C₁₆	C₂×Dic₅₉	1	4	yes	—
141	#97	—	59	2	no	—
142	#97	—	59	2	no	—
143	#97	—	118	2	no	—
144	#97	—	118	2	no	—
145	#97	—	118	1	no	—
146	#97	—	118	1	no	—
147	C₄×C₈	—	118	1	no	—
148	C₄×C₈	—	118	1	no	—
149	C₄×C₈	C₄×D₅₉	1	2	yes	—
150	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
151	C₂²×C₄	—	118	1	no	—
152	C₂²×C₄	—	118	1	no	—
153	C₂²×C₄	#11745659	1	2	yes	—
154	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	59	2	no	—
158	C₂×C₈	—	59	2	no	—
159	C₂×C₈	—	118	1	no	—
160	C₂×C₈	—	118	1	no	—
161	C₂×C₈	—	118	1	no	—
162	C₂×C₈	—	118	1	no	—
163	C₂×C₈	C₄×Dic₅₉	1	2	yes	—
164	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
165	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
166	C₂×C₈	#11745687	1	1	yes	characteristic
167	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
168	C₂×C₈	#11745689	1	1	yes	characteristic
169	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
170	C₄²	—	59	2	no	—
171	C₄²	—	59	2	no	—
172	C₄²	—	118	1	no	—
173	C₄²	—	118	1	no	—
174	C₄²	—	236	4	no	—
175	C₄²	#11745659	1	1	yes	characteristic
176	C₄²	#11745659	1	1	yes	characteristic
177	C₄²	C₄×Dic₅₉	1	2	yes	—
178	C₁₆	—	2	4	no	—
179	C₁₆	—	118	4	no	—
180	C₂³	C₂×#11745659	1	1	yes	characteristic
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	118	2	no	—
183	C₂×C₄	—	118	2	no	—
184	C₂×C₄	—	236	2	no	—
185	C₂×C₄	—	236	2	no	—
186	C₂×C₄	#14273637	1	2	yes	—
187	C₂×C₄	#14273637	1	2	yes	—
188	C₂×C₄	C₂×C₄×Dic₅₉	1	1	yes	characteristic
189	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
190	C₂×C₄	#14273685	1	1	yes	characteristic
191	C₂×C₄	M₁₆×D₅₉	1	1	yes	characteristic
192	C₂×C₄	M₁₆×D₅₉	1	1	yes	characteristic
193	C₂×C₄	C₂×#11745659	1	1	yes	characteristic
194	C₈	—	2	2	no	—
195	C₈	—	118	2	no	—
196	C₈	—	118	2	no	—
197	C₈	#14273689	1	1	yes	characteristic
198	C₈	D₄×Dic₅₉	1	1	yes	characteristic
199	C₈	#14273692	1	1	yes	characteristic
200	C₈	Q₈×Dic₅₉	1	1	yes	characteristic
201	C₂²	—	2	2	no	—
202	C₂²	C₂×#14273637	1	1	yes	characteristic
203	C₂²	#16837424	1	1	yes	characteristic
204	C₂²	#16837458	1	1	yes	characteristic
205	C₄	—	2	2	no	—
206	C₄	—	236	2	no	—
207	C₄	—	236	2	no	—
208	C₄	#16837179	1	4	yes	complemented
209	C₄	C₄⋊C₄×Dic₅₉	1	1	yes	characteristic
210	C₄	M₁₆×Dic₅₉	1	1	yes	characteristic
211	C₄	#16837424	1	1	yes	characteristic
212	C₄	#16837458	1	1	yes	characteristic
213	C₂	—	2	1	no	—
214	C₂	—	2	1	no	—
215	C₂	#21423951	1	1	yes	characteristic
216	C₂	C₂×#16837179	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₅₉
commutator	C₂×C₄×C₅₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#2150	C₂
socle	C₂²×C₅₉	C₂×#97
prime radical	C₂³×C₅₉	C₂×M₁₆
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#2150	D₅₉
hypocenter	C₅₉	#12658
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ #2150

Complements

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

N	H ≅ G/N	type
13. C₄⋊(C₅₉⋊C₁₆)	C₄	semidirect
17. C₅₉×C₄⋊C₁₆	C₄	semidirect
98. C₄×C₅₉	#347	semidirect
124. C₄⋊C₁₆	Dic₅₉	semidirect
126. C₅₉	#12658	semidirect
208. C₄	#16837179	semidirect

Maximal subgroups

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

subgroup	index
#21425852	2
C₂×#16837179	2
C₄×#14273637	2
#21422175	2
C₅₉×#2150	2
#12658	59