Subgroups — #28506779

Order 15104.

342 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837228	C₂	1	2	yes	maximal
2	#21424256	C₂	1	1	yes	maximal, characteristic
3	C₄×#14273661	C₂	1	1	yes	maximal, characteristic
4	#21422188	C₂	1	2	yes	maximal
5	C₂×C₅₉×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273661	—	2	1	no	—
7	C₂×#14273661	C₂²	1	2	yes	—
8	C₂×#14273661	C₂²	1	1	yes	characteristic
9	C₂×#14273661	C₂²	1	1	yes	characteristic
10	#16837272	—	2	1	no	—
11	C₄×C₄⋊Dic₅₉	—	2	1	no	—
12	#16837228	—	2	4	no	—
13	#16837165	—	2	1	no	—
14	C₂²×C₁₆×C₅₉	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₅₉×C₄⋊C₁₆	C₄	1	4	yes	complemented
17	C₂×C₄⋊Dic₅₉	—	2	1	no	—
18	C₂×C₄⋊Dic₅₉	—	2	1	no	—
19	C₂×C₄⋊Dic₅₉	—	2	2	no	—
20	C₂×C₄⋊Dic₅₉	—	4	1	no	—
21	C₂×C₄⋊Dic₅₉	—	4	1	no	—
22	C₂×C₄×Dic₅₉	—	4	1	no	—
23	#14273661	—	2	1	no	—
24	#14273661	—	2	1	no	—
25	#14273661	—	2	2	no	—
26	#14273661	—	2	2	no	—
27	#14273661	—	4	2	no	—
28	#14273661	D₄	1	2	yes	—
29	#14273661	Q₈	1	2	yes	—
30	C₅₉⋊#95	—	4	1	no	—
31	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
34	C₂×C₁₆×C₅₉	—	2	2	no	—
35	C₂×C₁₆×C₅₉	—	2	2	no	—
36	C₂×C₁₆×C₅₉	C₂×C₄	1	4	yes	—
37	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₅₉	Q₈	1	2	yes	—
39	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
40	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
41	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
42	C₂²×Dic₅₉	—	4	1	no	—
43	C₂²×Dic₅₉	—	4	1	no	—
44	C₂²×Dic₅₉	—	4	2	no	—
45	C₂×C₈×C₅₉	—	2	1	no	—
46	C₂×C₈×C₅₉	—	2	1	no	—
47	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
48	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
53	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
54	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
55	C₄²×C₅₉	Q₁₆	1	2	yes	—
56	C₄⋊Dic₅₉	—	2	2	no	—
57	C₄⋊Dic₅₉	—	2	2	no	—
58	C₄⋊Dic₅₉	—	4	1	no	—
59	C₄⋊Dic₅₉	—	4	1	no	—
60	C₄⋊Dic₅₉	—	4	2	no	—
61	C₄⋊Dic₅₉	—	4	2	no	—
62	C₄⋊Dic₅₉	—	8	2	no	—
63	C₄⋊Dic₅₉	—	8	2	no	—
64	C₄×Dic₅₉	—	8	2	no	—
65	C₁₆×C₅₉	—	2	4	no	—
66	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₅₉	—	2	1	no	—
68	C₂×C₄×C₅₉	—	2	1	no	—
69	C₂×C₄×C₅₉	#107	1	2	yes	—
70	C₂×C₄×C₅₉	#107	1	2	yes	—
71	C₂×C₄×C₅₉	D₁₆	1	2	yes	—
72	C₂×C₄×C₅₉	Q₃₂	1	2	yes	—
73	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
75	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
77	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
79	C₂×Dic₅₉	—	4	2	no	—
80	C₂×Dic₅₉	—	4	2	no	—
81	C₂×Dic₅₉	—	8	1	no	—
82	C₂×Dic₅₉	—	8	1	no	—
83	C₂×Dic₅₉	—	8	1	no	—
84	C₂×Dic₅₉	—	8	1	no	—
85	C₂×Dic₅₉	—	8	2	no	—
86	C₂×Dic₅₉	—	8	2	no	—
87	C₂×Dic₅₉	—	8	2	no	—
88	C₈×C₅₉	—	2	2	no	—
89	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
90	C₈×C₅₉	C₄×Q₈	1	1	yes	characteristic
91	C₈×C₅₉	#125	1	1	yes	characteristic
92	C₈×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
93	#13176	—	59	1	no	maximal, Sylow
94	C₂²×C₅₉	C₂×#107	1	1	yes	characteristic
95	C₂²×C₅₉	#459	1	1	yes	characteristic
96	C₂²×C₅₉	#474	1	1	yes	characteristic
97	C₂²×C₅₉	C₂×D₁₆	1	1	yes	characteristic
98	C₂²×C₅₉	C₂×Q₃₂	1	1	yes	characteristic
99	C₂²×C₅₉	#509	1	1	yes	characteristic
100	C₂²×C₅₉	#510	1	1	yes	characteristic
101	C₄×C₅₉	—	2	2	no	—
102	C₄×C₅₉	#366	1	4	yes	complemented
103	C₄×C₅₉	#389	1	1	yes	characteristic
104	C₄×C₅₉	#428	1	1	yes	characteristic
105	C₄×C₅₉	#462	1	1	yes	characteristic
106	C₄×C₅₉	#477	1	1	yes	characteristic
107	Dic₅₉	—	8	2	no	—
108	Dic₅₉	—	8	2	no	—
109	Dic₅₉	—	16	2	no	—
110	C₂×#366	—	59	2	no	—
111	C₂×C₄⋊C₁₆	D₅₉	1	1	yes	characteristic
112	#1845	—	59	1	no	—
113	C₄×#107	—	59	1	no	—
114	#1380	—	59	2	no	—
115	C₂×C₅₉	#1947	1	1	yes	characteristic
116	C₂×C₅₉	C₂×#366	1	1	yes	characteristic
117	C₂×C₅₉	#2159	1	1	yes	characteristic
118	C₂×C₅₉	#2206	1	1	yes	characteristic
119	C₂×C₅₉	#2209	1	1	yes	characteristic
120	C₂×C₅₉	#2224	1	1	yes	characteristic
121	C₂×C₅₉	#2227	1	1	yes	characteristic
122	C₂²×C₁₆	C₂×D₅₉	1	2	yes	—
123	C₂×#107	—	59	1	no	—
124	C₂×#107	—	59	1	no	—
125	C₂×#107	—	59	2	no	—
126	C₂×#107	—	118	1	no	—
127	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
128	#384	—	118	1	no	—
129	C₄×C₄⋊C₄	—	118	1	no	—
130	#366	—	118	4	no	—
131	C₄⋊C₁₆	Dic₅₉	1	4	yes	complemented
132	#340	—	118	1	no	—
133	C₅₉	#13176	1	1	yes	characteristic, complemented, Sylow
134	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
136	C₂×C₄⋊C₄	—	118	1	no	—
137	C₂×C₄⋊C₄	—	118	1	no	—
138	C₂×C₄⋊C₄	—	118	2	no	—
139	C₂×C₄⋊C₄	—	236	1	no	—
140	C₂×C₄⋊C₄	—	236	1	no	—
141	C₂×C₄²	—	236	1	no	—
142	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
143	C₂×C₁₆	—	2	2	no	—
144	C₂×C₁₆	—	2	2	no	—
145	C₂×C₁₆	C₂×Dic₅₉	1	4	yes	—
146	#107	—	59	2	no	—
147	#107	—	59	2	no	—
148	#107	—	118	1	no	—
149	#107	—	118	1	no	—
150	#107	—	118	2	no	—
151	#107	—	118	2	no	—
152	#107	—	236	2	no	—
153	C₄×C₈	Dic₁₁₈	1	2	yes	—
154	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
155	#95	—	236	1	no	—
156	C₂²×C₄	—	236	2	no	—
157	C₂²×C₄	—	236	1	no	—
158	C₂²×C₄	—	236	1	no	—
159	C₂²×C₄	D₄₇₂	1	1	yes	characteristic
160	C₂²×C₄	D₄₇₂	1	1	yes	characteristic
161	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	C₄⋊Dic₅₉	1	2	yes	—
165	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
166	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
167	C₂×C₈	#11745687	1	1	yes	characteristic
168	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
169	C₂×C₈	#11745689	1	1	yes	characteristic
170	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
171	C₄⋊C₄	—	118	2	no	—
172	C₄⋊C₄	—	118	2	no	—
173	C₄⋊C₄	—	472	2	no	—
174	C₄⋊C₄	—	472	2	no	—
175	C₄⋊C₄	—	236	2	no	—
176	C₄⋊C₄	—	236	2	no	—
177	C₄⋊C₄	—	236	1	no	—
178	C₄⋊C₄	—	236	1	no	—
179	C₄²	—	472	2	no	—
180	C₄²	Dic₂₃₆	1	2	yes	—
181	C₄²	C₄⋊Dic₅₉	1	2	yes	—
182	C₁₆	—	2	4	no	—
183	C₂³	C₂×D₄₇₂	1	1	yes	characteristic
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	472	2	no	—
187	C₂×C₄	—	472	2	no	—
188	C₂×C₄	—	472	2	no	—
189	C₂×C₄	—	472	1	no	—
190	C₂×C₄	—	472	1	no	—
191	C₂×C₄	—	472	1	no	—
192	C₂×C₄	—	472	1	no	—
193	C₂×C₄	—	236	2	no	—
194	C₂×C₄	—	236	2	no	—
195	C₂×C₄	#14273661	1	2	yes	—
196	C₂×C₄	#14273661	1	2	yes	—
197	C₂×C₄	D₉₄₄	1	2	yes	—
198	C₂×C₄	Dic₄₇₂	1	2	yes	—
199	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
200	C₂×C₄	#14273702	1	1	yes	characteristic
201	C₂×C₄	#14273707	1	1	yes	characteristic
202	C₂×C₄	C₂×Dic₂₃₆	1	1	yes	characteristic
203	C₂×C₄	#14273755	1	1	yes	characteristic
204	C₂×C₄	#14273760	1	1	yes	characteristic
205	C₈	—	2	2	no	—
206	C₈	D₄×Dic₅₉	1	1	yes	characteristic
207	C₈	Q₈×Dic₅₉	1	1	yes	characteristic
208	C₈	#14273718	1	1	yes	characteristic
209	C₈	#14273726	1	1	yes	characteristic
210	C₂²	C₂×#14273661	1	1	yes	characteristic
211	C₂²	#16837598	1	1	yes	characteristic
212	C₂²	#16837683	1	1	yes	characteristic
213	C₂²	C₂×D₉₄₄	1	1	yes	characteristic
214	C₂²	C₂×Dic₄₇₂	1	1	yes	characteristic
215	C₂²	#16837840	1	1	yes	characteristic
216	C₂²	#16837845	1	1	yes	characteristic
217	C₄	—	2	2	no	—
218	C₄	—	944	2	no	—
219	C₄	—	472	2	no	—
220	C₄	—	472	2	no	—
221	C₄	#16837228	1	4	yes	complemented
222	C₄	#16837301	1	1	yes	characteristic
223	C₄	#16837444	1	1	yes	characteristic
224	C₄	#16837613	1	1	yes	characteristic
225	C₄	#16837698	1	1	yes	characteristic
226	C₂	#21424855	1	1	yes	characteristic
227	C₂	C₂×#16837228	1	1	yes	characteristic
228	C₂	#21425884	1	1	yes	characteristic
229	C₂	#21426132	1	1	yes	characteristic
230	C₂	#21426147	1	1	yes	characteristic
231	C₂	#21426234	1	1	yes	characteristic
232	C₂	#21426249	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄₇₂
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂×C₅₉×C₄⋊C₁₆	C₂
socle, prime radical	C₂³×C₅₉	C₂×D₈
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₅₉
hypocenter	C₅₉	#13176
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
16. C₅₉×C₄⋊C₁₆	C₄	semidirect
102. C₄×C₅₉	#366	semidirect
131. C₄⋊C₁₆	Dic₅₉	semidirect
133. C₅₉	#13176	semidirect
221. C₄	#16837228	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837228	2
#21424256	2
C₄×#14273661	2
#21422188	2
C₂×C₅₉×C₄⋊C₁₆	2
#13176	59