Subgroups — #28487767

Order 15104.

486 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21424248	C₂	1	1	yes	maximal, characteristic
2	#21423949	C₂	1	1	yes	maximal, characteristic
3	C₂×#16837151	C₂	1	2	yes	maximal
4	C₂×#16837148	C₂	1	2	yes	maximal
5	C₅₉×#2862	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
7	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
9	C₂×#14273638	—	2	1	no	—
10	#16837157	—	2	1	no	—
11	#16837151	—	2	4	no	—
12	#16837148	—	2	4	no	—
13	C₂×C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₅₉×#391	—	2	2	no	—
16	C₅₉×#391	C₄	1	4	yes	—
17	C₅₉×#382	—	2	1	no	—
18	C₅₉×#382	—	2	1	no	—
19	C₄³×C₅₉	C₂²	1	1	yes	characteristic
20	C₂²×C₅₉⋊C₈	—	2	2	no	—
21	C₂²×C₅₉⋊C₈	—	2	2	no	—
22	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
23	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
24	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
25	C₄⋊(C₅₉⋊C₈)	—	4	4	no	—
26	C₄⋊(C₅₉⋊C₈)	D₄	1	4	yes	—
27	#14273638	—	4	2	no	—
28	C₂²×Q₈×C₅₉	—	2	1	no	—
29	C₂²×Q₈×C₅₉	C₂×C₄	1	2	yes	—
30	C₄⋊Q₈×C₅₉	—	2	2	no	—
31	C₄⋊Q₈×C₅₉	—	2	2	no	—
32	C₄⋊Q₈×C₅₉	—	2	2	no	—
33	C₄⋊Q₈×C₅₉	—	2	2	no	—
34	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
35	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
36	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
37	C₂×C₄⋊C₄×C₅₉	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₉	D₄	1	2	yes	—
41	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
43	#95×C₅₉	—	2	4	no	—
44	#95×C₅₉	—	2	4	no	—
45	C₂×Q₈×C₅₉	—	2	4	no	—
46	C₂×Q₈×C₅₉	—	4	4	no	—
47	C₂×Q₈×C₅₉	—	4	1	no	—
48	C₂×Q₈×C₅₉	—	4	1	no	—
49	C₂×Q₈×C₅₉	—	4	2	no	—
50	C₂×Q₈×C₅₉	—	4	2	no	—
51	C₂²×C₄×C₅₉	—	2	2	no	—
52	C₂²×C₄×C₅₉	—	2	2	no	—
53	C₂²×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
58	C₄⋊C₄×C₅₉	—	2	4	no	—
59	C₄⋊C₄×C₅₉	—	4	2	no	—
60	C₄⋊C₄×C₅₉	—	4	2	no	—
61	C₄⋊C₄×C₅₉	—	4	1	no	—
62	C₄⋊C₄×C₅₉	—	4	1	no	—
63	C₄²×C₅₉	—	2	2	no	—
64	C₄²×C₅₉	—	2	2	no	—
65	C₄²×C₅₉	—	2	2	no	—
66	C₄²×C₅₉	—	4	2	no	—
67	C₄²×C₅₉	SD₁₆	1	2	yes	—
68	C₄²×C₅₉	Q₁₆	1	2	yes	—
69	C₄²×C₅₉	C₂×D₄	1	2	yes	—
70	C₄²×C₅₉	#41	1	2	yes	—
71	C₂×C₅₉⋊C₈	—	2	4	no	—
72	C₂×C₅₉⋊C₈	—	4	2	no	—
73	C₂×C₅₉⋊C₈	—	4	2	no	—
74	C₂×C₅₉⋊C₈	—	4	2	no	—
75	C₂×C₅₉⋊C₈	—	4	4	no	—
76	C₂³×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	Q₈×C₅₉	—	8	4	no	—
78	Q₈×C₅₉	—	8	2	no	—
79	Q₈×C₅₉	—	8	1	no	—
80	Q₈×C₅₉	—	8	1	no	—
81	C₂×C₄×C₅₉	—	2	2	no	—
82	C₂×C₄×C₅₉	—	2	2	no	—
83	C₂×C₄×C₅₉	—	2	2	no	—
84	C₂×C₄×C₅₉	—	2	2	no	—
85	C₂×C₄×C₅₉	—	2	2	no	—
86	C₂×C₄×C₅₉	—	2	2	no	—
87	C₂×C₄×C₅₉	—	4	2	no	—
88	C₂×C₄×C₅₉	—	4	2	no	—
89	C₂×C₄×C₅₉	—	4	4	no	—
90	C₂×C₄×C₅₉	—	4	4	no	—
91	C₂×C₄×C₅₉	#103	1	4	yes	—
92	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
93	C₂×C₄×C₅₉	#121	1	2	yes	—
94	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
95	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
96	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
97	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
98	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
99	C₂×C₄×C₅₉	#137	1	2	yes	—
100	C₂×C₄×C₅₉	#137	1	2	yes	—
101	C₅₉⋊C₈	—	4	4	no	—
102	C₅₉⋊C₈	—	8	2	no	—
103	#9701	—	59	1	no	maximal, Sylow
104	C₂²×C₅₉	#390	1	1	yes	characteristic
105	C₂²×C₅₉	#413	1	2	yes	—
106	C₂²×C₅₉	C₂×#103	1	1	yes	characteristic
107	C₂²×C₅₉	#417	1	2	yes	—
108	C₂²×C₅₉	#419	1	1	yes	characteristic
109	C₄×C₅₉	—	2	4	no	—
110	C₄×C₅₉	—	2	2	no	—
111	C₄×C₅₉	—	4	2	no	—
112	C₄×C₅₉	—	8	2	no	—
113	C₄×C₅₉	—	8	2	no	—
114	C₄×C₅₉	#461	1	2	yes	—
115	C₄×C₅₉	#462	1	2	yes	—
116	C₄×C₅₉	#489	1	2	yes	—
117	C₄×C₅₉	#496	1	1	yes	characteristic
118	C₄×C₅₉	#497	1	1	yes	characteristic
119	#2862	D₅₉	1	1	yes	characteristic
120	#1842	—	59	1	no	—
121	#1767	—	59	1	no	—
122	C₂×#333	—	59	2	no	—
123	C₂×#332	—	59	2	no	—
124	C₂×C₅₉	#1550	1	2	yes	—
125	C₂×C₅₉	#1559	1	2	yes	—
126	C₂×C₅₉	#1959	1	1	yes	characteristic
127	C₂×C₅₉	#1967	1	1	yes	characteristic
128	C₂×C₅₉	#1981	1	1	yes	characteristic
129	C₂×C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
130	C₂×C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
131	C₂×C₄⋊C₈	—	59	2	no	—
132	C₂×C₄⋊C₈	—	59	2	no	—
133	C₂×C₄⋊C₈	—	118	2	no	—
134	C₂×#97	—	118	1	no	—
135	#391	—	2	2	no	—
136	#391	Dic₅₉	1	4	yes	—
137	#382	—	2	1	no	—
138	#382	—	2	1	no	—
139	C₄³	C₂×D₅₉	1	1	yes	characteristic
140	#336	—	118	1	no	—
141	#333	—	118	4	no	—
142	#332	—	118	4	no	—
143	C₅₉	#9701	1	1	yes	characteristic, complemented, Sylow
144	C₂²×Q₈	—	2	1	no	—
145	C₂²×Q₈	C₂×Dic₅₉	1	2	yes	—
146	C₂²×C₈	—	118	2	no	—
147	C₂²×C₈	—	118	2	no	—
148	C₄⋊Q₈	—	2	2	no	—
149	C₄⋊Q₈	—	2	2	no	—
150	C₄⋊Q₈	—	2	2	no	—
151	C₄⋊Q₈	—	2	2	no	—
152	C₂×C₄⋊C₄	—	2	1	no	—
153	C₂×C₄⋊C₄	—	2	1	no	—
154	C₂×C₄⋊C₄	—	2	1	no	—
155	C₂×C₄⋊C₄	C₂×Dic₅₉	1	2	yes	—
156	C₂×C₄²	C₂×Dic₅₉	1	2	yes	—
157	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
158	C₂×C₄²	C₂²⋊D₅₉	1	2	yes	—
159	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
160	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
161	C₄⋊C₈	—	59	4	no	—
162	C₄⋊C₈	—	118	2	no	—
163	C₄⋊C₈	—	118	2	no	—
164	C₄⋊C₈	—	118	2	no	—
165	C₄⋊C₈	—	236	4	no	—
166	#97	—	236	2	no	—
167	#95	—	2	4	no	—
168	#95	—	2	4	no	—
169	C₂×Q₈	—	2	4	no	—
170	C₂×Q₈	—	4	4	no	—
171	C₂×Q₈	—	4	1	no	—
172	C₂×Q₈	—	4	1	no	—
173	C₂×Q₈	—	4	2	no	—
174	C₂×Q₈	—	4	2	no	—
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	C₂²⋊Dic₅₉	1	2	yes	—
178	C₂²×C₄	C₂²⋊Dic₅₉	1	2	yes	—
179	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
180	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
181	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
182	C₂×C₈	—	118	4	no	—
183	C₂×C₈	—	236	2	no	—
184	C₂×C₈	—	236	4	no	—
185	C₂×C₈	—	236	2	no	—
186	C₂×C₈	—	236	2	no	—
187	C₄⋊C₄	—	2	4	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₄⋊C₄	—	4	1	no	—
191	C₄⋊C₄	—	4	1	no	—
192	C₄²	—	2	2	no	—
193	C₄²	—	2	2	no	—
194	C₄²	—	2	2	no	—
195	C₄²	—	4	2	no	—
196	C₄²	#11745671	1	2	yes	—
197	C₄²	#11745672	1	2	yes	—
198	C₄²	D₄×D₅₉	1	2	yes	—
199	C₄²	#11745687	1	2	yes	—
200	C₂³	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
201	Q₈	—	8	4	no	—
202	Q₈	—	8	2	no	—
203	Q₈	—	8	1	no	—
204	Q₈	—	8	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	—	4	4	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	#14273651	1	4	yes	—
216	C₂×C₄	D₄×Dic₅₉	1	2	yes	—
217	C₂×C₄	#14273698	1	2	yes	—
218	C₂×C₄	#14273714	1	1	yes	characteristic
219	C₂×C₄	#14273723	1	1	yes	characteristic
220	C₂×C₄	C₂×#11745671	1	1	yes	characteristic
221	C₂×C₄	C₂×#11745672	1	1	yes	characteristic
222	C₂×C₄	#14273752	1	2	yes	—
223	C₂×C₄	#14273757	1	2	yes	—
224	C₂×C₄	#14273758	1	2	yes	—
225	C₈	—	472	2	no	—
226	C₈	—	236	4	no	—
227	C₂²	#16837305	1	1	yes	characteristic
228	C₂²	#16837386	1	2	yes	—
229	C₂²	C₂×#14273651	1	1	yes	characteristic
230	C₂²	#16837403	1	2	yes	—
231	C₂²	#16837411	1	1	yes	characteristic
232	C₄	—	2	2	no	—
233	C₄	—	2	4	no	—
234	C₄	—	4	2	no	—
235	C₄	—	8	2	no	—
236	C₄	—	8	2	no	—
237	C₄	#16837604	1	2	yes	—
238	C₄	#16837609	1	2	yes	—
239	C₄	#16837761	1	2	yes	—
240	C₄	#16837792	1	1	yes	characteristic
241	C₄	#16837797	1	1	yes	characteristic
242	C₂	#21422786	1	2	yes	—
243	C₂	#21422826	1	2	yes	—
244	C₂	#21424913	1	1	yes	characteristic
245	C₂	#21424946	1	1	yes	characteristic
246	C₂	#21425020	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₅₉
commutator	C₂²×C₄×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#2862	C₂
socle, prime radical	C₂³×C₅₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#2862	D₅₉
hypocenter	C₅₉	#9701
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2862

Complements

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

N	H ≅ G/N	type
143. C₅₉	#9701	semidirect

Maximal subgroups

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

subgroup	index
#21424248	2
#21423949	2
C₂×#16837151	2
C₂×#16837148	2
C₅₉×#2862	2
#9701	59