Subgroups — #28487729

Order 15104.

478 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21424213	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₅₉⋊C₈)	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)	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄×C₅₉⋊C₈	—	2	1	no	—
9	#16837157	—	2	1	no	—
10	#16837157	—	2	2	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)	D₄	1	4	yes	—
26	C₄×C₅₉⋊C₈	—	4	2	no	—
27	C₂²×Q₈×C₅₉	—	2	1	no	—
28	C₂²×Q₈×C₅₉	C₂×C₄	1	2	yes	—
29	C₄⋊Q₈×C₅₉	—	2	2	no	—
30	C₄⋊Q₈×C₅₉	—	2	2	no	—
31	C₄⋊Q₈×C₅₉	—	2	2	no	—
32	C₄⋊Q₈×C₅₉	—	2	2	no	—
33	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
34	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
35	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
36	C₂×C₄⋊C₄×C₅₉	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₉	D₄	1	2	yes	—
40	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
42	#95×C₅₉	—	2	4	no	—
43	#95×C₅₉	—	2	4	no	—
44	C₂×Q₈×C₅₉	—	2	4	no	—
45	C₂×Q₈×C₅₉	—	4	4	no	—
46	C₂×Q₈×C₅₉	—	4	1	no	—
47	C₂×Q₈×C₅₉	—	4	1	no	—
48	C₂×Q₈×C₅₉	—	4	4	no	—
49	C₂²×C₄×C₅₉	—	2	2	no	—
50	C₂²×C₄×C₅₉	—	2	2	no	—
51	C₂²×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
56	C₄⋊C₄×C₅₉	—	2	4	no	—
57	C₄⋊C₄×C₅₉	—	4	2	no	—
58	C₄⋊C₄×C₅₉	—	4	2	no	—
59	C₄⋊C₄×C₅₉	—	4	1	no	—
60	C₄⋊C₄×C₅₉	—	4	1	no	—
61	C₄²×C₅₉	—	2	4	no	—
62	C₄²×C₅₉	—	2	2	no	—
63	C₄²×C₅₉	—	4	2	no	—
64	C₄²×C₅₉	SD₁₆	1	2	yes	—
65	C₄²×C₅₉	Q₁₆	1	2	yes	—
66	C₄²×C₅₉	C₂×D₄	1	2	yes	—
67	C₄²×C₅₉	#41	1	2	yes	—
68	C₂×C₅₉⋊C₈	—	2	4	no	—
69	C₂×C₅₉⋊C₈	—	4	2	no	—
70	C₂×C₅₉⋊C₈	—	4	2	no	—
71	C₂×C₅₉⋊C₈	—	4	2	no	—
72	C₂×C₅₉⋊C₈	—	4	4	no	—
73	C₂³×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	Q₈×C₅₉	—	8	4	no	—
75	Q₈×C₅₉	—	8	2	no	—
76	Q₈×C₅₉	—	8	2	no	—
77	C₂×C₄×C₅₉	—	2	2	no	—
78	C₂×C₄×C₅₉	—	2	2	no	—
79	C₂×C₄×C₅₉	—	2	2	no	—
80	C₂×C₄×C₅₉	—	2	2	no	—
81	C₂×C₄×C₅₉	—	2	2	no	—
82	C₂×C₄×C₅₉	—	2	2	no	—
83	C₂×C₄×C₅₉	—	4	2	no	—
84	C₂×C₄×C₅₉	—	4	2	no	—
85	C₂×C₄×C₅₉	—	4	4	no	—
86	C₂×C₄×C₅₉	—	4	4	no	—
87	C₂×C₄×C₅₉	#103	1	4	yes	—
88	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
89	C₂×C₄×C₅₉	#121	1	2	yes	—
90	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
91	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
92	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
93	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
94	C₂×C₄×C₅₉	#135	1	4	yes	—
95	C₂×C₄×C₅₉	#137	1	2	yes	—
96	C₅₉⋊C₈	—	4	4	no	—
97	C₅₉⋊C₈	—	8	2	no	—
98	#9693	—	59	1	no	maximal, Sylow
99	C₂²×C₅₉	#390	1	1	yes	characteristic
100	C₂²×C₅₉	#413	1	2	yes	—
101	C₂²×C₅₉	C₂×#103	1	1	yes	characteristic
102	C₂²×C₅₉	#416	1	2	yes	—
103	C₂²×C₅₉	#419	1	1	yes	characteristic
104	C₄×C₅₉	—	2	4	no	—
105	C₄×C₅₉	—	2	2	no	—
106	C₄×C₅₉	—	4	2	no	—
107	C₄×C₅₉	—	8	2	no	—
108	C₄×C₅₉	—	8	2	no	—
109	C₄×C₅₉	#461	1	2	yes	—
110	C₄×C₅₉	#462	1	2	yes	—
111	C₄×C₅₉	#488	1	2	yes	—
112	C₄×C₅₉	#495	1	2	yes	—
113	#2862	D₅₉	1	1	yes	characteristic
114	#1831	—	59	1	no	—
115	C₄×C₄⋊C₈	—	59	1	no	—
116	C₂×#333	—	59	2	no	—
117	C₂×#332	—	59	2	no	—
118	C₂×C₅₉	#1542	1	2	yes	—
119	C₂×C₅₉	#1554	1	2	yes	—
120	C₂×C₅₉	#1958	1	1	yes	characteristic
121	C₂×C₅₉	#1967	1	1	yes	characteristic
122	C₂×C₅₉	#1981	1	1	yes	characteristic
123	C₂×C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
124	C₂×C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
125	C₂×C₄⋊C₈	—	59	2	no	—
126	C₂×C₄⋊C₈	—	59	2	no	—
127	C₂×C₄×C₈	—	118	1	no	—
128	#391	—	2	2	no	—
129	#391	Dic₅₉	1	4	yes	—
130	#382	—	2	1	no	—
131	#382	—	2	1	no	—
132	C₄³	C₂×D₅₉	1	1	yes	characteristic
133	#336	—	118	2	no	—
134	#336	—	118	1	no	—
135	#333	—	118	4	no	—
136	#332	—	118	4	no	—
137	C₅₉	#9693	1	1	yes	characteristic, complemented, Sylow
138	C₂²×Q₈	—	2	1	no	—
139	C₂²×Q₈	C₂×Dic₅₉	1	2	yes	—
140	C₂²×C₈	—	118	2	no	—
141	C₂²×C₈	—	118	2	no	—
142	C₄⋊Q₈	—	2	2	no	—
143	C₄⋊Q₈	—	2	2	no	—
144	C₄⋊Q₈	—	2	2	no	—
145	C₄⋊Q₈	—	2	2	no	—
146	C₂×C₄⋊C₄	—	2	1	no	—
147	C₂×C₄⋊C₄	—	2	1	no	—
148	C₂×C₄⋊C₄	—	2	1	no	—
149	C₂×C₄⋊C₄	C₂×Dic₅₉	1	2	yes	—
150	C₂×C₄²	C₂×Dic₅₉	1	2	yes	—
151	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
152	C₂×C₄²	C₂²⋊D₅₉	1	2	yes	—
153	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
154	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
155	C₄⋊C₈	—	59	4	no	—
156	C₄⋊C₈	—	118	2	no	—
157	C₄⋊C₈	—	118	2	no	—
158	C₄⋊C₈	—	118	2	no	—
159	C₄×C₈	—	236	2	no	—
160	#95	—	2	4	no	—
161	#95	—	2	4	no	—
162	C₂×Q₈	—	2	4	no	—
163	C₂×Q₈	—	4	4	no	—
164	C₂×Q₈	—	4	1	no	—
165	C₂×Q₈	—	4	1	no	—
166	C₂×Q₈	—	4	4	no	—
167	C₂²×C₄	—	2	2	no	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	C₂²⋊Dic₅₉	1	2	yes	—
170	C₂²×C₄	C₂²⋊Dic₅₉	1	2	yes	—
171	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
172	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
173	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
174	C₂×C₈	—	118	4	no	—
175	C₂×C₈	—	236	2	no	—
176	C₂×C₈	—	236	4	no	—
177	C₂×C₈	—	236	2	no	—
178	C₂×C₈	—	236	2	no	—
179	C₄⋊C₄	—	2	4	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	4	1	no	—
183	C₄⋊C₄	—	4	1	no	—
184	C₄²	—	2	4	no	—
185	C₄²	—	2	2	no	—
186	C₄²	—	4	2	no	—
187	C₄²	#11745671	1	2	yes	—
188	C₄²	#11745672	1	2	yes	—
189	C₄²	D₄×D₅₉	1	2	yes	—
190	C₄²	#11745687	1	2	yes	—
191	C₂³	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
192	Q₈	—	8	4	no	—
193	Q₈	—	8	2	no	—
194	Q₈	—	8	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	4	4	no	—
202	C₂×C₄	—	4	4	no	—
203	C₂×C₄	—	4	2	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	#14273651	1	4	yes	—
206	C₂×C₄	D₄×Dic₅₉	1	2	yes	—
207	C₂×C₄	#14273698	1	2	yes	—
208	C₂×C₄	#14273714	1	1	yes	characteristic
209	C₂×C₄	#14273723	1	1	yes	characteristic
210	C₂×C₄	C₂×#11745671	1	1	yes	characteristic
211	C₂×C₄	C₂×#11745672	1	1	yes	characteristic
212	C₂×C₄	#14273747	1	4	yes	—
213	C₂×C₄	#14273758	1	2	yes	—
214	C₈	—	472	2	no	—
215	C₈	—	236	4	no	—
216	C₂²	#16837305	1	1	yes	characteristic
217	C₂²	#16837386	1	2	yes	—
218	C₂²	C₂×#14273651	1	1	yes	characteristic
219	C₂²	#16837398	1	2	yes	—
220	C₂²	#16837411	1	1	yes	characteristic
221	C₄	—	2	2	no	—
222	C₄	—	2	4	no	—
223	C₄	—	4	2	no	—
224	C₄	—	8	2	no	—
225	C₄	—	8	2	no	—
226	C₄	#16837604	1	2	yes	—
227	C₄	#16837609	1	2	yes	—
228	C₄	#16837756	1	2	yes	—
229	C₄	#16837784	1	2	yes	—
230	C₂	#21422753	1	2	yes	—
231	C₂	#21422806	1	2	yes	—
232	C₂	#21424908	1	1	yes	characteristic
233	C₂	#21424946	1	1	yes	characteristic
234	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	#9693
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
137. C₅₉	#9693	semidirect

Maximal subgroups

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

subgroup	index
#21424213	2
C₄×C₄⋊(C₅₉⋊C₈)	2
C₂×#16837151	2
C₂×#16837148	2
C₅₉×#2862	2
#9693	59