Subgroups — #28505161

Order 15104.

420 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14273667	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837223	C₂	1	2	yes	maximal, complemented
3	#21425707	C₂	1	1	yes	maximal, characteristic
4	#21422178	C₂	1	2	yes	maximal, complemented
5	C₈×C₂²⋊C₄×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273667	—	2	4	no	—
7	C₂×#14273667	—	2	1	no	—
8	C₂×#14273667	C₂²	1	1	yes	characteristic
9	C₂×#14273667	C₂²	1	1	yes	characteristic
10	C₂×#14273667	C₂²	1	2	yes	—
11	C₂²×C₅₉⋊C₁₆	—	2	1	no	—
12	#16837223	—	2	4	no	—
13	C₂²⋊(C₅₉⋊C₁₆)	—	2	1	no	—
14	C₂²⋊(C₅₉⋊C₁₆)	—	2	1	no	—
15	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊C₈×C₅₉	C₄	1	1	yes	characteristic
17	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
18	C₄×C₂²⋊C₄×C₅₉	C₄	1	1	yes	characteristic
19	C₅₉×#336	C₄	1	2	yes	—
20	#14273667	—	2	4	no	—
21	#14273667	—	2	2	no	—
22	#14273667	—	2	2	no	—
23	#14273667	—	2	2	no	—
24	#14273667	D₄	1	4	yes	—
25	C₂×C₅₉⋊C₁₆	—	2	4	no	—
26	C₂×C₅₉⋊C₁₆	—	2	2	no	—
27	C₂×C₅₉⋊C₁₆	—	2	1	no	—
28	C₂×C₅₉⋊C₁₆	—	2	1	no	—
29	C₂³×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₉	—	2	2	no	—
31	C₂²×C₈×C₅₉	—	2	2	no	—
32	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₅₉	Q₈	1	1	yes	characteristic
37	C₂²×C₈×C₅₉	Q₈	1	1	yes	characteristic
38	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
39	C₂×C₂²⋊C₄×C₅₉	—	2	1	no	—
40	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
41	C₂²⋊C₈×C₅₉	C₈	1	2	yes	—
42	C₂²⋊C₈×C₅₉	C₈	1	2	yes	—
43	C₄×C₈×C₅₉	—	2	2	no	—
44	C₄×C₈×C₅₉	—	2	2	no	—
45	#95×C₅₉	—	2	1	no	—
46	C₂⁴×C₅₉	M₁₆	1	1	yes	characteristic
47	C₂²×C₄×C₅₉	—	2	2	no	—
48	C₂²×C₄×C₅₉	—	2	2	no	—
49	C₂²×C₄×C₅₉	—	2	2	no	—
50	C₂²×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
53	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₅₉	M₁₆	1	1	yes	characteristic
55	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₅₉	—	2	2	no	—
57	C₂×C₈×C₅₉	—	2	2	no	—
58	C₂×C₈×C₅₉	—	2	2	no	—
59	C₂×C₈×C₅₉	—	2	1	no	—
60	C₂×C₈×C₅₉	—	2	1	no	—
61	C₂×C₈×C₅₉	—	2	2	no	—
62	C₂×C₈×C₅₉	—	2	2	no	—
63	C₂×C₈×C₅₉	—	2	1	no	—
64	C₂×C₈×C₅₉	—	2	1	no	—
65	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₅₉	C₂×C₈	1	4	yes	—
68	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
71	C₂×C₈×C₅₉	#41	1	2	yes	—
72	C₂²⋊C₄×C₅₉	—	2	4	no	—
73	C₄²×C₅₉	—	2	2	no	—
74	C₄²×C₅₉	—	2	2	no	—
75	C₅₉⋊C₁₆	—	2	4	no	—
76	C₅₉⋊C₁₆	—	4	1	no	—
77	C₅₉⋊C₁₆	—	4	1	no	—
78	C₂³×C₅₉	—	2	2	no	—
79	C₂³×C₅₉	—	2	2	no	—
80	C₂³×C₅₉	—	4	1	no	—
81	C₂³×C₅₉	C₄⋊C₈	1	1	yes	characteristic
82	C₂³×C₅₉	C₄⋊C₈	1	1	yes	characteristic
83	C₂³×C₅₉	C₂×M₁₆	1	1	yes	characteristic
84	C₂×C₄×C₅₉	—	2	2	no	—
85	C₂×C₄×C₅₉	—	2	1	no	—
86	C₂×C₄×C₅₉	—	2	1	no	—
87	C₂×C₄×C₅₉	—	2	1	no	—
88	C₂×C₄×C₅₉	—	2	1	no	—
89	C₂×C₄×C₅₉	—	2	2	no	—
90	C₂×C₄×C₅₉	—	2	2	no	—
91	C₂×C₄×C₅₉	—	2	4	no	—
92	C₂×C₄×C₅₉	—	4	2	no	—
93	C₂×C₄×C₅₉	—	4	2	no	—
94	C₂×C₄×C₅₉	C₄⋊C₈	1	1	yes	characteristic
95	C₂×C₄×C₅₉	C₄⋊C₈	1	1	yes	characteristic
96	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
98	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₄×C₅₉	#131	1	2	yes	—
100	C₈×C₅₉	—	2	1	no	—
101	C₈×C₅₉	—	2	1	no	—
102	C₈×C₅₉	—	2	4	no	—
103	C₈×C₅₉	#120	1	1	yes	characteristic
104	C₈×C₅₉	C₂²⋊Q₈	1	2	yes	—
105	C₈×C₅₉	#123	1	1	yes	characteristic
106	#12914	—	59	1	no	maximal, Sylow
107	C₂²×C₅₉	—	2	1	no	—
108	C₂²×C₅₉	—	2	1	no	—
109	C₂²×C₅₉	—	2	1	no	—
110	C₂²×C₅₉	—	2	1	no	—
111	C₂²×C₅₉	—	2	1	no	—
112	C₂²×C₅₉	—	2	1	no	—
113	C₂²×C₅₉	—	2	1	no	—
114	C₂²×C₅₉	—	4	4	no	—
115	C₂²×C₅₉	#364	1	2	yes	complemented
116	C₂²×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
117	C₂²×C₅₉	#435	1	2	yes	—
118	C₄×C₅₉	—	2	1	no	—
119	C₄×C₅₉	—	2	1	no	—
120	C₄×C₅₉	—	4	4	no	—
121	C₄×C₅₉	#385	1	1	yes	characteristic
122	C₄×C₅₉	#424	1	1	yes	characteristic
123	C₄×C₅₉	C₈×D₄	1	2	yes	—
124	C₂²×M₃₂	—	59	1	no	—
125	C₂×#364	—	59	2	no	—
126	#2112	—	59	1	no	—
127	C₈×C₂²⋊C₄	D₅₉	1	1	yes	characteristic
128	#1376	—	59	2	no	—
129	C₂×C₅₉	—	2	2	no	—
130	C₂×C₅₉	—	2	2	no	—
131	C₂×C₅₉	—	4	1	no	—
132	C₂×C₅₉	#1850	1	1	yes	characteristic
133	C₂×C₅₉	C₂×#364	1	1	yes	characteristic
134	C₂×C₅₉	#2153	1	1	yes	characteristic
135	C₂³×C₈	C₂×D₅₉	1	1	yes	characteristic
136	C₂×M₃₂	—	59	2	no	—
137	C₂×M₃₂	—	59	1	no	—
138	C₂×M₃₂	—	59	1	no	—
139	C₂×M₃₂	—	118	4	no	—
140	C₂×M₃₂	—	118	1	no	—
141	C₂²×C₁₆	—	118	1	no	—
142	C₂×C₂²⋊C₈	Dic₅₉	1	1	yes	characteristic
143	C₂×C₄×C₈	C₂×D₅₉	1	2	yes	—
144	C₄×C₂²⋊C₄	Dic₅₉	1	1	yes	characteristic
145	#364	—	118	4	no	—
146	C₂²⋊C₁₆	—	118	1	no	—
147	C₂²⋊C₁₆	—	118	1	no	—
148	#336	Dic₅₉	1	2	yes	—
149	C₅₉	#12914	1	1	yes	characteristic, complemented, Sylow
150	C₂³×C₄	C₂×Dic₅₉	1	1	yes	characteristic
151	C₂²×C₈	—	2	2	no	—
152	C₂²×C₈	—	2	2	no	—
153	C₂²×C₈	Dic₁₁₈	1	1	yes	characteristic
154	C₂²×C₈	Dic₁₁₈	1	1	yes	characteristic
155	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
156	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
157	C₂²×C₈	C₂×Dic₅₉	1	2	yes	—
158	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
159	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
160	C₂×C₂²⋊C₄	—	2	1	no	—
161	C₂×C₄²	C₂×Dic₅₉	1	2	yes	—
162	M₃₂	—	59	4	no	—
163	M₃₂	—	118	4	no	—
164	M₃₂	—	118	2	no	—
165	M₃₂	—	118	2	no	—
166	M₃₂	—	118	2	no	—
167	C₂×C₁₆	—	118	4	no	—
168	C₂×C₁₆	—	118	2	no	—
169	C₂×C₁₆	—	118	1	no	—
170	C₂×C₁₆	—	118	1	no	—
171	C₂²⋊C₈	C₅₉⋊C₈	1	2	yes	—
172	C₂²⋊C₈	C₅₉⋊C₈	1	2	yes	—
173	C₄×C₈	—	2	2	no	—
174	C₄×C₈	—	2	2	no	—
175	#95	—	2	1	no	—
176	C₂⁴	#11745664	1	1	yes	characteristic
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	C₂×C₅₉⋊C₈	1	1	yes	characteristic
181	C₂²×C₄	C₂×C₅₉⋊C₈	1	1	yes	characteristic
182	C₂²×C₄	#11745664	1	1	yes	characteristic
183	C₂²×C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
184	C₂²×C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
185	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	C₂×C₅₉⋊C₈	1	4	yes	—
196	C₂×C₈	C₄⋊Dic₅₉	1	1	yes	characteristic
197	C₂×C₈	C₄⋊Dic₅₉	1	1	yes	characteristic
198	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
199	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
200	C₂×C₈	D₄×D₅₉	1	2	yes	—
201	C₂×C₈	#11745687	1	2	yes	—
202	C₂²⋊C₄	—	2	4	no	—
203	C₄²	—	2	2	no	—
204	C₄²	—	2	2	no	—
205	C₁₆	—	118	4	no	—
206	C₁₆	—	236	1	no	—
207	C₁₆	—	236	1	no	—
208	C₂³	—	2	2	no	—
209	C₂³	—	2	2	no	—
210	C₂³	—	4	1	no	—
211	C₂³	C₄⋊(C₅₉⋊C₈)	1	1	yes	characteristic
212	C₂³	C₄⋊(C₅₉⋊C₈)	1	1	yes	characteristic
213	C₂³	C₂×#11745664	1	1	yes	characteristic
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	C₄⋊(C₅₉⋊C₈)	1	1	yes	characteristic
225	C₂×C₄	C₄⋊(C₅₉⋊C₈)	1	1	yes	characteristic
226	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
227	C₂×C₄	D₄×Dic₅₉	1	2	yes	—
228	C₂×C₄	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
229	C₂×C₄	#14273734	1	2	yes	—
230	C₈	—	2	4	no	—
231	C₈	—	2	1	no	—
232	C₈	—	2	1	no	—
233	C₈	#14273697	1	1	yes	characteristic
234	C₈	#14273704	1	2	yes	—
235	C₈	#14273711	1	1	yes	characteristic
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	4	4	no	—
244	C₂²	#16837223	1	2	yes	complemented
245	C₂²	C₂×C₄⋊(C₅₉⋊C₈)	1	1	yes	characteristic
246	C₂²	#16837474	1	2	yes	—
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	4	4	no	—
250	C₄	#16837277	1	1	yes	characteristic
251	C₄	#16837429	1	1	yes	characteristic
252	C₄	D₄×C₅₉⋊C₈	1	2	yes	—
253	C₂	—	2	2	no	—
254	C₂	—	2	2	no	—
255	C₂	—	4	1	no	—
256	C₂	#21424276	1	1	yes	characteristic
257	C₂	C₂×#16837223	1	1	yes	characteristic
258	C₂	#21425862	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₈×C₂²⋊C₄×C₅₉	C₂
socle	C₂²×C₅₉	C₂×C₄⋊C₈
prime radical	C₂⁴×C₅₉	M₁₆
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₈×C₂²⋊C₄	D₅₉
hypocenter	C₅₉	#12914
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ C₈×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂×#16837223	C₂	semidirect
4. #21422178	C₂	semidirect
115. C₂²×C₅₉	#364	semidirect
149. C₅₉	#12914	semidirect
244. C₂²	#16837223	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14273667	2
C₂×#16837223	2
#21425707	2
#21422178	2
C₈×C₂²⋊C₄×C₅₉	2
#12914	59