Subgroups — C₂×#21422057 #28498420

Order 15104.

470 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16837155	C₂	1	1	yes	maximal, characteristic, complemented
3	#21422057	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄×D₈×C₅₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₅₉⋊C₁₆	—	2	1	no	—
6	C₂×C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
7	C₄×C₅₉⋊C₁₆	—	2	1	no	—
8	C₄×C₅₉⋊C₁₆	C₂²	1	2	yes	complemented
9	#16837155	—	2	1	no	—
10	#16837155	C₂²	1	2	yes	complemented
11	C₂²×D₈×C₅₉	C₄	1	1	yes	characteristic
12	C₂×C₄×D₄×C₅₉	—	2	1	no	—
13	C₄×D₈×C₅₉	—	2	1	no	—
14	C₄×D₈×C₅₉	—	2	1	no	—
15	C₄×D₈×C₅₉	C₄	1	2	yes	—
16	C₄×D₈×C₅₉	C₂²	1	2	yes	—
17	C₂×#107×C₅₉	C₄	1	1	yes	characteristic
18	C₂×#102×C₅₉	—	2	1	no	—
19	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
20	C₂²×C₅₉⋊C₈	—	4	1	no	—
21	C₂×C₅₉⋊C₁₆	—	2	4	no	—
22	C₂×C₅₉⋊C₁₆	—	2	2	no	—
23	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
24	C₄⋊(C₅₉⋊C₈)	—	4	1	no	—
25	C₂²×D₄×C₅₉	—	2	1	no	—
26	C₂³×C₄×C₅₉	—	4	1	no	—
27	C₂×D₈×C₅₉	—	2	1	no	—
28	C₂×D₈×C₅₉	—	2	2	no	—
29	C₂×D₈×C₅₉	C₈	1	4	yes	complemented
30	C₂×D₈×C₅₉	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
33	C₄×D₄×C₅₉	—	2	2	no	—
34	C₄×D₄×C₅₉	—	2	2	no	—
35	C₄×D₄×C₅₉	—	4	1	no	—
36	C₄×D₄×C₅₉	—	4	1	no	—
37	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
38	C₂×C₂²⋊C₄×C₅₉	—	4	1	no	—
39	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
40	#107×C₅₉	—	2	1	no	—
41	#107×C₅₉	C₂×C₄	1	2	yes	—
42	#102×C₅₉	—	2	2	no	—
43	#102×C₅₉	—	2	2	no	—
44	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
46	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
47	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
48	C₂⁴×C₅₉	—	4	1	no	—
49	C₂×D₄×C₅₉	—	2	4	no	—
50	C₂×D₄×C₅₉	—	2	2	no	—
51	C₂×D₄×C₅₉	—	4	1	no	—
52	C₂×D₄×C₅₉	—	4	2	no	—
53	C₂²×C₄×C₅₉	—	4	1	no	—
54	C₂²×C₄×C₅₉	—	4	2	no	—
55	C₂²×C₄×C₅₉	—	4	2	no	—
56	C₂²×C₄×C₅₉	—	8	2	no	—
57	C₂²×C₄×C₅₉	—	8	2	no	—
58	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
61	D₈×C₅₉	—	2	4	no	—
62	D₈×C₅₉	—	2	2	no	—
63	D₈×C₅₉	C₂×C₈	1	4	yes	complemented
64	C₂×C₈×C₅₉	—	2	2	no	—
65	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₅₉	C₂×C₈	1	2	yes	—
68	C₂×C₈×C₅₉	M₁₆	1	2	yes	—
69	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
71	C₄⋊C₄×C₅₉	—	2	2	no	—
72	C₄⋊C₄×C₅₉	—	4	1	no	—
73	C₂²⋊C₄×C₅₉	—	4	2	no	—
74	C₂²⋊C₄×C₅₉	—	4	2	no	—
75	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
76	C₄²×C₅₉	D₈	1	1	yes	characteristic
77	C₄²×C₅₉	SD₁₆	1	1	yes	characteristic
78	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
79	C₂×C₅₉⋊C₈	—	4	2	no	—
80	C₂×C₅₉⋊C₈	—	8	2	no	—
81	C₅₉⋊C₁₆	—	2	4	no	—
82	C₂³×C₅₉	—	4	4	no	—
83	C₂³×C₅₉	—	4	2	no	—
84	C₂³×C₅₉	—	8	2	no	—
85	C₂³×C₅₉	—	8	2	no	—
86	C₂³×C₅₉	#102	1	1	yes	characteristic
87	D₄×C₅₉	—	2	4	no	—
88	D₄×C₅₉	—	4	4	no	—
89	D₄×C₅₉	—	4	2	no	—
90	C₂×C₄×C₅₉	—	2	2	no	—
91	C₂×C₄×C₅₉	—	4	2	no	—
92	C₂×C₄×C₅₉	—	8	2	no	—
93	C₂×C₄×C₅₉	—	8	4	no	—
94	C₂×C₄×C₅₉	—	8	2	no	—
95	C₂×C₄×C₅₉	—	8	2	no	—
96	C₂×C₄×C₅₉	C₂²⋊C₈	1	2	yes	—
97	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
98	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
99	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
100	C₂×C₄×C₅₉	#104	1	2	yes	—
101	C₂×C₄×C₅₉	D₁₆	1	2	yes	—
102	C₂×C₄×C₅₉	SD₃₂	1	2	yes	—
103	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
106	C₈×C₅₉	—	2	1	no	—
107	C₈×C₅₉	—	2	1	no	—
108	C₈×C₅₉	C₂²⋊C₈	1	2	yes	—
109	C₈×C₅₉	C₂²×C₈	1	1	yes	characteristic
110	C₈×C₅₉	C₂×M₁₆	1	1	yes	characteristic
111	C₅₉⋊C₈	—	8	2	no	—
112	C₂×#1331	—	59	1	no	maximal, Sylow
113	C₂²×C₅₉	—	4	4	no	—
114	C₂²×C₅₉	—	8	8	no	—
115	C₂²×C₅₉	—	8	2	no	—
116	C₂²×C₅₉	—	8	2	no	—
117	C₂²×C₅₉	D₄⋊C₈	1	2	yes	—
118	C₂²×C₅₉	#357	1	2	yes	—
119	C₂²×C₅₉	#359	1	2	yes	—
120	C₂²×C₅₉	C₂×#102	1	1	yes	characteristic
121	C₄×C₅₉	—	2	1	no	—
122	C₄×C₅₉	—	2	1	no	—
123	C₄×C₅₉	—	8	2	no	—
124	C₄×C₅₉	D₄⋊C₈	1	2	yes	—
125	C₄×C₅₉	#357	1	2	yes	—
126	C₄×C₅₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
127	C₄×C₅₉	C₂×#104	1	1	yes	characteristic
128	C₄×C₅₉	C₂×D₁₆	1	1	yes	characteristic
129	C₄×C₅₉	C₂×SD₃₂	1	1	yes	characteristic
130	C₂×C₄×D₈	D₅₉	1	1	yes	characteristic
131	C₂×C₄×C₁₆	—	59	1	no	—
132	C₂×#335	—	59	1	no	—
133	#1331	—	59	4	no	—
134	C₂×C₅₉	—	8	4	no	—
135	C₂×C₅₉	#1331	1	4	yes	complemented
136	C₂×C₅₉	C₂×D₄⋊C₈	1	1	yes	characteristic
137	C₂×C₅₉	C₂×#357	1	1	yes	characteristic
138	C₂×C₅₉	C₂×#359	1	1	yes	characteristic
139	C₂²×D₈	Dic₅₉	1	1	yes	characteristic
140	C₂×C₄×D₄	—	2	1	no	—
141	C₂²×C₁₆	—	118	1	no	—
142	C₄×D₈	—	2	1	no	—
143	C₄×D₈	—	2	1	no	—
144	C₄×D₈	Dic₅₉	1	2	yes	—
145	C₄×D₈	C₂×D₅₉	1	2	yes	—
146	C₂×#107	Dic₅₉	1	1	yes	characteristic
147	C₂×C₄⋊C₈	—	118	1	no	—
148	C₂×#102	—	2	1	no	—
149	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
150	C₄×C₁₆	—	59	2	no	—
151	C₄×C₁₆	—	118	1	no	—
152	#335	—	59	2	no	—
153	#335	—	118	1	no	—
154	C₅₉	C₂×#1331	1	1	yes	characteristic, complemented, Sylow
155	C₂²×D₄	—	2	1	no	—
156	C₂³×C₄	—	4	1	no	—
157	C₂×D₈	—	2	2	no	—
158	C₂×D₈	—	2	1	no	—
159	C₂×D₈	C₅₉⋊C₈	1	4	yes	complemented
160	C₂×D₈	C₂×Dic₅₉	1	2	yes	—
161	C₂²×C₈	—	236	1	no	—
162	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
163	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
164	C₄×D₄	—	2	2	no	—
165	C₄×D₄	—	2	2	no	—
166	C₄×D₄	—	4	1	no	—
167	C₄×D₄	—	4	1	no	—
168	C₂×C₄⋊C₄	—	2	1	no	—
169	C₂×C₂²⋊C₄	—	4	1	no	—
170	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
171	C₂×C₁₆	—	118	4	no	—
172	C₂×C₁₆	—	118	2	no	—
173	#107	—	2	1	no	—
174	#107	C₂×Dic₅₉	1	2	yes	—
175	C₄⋊C₈	—	118	2	no	—
176	C₄⋊C₈	—	236	1	no	—
177	#102	—	2	2	no	—
178	#102	—	2	2	no	—
179	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
180	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
181	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
182	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
183	C₂⁴	—	4	1	no	—
184	C₂×D₄	—	2	4	no	—
185	C₂×D₄	—	2	2	no	—
186	C₂×D₄	—	4	1	no	—
187	C₂×D₄	—	4	2	no	—
188	C₂²×C₄	—	4	1	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	—	4	2	no	—
191	C₂²×C₄	—	8	2	no	—
192	C₂²×C₄	—	8	2	no	—
193	C₂²×C₄	#11745669	1	1	yes	characteristic
194	C₂²×C₄	#11745670	1	1	yes	characteristic
195	C₂²×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
196	D₈	—	2	4	no	—
197	D₈	—	2	2	no	—
198	D₈	C₂×C₅₉⋊C₈	1	4	yes	complemented
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	472	2	no	—
201	C₂×C₈	—	236	2	no	—
202	C₂×C₈	C₂×C₅₉⋊C₈	1	2	yes	—
203	C₂×C₈	#11745664	1	2	yes	—
204	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
205	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
206	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
207	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	4	1	no	—
210	C₂²⋊C₄	—	4	2	no	—
211	C₂²⋊C₄	—	4	2	no	—
212	C₄²	#11745669	1	1	yes	characteristic
213	C₄²	#11745670	1	1	yes	characteristic
214	C₄²	C₂²⋊Dic₅₉	1	1	yes	characteristic
215	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
216	C₁₆	—	118	4	no	—
217	C₂³	—	4	4	no	—
218	C₂³	—	4	2	no	—
219	C₂³	—	8	2	no	—
220	C₂³	—	8	2	no	—
221	C₂³	#14273648	1	1	yes	characteristic
222	D₄	—	2	4	no	—
223	D₄	—	4	4	no	—
224	D₄	—	4	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	8	2	no	—
228	C₂×C₄	—	8	4	no	—
229	C₂×C₄	—	8	2	no	—
230	C₂×C₄	—	8	2	no	—
231	C₂×C₄	C₂²⋊(C₅₉⋊C₈)	1	2	yes	—
232	C₂×C₄	#14273648	1	1	yes	characteristic
233	C₂×C₄	#14273648	1	1	yes	characteristic
234	C₂×C₄	#14273648	1	1	yes	characteristic
235	C₂×C₄	#14273654	1	2	yes	—
236	C₂×C₄	#14273668	1	2	yes	—
237	C₂×C₄	#14273671	1	2	yes	—
238	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
239	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
240	C₂×C₄	C₂×#11745670	1	1	yes	characteristic
241	C₈	—	2	1	no	—
242	C₈	—	2	1	no	—
243	C₈	—	472	2	no	—
244	C₈	C₂²⋊(C₅₉⋊C₈)	1	2	yes	—
245	C₈	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
246	C₈	C₂×#11745664	1	1	yes	characteristic
247	C₂²	—	4	4	no	—
248	C₂²	—	8	2	no	—
249	C₂²	—	8	2	no	—
250	C₂²	—	8	8	no	—
251	C₂²	D₄⋊(C₅₉⋊C₈)	1	2	yes	—
252	C₂²	#16837203	1	2	yes	—
253	C₂²	#16837209	1	2	yes	—
254	C₂²	C₂×#14273648	1	1	yes	characteristic
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	8	2	no	—
258	C₄	D₄⋊(C₅₉⋊C₈)	1	2	yes	—
259	C₄	#16837203	1	2	yes	—
260	C₄	C₂×C₂²⋊(C₅₉⋊C₈)	1	1	yes	characteristic
261	C₄	C₂×#14273654	1	1	yes	characteristic
262	C₄	C₂×#14273668	1	1	yes	characteristic
263	C₄	C₂×#14273671	1	1	yes	characteristic
264	C₂	—	8	4	no	—
265	C₂	#21422057	1	4	yes	direct product factor
266	C₂	C₂×D₄⋊(C₅₉⋊C₈)	1	1	yes	characteristic
267	C₂	C₂×#16837203	1	1	yes	characteristic
268	C₂	C₂×#16837209	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#11745669
commutator	C₈×C₅₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₂×C₄×D₈×C₅₉	C₂
socle	C₂³×C₅₉	#102
prime radical	C₂²×D₈×C₅₉	C₄
cosocle	C₄×C₈×C₅₉	C₂³
hypercenter	C₂×C₄×D₈	D₅₉
hypocenter	C₅₉	C₂×#1331
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₄×C₅₉⋊C₁₆	C₂	semidirect
2. C₂×#16837155	C₂	semidirect
3. #21422057	C₂	direct (N × H)
8. C₄×C₅₉⋊C₁₆	C₂²	semidirect
10. #16837155	C₂²	semidirect
29. C₂×D₈×C₅₉	C₈	semidirect
63. D₈×C₅₉	C₂×C₈	semidirect
135. C₂×C₅₉	#1331	semidirect
154. C₅₉	C₂×#1331	semidirect
159. C₂×D₈	C₅₉⋊C₈	semidirect
198. D₈	C₂×C₅₉⋊C₈	semidirect
265. C₂	#21422057	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×C₅₉⋊C₁₆	2
C₂×#16837155	2
#21422057	2
C₂×C₄×D₈×C₅₉	2
C₂×#1331	59

Subgroups — C2×#21422057 #28498420

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21422057 #28498420