Subgroups — C₅₉×#35446 #28447555

Order 15104.

5398 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×H(5,𝔽₂)×C₅₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×#123×C₅₉	C₂	1	3	yes	maximal, complemented
3	C₅₉×#2406	C₂	1	12	yes	maximal, complemented
4	C₅₉×#2405	C₂	1	12	yes	maximal, complemented
5	C₅₉×#2308	C₂	1	3	yes	maximal, complemented
6	C₂×H(5,𝔽₂)×C₅₉	—	2	24	no	—
7	C₂²×#41×C₅₉	C₂²	1	6	yes	—
8	C₂³×D₄×C₅₉	C₂²	1	6	yes	—
9	C₂³×D₄×C₅₉	C₂²	1	3	yes	—
10	C₂⁴×C₄×C₅₉	C₂²	1	1	yes	characteristic, complemented
11	C₂×#123×C₅₉	—	2	24	no	—
12	C₂×#123×C₅₉	C₂²	1	24	yes	complemented
13	C₂²×C₄⋊C₄×C₅₉	C₂²	1	6	yes	complemented
14	C₂²×C₂²⋊C₄×C₅₉	C₂²	1	3	yes	complemented
15	C₂²×C₂²⋊C₄×C₅₉	C₂²	1	6	yes	complemented
16	C₅₉×#395	C₂²	1	8	yes	complemented
17	C₅₉×#394	C₂²	1	24	yes	complemented
18	C₅₉×#386	C₂²	1	24	yes	complemented
19	C₅₉×#385	C₂²	1	24	yes	complemented
20	C₅₉×#381	C₂²	1	6	yes	complemented
21	C₅₉×#379	C₂²	1	6	yes	complemented
22	C₅₉×#379	C₂²	1	8	yes	complemented
23	C₂⁵×C₅₉	C₂³	1	2	yes	—
24	C₂⁵×C₅₉	C₂³	1	3	yes	—
25	C₂⁵×C₅₉	C₂³	1	1	yes	characteristic
26	H(5,𝔽₂)×C₅₉	—	2	128	no	—
27	C₂×#41×C₅₉	—	2	48	no	—
28	C₂×#41×C₅₉	—	2	24	no	—
29	C₂²×Q₈×C₅₉	C₂³	1	2	yes	—
30	C₂²×D₄×C₅₉	—	2	24	no	—
31	C₂²×D₄×C₅₉	—	2	24	no	—
32	C₂²×D₄×C₅₉	—	2	48	no	—
33	C₂²×D₄×C₅₉	D₄	1	24	yes	—
34	C₂²×D₄×C₅₉	C₂³	1	6	yes	—
35	C₂²×D₄×C₅₉	C₂³	1	12	yes	—
36	C₂³×C₄×C₅₉	—	2	12	no	—
37	C₂³×C₄×C₅₉	C₂³	1	6	yes	—
38	C₂³×C₄×C₅₉	C₂³	1	6	yes	—
39	C₂³×C₄×C₅₉	C₂³	1	3	yes	—
40	C₂³×C₄×C₅₉	C₂³	1	6	yes	—
41	#123×C₅₉	—	2	96	no	—
42	C₂×C₄⋊C₄×C₅₉	—	2	24	no	—
43	C₂×C₄⋊C₄×C₅₉	C₂³	1	24	yes	—
44	C₂×C₂²⋊C₄×C₅₉	—	2	12	no	—
45	C₂×C₂²⋊C₄×C₅₉	—	2	24	no	—
46	C₂×C₂²⋊C₄×C₅₉	C₂³	1	12	yes	—
47	C₂×C₂²⋊C₄×C₅₉	C₂³	1	24	yes	—
48	C₂×C₂²⋊C₄×C₅₉	C₂³	1	24	yes	—
49	#95×C₅₉	C₂³	1	24	yes	—
50	C₂⁴×C₅₉	—	2	12	no	—
51	C₂⁴×C₅₉	—	2	12	no	—
52	C₂⁴×C₅₉	—	4	4	no	—
53	C₂⁴×C₅₉	—	4	12	no	—
54	C₂⁴×C₅₉	—	4	8	no	—
55	C₂⁴×C₅₉	C₂×D₄	1	12	yes	—
56	C₂⁴×C₅₉	C₂×D₄	1	12	yes	—
57	C₂⁴×C₅₉	C₂⁴	1	6	yes	—
58	C₂⁴×C₅₉	C₂⁴	1	3	yes	—
59	#41×C₅₉	—	2	192	no	—
60	C₂×Q₈×C₅₉	—	2	12	no	—
61	C₂×D₄×C₅₉	—	2	192	no	—
62	C₂×D₄×C₅₉	—	2	96	no	—
63	C₂×D₄×C₅₉	—	2	24	no	—
64	C₂×D₄×C₅₉	—	2	24	no	—
65	C₂×D₄×C₅₉	—	2	24	no	—
66	C₂×D₄×C₅₉	—	2	12	no	—
67	C₂×D₄×C₅₉	C₂×D₄	1	48	yes	—
68	C₂²×C₄×C₅₉	—	2	12	no	—
69	C₂²×C₄×C₅₉	—	2	12	no	—
70	C₂²×C₄×C₅₉	—	2	24	no	—
71	C₂²×C₄×C₅₉	—	2	12	no	—
72	C₂²×C₄×C₅₉	—	2	12	no	—
73	C₂²×C₄×C₅₉	—	2	24	no	—
74	C₂²×C₄×C₅₉	—	4	12	no	—
75	C₂²×C₄×C₅₉	—	4	16	no	—
76	C₂²×C₄×C₅₉	C₂×D₄	1	12	yes	—
77	C₂²×C₄×C₅₉	C₂⁴	1	12	yes	—
78	C₂²×C₄×C₅₉	C₂⁴	1	4	yes	—
79	C₂²×C₄×C₅₉	C₂⁴	1	6	yes	—
80	C₄⋊C₄×C₅₉	—	2	48	no	—
81	C₂²⋊C₄×C₅₉	—	2	24	no	—
82	C₂²⋊C₄×C₅₉	—	2	48	no	—
83	C₂²⋊C₄×C₅₉	—	4	24	no	—
84	C₂³×C₅₉	—	2	12	no	—
85	C₂³×C₅₉	—	2	16	no	—
86	C₂³×C₅₉	—	2	12	no	—
87	C₂³×C₅₉	—	2	24	no	—
88	C₂³×C₅₉	—	2	24	no	—
89	C₂³×C₅₉	—	4	24	no	—
90	C₂³×C₅₉	—	4	24	no	—
91	C₂³×C₅₉	—	8	24	no	—
92	C₂³×C₅₉	—	8	24	no	—
93	C₂³×C₅₉	#120	1	16	yes	—
94	C₂³×C₅₉	C₂²×D₄	1	12	yes	—
95	C₂³×C₅₉	C₂²×D₄	1	12	yes	—
96	C₂³×C₅₉	C₂²×D₄	1	6	yes	—
97	C₂³×C₅₉	C₂⁵	1	1	yes	characteristic
98	Q₈×C₅₉	—	2	16	no	—
99	D₄×C₅₉	—	2	48	no	—
100	D₄×C₅₉	—	2	96	no	—
101	C₂×C₄×C₅₉	—	2	48	no	—
102	C₂×C₄×C₅₉	—	2	24	no	—
103	C₂×C₄×C₅₉	—	2	12	no	—
104	C₂×C₄×C₅₉	—	2	12	no	—
105	C₂×C₄×C₅₉	—	2	12	no	—
106	C₂×C₄×C₅₉	—	2	24	no	—
107	C₂×C₄×C₅₉	—	4	24	no	—
108	C₂×C₄×C₅₉	—	4	24	no	—
109	C₂×C₄×C₅₉	C₂²×D₄	1	12	yes	—
110	#35446	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂²×C₅₉	—	2	12	no	—
112	C₂²×C₅₉	—	2	24	no	—
113	C₂²×C₅₉	—	2	12	no	—
114	C₂²×C₅₉	—	2	24	no	—
115	C₂²×C₅₉	—	4	16	no	—
116	C₂²×C₅₉	—	4	24	no	—
117	C₂²×C₅₉	—	8	24	no	—
118	C₂²×C₅₉	—	8	16	no	—
119	C₂²×C₅₉	C₂×#120	1	12	yes	—
120	C₂²×C₅₉	C₂³×D₄	1	3	yes	—
121	C₂²×C₅₉	#585	1	4	yes	—
122	C₄×C₅₉	—	2	12	no	—
123	C₄×C₅₉	—	4	4	no	—
124	C₄×C₅₉	—	4	12	no	—
125	C₂²×H(5,𝔽₂)	C₂×C₅₉	1	1	yes	characteristic
126	C₂²×#123	C₂×C₅₉	1	3	yes	complemented
127	#2406	C₂×C₅₉	1	12	yes	complemented
128	#2405	C₂×C₅₉	1	12	yes	complemented
129	#2308	C₂×C₅₉	1	3	yes	complemented
130	C₂×C₅₉	—	2	12	no	—
131	C₂×C₅₉	—	4	12	no	—
132	C₂×C₅₉	C₂²×#120	1	1	yes	characteristic
133	C₂×C₅₉	#3485	1	6	yes	—
134	C₂×H(5,𝔽₂)	—	2	24	no	—
135	C₂²×#41	C₂²×C₅₉	1	6	yes	—
136	C₂³×D₄	C₂²×C₅₉	1	6	yes	—
137	C₂³×D₄	C₂²×C₅₉	1	3	yes	—
138	C₂⁴×C₄	C₂²×C₅₉	1	1	yes	characteristic, complemented
139	C₂×#123	—	2	24	no	—
140	C₂×#123	C₂²×C₅₉	1	24	yes	complemented
141	C₂²×C₄⋊C₄	C₂²×C₅₉	1	6	yes	complemented
142	C₂²×C₂²⋊C₄	C₂²×C₅₉	1	3	yes	complemented
143	C₂²×C₂²⋊C₄	C₂²×C₅₉	1	6	yes	complemented
144	#395	C₂²×C₅₉	1	8	yes	complemented
145	#394	C₂²×C₅₉	1	24	yes	complemented
146	#386	C₂²×C₅₉	1	24	yes	complemented
147	#385	C₂²×C₅₉	1	24	yes	complemented
148	#381	C₂²×C₅₉	1	6	yes	complemented
149	#379	C₂²×C₅₉	1	6	yes	complemented
150	#379	C₂²×C₅₉	1	8	yes	complemented
151	C₅₉	#35446	1	1	yes	characteristic, direct product factor, Sylow
152	C₂⁵	C₂³×C₅₉	1	2	yes	—
153	C₂⁵	C₂³×C₅₉	1	3	yes	—
154	C₂⁵	C₂³×C₅₉	1	1	yes	characteristic
155	H(5,𝔽₂)	—	2	128	no	—
156	C₂×#41	—	2	48	no	—
157	C₂×#41	—	2	24	no	—
158	C₂²×Q₈	C₂³×C₅₉	1	2	yes	—
159	C₂²×D₄	—	2	24	no	—
160	C₂²×D₄	—	2	24	no	—
161	C₂²×D₄	—	2	48	no	—
162	C₂²×D₄	D₄×C₅₉	1	24	yes	—
163	C₂²×D₄	C₂³×C₅₉	1	6	yes	—
164	C₂²×D₄	C₂³×C₅₉	1	12	yes	—
165	C₂³×C₄	—	2	12	no	—
166	C₂³×C₄	C₂³×C₅₉	1	6	yes	—
167	C₂³×C₄	C₂³×C₅₉	1	6	yes	—
168	C₂³×C₄	C₂³×C₅₉	1	3	yes	—
169	C₂³×C₄	C₂³×C₅₉	1	6	yes	—
170	#123	—	2	96	no	—
171	C₂×C₄⋊C₄	—	2	24	no	—
172	C₂×C₄⋊C₄	C₂³×C₅₉	1	24	yes	—
173	C₂×C₂²⋊C₄	—	2	12	no	—
174	C₂×C₂²⋊C₄	—	2	24	no	—
175	C₂×C₂²⋊C₄	C₂³×C₅₉	1	12	yes	—
176	C₂×C₂²⋊C₄	C₂³×C₅₉	1	24	yes	—
177	C₂×C₂²⋊C₄	C₂³×C₅₉	1	24	yes	—
178	#95	C₂³×C₅₉	1	24	yes	—
179	C₂⁴	—	2	12	no	—
180	C₂⁴	—	2	12	no	—
181	C₂⁴	—	4	4	no	—
182	C₂⁴	—	4	12	no	—
183	C₂⁴	—	4	8	no	—
184	C₂⁴	C₂×D₄×C₅₉	1	12	yes	—
185	C₂⁴	C₂×D₄×C₅₉	1	12	yes	—
186	C₂⁴	C₂⁴×C₅₉	1	6	yes	—
187	C₂⁴	C₂⁴×C₅₉	1	3	yes	—
188	#41	—	2	192	no	—
189	C₂×Q₈	—	2	12	no	—
190	C₂×D₄	—	2	192	no	—
191	C₂×D₄	—	2	96	no	—
192	C₂×D₄	—	2	24	no	—
193	C₂×D₄	—	2	24	no	—
194	C₂×D₄	—	2	24	no	—
195	C₂×D₄	—	2	12	no	—
196	C₂×D₄	C₂×D₄×C₅₉	1	48	yes	—
197	C₂²×C₄	—	2	12	no	—
198	C₂²×C₄	—	2	12	no	—
199	C₂²×C₄	—	2	24	no	—
200	C₂²×C₄	—	2	12	no	—
201	C₂²×C₄	—	2	12	no	—
202	C₂²×C₄	—	2	24	no	—
203	C₂²×C₄	—	4	12	no	—
204	C₂²×C₄	—	4	16	no	—
205	C₂²×C₄	C₂×D₄×C₅₉	1	12	yes	—
206	C₂²×C₄	C₂⁴×C₅₉	1	12	yes	—
207	C₂²×C₄	C₂⁴×C₅₉	1	4	yes	—
208	C₂²×C₄	C₂⁴×C₅₉	1	6	yes	—
209	C₄⋊C₄	—	2	48	no	—
210	C₂²⋊C₄	—	2	24	no	—
211	C₂²⋊C₄	—	2	48	no	—
212	C₂²⋊C₄	—	4	24	no	—
213	C₂³	—	2	12	no	—
214	C₂³	—	2	16	no	—
215	C₂³	—	2	12	no	—
216	C₂³	—	2	24	no	—
217	C₂³	—	2	24	no	—
218	C₂³	—	4	24	no	—
219	C₂³	—	4	24	no	—
220	C₂³	—	8	24	no	—
221	C₂³	—	8	24	no	—
222	C₂³	#120×C₅₉	1	16	yes	—
223	C₂³	C₂²×D₄×C₅₉	1	12	yes	—
224	C₂³	C₂²×D₄×C₅₉	1	12	yes	—
225	C₂³	C₂²×D₄×C₅₉	1	6	yes	—
226	C₂³	C₂⁵×C₅₉	1	1	yes	characteristic
227	Q₈	—	2	16	no	—
228	D₄	—	2	48	no	—
229	D₄	—	2	96	no	—
230	C₂×C₄	—	2	48	no	—
231	C₂×C₄	—	2	24	no	—
232	C₂×C₄	—	2	12	no	—
233	C₂×C₄	—	2	12	no	—
234	C₂×C₄	—	2	12	no	—
235	C₂×C₄	—	2	24	no	—
236	C₂×C₄	—	4	24	no	—
237	C₂×C₄	—	4	24	no	—
238	C₂×C₄	C₂²×D₄×C₅₉	1	12	yes	—
239	C₂²	—	2	12	no	—
240	C₂²	—	2	24	no	—
241	C₂²	—	2	12	no	—
242	C₂²	—	2	24	no	—
243	C₂²	—	4	16	no	—
244	C₂²	—	4	24	no	—
245	C₂²	—	8	24	no	—
246	C₂²	—	8	16	no	—
247	C₂²	C₂×#120×C₅₉	1	12	yes	—
248	C₂²	C₂³×D₄×C₅₉	1	3	yes	—
249	C₂²	C₅₉×#585	1	4	yes	—
250	C₄	—	2	12	no	—
251	C₄	—	4	4	no	—
252	C₄	—	4	12	no	—
253	C₂	—	2	12	no	—
254	C₂	—	4	12	no	—
255	C₂	C₂²×#120×C₅₉	1	1	yes	characteristic
256	C₂	C₅₉×#3485	1	6	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₉	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅₉
prime radical	C₂²×H(5,𝔽₂)×C₅₉	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×#123×C₅₉	C₂	semidirect
3. C₅₉×#2406	C₂	semidirect
4. C₅₉×#2405	C₂	semidirect
5. C₅₉×#2308	C₂	semidirect
10. C₂⁴×C₄×C₅₉	C₂²	semidirect
12. C₂×#123×C₅₉	C₂²	semidirect
13. C₂²×C₄⋊C₄×C₅₉	C₂²	semidirect
14. C₂²×C₂²⋊C₄×C₅₉	C₂²	semidirect
15. C₂²×C₂²⋊C₄×C₅₉	C₂²	semidirect
16. C₅₉×#395	C₂²	semidirect
17. C₅₉×#394	C₂²	semidirect
18. C₅₉×#386	C₂²	semidirect
19. C₅₉×#385	C₂²	semidirect
20. C₅₉×#381	C₂²	semidirect
21. C₅₉×#379	C₂²	semidirect
22. C₅₉×#379	C₂²	semidirect
110. #35446	C₅₉	direct (N × H)
126. C₂²×#123	C₂×C₅₉	semidirect
127. #2406	C₂×C₅₉	semidirect
128. #2405	C₂×C₅₉	semidirect
129. #2308	C₂×C₅₉	semidirect
138. C₂⁴×C₄	C₂²×C₅₉	semidirect
140. C₂×#123	C₂²×C₅₉	semidirect
141. C₂²×C₄⋊C₄	C₂²×C₅₉	semidirect
142. C₂²×C₂²⋊C₄	C₂²×C₅₉	semidirect
143. C₂²×C₂²⋊C₄	C₂²×C₅₉	semidirect
144. #395	C₂²×C₅₉	semidirect
145. #394	C₂²×C₅₉	semidirect
146. #386	C₂²×C₅₉	semidirect
147. #385	C₂²×C₅₉	semidirect
148. #381	C₂²×C₅₉	semidirect
149. #379	C₂²×C₅₉	semidirect
150. #379	C₂²×C₅₉	semidirect
151. C₅₉	#35446	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×H(5,𝔽₂)×C₅₉	2
C₂²×#123×C₅₉	2
C₅₉×#2406	2
C₅₉×#2405	2
C₅₉×#2308	2
#35446	59

Subgroups — C59×#35446 #28447555

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#35446 #28447555