Subgroups — C₅₉×#2432 #21420750

Order 7552.

836 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#41×C₅₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Q₈×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#124×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂²⋊Q₈×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₉×#394	C₂	1	4	yes	maximal, complemented
6	C₅₉×#391	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#390	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₉×#385	C₂	1	4	yes	maximal, complemented
9	C₄×C₂²⋊C₄×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41×C₅₉	—	2	4	no	—
11	C₂×#41×C₅₉	—	2	4	no	—
12	C₂×#41×C₅₉	—	2	4	no	—
13	C₂²×Q₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₂²×Q₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₂²×D₄×C₅₉	C₂²	1	2	yes	—
16	C₂²×D₄×C₅₉	C₂²	1	1	yes	characteristic
17	C₂³×C₄×C₅₉	C₂²	1	2	yes	—
18	C₂³×C₄×C₅₉	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₅₉	—	2	4	no	—
20	#124×C₅₉	—	2	4	no	—
21	C₂²⋊Q₈×C₅₉	—	2	4	no	—
22	C₂×C₄⋊C₄×C₅₉	C₂²	1	2	yes	—
23	C₂×C₄⋊C₄×C₅₉	C₂²	1	4	yes	—
24	C₂×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₄×C₅₉	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₄×C₅₉	C₂²	1	4	yes	—
27	C₂×C₂²⋊C₄×C₅₉	C₂²	1	4	yes	—
28	C₂×C₂²⋊C₄×C₅₉	C₂²	1	2	yes	—
29	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
30	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
31	#95×C₅₉	C₂²	1	2	yes	—
32	#95×C₅₉	C₂²	1	4	yes	—
33	C₂⁴×C₅₉	C₂³	1	2	yes	—
34	C₂⁴×C₅₉	C₂³	1	1	yes	characteristic
35	#41×C₅₉	—	2	32	no	—
36	C₂×Q₈×C₅₉	—	2	2	no	—
37	C₂×Q₈×C₅₉	—	2	2	no	—
38	C₂×Q₈×C₅₉	—	2	2	no	—
39	C₂×Q₈×C₅₉	—	2	2	no	—
40	C₂×Q₈×C₅₉	—	2	2	no	—
41	C₂×Q₈×C₅₉	D₄	1	4	yes	—
42	C₂×D₄×C₅₉	—	2	4	no	—
43	C₂×D₄×C₅₉	—	2	4	no	—
44	C₂×D₄×C₅₉	—	2	4	no	—
45	C₂×D₄×C₅₉	—	2	2	no	—
46	C₂×D₄×C₅₉	—	2	2	no	—
47	C₂×D₄×C₅₉	D₄	1	4	yes	—
48	C₂²×C₄×C₅₉	—	2	4	no	—
49	C₂²×C₄×C₅₉	—	2	4	no	—
50	C₂²×C₄×C₅₉	—	2	4	no	—
51	C₂²×C₄×C₅₉	—	2	2	no	—
52	C₂²×C₄×C₅₉	—	2	2	no	—
53	C₂²×C₄×C₅₉	D₄	1	4	yes	—
54	C₂²×C₄×C₅₉	C₂³	1	2	yes	—
55	C₂²×C₄×C₅₉	C₂³	1	2	yes	—
56	C₂²×C₄×C₅₉	C₂³	1	2	yes	—
57	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₅₉	C₂³	1	2	yes	—
59	C₂²×C₄×C₅₉	C₂³	1	2	yes	—
60	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₅₉	—	2	2	no	—
62	C₄⋊C₄×C₅₉	—	2	4	no	—
63	C₄⋊C₄×C₅₉	—	2	8	no	—
64	C₂²⋊C₄×C₅₉	—	2	4	no	—
65	C₂²⋊C₄×C₅₉	—	2	8	no	—
66	C₂²⋊C₄×C₅₉	—	2	4	no	—
67	C₂²⋊C₄×C₅₉	—	2	8	no	—
68	C₄²×C₅₉	—	2	2	no	—
69	C₄²×C₅₉	—	2	2	no	—
70	C₂³×C₅₉	—	2	2	no	—
71	C₂³×C₅₉	—	2	1	no	—
72	C₂³×C₅₉	—	2	2	no	—
73	C₂³×C₅₉	—	2	1	no	—
74	C₂³×C₅₉	—	2	2	no	—
75	C₂³×C₅₉	—	2	4	no	—
76	C₂³×C₅₉	—	2	2	no	—
77	C₂³×C₅₉	—	2	4	no	—
78	C₂³×C₅₉	C₂×D₄	1	4	yes	—
79	C₂³×C₅₉	C₂×D₄	1	2	yes	—
80	C₂³×C₅₉	C₂⁴	1	1	yes	characteristic
81	Q₈×C₅₉	—	2	8	no	—
82	Q₈×C₅₉	—	4	4	no	—
83	D₄×C₅₉	—	2	16	no	—
84	D₄×C₅₉	—	2	8	no	—
85	C₂×C₄×C₅₉	—	2	16	no	—
86	C₂×C₄×C₅₉	—	2	2	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	2	no	—
92	C₂×C₄×C₅₉	—	2	8	no	—
93	C₂×C₄×C₅₉	—	2	2	no	—
94	C₂×C₄×C₅₉	—	2	2	no	—
95	C₂×C₄×C₅₉	—	2	2	no	—
96	C₂×C₄×C₅₉	—	2	2	no	—
97	C₂×C₄×C₅₉	—	2	2	no	—
98	C₂×C₄×C₅₉	—	2	4	no	—
99	C₂×C₄×C₅₉	—	2	2	no	—
100	C₂×C₄×C₅₉	—	2	2	no	—
101	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
102	C₂×C₄×C₅₉	C₂×D₄	1	4	yes	—
103	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
104	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
105	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
106	C₂×C₄×C₅₉	#41	1	4	yes	complemented
107	C₂²×C₅₉	—	2	4	no	—
108	C₂²×C₅₉	—	2	2	no	—
109	C₂²×C₅₉	—	2	2	no	—
110	C₂²×C₅₉	—	2	2	no	—
111	C₂²×C₅₉	—	2	8	no	—
112	C₂²×C₅₉	—	4	2	no	—
113	C₂²×C₅₉	—	4	2	no	—
114	C₂²×C₅₉	—	4	2	no	—
115	C₂²×C₅₉	—	4	2	no	—
116	C₂²×C₅₉	—	4	4	no	—
117	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
118	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
119	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
120	C₂²×C₅₉	C₂×#41	1	2	yes	—
121	C₂²×C₅₉	2₋¹⁺⁴	1	2	yes	—
122	C₄×C₅₉	—	2	2	no	—
123	C₄×C₅₉	—	2	4	no	—
124	C₄×C₅₉	—	2	4	no	—
125	C₄×C₅₉	—	4	2	no	—
126	C₄×C₅₉	—	4	2	no	—
127	C₄×C₅₉	—	4	2	no	—
128	C₄×C₅₉	#120	1	4	yes	—
129	#2432	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂×C₅₉	—	4	2	no	—
131	C₂×C₅₉	—	4	4	no	—
132	C₂×C₅₉	C₂×#120	1	1	yes	characteristic
133	C₂×C₅₉	#532	1	1	yes	characteristic
134	C₂×C₅₉	#536	1	1	yes	characteristic
135	C₂×C₅₉	#547	1	2	yes	—
136	C₂×C₅₉	#548	1	2	yes	—
137	C₂²×#41	C₂×C₅₉	1	1	yes	characteristic
138	C₂×C₄⋊Q₈	C₂×C₅₉	1	1	yes	characteristic, complemented
139	C₂×#124	C₂×C₅₉	1	1	yes	characteristic, complemented
140	C₂×C₂²⋊Q₈	C₂×C₅₉	1	1	yes	characteristic, complemented
141	#394	C₂×C₅₉	1	4	yes	complemented
142	#391	C₂×C₅₉	1	1	yes	characteristic, complemented
143	#390	C₂×C₅₉	1	1	yes	characteristic, complemented
144	#385	C₂×C₅₉	1	4	yes	complemented
145	C₄×C₂²⋊C₄	C₂×C₅₉	1	1	yes	characteristic, complemented
146	C₅₉	#2432	1	1	yes	characteristic, direct product factor, Sylow
147	C₂×#41	—	2	4	no	—
148	C₂×#41	—	2	4	no	—
149	C₂×#41	—	2	4	no	—
150	C₂²×Q₈	C₂²×C₅₉	1	1	yes	characteristic
151	C₂²×Q₈	C₂²×C₅₉	1	1	yes	characteristic
152	C₂²×D₄	C₂²×C₅₉	1	2	yes	—
153	C₂²×D₄	C₂²×C₅₉	1	1	yes	characteristic
154	C₂³×C₄	C₂²×C₅₉	1	2	yes	—
155	C₂³×C₄	C₂²×C₅₉	1	1	yes	characteristic
156	C₄⋊Q₈	—	2	4	no	—
157	#124	—	2	4	no	—
158	C₂²⋊Q₈	—	2	4	no	—
159	C₂×C₄⋊C₄	C₂²×C₅₉	1	2	yes	—
160	C₂×C₄⋊C₄	C₂²×C₅₉	1	1	yes	characteristic
161	C₂×C₄⋊C₄	C₂²×C₅₉	1	4	yes	—
162	C₂×C₂²⋊C₄	C₂²×C₅₉	1	2	yes	—
163	C₂×C₂²⋊C₄	C₂²×C₅₉	1	4	yes	—
164	C₂×C₂²⋊C₄	C₂²×C₅₉	1	2	yes	—
165	C₂×C₂²⋊C₄	C₂²×C₅₉	1	4	yes	—
166	C₂×C₄²	C₂²×C₅₉	1	1	yes	characteristic
167	C₂×C₄²	C₂²×C₅₉	1	1	yes	characteristic
168	#95	C₂²×C₅₉	1	2	yes	—
169	#95	C₂²×C₅₉	1	4	yes	—
170	C₂⁴	C₂³×C₅₉	1	2	yes	—
171	C₂⁴	C₂³×C₅₉	1	1	yes	characteristic
172	#41	—	2	32	no	—
173	C₂×Q₈	—	2	2	no	—
174	C₂×Q₈	—	2	2	no	—
175	C₂×Q₈	—	2	2	no	—
176	C₂×Q₈	—	2	2	no	—
177	C₂×Q₈	—	2	2	no	—
178	C₂×Q₈	D₄×C₅₉	1	4	yes	—
179	C₂×D₄	—	2	4	no	—
180	C₂×D₄	—	2	4	no	—
181	C₂×D₄	—	2	4	no	—
182	C₂×D₄	—	2	2	no	—
183	C₂×D₄	—	2	2	no	—
184	C₂×D₄	D₄×C₅₉	1	4	yes	—
185	C₂²×C₄	—	2	4	no	—
186	C₂²×C₄	—	2	4	no	—
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	D₄×C₅₉	1	4	yes	—
191	C₂²×C₄	C₂³×C₅₉	1	2	yes	—
192	C₂²×C₄	C₂³×C₅₉	1	2	yes	—
193	C₂²×C₄	C₂³×C₅₉	1	2	yes	—
194	C₂²×C₄	C₂³×C₅₉	1	1	yes	characteristic
195	C₂²×C₄	C₂³×C₅₉	1	2	yes	—
196	C₂²×C₄	C₂³×C₅₉	1	2	yes	—
197	C₂²×C₄	C₂³×C₅₉	1	1	yes	characteristic
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	—	2	4	no	—
200	C₄⋊C₄	—	2	8	no	—
201	C₂²⋊C₄	—	2	4	no	—
202	C₂²⋊C₄	—	2	8	no	—
203	C₂²⋊C₄	—	2	4	no	—
204	C₂²⋊C₄	—	2	8	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	2	2	no	—
207	C₂³	—	2	2	no	—
208	C₂³	—	2	1	no	—
209	C₂³	—	2	2	no	—
210	C₂³	—	2	1	no	—
211	C₂³	—	2	2	no	—
212	C₂³	—	2	4	no	—
213	C₂³	—	2	2	no	—
214	C₂³	—	2	4	no	—
215	C₂³	C₂×D₄×C₅₉	1	4	yes	—
216	C₂³	C₂×D₄×C₅₉	1	2	yes	—
217	C₂³	C₂⁴×C₅₉	1	1	yes	characteristic
218	Q₈	—	2	8	no	—
219	Q₈	—	4	4	no	—
220	D₄	—	2	16	no	—
221	D₄	—	2	8	no	—
222	C₂×C₄	—	2	16	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	8	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	C₂×D₄×C₅₉	1	2	yes	—
239	C₂×C₄	C₂×D₄×C₅₉	1	4	yes	—
240	C₂×C₄	C₂×D₄×C₅₉	1	2	yes	—
241	C₂×C₄	C₂×D₄×C₅₉	1	2	yes	—
242	C₂×C₄	C₂×D₄×C₅₉	1	2	yes	—
243	C₂×C₄	#41×C₅₉	1	4	yes	complemented
244	C₂²	—	2	4	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	2	8	no	—
249	C₂²	—	4	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	4	2	no	—
252	C₂²	—	4	2	no	—
253	C₂²	—	4	4	no	—
254	C₂²	C₂²×D₄×C₅₉	1	1	yes	characteristic
255	C₂²	C₂²×D₄×C₅₉	1	1	yes	characteristic
256	C₂²	C₂²×D₄×C₅₉	1	1	yes	characteristic
257	C₂²	C₂×#41×C₅₉	1	2	yes	—
258	C₂²	2₋¹⁺⁴×C₅₉	1	2	yes	—
259	C₄	—	2	2	no	—
260	C₄	—	2	4	no	—
261	C₄	—	2	4	no	—
262	C₄	—	4	2	no	—
263	C₄	—	4	2	no	—
264	C₄	—	4	2	no	—
265	C₄	#120×C₅₉	1	4	yes	—
266	C₂	—	4	2	no	—
267	C₂	—	4	4	no	—
268	C₂	C₂×#120×C₅₉	1	1	yes	characteristic
269	C₂	C₅₉×#532	1	1	yes	characteristic
270	C₂	C₅₉×#536	1	1	yes	characteristic
271	C₂	C₅₉×#547	1	2	yes	—
272	C₂	C₅₉×#548	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₉	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₅₉
prime radical	C₂²×#41×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. C₂×C₄⋊Q₈×C₅₉	C₂	semidirect
3. C₂×#124×C₅₉	C₂	semidirect
4. C₂×C₂²⋊Q₈×C₅₉	C₂	semidirect
5. C₅₉×#394	C₂	semidirect
6. C₅₉×#391	C₂	semidirect
7. C₅₉×#390	C₂	semidirect
8. C₅₉×#385	C₂	semidirect
9. C₄×C₂²⋊C₄×C₅₉	C₂	semidirect
106. C₂×C₄×C₅₉	#41	semidirect
129. #2432	C₅₉	direct (N × H)
138. C₂×C₄⋊Q₈	C₂×C₅₉	semidirect
139. C₂×#124	C₂×C₅₉	semidirect
140. C₂×C₂²⋊Q₈	C₂×C₅₉	semidirect
141. #394	C₂×C₅₉	semidirect
142. #391	C₂×C₅₉	semidirect
143. #390	C₂×C₅₉	semidirect
144. #385	C₂×C₅₉	semidirect
145. C₄×C₂²⋊C₄	C₂×C₅₉	semidirect
146. C₅₉	#2432	direct (N × H)
243. C₂×C₄	#41×C₅₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#41×C₅₉	2
C₂×C₄⋊Q₈×C₅₉	2
C₂×#124×C₅₉	2
C₂×C₂²⋊Q₈×C₅₉	2
C₅₉×#394	2
C₅₉×#391	2
C₅₉×#390	2
C₅₉×#385	2
C₄×C₂²⋊C₄×C₅₉	2
#2432	59

Subgroups — C59×#2432 #21420750

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#2432 #21420750