Subgroups — C₅₉×#3098 #21421416

Order 7552.

398 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₉×#537	C₂	1	1	yes	maximal, characteristic
2	C₅₉×#520	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₉×#485	C₂	1	2	yes	maximal, complemented
4	C₅₉×#484	C₂	1	2	yes	maximal, complemented
5	C₅₉×#482	C₂	1	2	yes	maximal, complemented
6	C₅₉×#480	C₂	1	2	yes	maximal, complemented
7	C₅₉×#428	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#107×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₉×#417	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₉×#416	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₉×#408	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₅₉	C₂²	1	1	yes	characteristic
13	C₂×#41×C₅₉	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₂²⋊Q₈×C₅₉	—	2	1	no	—
17	C₂²⋊Q₈×C₅₉	C₂²	1	2	yes	—
18	#121×C₅₉	—	2	1	no	—
19	#121×C₅₉	—	2	2	no	—
20	#121×C₅₉	C₂²	1	2	yes	—
21	C₄×Q₈×C₅₉	—	2	1	no	—
22	C₄×D₄×C₅₉	—	2	2	no	—
23	C₄×D₄×C₅₉	—	2	1	no	—
24	#117×C₅₉	C₂²	1	1	yes	characteristic
25	#117×C₅₉	C₂²	1	2	yes	—
26	C₂×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₅₉	C₂²	1	2	yes	—
29	#107×C₅₉	C₂²	1	2	yes	—
30	#107×C₅₉	C₂²	1	2	yes	—
31	#107×C₅₉	C₂²	1	2	yes	—
32	#106×C₅₉	C₂²	1	2	yes	—
33	#103×C₅₉	C₂²	1	1	yes	characteristic
34	#103×C₅₉	C₂²	1	1	yes	characteristic
35	#103×C₅₉	C₂²	1	2	yes	—
36	#102×C₅₉	C₂²	1	1	yes	characteristic
37	#102×C₅₉	C₂²	1	1	yes	characteristic
38	#102×C₅₉	C₂²	1	2	yes	—
39	C₂²⋊C₈×C₅₉	C₂²	1	2	yes	—
40	C₂²⋊C₈×C₅₉	C₂²	1	2	yes	—
41	#41×C₅₉	—	2	2	no	—
42	#41×C₅₉	—	2	2	no	—
43	#41×C₅₉	—	2	2	no	—
44	#41×C₅₉	—	2	2	no	—
45	C₂×Q₈×C₅₉	D₄	1	1	yes	characteristic
46	C₂×Q₈×C₅₉	C₂³	1	1	yes	characteristic
47	C₂×D₄×C₅₉	—	2	1	no	—
48	C₂×D₄×C₅₉	—	2	2	no	—
49	C₂×D₄×C₅₉	D₄	1	2	yes	—
50	C₂×D₄×C₅₉	D₄	1	1	yes	characteristic
51	C₂×D₄×C₅₉	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₅₉	—	2	2	no	—
53	C₂²×C₄×C₅₉	—	2	1	no	—
54	C₂²×C₄×C₅₉	—	2	1	no	—
55	C₂²×C₄×C₅₉	—	2	1	no	—
56	C₂²×C₄×C₅₉	C₂³	1	2	yes	—
57	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
58	M₁₆×C₅₉	—	2	2	no	—
59	C₂×C₈×C₅₉	—	2	2	no	—
60	C₂×C₈×C₅₉	C₂³	1	2	yes	—
61	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
62	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
63	C₄⋊C₄×C₅₉	—	2	2	no	—
64	C₄⋊C₄×C₅₉	—	2	1	no	—
65	C₄⋊C₄×C₅₉	—	2	1	no	—
66	C₄⋊C₄×C₅₉	—	2	2	no	—
67	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
68	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
70	C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
71	C₂²⋊C₄×C₅₉	—	2	2	no	—
72	C₂²⋊C₄×C₅₉	—	2	1	no	—
73	C₂²⋊C₄×C₅₉	—	2	2	no	—
74	C₂²⋊C₄×C₅₉	—	2	2	no	—
75	C₄²×C₅₉	—	2	2	no	—
76	C₄²×C₅₉	—	2	1	no	—
77	C₂³×C₅₉	—	2	1	no	—
78	C₂³×C₅₉	C₂×D₄	1	2	yes	—
79	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
80	Q₈×C₅₉	—	2	2	no	—
81	Q₈×C₅₉	—	2	1	no	—
82	Q₈×C₅₉	—	2	1	no	—
83	D₄×C₅₉	—	2	4	no	—
84	D₄×C₅₉	—	2	2	no	—
85	D₄×C₅₉	—	2	1	no	—
86	D₄×C₅₉	—	2	1	no	—
87	D₄×C₅₉	—	4	2	no	—
88	D₄×C₅₉	—	8	2	no	—
89	C₂×C₄×C₅₉	—	2	2	no	—
90	C₂×C₄×C₅₉	—	2	2	no	—
91	C₂×C₄×C₅₉	—	2	1	no	—
92	C₂×C₄×C₅₉	—	2	2	no	—
93	C₂×C₄×C₅₉	—	2	1	no	—
94	C₂×C₄×C₅₉	—	2	2	no	—
95	C₂×C₄×C₅₉	—	2	1	no	—
96	C₂×C₄×C₅₉	—	4	2	no	—
97	C₂×C₄×C₅₉	—	8	1	no	—
98	C₂×C₄×C₅₉	—	8	2	no	—
99	C₂×C₄×C₅₉	—	8	1	no	—
100	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
101	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₅₉	#41	1	2	yes	—
103	C₂×C₄×C₅₉	#41	1	2	yes	—
104	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
105	C₈×C₅₉	—	2	1	no	—
106	C₈×C₅₉	—	2	1	no	—
107	C₈×C₅₉	—	2	2	no	—
108	C₂²×C₅₉	—	2	1	no	—
109	C₂²×C₅₉	—	2	1	no	—
110	C₂²×C₅₉	—	2	2	no	—
111	C₂²×C₅₉	—	2	2	no	—
112	C₂²×C₅₉	—	2	2	no	—
113	C₂²×C₅₉	—	4	1	no	—
114	C₂²×C₅₉	—	4	1	no	—
115	C₂²×C₅₉	—	4	1	no	—
116	C₂²×C₅₉	#123	1	2	yes	—
117	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
118	C₄×C₅₉	—	2	2	no	—
119	C₄×C₅₉	—	4	1	no	—
120	C₄×C₅₉	—	4	2	no	—
121	C₄×C₅₉	—	4	1	no	—
122	C₄×C₅₉	—	4	2	no	—
123	C₄×C₅₉	—	4	1	no	—
124	C₄×C₅₉	#123	1	2	yes	—
125	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
126	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
127	#3098	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₂×C₅₉	—	2	2	no	—
129	C₂×C₅₉	—	4	2	no	—
130	C₂×C₅₉	—	8	1	no	—
131	C₂×C₅₉	C₂×#123	1	1	yes	characteristic
132	C₂×C₅₉	#576	1	1	yes	characteristic
133	C₂×C₅₉	#578	1	1	yes	characteristic
134	#537	C₂×C₅₉	1	1	yes	characteristic
135	#520	C₂×C₅₉	1	1	yes	characteristic, complemented
136	#485	C₂×C₅₉	1	2	yes	complemented
137	#484	C₂×C₅₉	1	2	yes	complemented
138	#482	C₂×C₅₉	1	2	yes	complemented
139	#480	C₂×C₅₉	1	2	yes	complemented
140	#428	C₂×C₅₉	1	1	yes	characteristic, complemented
141	C₂×#107	C₂×C₅₉	1	1	yes	characteristic, complemented
142	#417	C₂×C₅₉	1	1	yes	characteristic, complemented
143	#416	C₂×C₅₉	1	1	yes	characteristic, complemented
144	#408	C₂×C₅₉	1	1	yes	characteristic, complemented
145	C₅₉	#3098	1	1	yes	characteristic, direct product factor, Sylow
146	C₂×#41	C₂²×C₅₉	1	1	yes	characteristic
147	C₂×#41	C₂²×C₅₉	1	1	yes	characteristic
148	C₂×M₁₆	C₂²×C₅₉	1	1	yes	characteristic
149	C₂²×C₈	C₂²×C₅₉	1	1	yes	characteristic
150	C₂²⋊Q₈	—	2	1	no	—
151	C₂²⋊Q₈	C₂²×C₅₉	1	2	yes	—
152	#121	—	2	1	no	—
153	#121	—	2	2	no	—
154	#121	C₂²×C₅₉	1	2	yes	—
155	C₄×Q₈	—	2	1	no	—
156	C₄×D₄	—	2	2	no	—
157	C₄×D₄	—	2	1	no	—
158	#117	C₂²×C₅₉	1	1	yes	characteristic
159	#117	C₂²×C₅₉	1	2	yes	—
160	C₂×C₄⋊C₄	C₂²×C₅₉	1	1	yes	characteristic
161	C₂×C₄⋊C₄	C₂²×C₅₉	1	2	yes	—
162	C₂×C₄⋊C₄	C₂²×C₅₉	1	1	yes	characteristic
163	#107	C₂²×C₅₉	1	2	yes	—
164	#107	C₂²×C₅₉	1	2	yes	—
165	#107	C₂²×C₅₉	1	2	yes	—
166	#106	C₂²×C₅₉	1	2	yes	—
167	#103	C₂²×C₅₉	1	1	yes	characteristic
168	#103	C₂²×C₅₉	1	1	yes	characteristic
169	#103	C₂²×C₅₉	1	2	yes	—
170	#102	C₂²×C₅₉	1	1	yes	characteristic
171	#102	C₂²×C₅₉	1	1	yes	characteristic
172	#102	C₂²×C₅₉	1	2	yes	—
173	C₂²⋊C₈	C₂²×C₅₉	1	2	yes	—
174	C₂²⋊C₈	C₂²×C₅₉	1	2	yes	—
175	#41	—	2	2	no	—
176	#41	—	2	2	no	—
177	#41	—	2	2	no	—
178	#41	—	2	2	no	—
179	C₂×Q₈	D₄×C₅₉	1	1	yes	characteristic
180	C₂×Q₈	C₂³×C₅₉	1	1	yes	characteristic
181	C₂×D₄	—	2	1	no	—
182	C₂×D₄	—	2	2	no	—
183	C₂×D₄	D₄×C₅₉	1	2	yes	—
184	C₂×D₄	D₄×C₅₉	1	1	yes	characteristic
185	C₂×D₄	C₂³×C₅₉	1	1	yes	characteristic
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	C₂³×C₅₉	1	2	yes	—
191	C₂²×C₄	C₂³×C₅₉	1	1	yes	characteristic
192	M₁₆	—	2	2	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	C₂³×C₅₉	1	2	yes	—
195	C₂×C₈	C₂³×C₅₉	1	1	yes	characteristic
196	C₂×C₈	C₂³×C₅₉	1	1	yes	characteristic
197	C₄⋊C₄	—	2	2	no	—
198	C₄⋊C₄	—	2	1	no	—
199	C₄⋊C₄	—	2	1	no	—
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	C₂³×C₅₉	1	1	yes	characteristic
202	C₄⋊C₄	C₂³×C₅₉	1	1	yes	characteristic
203	C₄⋊C₄	C₂³×C₅₉	1	2	yes	—
204	C₄⋊C₄	C₂³×C₅₉	1	2	yes	—
205	C₂²⋊C₄	—	2	2	no	—
206	C₂²⋊C₄	—	2	1	no	—
207	C₂²⋊C₄	—	2	2	no	—
208	C₂²⋊C₄	—	2	2	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	1	no	—
211	C₂³	—	2	1	no	—
212	C₂³	C₂×D₄×C₅₉	1	2	yes	—
213	C₂³	C₂×D₄×C₅₉	1	1	yes	characteristic
214	Q₈	—	2	2	no	—
215	Q₈	—	2	1	no	—
216	Q₈	—	2	1	no	—
217	D₄	—	2	4	no	—
218	D₄	—	2	2	no	—
219	D₄	—	2	1	no	—
220	D₄	—	2	1	no	—
221	D₄	—	4	2	no	—
222	D₄	—	8	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	8	1	no	—
232	C₂×C₄	—	8	2	no	—
233	C₂×C₄	—	8	1	no	—
234	C₂×C₄	C₂×D₄×C₅₉	1	2	yes	—
235	C₂×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
236	C₂×C₄	#41×C₅₉	1	2	yes	—
237	C₂×C₄	#41×C₅₉	1	2	yes	—
238	C₂×C₄	C₂⁴×C₅₉	1	1	yes	characteristic
239	C₈	—	2	1	no	—
240	C₈	—	2	1	no	—
241	C₈	—	2	2	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	4	1	no	—
248	C₂²	—	4	1	no	—
249	C₂²	—	4	1	no	—
250	C₂²	#123×C₅₉	1	2	yes	—
251	C₂²	C₂²×D₄×C₅₉	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	4	1	no	—
254	C₄	—	4	2	no	—
255	C₄	—	4	1	no	—
256	C₄	—	4	2	no	—
257	C₄	—	4	1	no	—
258	C₄	#123×C₅₉	1	2	yes	—
259	C₄	C₂×#41×C₅₉	1	1	yes	characteristic
260	C₄	C₂×#41×C₅₉	1	1	yes	characteristic
261	C₂	—	2	2	no	—
262	C₂	—	4	2	no	—
263	C₂	—	8	1	no	—
264	C₂	C₂×#123×C₅₉	1	1	yes	characteristic
265	C₂	C₅₉×#576	1	1	yes	characteristic
266	C₂	C₅₉×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₅₉
prime radical	C₅₉×#537	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₉ ◁ C₂×#41×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₉×#520	C₂	semidirect
3. C₅₉×#485	C₂	semidirect
4. C₅₉×#484	C₂	semidirect
5. C₅₉×#482	C₂	semidirect
6. C₅₉×#480	C₂	semidirect
7. C₅₉×#428	C₂	semidirect
8. C₂×#107×C₅₉	C₂	semidirect
9. C₅₉×#417	C₂	semidirect
10. C₅₉×#416	C₂	semidirect
11. C₅₉×#408	C₂	semidirect
127. #3098	C₅₉	direct (N × H)
135. #520	C₂×C₅₉	semidirect
136. #485	C₂×C₅₉	semidirect
137. #484	C₂×C₅₉	semidirect
138. #482	C₂×C₅₉	semidirect
139. #480	C₂×C₅₉	semidirect
140. #428	C₂×C₅₉	semidirect
141. C₂×#107	C₂×C₅₉	semidirect
142. #417	C₂×C₅₉	semidirect
143. #416	C₂×C₅₉	semidirect
144. #408	C₂×C₅₉	semidirect
145. C₅₉	#3098	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₉×#537	2
C₅₉×#520	2
C₅₉×#485	2
C₅₉×#484	2
C₅₉×#482	2
C₅₉×#480	2
C₅₉×#428	2
C₂×#107×C₅₉	2
C₅₉×#417	2
C₅₉×#416	2
C₅₉×#408	2
#3098	59

Subgroups — C59×#3098 #21421416

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#3098 #21421416