Subgroups — #21432760

Order 7552.

392 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16837870	C₂	1	1	yes	maximal, characteristic
2	#16837464	C₂	1	2	yes	maximal, complemented
3	#16837461	C₂	1	2	yes	maximal, complemented
4	#16837457	C₂	1	2	yes	maximal, complemented
5	#16837453	C₂	1	2	yes	maximal, complemented
6	#16837425	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×Dic₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837370	C₂	1	1	yes	maximal, characteristic
9	#16837368	C₂	1	1	yes	maximal, characteristic
10	#16837358	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₉×#405	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11745685	C₂²	1	1	yes	characteristic
13	C₂×#11745664	C₂²	1	1	yes	characteristic
14	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
15	#14273692	C₄	1	1	yes	characteristic
16	#14273692	C₄	1	1	yes	characteristic
17	#14273689	C₄	1	1	yes	characteristic
18	#14273689	C₄	1	1	yes	characteristic
19	(C₂²⋊D₅₉)⋊C₄	C₄	1	2	yes	—
20	(C₂²⋊D₅₉)⋊C₄	C₄	1	2	yes	—
21	#14273684	C₂²	1	2	yes	—
22	C₂×C₄×Dic₅₉	C₂²	1	1	yes	characteristic
23	C₄²×D₅₉	C₂²	1	2	yes	—
24	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
25	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
26	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
27	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
28	Dic₅₉⋊C₈	C₂²	1	2	yes	—
29	Dic₅₉⋊C₈	C₂²	1	2	yes	—
30	(C₂×D₅₉)⋊C₈	C₂²	1	2	yes	—
31	(C₂×D₅₉)⋊C₈	C₂²	1	2	yes	—
32	(C₂×D₅₉)⋊C₈	C₂²	1	2	yes	—
33	(C₂×D₅₉)⋊C₈	C₂²	1	2	yes	—
34	#14273638	C₂²	1	2	yes	—
35	C₄×C₅₉⋊C₈	C₂²	1	2	yes	—
36	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
37	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
38	#117×C₅₉	C₂²	1	1	yes	characteristic
39	#97×C₅₉	C₂²	1	2	yes	—
40	C₄×C₈×C₅₉	C₂²	1	2	yes	—
41	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
42	C₂×C₂²⋊D₅₉	C₂×C₄	1	2	yes	—
43	C₂²×Dic₅₉	C₂×C₄	1	1	yes	characteristic
44	C₂²×Dic₅₉	C₂×C₄	1	1	yes	characteristic
45	#11745685	—	2	2	no	—
46	#11745685	—	2	2	no	—
47	C₂×D₂₃₆	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₅₉	C₂×C₄	1	2	yes	—
49	C₂×C₄×D₅₉	C₂×C₄	1	2	yes	—
50	C₂×C₄×D₅₉	C₂³	1	2	yes	—
51	C₂×Dic₁₁₈	C₂×C₄	1	1	yes	characteristic
52	M₁₆×C₅₉	—	2	2	no	—
53	C₂×C₈×C₅₉	—	2	2	no	—
54	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
55	C₂×C₈×C₅₉	C₂³	1	2	yes	—
56	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
57	C₄⋊C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
58	C₄⋊C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
59	C₂²⋊C₄×C₅₉	C₂×C₄	1	2	yes	—
60	C₄²×C₅₉	C₂³	1	2	yes	—
61	(C₂×D₅₉)⋊C₄	C₂×C₄	1	2	yes	—
62	(C₂×D₅₉)⋊C₄	C₂×C₄	1	2	yes	—
63	Dic₅₉⋊C₄	C₂×C₄	1	2	yes	—
64	Dic₅₉⋊C₄	C₂×C₄	1	2	yes	—
65	C₄×Dic₅₉	C₂×C₄	1	2	yes	—
66	C₄×Dic₅₉	C₂×C₄	1	2	yes	—
67	C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
68	C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
69	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
70	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
71	#11745664	—	2	2	no	—
72	C₂×C₅₉⋊C₈	—	2	2	no	—
73	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
74	C₂×C₅₉⋊C₈	C₂³	1	2	yes	—
75	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
76	C₂³×C₅₉	C₂²×C₄	1	1	yes	characteristic
77	C₂²×D₅₉	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₅₉	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₅₉	C₂²×C₄	1	2	yes	—
80	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₅₉	#41	1	2	yes	—
82	C₂×C₄×C₅₉	#41	1	2	yes	—
83	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
84	C₂²⋊D₅₉	—	2	4	no	—
85	C₂×Dic₅₉	—	2	2	no	—
86	C₂×Dic₅₉	—	4	1	no	—
87	C₂×Dic₅₉	C₂²×C₄	1	1	yes	characteristic
88	C₂×Dic₅₉	C₂²×C₄	1	2	yes	—
89	C₂×Dic₅₉	C₂²×C₄	1	1	yes	characteristic
90	C₂×Dic₅₉	C₂²×C₄	1	1	yes	characteristic
91	C₂×Dic₅₉	C₂²×C₄	1	1	yes	characteristic
92	D₂₃₆	—	2	2	no	—
93	C₄×D₅₉	—	2	2	no	—
94	C₄×D₅₉	—	2	2	no	—
95	C₄×D₅₉	—	2	4	no	—
96	C₄×D₅₉	—	4	2	no	—
97	Dic₁₁₈	—	2	2	no	—
98	C₈×C₅₉	—	2	1	no	—
99	C₈×C₅₉	—	2	1	no	—
100	C₈×C₅₉	—	2	2	no	—
101	C₅₉⋊C₈	—	2	1	no	—
102	C₅₉⋊C₈	—	2	1	no	—
103	C₅₉⋊C₈	—	2	2	no	—
104	C₂²×C₅₉	—	2	1	no	—
105	C₂²×C₅₉	—	2	1	no	—
106	C₂²×C₅₉	#117	1	2	yes	—
107	C₂²×C₅₉	C₂³×C₄	1	1	yes	characteristic
108	C₂×D₅₉	—	2	2	no	—
109	C₂×D₅₉	—	2	2	no	—
110	C₂×D₅₉	—	2	2	no	—
111	C₄×C₅₉	—	2	2	no	—
112	C₄×C₅₉	—	2	2	no	—
113	C₄×C₅₉	#117	1	2	yes	—
114	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
115	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
116	Dic₅₉	—	2	2	no	—
117	Dic₅₉	—	2	1	no	—
118	Dic₅₉	—	2	1	no	—
119	Dic₅₉	—	2	1	no	—
120	Dic₅₉	#131	1	2	yes	—
121	#2925	—	59	1	no	maximal, Sylow
122	C₂×C₅₉	—	2	2	no	—
123	C₂×C₅₉	C₂×#117	1	1	yes	characteristic
124	C₂×C₅₉	C₂×#131	1	1	yes	characteristic
125	C₂×C₅₉	#568	1	1	yes	characteristic
126	D₅₉	—	4	2	no	—
127	C₄×#41	—	59	1	no	—
128	#433	—	59	2	no	—
129	#433	—	59	2	no	—
130	#432	—	59	2	no	—
131	#431	—	59	2	no	—
132	#423	—	59	1	no	—
133	C₂×C₄⋊C₈	—	59	1	no	—
134	#408	—	59	1	no	—
135	#408	—	59	1	no	—
136	#405	—	59	1	no	—
137	#405	D₅₉	1	1	yes	characteristic, complemented
138	C₅₉	#2925	1	1	yes	characteristic, complemented, Sylow
139	C₂×#41	—	59	1	no	—
140	C₂×M₁₆	—	59	1	no	—
141	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
142	C₂²×C₈	—	59	1	no	—
143	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
144	C₄×Q₈	—	59	1	no	—
145	C₄×Q₈	—	59	1	no	—
146	C₄×D₄	—	59	1	no	—
147	C₄×D₄	—	59	1	no	—
148	C₄×D₄	—	59	2	no	—
149	C₄×D₄	—	59	2	no	—
150	#117	—	59	2	no	—
151	#117	C₂×D₅₉	1	1	yes	characteristic
152	C₂×C₄²	—	59	1	no	—
153	C₂×C₄²	—	59	2	no	—
154	C₄⋊C₈	—	59	1	no	—
155	C₄⋊C₈	—	59	1	no	—
156	C₄⋊C₈	—	59	1	no	—
157	C₄⋊C₈	—	59	1	no	—
158	C₄⋊C₈	—	59	2	no	—
159	C₄⋊C₈	—	59	2	no	—
160	C₂²⋊C₈	—	59	2	no	—
161	C₂²⋊C₈	—	59	2	no	—
162	C₂²⋊C₈	—	59	2	no	—
163	C₂²⋊C₈	—	59	2	no	—
164	#97	—	59	2	no	—
165	#97	C₂×D₅₉	1	2	yes	—
166	C₄×C₈	—	59	2	no	—
167	C₄×C₈	C₂×D₅₉	1	2	yes	—
168	#41	—	118	2	no	—
169	#41	—	118	2	no	—
170	C₂×Q₈	—	59	1	no	—
171	C₂×D₄	—	59	2	no	—
172	C₂×D₄	—	59	1	no	—
173	C₂²×C₄	—	59	1	no	—
174	C₂²×C₄	—	59	1	no	—
175	C₂²×C₄	—	59	2	no	—
176	C₂²×C₄	—	59	2	no	—
177	C₂²×C₄	—	59	2	no	—
178	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
179	M₁₆	—	2	2	no	—
180	M₁₆	—	118	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	59	1	no	—
183	C₂×C₈	—	59	2	no	—
184	C₂×C₈	—	59	1	no	—
185	C₂×C₈	—	118	2	no	—
186	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
187	C₂×C₈	C₂²×D₅₉	1	2	yes	—
188	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
189	C₄⋊C₄	—	59	2	no	—
190	C₄⋊C₄	—	59	2	no	—
191	C₄⋊C₄	C₄×D₅₉	1	1	yes	characteristic
192	C₄⋊C₄	C₄×D₅₉	1	1	yes	characteristic
193	C₂²⋊C₄	—	59	2	no	—
194	C₂²⋊C₄	—	59	2	no	—
195	C₂²⋊C₄	C₄×D₅₉	1	2	yes	—
196	C₄²	—	59	1	no	—
197	C₄²	—	59	2	no	—
198	C₄²	—	59	1	no	—
199	C₄²	—	59	2	no	—
200	C₄²	—	59	1	no	—
201	C₄²	—	59	1	no	—
202	C₄²	C₂²×D₅₉	1	2	yes	—
203	C₂³	—	59	2	no	—
204	C₂³	C₂×C₄×D₅₉	1	1	yes	characteristic
205	Q₈	—	118	2	no	—
206	D₄	—	118	4	no	—
207	D₄	—	118	2	no	—
208	C₂×C₄	—	59	1	no	—
209	C₂×C₄	—	59	1	no	—
210	C₂×C₄	—	59	1	no	—
211	C₂×C₄	—	59	1	no	—
212	C₂×C₄	—	59	2	no	—
213	C₂×C₄	—	118	4	no	—
214	C₂×C₄	—	118	2	no	—
215	C₂×C₄	—	118	2	no	—
216	C₂×C₄	—	118	2	no	—
217	C₂×C₄	—	236	2	no	—
218	C₂×C₄	—	236	1	no	—
219	C₂×C₄	C₂×C₄×D₅₉	1	2	yes	—
220	C₂×C₄	C₂×C₄×D₅₉	1	2	yes	—
221	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
222	C₂×C₄	#11745685	1	2	yes	—
223	C₂×C₄	#11745685	1	2	yes	—
224	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
225	C₈	—	2	1	no	—
226	C₈	—	2	1	no	—
227	C₈	—	2	2	no	—
228	C₈	—	118	1	no	—
229	C₈	—	118	1	no	—
230	C₈	—	118	2	no	—
231	C₂²	—	2	1	no	—
232	C₂²	—	2	1	no	—
233	C₂²	—	118	2	no	—
234	C₂²	—	118	2	no	—
235	C₂²	—	118	2	no	—
236	C₂²	#14273684	1	2	yes	—
237	C₂²	C₂²×C₄×D₅₉	1	1	yes	characteristic
238	C₄	—	2	2	no	—
239	C₄	—	2	2	no	—
240	C₄	—	59	2	no	—
241	C₄	—	118	1	no	—
242	C₄	—	118	1	no	—
243	C₄	—	118	1	no	—
244	C₄	—	118	2	no	—
245	C₄	#14273684	1	2	yes	—
246	C₄	C₂×#11745685	1	1	yes	characteristic
247	C₄	C₂×#11745685	1	1	yes	characteristic
248	C₂	—	2	2	no	—
249	C₂	—	236	2	no	—
250	C₂	C₂×#14273684	1	1	yes	characteristic
251	C₂	#131×D₅₉	1	1	yes	characteristic
252	C₂	#16838167	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₉
commutator, socle	C₂²×C₅₉	C₂³×C₄
Fitting	C₅₉×#405	C₂
hypercenter	#405	D₅₉
hypocenter	C₅₉	#2925
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ #405

Complements

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

N	H ≅ G/N	type
2. #16837464	C₂	semidirect
3. #16837461	C₂	semidirect
4. #16837457	C₂	semidirect
5. #16837453	C₂	semidirect
6. #16837425	C₂	semidirect
7. C₂×Dic₅₉⋊C₈	C₂	semidirect
10. #16837358	C₂	semidirect
11. C₅₉×#405	C₂	semidirect
137. #405	D₅₉	semidirect
138. C₅₉	#2925	semidirect