Subgroups — #21435433

Order 7552.

354 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	#16837959	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837706	C₂	1	2	yes	maximal, complemented
3	#16837687	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837682	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837641	C₂	1	2	yes	maximal, complemented
6	#16837629	C₂	1	2	yes	maximal
7	#16837621	C₂	1	2	yes	maximal, complemented
8	#16837612	C₂	1	1	yes	maximal, characteristic, complemented
9	#16837607	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄⋊(C₅₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₉×#529	C₂	1	1	yes	maximal, characteristic
12	C₂×#11745672	C₂²	1	2	yes	—
13	C₂×#11745670	C₂²	1	2	yes	—
14	C₂²×C₅₉⋊C₈	C₂²	1	2	yes	—
15	#14273727	C₂²	1	1	yes	characteristic
16	#14273727	C₂²	1	1	yes	characteristic
17	#14273719	—	2	1	no	—
18	#14273706	—	2	1	no	—
19	#14273706	C₂²	1	2	yes	—
20	C₄×Dic₁₁₈	—	2	1	no	—
21	C₄×Dic₁₁₈	C₂²	1	2	yes	—
22	#14273686	—	2	1	no	—
23	#14273660	C₂²	1	2	yes	—
24	#14273658	C₂²	1	2	yes	—
25	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
26	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
27	C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
28	#14273651	C₂²	1	2	yes	—
29	#14273650	C₂²	1	2	yes	—
30	#14273650	C₂²	1	2	yes	—
31	#14273648	C₂²	1	2	yes	—
32	#126×C₅₉	—	2	1	no	—
33	#125×C₅₉	C₂²	1	1	yes	characteristic
34	#124×C₅₉	C₂²	1	1	yes	characteristic
35	#123×C₅₉	—	2	1	no	—
36	C₂²⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
37	#121×C₅₉	C₂²	1	1	yes	characteristic
38	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
39	C₄×D₄×C₅₉	—	2	1	no	—
40	C₄×D₄×C₅₉	C₂²	1	1	yes	characteristic
41	#117×C₅₉	—	2	1	no	—
42	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
43	C₂×Q₈×C₅₉	C₂³	1	1	yes	characteristic
44	C₂×D₄×C₅₉	—	2	1	no	—
45	C₂×D₄×C₅₉	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₅₉	—	2	1	no	—
47	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
50	C₂×Dic₁₁₈	—	2	1	no	—
51	C₂×Dic₁₁₈	C₂³	1	2	yes	—
52	C₄⋊C₄×C₅₉	—	2	1	no	—
53	C₄⋊C₄×C₅₉	—	2	1	no	—
54	C₄⋊C₄×C₅₉	—	2	1	no	—
55	C₄⋊C₄×C₅₉	—	2	1	no	—
56	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
57	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
58	C₂²⋊C₄×C₅₉	—	2	1	no	—
59	C₂²⋊C₄×C₅₉	—	2	1	no	—
60	C₂²⋊C₄×C₅₉	—	2	1	no	—
61	C₂²⋊C₄×C₅₉	—	2	1	no	—
62	C₂²⋊C₄×C₅₉	—	2	1	no	—
63	C₄²×C₅₉	—	2	1	no	—
64	C₄²×C₅₉	D₄	1	1	yes	characteristic
65	C₄²×C₅₉	D₄	1	1	yes	characteristic
66	C₄²×C₅₉	C₂³	1	1	yes	characteristic
67	C₄²×C₅₉	C₂³	1	1	yes	characteristic
68	C₂²⋊Dic₅₉	—	2	2	no	—
69	#11745672	—	4	2	no	—
70	#11745670	—	4	2	no	—
71	C₄⋊Dic₅₉	—	2	1	no	—
72	C₄⋊Dic₅₉	—	2	1	no	—
73	C₄⋊Dic₅₉	—	2	1	no	—
74	C₄⋊Dic₅₉	C₂³	1	2	yes	—
75	Dic₅₉⋊C₄	—	2	2	no	—
76	Dic₅₉⋊C₄	—	2	2	no	—
77	C₄×Dic₅₉	—	2	2	no	—
78	C₂×C₅₉⋊C₈	—	4	2	no	—
79	C₂×C₅₉⋊C₈	C₂³	1	2	yes	—
80	C₂×C₅₉⋊C₈	C₂³	1	2	yes	—
81	C₂³×C₅₉	—	2	1	no	—
82	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
83	Q₈×C₅₉	—	2	1	no	—
84	Q₈×C₅₉	—	2	1	no	—
85	D₄×C₅₉	—	2	1	no	—
86	D₄×C₅₉	—	2	1	no	—
87	D₄×C₅₉	—	4	2	no	—
88	C₂×C₄×C₅₉	—	2	2	no	—
89	C₂×C₄×C₅₉	—	2	2	no	—
90	C₂×C₄×C₅₉	—	2	1	no	—
91	C₂×C₄×C₅₉	—	2	1	no	—
92	C₂×C₄×C₅₉	—	2	1	no	—
93	C₂×C₄×C₅₉	—	2	1	no	—
94	C₂×C₄×C₅₉	—	2	1	no	—
95	C₂×C₄×C₅₉	—	4	2	no	—
96	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
102	C₂×Dic₅₉	—	2	2	no	—
103	C₂×Dic₅₉	—	2	2	no	—
104	Dic₁₁₈	—	2	2	no	—
105	Dic₁₁₈	—	2	2	no	—
106	Dic₁₁₈	—	4	2	no	—
107	C₅₉⋊C₈	—	2	2	no	—
108	C₅₉⋊C₈	—	2	2	no	—
109	C₂²×C₅₉	—	2	1	no	—
110	C₂²×C₅₉	—	2	1	no	—
111	C₂²×C₅₉	—	4	1	no	—
112	C₂²×C₅₉	—	4	1	no	—
113	C₂²×C₅₉	—	4	1	no	—
114	C₂²×C₅₉	#137	1	2	yes	—
115	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
116	C₄×C₅₉	—	2	1	no	—
117	C₄×C₅₉	—	2	1	no	—
118	C₄×C₅₉	—	2	1	no	—
119	C₄×C₅₉	—	2	1	no	—
120	C₄×C₅₉	—	4	1	no	—
121	C₄×C₅₉	—	4	1	no	—
122	C₄×C₅₉	—	4	1	no	—
123	C₄×C₅₉	#135	1	2	yes	—
124	C₄×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
125	C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
126	Dic₅₉	—	4	2	no	—
127	Dic₅₉	—	4	2	no	—
128	#3248	—	59	1	no	maximal, Sylow
129	C₂×C₅₉	—	2	2	no	—
130	C₂×C₅₉	—	8	1	no	—
131	C₂×C₅₉	#537	1	1	yes	characteristic
132	C₂×C₅₉	C₂×#135	1	1	yes	characteristic
133	C₂×C₅₉	C₂×#137	1	1	yes	characteristic
134	#533	—	59	1	no	—
135	#529	D₅₉	1	1	yes	characteristic
136	#479	—	59	2	no	—
137	#475	—	59	1	no	—
138	#474	—	59	1	no	—
139	#467	—	59	2	no	—
140	#465	—	59	2	no	—
141	#464	—	59	2	no	—
142	#462	—	59	1	no	—
143	#461	—	59	1	no	—
144	C₂×C₄⋊C₈	—	59	1	no	—
145	C₅₉	#3248	1	1	yes	characteristic, complemented, Sylow
146	C₂×Q₁₆	—	59	2	no	—
147	C₂×SD₁₆	—	59	2	no	—
148	C₂²×C₈	—	59	2	no	—
149	C₄⋊Q₈	—	59	1	no	—
150	C₄⋊Q₈	—	59	1	no	—
151	#126	—	2	1	no	—
152	#125	—	118	1	no	—
153	#125	C₂×D₅₉	1	1	yes	characteristic
154	#124	C₂×D₅₉	1	1	yes	characteristic
155	#123	—	2	1	no	—
156	C₂²⋊Q₈	—	59	2	no	—
157	C₂²⋊Q₈	—	118	1	no	—
158	C₂²⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
159	#121	C₂×D₅₉	1	1	yes	characteristic
160	C₄×Q₈	—	59	2	no	—
161	C₄×Q₈	—	118	1	no	—
162	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
163	C₄×D₄	—	2	1	no	—
164	C₄×D₄	C₂×D₅₉	1	1	yes	characteristic
165	#117	—	2	1	no	—
166	#117	—	118	1	no	—
167	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
168	#107	—	59	2	no	—
169	#106	—	59	2	no	—
170	C₄⋊C₈	—	59	1	no	—
171	C₄⋊C₈	—	59	1	no	—
172	C₄⋊C₈	—	59	2	no	—
173	#103	—	59	2	no	—
174	#103	—	59	2	no	—
175	#103	—	59	2	no	—
176	#102	—	59	2	no	—
177	C₂×Q₈	—	59	2	no	—
178	C₂×Q₈	—	118	1	no	—
179	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
180	C₂×D₄	—	2	1	no	—
181	C₂×D₄	C₂²×D₅₉	1	1	yes	characteristic
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
184	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
185	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
186	Q₁₆	—	236	2	no	—
187	SD₁₆	—	236	2	no	—
188	C₂×C₈	—	59	2	no	—
189	C₂×C₈	—	59	2	no	—
190	C₂×C₈	—	236	2	no	—
191	C₄⋊C₄	—	2	1	no	—
192	C₄⋊C₄	—	2	1	no	—
193	C₄⋊C₄	—	2	1	no	—
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	—	59	2	no	—
196	C₄⋊C₄	—	118	1	no	—
197	C₄⋊C₄	—	118	1	no	—
198	C₄⋊C₄	—	118	2	no	—
199	C₄⋊C₄	—	118	2	no	—
200	C₄⋊C₄	—	118	1	no	—
201	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
202	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
203	C₂²⋊C₄	—	2	1	no	—
204	C₂²⋊C₄	—	2	1	no	—
205	C₂²⋊C₄	—	2	1	no	—
206	C₂²⋊C₄	—	2	1	no	—
207	C₂²⋊C₄	—	2	1	no	—
208	C₂²⋊C₄	—	118	2	no	—
209	C₄²	—	2	1	no	—
210	C₄²	—	118	2	no	—
211	C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
212	C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
213	C₄²	C₂²×D₅₉	1	1	yes	characteristic
214	C₄²	C₂²×D₅₉	1	1	yes	characteristic
215	C₂³	—	2	1	no	—
216	C₂³	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
217	Q₈	—	2	1	no	—
218	Q₈	—	2	1	no	—
219	Q₈	—	118	2	no	—
220	Q₈	—	118	2	no	—
221	Q₈	—	236	2	no	—
222	D₄	—	2	1	no	—
223	D₄	—	2	1	no	—
224	D₄	—	4	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	118	2	no	—
234	C₂×C₄	—	118	2	no	—
235	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
236	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
237	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
238	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
239	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
240	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
241	C₈	—	118	2	no	—
242	C₈	—	118	2	no	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	4	1	no	—
246	C₂²	—	4	1	no	—
247	C₂²	—	4	1	no	—
248	C₂²	#14273757	1	2	yes	—
249	C₂²	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	4	1	no	—
255	C₄	—	4	1	no	—
256	C₄	—	4	1	no	—
257	C₄	—	236	2	no	—
258	C₄	—	236	2	no	—
259	C₄	#14273747	1	2	yes	—
260	C₄	#14273773	1	1	yes	characteristic
261	C₄	#14273774	1	1	yes	characteristic
262	C₂	—	2	2	no	—
263	C₂	—	8	1	no	—
264	C₂	#16837980	1	1	yes	characteristic
265	C₂	C₂×#14273747	1	1	yes	characteristic
266	C₂	C₂×#14273757	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#529	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#529	D₅₉
hypocenter	C₅₉	#3248
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #529

Complements

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

N	H ≅ G/N	type
1. #16837959	C₂	semidirect
2. #16837706	C₂	semidirect
3. #16837687	C₂	semidirect
4. #16837682	C₂	semidirect
5. #16837641	C₂	semidirect
7. #16837621	C₂	semidirect
8. #16837612	C₂	semidirect
9. #16837607	C₂	semidirect
10. C₂×C₄⋊(C₅₉⋊C₈)	C₂	semidirect
145. C₅₉	#3248	semidirect