Subgroups — #21433002

Order 7552.

458 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16838200	C₂	1	1	yes	maximal, characteristic
2	#16838169	C₂	1	1	yes	maximal, characteristic, complemented
3	#16838168	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837521	C₂	1	4	yes	maximal, complemented
5	#16837516	C₂	1	4	yes	maximal, complemented
6	#16837450	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14273655	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#11745664	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#104×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#11745685	C₂²	1	1	yes	characteristic
11	#14273758	C₄	1	2	yes	complemented
12	#14273757	C₄	1	2	yes	complemented
13	#14273753	C₄	1	2	yes	complemented
14	#14273752	C₄	1	2	yes	complemented
15	#14273747	C₂²	1	4	yes	—
16	#14273735	C₂²	1	4	yes	complemented
17	#14273734	—	2	2	no	—
18	#14273734	C₂²	1	4	yes	complemented
19	#14273733	—	2	2	no	—
20	C₂×#11745664	—	2	2	no	—
21	C₂×#11745664	C₂²	1	1	yes	characteristic
22	C₂×#11745664	C₂²	1	1	yes	characteristic
23	C₂×#11745664	C₂²	1	1	yes	characteristic
24	M₁₆×D₅₉	—	2	2	no	—
25	#14273662	C₂²	1	4	yes	complemented
26	#14273655	C₂²	1	2	yes	complemented
27	#14273655	C₂²	1	2	yes	complemented
28	#14273638	C₂²	1	2	yes	complemented
29	C₄×C₅₉⋊C₈	C₂²	1	2	yes	complemented
30	C₂×#41×C₅₉	C₂²	1	1	yes	characteristic
31	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
32	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
33	#104×C₅₉	C₂²	1	2	yes	complemented
34	#104×C₅₉	C₂²	1	2	yes	complemented
35	#41×C₅₉	—	2	1	no	—
36	#41×C₅₉	—	2	1	no	—
37	#41×C₅₉	C₂×C₄	1	2	yes	—
38	#41×C₅₉	C₂³	1	2	yes	—
39	C₂×Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
40	C₂×D₄×C₅₉	—	2	1	no	—
41	C₂×D₄×C₅₉	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₉	—	2	1	no	—
43	C₂²×C₄×C₅₉	—	2	1	no	—
44	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
45	C₂×C₂²⋊D₅₉	—	2	1	no	—
46	#11745685	—	2	1	no	—
47	#11745685	—	2	1	no	—
48	#11745685	C₂×C₄	1	2	yes	—
49	#11745685	C₂³	1	2	yes	—
50	C₂×D₂₃₆	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₅₉	—	2	1	no	—
52	C₂×Dic₁₁₈	C₂×C₄	1	1	yes	characteristic
53	M₁₆×C₅₉	—	2	1	no	—
54	M₁₆×C₅₉	C₂³	1	2	yes	—
55	C₂×C₈×C₅₉	—	2	1	no	—
56	C₄²×C₅₉	—	2	1	no	—
57	C₄²×C₅₉	C₂³	1	1	yes	characteristic
58	C₄²×C₅₉	C₂³	1	1	yes	characteristic
59	#11745672	C₂×C₄	1	4	yes	complemented
60	#11745671	C₂×C₄	1	4	yes	complemented
61	#11745670	C₂×C₄	1	4	yes	complemented
62	#11745669	C₂×C₄	1	4	yes	complemented
63	#11745664	—	2	1	no	—
64	#11745664	—	2	4	no	—
65	#11745664	C₂×C₄	1	2	yes	—
66	#11745664	C₂×C₄	1	2	yes	—
67	#11745664	D₄	1	2	yes	—
68	#11745664	D₄	1	2	yes	—
69	#11745664	C₂³	1	2	yes	—
70	C₂×C₅₉⋊C₈	—	2	1	no	—
71	C₂×C₅₉⋊C₈	—	2	4	no	—
72	C₂×C₅₉⋊C₈	C₂³	1	2	yes	—
73	C₂×C₅₉⋊C₈	C₂³	1	2	yes	—
74	#11745659	—	2	4	no	—
75	C₈×D₅₉	—	2	4	no	—
76	C₂³×C₅₉	—	2	1	no	—
77	C₂³×C₅₉	#41	1	1	yes	characteristic
78	C₂²×D₅₉	—	2	1	no	—
79	Q₈×C₅₉	—	2	1	no	—
80	Q₈×C₅₉	C₂²×C₄	1	2	yes	—
81	D₄×C₅₉	—	2	2	no	—
82	D₄×C₅₉	—	2	2	no	—
83	D₄×C₅₉	—	2	1	no	—
84	D₄×C₅₉	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₅₉	—	2	1	no	—
86	C₂×C₄×C₅₉	—	2	1	no	—
87	C₂×C₄×C₅₉	—	2	2	no	—
88	C₂×C₄×C₅₉	—	2	2	no	—
89	C₂×C₄×C₅₉	—	2	1	no	—
90	C₂×C₄×C₅₉	—	4	2	no	—
91	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
96	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
97	C₂²⋊D₅₉	—	2	2	no	—
98	C₂²⋊D₅₉	—	2	2	no	—
99	C₂×Dic₅₉	—	2	1	no	—
100	D₂₃₆	—	2	1	no	—
101	D₂₃₆	C₂²×C₄	1	2	yes	—
102	C₄×D₅₉	—	2	2	no	—
103	C₄×D₅₉	—	2	2	no	—
104	Dic₁₁₈	—	2	1	no	—
105	Dic₁₁₈	C₂²×C₄	1	2	yes	—
106	C₈×C₅₉	—	2	2	no	—
107	C₅₉⋊C₈	—	2	2	no	—
108	C₅₉⋊C₈	C₂²×C₄	1	4	yes	—
109	C₅₉⋊C₈	C₂×D₄	1	4	yes	—
110	C₂²×C₅₉	—	2	2	no	—
111	C₂²×C₅₉	—	4	1	no	—
112	C₂²×C₅₉	—	8	1	no	—
113	C₂²×C₅₉	C₄×D₄	1	1	yes	characteristic
114	C₂²×C₅₉	C₄×D₄	1	1	yes	characteristic
115	C₂²×C₅₉	C₂×#41	1	1	yes	characteristic
116	C₂×D₅₉	—	2	2	no	—
117	C₂×D₅₉	—	8	1	no	—
118	C₄×C₅₉	—	2	2	no	—
119	C₄×C₅₉	—	2	2	no	—
120	C₄×C₅₉	—	4	1	no	—
121	C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
122	C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
123	C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
124	C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
125	Dic₅₉	—	2	2	no	—
126	#2956	—	59	1	no	maximal, Sylow
127	C₂×C₅₉	—	2	1	no	—
128	C₂×C₅₉	—	2	1	no	—
129	C₂×C₅₉	—	2	1	no	—
130	C₂×C₅₉	—	4	2	no	—
131	C₂×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
132	D₅₉	—	4	2	no	—
133	#575	—	59	1	no	—
134	#568	—	59	1	no	—
135	#568	—	59	1	no	—
136	#444	—	59	4	no	—
137	#443	—	59	4	no	—
138	#430	—	59	1	no	—
139	C₂×#104	—	59	1	no	—
140	C₂×#104	D₅₉	1	1	yes	characteristic, complemented
141	C₄×M₁₆	—	59	1	no	—
142	C₅₉	#2956	1	1	yes	characteristic, complemented, Sylow
143	C₂×#41	—	59	1	no	—
144	C₂×#41	C₂×D₅₉	1	1	yes	characteristic
145	#137	—	59	2	no	—
146	#137	—	59	2	no	—
147	AGL(1,ℤ/8)	—	59	2	no	—
148	AGL(1,ℤ/8)	—	59	2	no	—
149	#135	—	59	4	no	—
150	#131	—	59	4	no	—
151	#131	—	59	4	no	—
152	#131	—	118	2	no	—
153	#131	—	118	2	no	—
154	C₂×M₁₆	—	59	1	no	—
155	C₂×M₁₆	—	59	1	no	—
156	C₂×M₁₆	—	59	1	no	—
157	C₂×M₁₆	—	118	2	no	—
158	C₂×M₁₆	—	118	2	no	—
159	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
160	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
161	#108	—	59	4	no	—
162	#104	—	59	2	no	—
163	#104	—	59	2	no	—
164	#104	C₂×D₅₉	1	2	yes	complemented
165	#104	C₂×D₅₉	1	2	yes	complemented
166	#97	—	59	2	no	—
167	C₄×C₈	—	59	2	no	—
168	#41	—	2	1	no	—
169	#41	—	2	1	no	—
170	#41	—	59	2	no	—
171	#41	—	59	2	no	—
172	#41	—	118	1	no	—
173	#41	—	118	1	no	—
174	#41	C₄×D₅₉	1	2	yes	—
175	#41	C₂²×D₅₉	1	2	yes	—
176	C₂×Q₈	—	59	1	no	—
177	C₂×Q₈	C₄×D₅₉	1	1	yes	characteristic
178	C₂×D₄	—	2	1	no	—
179	C₂×D₄	—	59	1	no	—
180	C₂×D₄	—	118	1	no	—
181	C₂×D₄	C₄×D₅₉	1	1	yes	characteristic
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	118	1	no	—
185	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
186	Q₁₆	—	59	4	no	—
187	SD₁₆	—	59	4	no	—
188	SD₁₆	—	59	4	no	—
189	D₈	—	59	4	no	—
190	M₁₆	—	2	1	no	—
191	M₁₆	—	59	2	no	—
192	M₁₆	—	59	2	no	—
193	M₁₆	—	59	2	no	—
194	M₁₆	—	59	2	no	—
195	M₁₆	—	59	2	no	—
196	M₁₆	—	118	4	no	—
197	M₁₆	—	118	4	no	—
198	M₁₆	—	118	1	no	—
199	M₁₆	C₂²×D₅₉	1	2	yes	—
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	59	2	no	—
202	C₂×C₈	—	59	2	no	—
203	C₂×C₈	—	118	4	no	—
204	C₂×C₈	—	118	4	no	—
205	C₂×C₈	—	118	1	no	—
206	C₄²	—	2	1	no	—
207	C₄²	C₂²×D₅₉	1	1	yes	characteristic
208	C₄²	C₂²×D₅₉	1	1	yes	characteristic
209	C₂³	—	2	1	no	—
210	C₂³	—	118	1	no	—
211	C₂³	#11745687	1	1	yes	characteristic
212	Q₈	—	2	1	no	—
213	Q₈	—	59	2	no	—
214	Q₈	—	118	1	no	—
215	Q₈	C₂×C₄×D₅₉	1	2	yes	—
216	D₄	—	2	2	no	—
217	D₄	—	2	2	no	—
218	D₄	—	2	1	no	—
219	D₄	—	59	2	no	—
220	D₄	—	118	2	no	—
221	D₄	—	118	2	no	—
222	D₄	—	118	1	no	—
223	D₄	C₂×C₄×D₅₉	1	2	yes	—
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	1	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	118	1	no	—
231	C₂×C₄	—	118	2	no	—
232	C₂×C₄	—	118	2	no	—
233	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
234	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
235	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
236	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
237	C₂×C₄	#11745687	1	1	yes	characteristic
238	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
239	C₈	—	2	2	no	—
240	C₈	—	59	4	no	—
241	C₈	—	59	4	no	—
242	C₈	—	118	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	4	1	no	—
245	C₂²	—	8	1	no	—
246	C₂²	—	118	2	no	—
247	C₂²	—	472	1	no	—
248	C₂²	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
249	C₂²	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
250	C₂²	C₂×#11745687	1	1	yes	characteristic
251	C₄	—	2	2	no	—
252	C₄	—	2	2	no	—
253	C₄	—	4	1	no	—
254	C₄	—	118	2	no	—
255	C₄	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
256	C₄	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
257	C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
258	C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
259	C₂	—	2	1	no	—
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	—	4	2	no	—
263	C₂	—	236	2	no	—
264	C₂	C₂×(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₅₉
commutator	C₄×C₅₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₂×#104×C₅₉	C₂
socle	C₂×C₅₉	C₂×C₄×D₄
hypercenter	C₂×#104	D₅₉
hypocenter	C₅₉	#2956
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#104

Complements

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

N	H ≅ G/N	type
2. #16838169	C₂	semidirect
3. #16838168	C₂	semidirect
4. #16837521	C₂	semidirect
5. #16837516	C₂	semidirect
6. #16837450	C₂	semidirect
7. C₂×#14273655	C₂	semidirect
8. C₄×#11745664	C₂	semidirect
9. C₂×#104×C₅₉	C₂	semidirect
11. #14273758	C₄	semidirect
12. #14273757	C₄	semidirect
13. #14273753	C₄	semidirect
14. #14273752	C₄	semidirect
16. #14273735	C₂²	semidirect
18. #14273734	C₂²	semidirect
25. #14273662	C₂²	semidirect
26. #14273655	C₂²	semidirect
27. #14273655	C₂²	semidirect
28. #14273638	C₂²	semidirect
29. C₄×C₅₉⋊C₈	C₂²	semidirect
33. #104×C₅₉	C₂²	semidirect
34. #104×C₅₉	C₂²	semidirect
59. #11745672	C₂×C₄	semidirect
60. #11745671	C₂×C₄	semidirect
61. #11745670	C₂×C₄	semidirect
62. #11745669	C₂×C₄	semidirect
140. C₂×#104	D₅₉	semidirect
142. C₅₉	#2956	semidirect
164. #104	C₂×D₅₉	semidirect
165. #104	C₂×D₅₉	semidirect