Subgroups — #21437043

Order 7552.

334 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16838130	C₂	1	1	yes	maximal, characteristic, complemented
2	#16838112	C₂	1	1	yes	maximal, characteristic
3	#16837811	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837811	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837804	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837710	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837710	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837703	C₂	1	1	yes	maximal, characteristic
9	#16837703	C₂	1	1	yes	maximal, characteristic
10	#16837693	C₂	1	1	yes	maximal, characteristic
11	#16837688	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837504	C₂	1	1	yes	maximal, characteristic
13	#16837504	C₂	1	1	yes	maximal, characteristic
14	C₄×#11745660	C₂	1	1	yes	maximal, characteristic
15	C₅₉×#446	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11745660	C₂²	1	1	yes	characteristic
17	#14273727	C₂²	1	1	yes	characteristic
18	#14273727	C₂²	1	1	yes	characteristic
19	#14273725	—	2	1	no	—
20	#14273719	C₂²	1	1	yes	characteristic
21	#14273719	C₂²	1	1	yes	characteristic
22	#14273719	C₂²	1	1	yes	characteristic
23	#14273719	C₂²	1	1	yes	characteristic
24	#14273717	—	2	1	no	—
25	#14273717	—	2	1	no	—
26	#14273705	—	2	1	no	—
27	#14273705	—	2	1	no	—
28	#14273703	—	2	1	no	—
29	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
30	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
31	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
32	#14273692	—	2	1	no	—
33	C₄×D₂₃₆	C₂²	1	1	yes	characteristic
34	C₄×D₂₃₆	C₂²	1	1	yes	characteristic
35	C₄×D₂₃₆	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₅₉	—	2	1	no	—
37	#14273661	C₂²	1	1	yes	characteristic
38	#14273661	C₂²	1	1	yes	characteristic
39	#14273661	C₂²	1	1	yes	characteristic
40	#14273661	C₂²	1	1	yes	characteristic
41	#14273659	C₂²	1	1	yes	characteristic
42	#14273659	C₂²	1	1	yes	characteristic
43	#14273659	C₂²	1	1	yes	characteristic
44	#14273659	C₂²	1	1	yes	characteristic
45	#14273659	C₂²	1	1	yes	characteristic
46	#14273652	C₂²	1	1	yes	characteristic
47	#14273652	C₂²	1	1	yes	characteristic
48	#14273652	C₂²	1	1	yes	characteristic
49	#14273649	C₂²	1	1	yes	characteristic
50	#14273649	C₂²	1	1	yes	characteristic
51	#14273649	C₂²	1	1	yes	characteristic
52	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
53	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
54	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
55	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
56	#97×C₅₉	C₂²	1	1	yes	characteristic
57	#97×C₅₉	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₅₉	D₄	1	1	yes	characteristic
60	C₂×D₂₃₆	C₂³	1	1	yes	characteristic
61	C₂×C₄×D₅₉	—	2	1	no	—
62	C₂×C₄×D₅₉	—	2	1	no	—
63	C₂×C₄×D₅₉	—	2	1	no	—
64	C₂×Dic₁₁₈	—	2	1	no	—
65	C₂×Dic₁₁₈	—	2	1	no	—
66	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
67	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
72	C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
73	C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
74	C₄²×C₅₉	C₂³	1	1	yes	characteristic
75	C₄²×C₅₉	C₂³	1	1	yes	characteristic
76	C₄²×C₅₉	C₂³	1	1	yes	characteristic
77	(C₂×D₅₉)⋊C₄	—	2	1	no	—
78	(C₂×D₅₉)⋊C₄	—	2	1	no	—
79	(C₂×D₅₉)⋊C₄	—	2	1	no	—
80	C₄⋊Dic₅₉	—	2	1	no	—
81	C₄⋊Dic₅₉	—	2	1	no	—
82	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
83	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
84	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
85	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
86	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
87	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
88	Dic₅₉⋊C₄	—	2	1	no	—
89	Dic₅₉⋊C₄	—	2	1	no	—
90	Dic₅₉⋊C₄	—	2	1	no	—
91	Dic₅₉⋊C₄	—	2	1	no	—
92	Dic₅₉⋊C₄	—	2	1	no	—
93	Dic₅₉⋊C₄	—	2	1	no	—
94	Dic₅₉⋊C₄	—	2	1	no	—
95	Dic₅₉⋊C₄	—	2	1	no	—
96	Dic₅₉⋊C₄	—	2	1	no	—
97	Dic₅₉⋊C₄	—	2	1	no	—
98	Dic₅₉⋊C₄	—	2	1	no	—
99	C₄×Dic₅₉	—	2	1	no	—
100	C₄×Dic₅₉	—	2	1	no	—
101	C₄×Dic₅₉	—	2	1	no	—
102	#11745660	Q₈	1	4	yes	complemented
103	C₂²×D₅₉	—	2	1	no	—
104	Q₈×C₅₉	—	4	1	no	—
105	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
112	C₂×Dic₅₉	—	2	1	no	—
113	C₂×Dic₅₉	—	2	1	no	—
114	C₂×Dic₅₉	—	2	1	no	—
115	C₂×Dic₅₉	—	2	1	no	—
116	C₂×Dic₅₉	—	2	1	no	—
117	C₂×Dic₅₉	—	2	1	no	—
118	C₂×Dic₅₉	—	2	1	no	—
119	D₂₃₆	—	2	1	no	—
120	D₂₃₆	C₂×Q₈	1	2	yes	—
121	C₄×D₅₉	—	4	2	no	—
122	C₄×D₅₉	—	8	1	no	—
123	C₄×D₅₉	—	8	1	no	—
124	Dic₁₁₈	—	2	1	no	—
125	Dic₁₁₈	—	4	2	no	—
126	Dic₁₁₈	—	8	1	no	—
127	Dic₁₁₈	C₂×Q₈	1	2	yes	—
128	C₈×C₅₉	—	2	1	no	—
129	C₈×C₅₉	—	2	1	no	—
130	C₈×C₅₉	—	2	1	no	—
131	C₈×C₅₉	C₂×Q₈	1	2	yes	—
132	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
133	C₂×D₅₉	—	2	2	no	—
134	C₂×D₅₉	—	4	1	no	—
135	C₂×D₅₉	—	4	1	no	—
136	C₄×C₅₉	—	2	1	no	—
137	C₄×C₅₉	—	2	1	no	—
138	C₄×C₅₉	—	2	1	no	—
139	C₄×C₅₉	—	2	1	no	—
140	C₄×C₅₉	—	2	1	no	—
141	C₄×C₅₉	—	2	1	no	—
142	C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
143	C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
144	Dic₅₉	—	2	2	no	—
145	Dic₅₉	—	4	1	no	—
146	Dic₅₉	—	4	1	no	—
147	Dic₅₉	—	4	1	no	—
148	Dic₅₉	—	4	1	no	—
149	Dic₅₉	—	4	1	no	—
150	Dic₅₉	—	4	1	no	—
151	#3388	—	59	1	no	maximal, Sylow
152	C₂×C₅₉	D₄×Q₈	1	1	yes	characteristic
153	C₂×C₅₉	#575	1	1	yes	characteristic
154	C₂×C₅₉	#577	1	1	yes	characteristic
155	D₅₉	—	4	2	no	—
156	#557	—	59	1	no	—
157	#554	—	59	1	no	—
158	#501	—	59	1	no	—
159	#501	—	59	1	no	—
160	#499	—	59	1	no	—
161	#479	—	59	1	no	—
162	#479	—	59	1	no	—
163	#478	—	59	1	no	—
164	#478	—	59	1	no	—
165	#476	—	59	1	no	—
166	#475	—	59	1	no	—
167	#446	D₅₉	1	1	yes	characteristic, complemented
168	#440	—	59	1	no	—
169	#440	—	59	1	no	—
170	C₄×SD₁₆	—	59	1	no	—
171	C₅₉	#3388	1	1	yes	characteristic, complemented, Sylow
172	C₂×SD₁₆	—	59	1	no	—
173	C₄⋊Q₈	—	59	1	no	—
174	C₄⋊Q₈	—	59	1	no	—
175	C₄⋊Q₈	—	118	1	no	—
176	#125	—	59	1	no	—
177	#125	—	59	1	no	—
178	#125	—	59	1	no	—
179	#125	—	59	1	no	—
180	#125	—	118	1	no	—
181	#125	—	118	1	no	—
182	C₂²⋊Q₈	—	118	1	no	—
183	C₂²⋊Q₈	—	118	1	no	—
184	C₂²⋊Q₈	—	118	1	no	—
185	C₄×Q₈	—	59	1	no	—
186	C₄×Q₈	—	59	1	no	—
187	C₄×Q₈	—	59	1	no	—
188	C₄×Q₈	—	118	1	no	—
189	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
190	C₄×D₄	—	59	1	no	—
191	C₄×D₄	—	59	1	no	—
192	C₄×D₄	—	59	1	no	—
193	C₂×C₄⋊C₄	—	118	1	no	—
194	#107	—	59	1	no	—
195	#107	—	59	1	no	—
196	#107	—	59	1	no	—
197	#107	—	59	1	no	—
198	#106	—	59	1	no	—
199	#106	—	59	1	no	—
200	#106	—	59	1	no	—
201	#106	—	59	1	no	—
202	#106	—	59	1	no	—
203	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
204	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
205	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
206	#103	—	59	1	no	—
207	#103	—	59	1	no	—
208	#103	—	59	1	no	—
209	#102	—	59	1	no	—
210	#102	—	59	1	no	—
211	#102	—	59	1	no	—
212	#97	C₂×D₅₉	1	1	yes	characteristic
213	#97	C₂×D₅₉	1	1	yes	characteristic
214	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
215	C₂×Q₈	—	59	1	no	—
216	C₂×Q₈	—	118	1	no	—
217	C₂×Q₈	—	118	1	no	—
218	C₂×Q₈	D₂₃₆	1	1	yes	characteristic
219	C₂×D₄	—	59	1	no	—
220	C₂²×C₄	—	118	1	no	—
221	C₂²×C₄	—	118	1	no	—
222	C₂²×C₄	—	118	1	no	—
223	SD₁₆	—	59	4	no	—
224	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
225	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
226	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
227	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
228	C₄⋊C₄	—	59	1	no	—
229	C₄⋊C₄	—	59	1	no	—
230	C₄⋊C₄	—	59	1	no	—
231	C₄⋊C₄	—	59	1	no	—
232	C₄⋊C₄	—	59	1	no	—
233	C₄⋊C₄	—	59	1	no	—
234	C₄⋊C₄	—	118	1	no	—
235	C₄⋊C₄	—	118	1	no	—
236	C₄⋊C₄	—	118	1	no	—
237	C₄⋊C₄	—	118	1	no	—
238	C₄⋊C₄	—	118	1	no	—
239	C₄⋊C₄	—	118	1	no	—
240	C₄⋊C₄	—	118	1	no	—
241	C₄⋊C₄	—	118	1	no	—
242	C₄⋊C₄	—	118	1	no	—
243	C₄⋊C₄	—	118	1	no	—
244	C₄⋊C₄	—	118	1	no	—
245	C₄⋊C₄	—	118	1	no	—
246	C₄⋊C₄	—	118	1	no	—
247	C₄⋊C₄	D₂₃₆	1	1	yes	characteristic
248	C₄⋊C₄	D₂₃₆	1	1	yes	characteristic
249	C₄⋊C₄	D₂₃₆	1	1	yes	characteristic
250	C₂²⋊C₄	—	118	1	no	—
251	C₂²⋊C₄	—	118	1	no	—
252	C₂²⋊C₄	—	118	1	no	—
253	C₄²	—	118	1	no	—
254	C₄²	—	118	1	no	—
255	C₄²	—	118	1	no	—
256	C₄²	C₂²×D₅₉	1	1	yes	characteristic
257	C₄²	C₂²×D₅₉	1	1	yes	characteristic
258	C₄²	C₂²×D₅₉	1	1	yes	characteristic
259	C₂³	—	118	1	no	—
260	Q₈	—	4	1	no	—
261	Q₈	—	59	2	no	—
262	Q₈	—	118	1	no	—
263	Q₈	—	472	1	no	—
264	Q₈	—	236	2	no	—
265	D₄	—	59	2	no	—
266	D₄	—	118	1	no	—
267	C₂×C₄	—	118	1	no	—
268	C₂×C₄	—	118	1	no	—
269	C₂×C₄	—	118	1	no	—
270	C₂×C₄	—	118	1	no	—
271	C₂×C₄	—	118	1	no	—
272	C₂×C₄	—	118	1	no	—
273	C₂×C₄	—	118	1	no	—
274	C₂×C₄	—	472	1	no	—
275	C₂×C₄	—	472	1	no	—
276	C₂×C₄	—	236	2	no	—
277	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
278	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
279	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
280	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
281	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
282	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
283	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	—	2	1	no	—
287	C₈	Q₈×D₅₉	1	2	yes	—
288	C₂²	—	118	2	no	—
289	C₂²	—	236	1	no	—
290	C₂²	—	236	1	no	—
291	C₂²	C₂²×D₂₃₆	1	1	yes	characteristic
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	118	2	no	—
299	C₄	—	236	1	no	—
300	C₄	—	236	1	no	—
301	C₄	—	236	1	no	—
302	C₄	—	236	1	no	—
303	C₄	—	236	1	no	—
304	C₄	—	236	1	no	—
305	C₄	C₂×Q₈×D₅₉	1	1	yes	characteristic
306	C₄	#14273774	1	1	yes	characteristic
307	C₂	—	236	2	no	—
308	C₂	Q₈×D₂₃₆	1	1	yes	characteristic
309	C₂	#16838202	1	1	yes	characteristic
310	C₂	#16838214	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#446	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#446	D₅₉
hypocenter	C₅₉	#3388
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #446

Complements

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

N	H ≅ G/N	type
1. #16838130	C₂	semidirect
3. #16837811	C₂	semidirect
4. #16837811	C₂	semidirect
5. #16837804	C₂	semidirect
6. #16837710	C₂	semidirect
7. #16837710	C₂	semidirect
11. #16837688	C₂	semidirect
15. C₅₉×#446	C₂	semidirect
102. #11745660	Q₈	semidirect
167. #446	D₅₉	semidirect
171. C₅₉	#3388	semidirect