Subgroups — #21436992

Order 7552.

374 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16838108	C₂	1	1	yes	maximal, characteristic, complemented
2	#16838067	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837810	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837799	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837705	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837699	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837691	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837691	C₂	1	1	yes	maximal, characteristic, complemented
9	#16837689	C₂	1	1	yes	maximal, characteristic, complemented
10	#16837681	C₂	1	1	yes	maximal, characteristic, complemented
11	#16837530	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837515	C₂	1	1	yes	maximal, characteristic
13	#16837513	C₂	1	1	yes	maximal, characteristic
14	#16837484	C₂	1	1	yes	maximal, characteristic
15	C₅₉×#501	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₅₉	—	2	1	no	—
17	C₂×D₄₇₂	C₂²	1	1	yes	characteristic
18	#14273726	C₂²	1	1	yes	characteristic
19	#14273726	C₂²	1	1	yes	characteristic
20	#14273726	C₂²	1	1	yes	characteristic
21	#14273718	C₂²	1	1	yes	characteristic
22	#14273707	—	2	1	no	—
23	#14273705	—	2	1	no	—
24	#14273705	—	2	1	no	—
25	#14273705	—	2	1	no	—
26	#14273705	—	2	1	no	—
27	#14273703	—	2	1	no	—
28	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
29	C₄×D₂₃₆	C₂²	1	1	yes	characteristic
30	C₄×D₂₃₆	C₂²	1	1	yes	characteristic
31	#14273689	C₂²	1	1	yes	characteristic
32	#14273689	C₂²	1	1	yes	characteristic
33	#14273689	C₂²	1	1	yes	characteristic
34	#14273689	C₂²	1	1	yes	characteristic
35	C₄⋊C₄×D₅₉	—	2	1	no	—
36	#14273660	C₂²	1	1	yes	characteristic
37	#14273659	C₂²	1	1	yes	characteristic
38	#14273658	C₂²	1	1	yes	characteristic
39	#14273658	C₂²	1	1	yes	characteristic
40	#14273658	C₂²	1	1	yes	characteristic
41	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
42	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
43	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
44	#14273649	C₂²	1	1	yes	characteristic
45	#14273649	C₂²	1	1	yes	characteristic
46	#14273647	C₂²	1	1	yes	characteristic
47	#14273647	C₂²	1	1	yes	characteristic
48	#14273647	C₂²	1	1	yes	characteristic
49	#14273647	C₂²	1	1	yes	characteristic
50	#14273637	C₂²	1	1	yes	characteristic
51	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
52	C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
53	#125×C₅₉	C₂²	1	1	yes	characteristic
54	#107×C₅₉	C₂²	1	1	yes	characteristic
55	#107×C₅₉	C₂²	1	1	yes	characteristic
56	#106×C₅₉	C₂²	1	1	yes	characteristic
57	#106×C₅₉	C₂²	1	1	yes	characteristic
58	#97×C₅₉	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₅₉	—	2	1	no	—
60	Q₈×D₅₉	—	4	4	no	—
61	C₂×D₂₃₆	C₂³	1	1	yes	characteristic
62	C₂×D₂₃₆	C₂³	1	1	yes	characteristic
63	C₂×C₄×D₅₉	—	2	1	no	—
64	C₂×C₄×D₅₉	—	2	1	no	—
65	C₂×C₄×D₅₉	—	2	1	no	—
66	C₂×C₄×D₅₉	—	2	1	no	—
67	C₂×C₄×D₅₉	—	2	1	no	—
68	C₂×C₄×D₅₉	—	2	1	no	—
69	C₂×Dic₁₁₈	—	2	1	no	—
70	C₂×Dic₁₁₈	—	2	1	no	—
71	C₂×Dic₁₁₈	—	2	1	no	—
72	C₂×Dic₁₁₈	D₄	1	1	yes	characteristic
73	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₅₉	—	2	1	no	—
76	C₄⋊C₄×C₅₉	—	2	1	no	—
77	C₄⋊C₄×C₅₉	—	2	1	no	—
78	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
79	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
80	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
81	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
82	C₄²×C₅₉	C₂³	1	1	yes	characteristic
83	(C₂×D₅₉)⋊C₄	—	2	1	no	—
84	(C₂×D₅₉)⋊C₄	—	2	1	no	—
85	(C₂×D₅₉)⋊C₄	—	2	1	no	—
86	(C₂×D₅₉)⋊C₄	—	2	1	no	—
87	(C₂×D₅₉)⋊C₄	—	2	1	no	—
88	(C₂×D₅₉)⋊C₄	—	2	1	no	—
89	C₄⋊Dic₅₉	—	2	1	no	—
90	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
91	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
92	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
98	Dic₅₉⋊C₄	D₄	1	1	yes	characteristic
99	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
100	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
101	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
102	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
103	D₄₇₂	Q₈	1	4	yes	complemented
104	C₂²×D₅₉	—	2	1	no	—
105	C₂²×D₅₉	—	2	1	no	—
106	Q₈×C₅₉	—	4	2	no	—
107	C₂×C₄×C₅₉	—	2	1	no	—
108	C₂×C₄×C₅₉	—	2	1	no	—
109	C₂×C₄×C₅₉	—	2	1	no	—
110	C₂×C₄×C₅₉	—	2	1	no	—
111	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
114	C₂×Dic₅₉	—	2	1	no	—
115	C₂×Dic₅₉	—	2	1	no	—
116	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
117	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
118	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
119	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
120	D₂₃₆	—	2	1	no	—
121	D₂₃₆	—	2	1	no	—
122	D₂₃₆	C₂×Q₈	1	2	yes	—
123	D₂₃₆	C₂×Q₈	1	2	yes	complemented
124	C₄×D₅₉	—	4	2	no	—
125	C₄×D₅₉	—	4	2	no	—
126	C₄×D₅₉	—	4	2	no	—
127	C₄×D₅₉	—	4	2	no	—
128	C₄×D₅₉	—	8	1	no	—
129	C₄×D₅₉	—	8	1	no	—
130	Dic₁₁₈	—	2	2	no	—
131	Dic₁₁₈	—	4	2	no	—
132	Dic₁₁₈	—	4	2	no	—
133	Dic₁₁₈	—	8	1	no	—
134	C₈×C₅₉	—	2	1	no	—
135	C₈×C₅₉	C₂×Q₈	1	2	yes	complemented
136	C₅₉⋊C₈	—	2	1	no	—
137	C₅₉⋊C₈	—	2	1	no	—
138	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
139	C₂×D₅₉	—	2	2	no	—
140	C₂×D₅₉	—	2	2	no	—
141	C₂×D₅₉	—	4	1	no	—
142	C₂×D₅₉	—	4	1	no	—
143	C₂×D₅₉	—	4	1	no	—
144	C₂×D₅₉	—	4	1	no	—
145	C₄×C₅₉	—	2	1	no	—
146	C₄×C₅₉	—	2	1	no	—
147	C₄×C₅₉	—	4	1	no	—
148	C₄×C₅₉	—	4	1	no	—
149	C₄×C₅₉	—	4	1	no	—
150	C₄×C₅₉	—	4	1	no	—
151	C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
152	C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
153	Dic₅₉	—	2	1	no	—
154	Dic₅₉	—	2	1	no	—
155	Dic₅₉	—	2	1	no	—
156	Dic₅₉	—	4	1	no	—
157	Dic₅₉	—	4	1	no	—
158	Dic₅₉	AGL(1,ℤ/8)	1	2	yes	—
159	#3384	—	59	1	no	maximal, Sylow
160	C₂×C₅₉	D₄×Q₈	1	1	yes	characteristic
161	C₂×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
162	C₂×C₅₉	#577	1	1	yes	characteristic
163	D₅₉	—	4	2	no	—
164	D₅₉	—	4	2	no	—
165	#554	—	59	1	no	—
166	D₄×Q₈	—	59	1	no	—
167	#501	—	59	1	no	—
168	#501	D₅₉	1	1	yes	characteristic, complemented
169	#498	—	59	1	no	—
170	#478	—	59	1	no	—
171	#478	—	59	1	no	—
172	#476	—	59	1	no	—
173	#476	—	59	1	no	—
174	#476	—	59	1	no	—
175	#474	—	59	1	no	—
176	#446	—	59	1	no	—
177	#442	—	59	1	no	—
178	#442	—	59	1	no	—
179	C₄×D₈	—	59	1	no	—
180	C₅₉	#3384	1	1	yes	characteristic, complemented, Sylow
181	C₂²×Q₈	—	118	1	no	—
182	C₂×D₈	—	59	1	no	—
183	C₄⋊Q₈	—	59	1	no	—
184	C₄⋊Q₈	—	59	1	no	—
185	C₄⋊Q₈	—	59	1	no	—
186	C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
187	#125	—	59	1	no	—
188	#125	C₂×D₅₉	1	1	yes	characteristic
189	C₂²⋊Q₈	—	118	1	no	—
190	C₂²⋊Q₈	—	118	1	no	—
191	C₂²⋊Q₈	—	118	1	no	—
192	C₂²⋊Q₈	—	118	1	no	—
193	C₂²⋊Q₈	—	118	1	no	—
194	C₂²⋊Q₈	—	118	1	no	—
195	C₄×Q₈	—	59	1	no	—
196	C₄×D₄	—	59	1	no	—
197	C₄×D₄	—	59	1	no	—
198	C₄×D₄	—	59	1	no	—
199	C₄×D₄	—	59	1	no	—
200	C₄×D₄	—	59	1	no	—
201	C₄×D₄	—	59	1	no	—
202	C₂×C₄⋊C₄	—	118	1	no	—
203	#107	—	59	1	no	—
204	#107	C₂×D₅₉	1	1	yes	characteristic
205	#107	C₂×D₅₉	1	1	yes	characteristic
206	#106	—	59	1	no	—
207	#106	—	59	1	no	—
208	#106	—	59	1	no	—
209	#106	—	59	1	no	—
210	#106	C₂×D₅₉	1	1	yes	characteristic
211	#106	C₂×D₅₉	1	1	yes	characteristic
212	C₄⋊C₈	—	59	1	no	—
213	C₄⋊C₈	—	59	1	no	—
214	C₄⋊C₈	—	59	1	no	—
215	#102	—	59	1	no	—
216	#102	—	59	1	no	—
217	#102	—	59	1	no	—
218	#102	—	59	1	no	—
219	#102	—	59	1	no	—
220	#102	—	59	1	no	—
221	#97	—	59	1	no	—
222	#97	C₂×D₅₉	1	1	yes	characteristic
223	C₄×C₈	—	59	1	no	—
224	C₂×Q₈	—	2	1	no	—
225	C₂×Q₈	—	59	1	no	—
226	C₂×Q₈	—	118	1	no	—
227	C₂×Q₈	—	118	1	no	—
228	C₂×Q₈	—	118	1	no	—
229	C₂×Q₈	—	236	4	no	—
230	C₂×D₄	—	59	1	no	—
231	C₂×D₄	—	59	1	no	—
232	C₂²×C₄	—	118	1	no	—
233	C₂²×C₄	—	118	1	no	—
234	C₂²×C₄	—	118	1	no	—
235	C₂²×C₄	—	118	1	no	—
236	C₂²×C₄	—	118	1	no	—
237	C₂²×C₄	—	118	1	no	—
238	D₈	—	59	4	no	—
239	C₂×C₈	—	59	1	no	—
240	C₂×C₈	—	59	1	no	—
241	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
242	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
243	C₄⋊C₄	—	2	1	no	—
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	59	1	no	—
247	C₄⋊C₄	—	59	1	no	—
248	C₄⋊C₄	—	59	1	no	—
249	C₄⋊C₄	—	59	1	no	—
250	C₄⋊C₄	—	59	1	no	—
251	C₄⋊C₄	—	118	1	no	—
252	C₄⋊C₄	—	118	1	no	—
253	C₄⋊C₄	—	118	1	no	—
254	C₄⋊C₄	—	118	1	no	—
255	C₄⋊C₄	—	118	1	no	—
256	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
257	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
258	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
259	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
260	C₂²⋊C₄	—	118	1	no	—
261	C₂²⋊C₄	—	118	1	no	—
262	C₂²⋊C₄	—	118	1	no	—
263	C₂²⋊C₄	—	118	1	no	—
264	C₂²⋊C₄	—	118	1	no	—
265	C₂²⋊C₄	—	118	1	no	—
266	C₄²	—	59	1	no	—
267	C₄²	—	59	1	no	—
268	C₄²	C₂²×D₅₉	1	1	yes	characteristic
269	C₂³	—	118	1	no	—
270	C₂³	—	118	1	no	—
271	Q₈	—	4	2	no	—
272	Q₈	—	118	2	no	—
273	Q₈	—	472	1	no	—
274	Q₈	—	236	2	no	—
275	Q₈	—	236	2	no	—
276	D₄	—	59	2	no	—
277	D₄	—	59	2	no	—
278	D₄	—	118	1	no	—
279	D₄	—	118	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	59	1	no	—
285	C₂×C₄	—	59	1	no	—
286	C₂×C₄	—	59	1	no	—
287	C₂×C₄	—	59	1	no	—
288	C₂×C₄	—	118	1	no	—
289	C₂×C₄	—	118	1	no	—
290	C₂×C₄	—	472	1	no	—
291	C₂×C₄	—	472	1	no	—
292	C₂×C₄	—	236	2	no	—
293	C₂×C₄	—	236	2	no	—
294	C₂×C₄	—	236	2	no	—
295	C₂×C₄	—	236	2	no	—
296	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
297	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
298	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	118	1	no	—
301	C₈	—	118	1	no	—
302	C₈	Q₈×D₅₉	1	2	yes	complemented
303	C₂²	—	118	2	no	—
304	C₂²	—	118	2	no	—
305	C₂²	—	236	1	no	—
306	C₂²	—	236	1	no	—
307	C₂²	—	236	1	no	—
308	C₂²	—	236	1	no	—
309	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	4	1	no	—
316	C₄	—	59	2	no	—
317	C₄	—	118	1	no	—
318	C₄	—	118	1	no	—
319	C₄	—	118	1	no	—
320	C₄	—	236	1	no	—
321	C₄	—	236	1	no	—
322	C₄	C₂×Q₈×D₅₉	1	1	yes	characteristic
323	C₄	#14273775	1	1	yes	characteristic
324	C₂	—	236	2	no	—
325	C₂	—	236	2	no	—
326	C₂	#16838067	1	1	yes	characteristic
327	C₂	AGL(1,ℤ/8)×D₅₉	1	1	yes	characteristic
328	C₂	#16838211	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#501	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#501	D₅₉
hypocenter	C₅₉	#3384
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #501

Complements

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

N	H ≅ G/N	type
1. #16838108	C₂	semidirect
2. #16838067	C₂	semidirect
3. #16837810	C₂	semidirect
4. #16837799	C₂	semidirect
5. #16837705	C₂	semidirect
6. #16837699	C₂	semidirect
7. #16837691	C₂	semidirect
8. #16837691	C₂	semidirect
9. #16837689	C₂	semidirect
10. #16837681	C₂	semidirect
11. #16837530	C₂	semidirect
15. C₅₉×#501	C₂	semidirect
103. D₄₇₂	Q₈	semidirect
123. D₂₃₆	C₂×Q₈	semidirect
135. C₈×C₅₉	C₂×Q₈	semidirect
168. #501	D₅₉	semidirect
180. C₅₉	#3384	semidirect
302. C₈	Q₈×D₅₉	semidirect