Subgroups — #21434154

Order 7552.

434 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16837949	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#11745689	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837728	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837724	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837695	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837691	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837600	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837596	C₂	1	1	yes	maximal, characteristic, complemented
9	#16837550	C₂	1	1	yes	maximal, characteristic, complemented
10	#16837546	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#11745671	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×D₄₇₂	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×#11745669	C₂	1	1	yes	maximal, characteristic, complemented
14	#16837453	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×SD₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11745689	C₂²	1	1	yes	characteristic
17	C₂×#11745685	—	2	1	no	—
18	C₂×#11745671	C₂²	1	1	yes	characteristic
19	C₂×D₄₇₂	C₂²	1	1	yes	characteristic
20	C₂×#11745669	C₂²	1	1	yes	characteristic
21	#14273726	C₂²	1	1	yes	characteristic
22	#14273723	C₂²	1	1	yes	characteristic
23	#14273713	—	2	1	no	—
24	#14273702	C₂²	1	1	yes	characteristic
25	#14273700	—	2	1	no	—
26	#14273698	C₂²	1	1	yes	characteristic
27	Q₈×Dic₅₉	C₂²	1	1	yes	characteristic
28	C₄×D₂₃₆	—	2	1	no	—
29	C₄×D₂₃₆	C₂²	1	1	yes	characteristic
30	C₄×D₂₃₆	C₂²	1	1	yes	characteristic
31	#14273689	—	2	1	no	—
32	#14273689	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊D₅₉	—	2	1	no	—
34	#14273685	—	2	1	no	—
35	#14273685	C₂²	1	1	yes	characteristic
36	C₄²×D₅₉	—	2	1	no	—
37	C₄²×D₅₉	C₂²	1	1	yes	characteristic
38	#14273661	C₂²	1	1	yes	characteristic
39	#14273660	C₂²	1	1	yes	characteristic
40	#14273658	C₂²	1	1	yes	characteristic
41	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
42	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
43	#14273651	C₂²	1	1	yes	characteristic
44	#14273649	C₂²	1	1	yes	characteristic
45	#14273649	C₂²	1	1	yes	characteristic
46	#14273648	C₂²	1	1	yes	characteristic
47	#14273647	C₂²	1	1	yes	characteristic
48	#14273647	C₂²	1	1	yes	characteristic
49	(C₂×D₅₉)⋊C₈	C₂²	1	1	yes	characteristic
50	(C₂×D₅₉)⋊C₈	C₂²	1	1	yes	characteristic
51	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₅₉	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₅₉	C₂²	1	1	yes	characteristic
55	#106×C₅₉	C₂²	1	1	yes	characteristic
56	#103×C₅₉	C₂²	1	1	yes	characteristic
57	#102×C₅₉	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₅₉	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₅₉	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₅₉	—	2	1	no	—
62	C₂×C₂²⋊D₅₉	—	2	1	no	—
63	C₂×C₂²⋊D₅₉	—	2	1	no	—
64	C₂×C₂²⋊D₅₉	—	2	1	no	—
65	#11745689	—	2	2	no	—
66	#11745689	—	2	2	no	—
67	#11745685	—	4	4	no	—
68	C₂×D₂₃₆	—	2	1	no	—
69	C₂×D₂₃₆	—	2	1	no	—
70	C₂×D₂₃₆	C₂³	1	1	yes	characteristic
71	C₂×D₂₃₆	C₂³	1	1	yes	characteristic
72	C₂×C₄×D₅₉	—	2	1	no	—
73	C₂×C₄×D₅₉	—	2	1	no	—
74	C₂×C₄×D₅₉	—	2	1	no	—
75	C₂×C₄×D₅₉	—	2	1	no	—
76	C₂×C₄×D₅₉	D₄	1	1	yes	characteristic
77	C₂×C₄×D₅₉	D₄	1	1	yes	characteristic
78	C₂×C₄×D₅₉	C₂³	1	1	yes	characteristic
79	C₂×Dic₁₁₈	—	2	1	no	—
80	SD₁₆×C₅₉	—	2	2	no	—
81	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
82	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
83	C₄⋊C₄×C₅₉	—	2	1	no	—
84	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
85	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
86	C₂²⋊C₄×C₅₉	—	2	1	no	—
87	C₄²×C₅₉	—	2	1	no	—
88	C₄²×C₅₉	C₂³	1	1	yes	characteristic
89	C₂²⋊Dic₅₉	—	2	1	no	—
90	#11745671	—	2	2	no	—
91	#11745669	—	2	2	no	—
92	(C₂×D₅₉)⋊C₄	—	2	1	no	—
93	(C₂×D₅₉)⋊C₄	—	2	1	no	—
94	(C₂×D₅₉)⋊C₄	—	2	1	no	—
95	(C₂×D₅₉)⋊C₄	—	2	1	no	—
96	(C₂×D₅₉)⋊C₄	—	2	1	no	—
97	C₄⋊Dic₅₉	—	2	1	no	—
98	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
99	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
100	Dic₅₉⋊C₄	—	2	1	no	—
101	C₄×Dic₅₉	—	2	1	no	—
102	C₄×Dic₅₉	—	2	1	no	—
103	C₄×Dic₅₉	D₄	1	1	yes	characteristic
104	C₄×Dic₅₉	D₄	1	1	yes	characteristic
105	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
106	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
107	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
108	D₄₇₂	—	2	2	no	—
109	C₂³×C₅₉	—	2	1	no	—
110	C₂²×D₅₉	—	2	1	no	—
111	C₂²×D₅₉	—	2	1	no	—
112	C₂²×D₅₉	C₂×D₄	1	1	yes	characteristic
113	Q₈×C₅₉	—	2	1	no	—
114	Q₈×C₅₉	#41	1	2	yes	complemented
115	D₄×C₅₉	—	2	1	no	—
116	D₄×C₅₉	—	2	1	no	—
117	C₂×C₄×C₅₉	—	2	1	no	—
118	C₂×C₄×C₅₉	—	2	1	no	—
119	C₂×C₄×C₅₉	—	2	1	no	—
120	C₂×C₄×C₅₉	—	4	2	no	—
121	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
124	C₂²⋊D₅₉	—	4	2	no	—
125	C₂²⋊D₅₉	—	4	2	no	—
126	C₂²⋊D₅₉	—	4	2	no	—
127	C₂×Dic₅₉	—	2	1	no	—
128	C₂×Dic₅₉	—	2	1	no	—
129	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
131	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
132	D₂₃₆	—	2	4	no	—
133	D₂₃₆	—	2	1	no	—
134	D₂₃₆	—	2	1	no	—
135	D₂₃₆	—	2	1	no	—
136	D₂₃₆	—	4	2	no	—
137	D₂₃₆	#41	1	2	yes	complemented
138	C₄×D₅₉	—	2	4	no	—
139	C₄×D₅₉	—	2	2	no	—
140	C₄×D₅₉	—	2	1	no	—
141	C₄×D₅₉	—	2	1	no	—
142	C₄×D₅₉	—	2	2	no	—
143	C₄×D₅₉	—	4	2	no	—
144	C₄×D₅₉	—	4	2	no	—
145	C₄×D₅₉	—	4	2	no	—
146	Dic₁₁₈	—	4	2	no	—
147	C₈×C₅₉	—	2	1	no	—
148	C₈×C₅₉	—	2	1	no	—
149	C₅₉⋊C₈	—	2	1	no	—
150	C₅₉⋊C₈	—	2	1	no	—
151	C₂²×C₅₉	—	4	1	no	—
152	C₂²×C₅₉	—	4	1	no	—
153	C₂²×C₅₉	—	4	1	no	—
154	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
155	C₂×D₅₉	—	2	1	no	—
156	C₂×D₅₉	—	2	2	no	—
157	C₂×D₅₉	—	2	1	no	—
158	C₂×D₅₉	—	2	1	no	—
159	C₂×D₅₉	—	4	1	no	—
160	C₂×D₅₉	—	4	1	no	—
161	C₂×D₅₉	—	4	1	no	—
162	C₂×D₅₉	—	4	1	no	—
163	C₂×D₅₉	—	4	1	no	—
164	C₄×C₅₉	—	2	2	no	—
165	C₄×C₅₉	—	2	1	no	—
166	C₄×C₅₉	—	4	1	no	—
167	C₄×C₅₉	—	4	1	no	—
168	C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
169	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
170	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
171	Dic₅₉	—	2	1	no	—
172	Dic₅₉	—	2	1	no	—
173	Dic₅₉	—	4	1	no	—
174	Dic₅₉	—	4	1	no	—
175	Dic₅₉	#135	1	2	yes	—
176	#3106	—	59	1	no	maximal, Sylow
177	C₂×C₅₉	—	8	1	no	—
178	C₂×C₅₉	#525	1	1	yes	characteristic
179	C₂×C₅₉	C₂×#135	1	1	yes	characteristic
180	C₂×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
181	D₅₉	—	4	1	no	—
182	D₅₉	—	4	2	no	—
183	D₅₉	—	8	1	no	—
184	#532	—	59	1	no	—
185	C₄×#41	—	59	1	no	—
186	#482	—	59	1	no	—
187	#482	—	59	1	no	—
188	#477	—	59	1	no	—
189	#476	—	59	1	no	—
190	#460	—	59	1	no	—
191	#459	—	59	1	no	—
192	#449	—	59	1	no	—
193	#449	—	59	1	no	—
194	C₄×SD₁₆	—	59	1	no	—
195	C₄×SD₁₆	D₅₉	1	1	yes	characteristic, complemented
196	C₄×D₈	—	59	1	no	—
197	C₄×D₈	—	59	1	no	—
198	#431	—	59	1	no	—
199	C₅₉	#3106	1	1	yes	characteristic, complemented, Sylow
200	C₂×#41	—	59	1	no	—
201	C₂×#41	—	118	1	no	—
202	C₂×SD₁₆	—	59	1	no	—
203	C₂×SD₁₆	C₂×D₅₉	1	1	yes	characteristic
204	C₂×D₈	—	59	1	no	—
205	C₂×D₈	—	59	1	no	—
206	C₄⋊Q₈	—	59	1	no	—
207	#127	—	59	1	no	—
208	#124	—	118	1	no	—
209	#121	—	59	1	no	—
210	#121	—	59	1	no	—
211	#121	—	118	1	no	—
212	C₄×Q₈	—	59	1	no	—
213	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
214	C₄×D₄	—	59	1	no	—
215	C₄×D₄	—	59	1	no	—
216	C₄×D₄	—	59	1	no	—
217	C₄×D₄	—	118	1	no	—
218	C₄×D₄	—	118	1	no	—
219	C₄×D₄	—	118	1	no	—
220	C₄×D₄	C₂×D₅₉	1	1	yes	characteristic
221	#117	—	59	1	no	—
222	#117	—	118	1	no	—
223	C₂×C₄²	—	59	1	no	—
224	C₂×C₄²	—	118	1	no	—
225	#107	—	59	1	no	—
226	#107	—	59	1	no	—
227	#106	—	59	1	no	—
228	#106	C₂×D₅₉	1	1	yes	characteristic
229	C₄⋊C₈	—	59	1	no	—
230	C₄⋊C₈	—	59	1	no	—
231	#103	—	59	1	no	—
232	#103	C₂×D₅₉	1	1	yes	characteristic
233	#102	—	59	1	no	—
234	#102	—	59	1	no	—
235	#102	—	59	1	no	—
236	#102	—	59	1	no	—
237	#102	—	59	1	no	—
238	#102	C₂×D₅₉	1	1	yes	characteristic
239	C₂²⋊C₈	—	59	1	no	—
240	C₂²⋊C₈	—	59	1	no	—
241	C₄×C₈	—	59	1	no	—
242	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
243	#41	—	118	2	no	—
244	#41	—	118	2	no	—
245	#41	—	236	4	no	—
246	C₂×Q₈	—	118	1	no	—
247	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
248	C₂×D₄	—	59	1	no	—
249	C₂×D₄	—	59	1	no	—
250	C₂×D₄	—	118	1	no	—
251	C₂×D₄	—	118	1	no	—
252	C₂×D₄	—	118	1	no	—
253	C₂×D₄	—	118	1	no	—
254	C₂×D₄	—	118	1	no	—
255	C₂×D₄	C₂²×D₅₉	1	1	yes	characteristic
256	C₂²×C₄	—	2	1	no	—
257	C₂²×C₄	—	59	1	no	—
258	C₂²×C₄	—	59	1	no	—
259	C₂²×C₄	—	59	1	no	—
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	SD₁₆	—	2	2	no	—
265	SD₁₆	—	118	2	no	—
266	D₈	—	118	2	no	—
267	D₈	—	118	2	no	—
268	C₂×C₈	—	59	1	no	—
269	C₂×C₈	—	59	1	no	—
270	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
271	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	59	1	no	—
274	C₄⋊C₄	—	59	1	no	—
275	C₄⋊C₄	—	118	1	no	—
276	C₄⋊C₄	—	118	1	no	—
277	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
278	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
279	C₂²⋊C₄	—	2	1	no	—
280	C₂²⋊C₄	—	118	1	no	—
281	C₂²⋊C₄	—	118	1	no	—
282	C₂²⋊C₄	—	118	1	no	—
283	C₂²⋊C₄	—	118	1	no	—
284	C₂²⋊C₄	—	118	1	no	—
285	C₂²⋊C₄	—	118	1	no	—
286	C₄²	—	2	1	no	—
287	C₄²	—	59	1	no	—
288	C₄²	—	59	1	no	—
289	C₄²	—	59	1	no	—
290	C₄²	—	118	1	no	—
291	C₄²	—	118	1	no	—
292	C₄²	C₂²×D₅₉	1	1	yes	characteristic
293	C₂³	—	2	1	no	—
294	C₂³	—	59	1	no	—
295	C₂³	—	118	1	no	—
296	C₂³	—	118	1	no	—
297	Q₈	—	2	1	no	—
298	Q₈	—	236	2	no	—
299	Q₈	#11745685	1	2	yes	complemented
300	D₄	—	2	1	no	—
301	D₄	—	2	1	no	—
302	D₄	—	59	2	no	—
303	D₄	—	118	4	no	—
304	D₄	—	118	1	no	—
305	D₄	—	118	1	no	—
306	D₄	—	118	1	no	—
307	D₄	—	236	2	no	—
308	D₄	—	236	2	no	—
309	D₄	—	236	2	no	—
310	D₄	—	236	2	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	59	1	no	—
316	C₂×C₄	—	59	1	no	—
317	C₂×C₄	—	59	1	no	—
318	C₂×C₄	—	118	2	no	—
319	C₂×C₄	—	118	4	no	—
320	C₂×C₄	—	118	2	no	—
321	C₂×C₄	—	118	1	no	—
322	C₂×C₄	—	118	1	no	—
323	C₂×C₄	—	118	1	no	—
324	C₂×C₄	—	118	1	no	—
325	C₂×C₄	—	236	2	no	—
326	C₂×C₄	—	236	2	no	—
327	C₂×C₄	—	236	2	no	—
328	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
329	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
330	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
331	C₈	—	2	1	no	—
332	C₈	—	2	1	no	—
333	C₈	—	118	1	no	—
334	C₈	—	118	1	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	4	1	no	—
337	C₂²	—	4	1	no	—
338	C₂²	—	118	2	no	—
339	C₂²	—	118	1	no	—
340	C₂²	—	118	1	no	—
341	C₂²	—	118	1	no	—
342	C₂²	—	236	1	no	—
343	C₂²	—	236	1	no	—
344	C₂²	—	236	1	no	—
345	C₂²	—	236	1	no	—
346	C₂²	—	236	1	no	—
347	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
348	C₄	—	2	2	no	—
349	C₄	—	2	1	no	—
350	C₄	—	4	1	no	—
351	C₄	—	4	1	no	—
352	C₄	—	59	2	no	—
353	C₄	—	118	1	no	—
354	C₄	—	118	1	no	—
355	C₄	—	236	1	no	—
356	C₄	—	236	1	no	—
357	C₄	#14273754	1	2	yes	—
358	C₄	#41×D₅₉	1	1	yes	characteristic
359	C₄	C₂×#11745685	1	1	yes	characteristic
360	C₂	—	8	1	no	—
361	C₂	—	472	1	no	—
362	C₂	—	236	1	no	—
363	C₂	—	236	2	no	—
364	C₂	#16837911	1	1	yes	characteristic
365	C₂	#135×D₅₉	1	1	yes	characteristic
366	C₂	C₂×#14273754	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₄×SD₁₆×C₅₉	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	C₄×SD₁₆	D₅₉
hypocenter	C₅₉	#3106
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×SD₁₆

Complements

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

N	H ≅ G/N	type
1. #16837949	C₂	semidirect
2. C₄×#11745689	C₂	semidirect
3. #16837728	C₂	semidirect
4. #16837724	C₂	semidirect
5. #16837695	C₂	semidirect
6. #16837691	C₂	semidirect
7. #16837600	C₂	semidirect
8. #16837596	C₂	semidirect
9. #16837550	C₂	semidirect
10. #16837546	C₂	semidirect
11. C₄×#11745671	C₂	semidirect
12. C₄×D₄₇₂	C₂	semidirect
13. C₄×#11745669	C₂	semidirect
14. #16837453	C₂	semidirect
15. C₄×SD₁₆×C₅₉	C₂	semidirect
114. Q₈×C₅₉	#41	semidirect
137. D₂₃₆	#41	semidirect
195. C₄×SD₁₆	D₅₉	semidirect
199. C₅₉	#3106	semidirect
299. Q₈	#11745685	semidirect