Subgroups — #21434273

Order 7552.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16838019	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837877	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837744	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837723	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837690	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837680	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837615	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837615	C₂	1	1	yes	maximal, characteristic, complemented
9	#16837553	C₂	1	1	yes	maximal, characteristic, complemented
10	#16837514	C₂	1	1	yes	maximal, characteristic, complemented
11	#16837499	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837493	C₂	1	1	yes	maximal, characteristic, complemented
13	D₈×Dic₅₉	C₂	1	1	yes	maximal, characteristic, complemented
14	#16837460	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₉×#447	C₂	1	1	yes	maximal, characteristic
16	C₂×#11745670	C₂²	1	1	yes	characteristic
17	C₂×#11745670	C₂²	1	1	yes	characteristic
18	#14273726	C₂²	1	1	yes	characteristic, complemented
19	#14273720	—	2	1	no	—
20	#14273714	—	2	1	no	—
21	#14273714	C₂²	1	1	yes	characteristic, complemented
22	#14273708	—	2	1	no	—
23	#14273706	C₂²	1	1	yes	characteristic
24	#14273704	—	2	1	no	—
25	#14273692	C₂²	1	1	yes	characteristic, complemented
26	D₄×Dic₅₉	—	2	1	no	—
27	D₄×Dic₅₉	—	2	1	no	—
28	D₄×Dic₅₉	C₂²	1	1	yes	characteristic
29	D₄×Dic₅₉	C₂²	1	1	yes	characteristic
30	D₄×Dic₅₉	C₂²	1	1	yes	characteristic
31	D₄×Dic₅₉	C₂²	1	1	yes	characteristic
32	D₄×Dic₅₉	C₂²	1	1	yes	characteristic, complemented
33	#14273686	C₂²	1	1	yes	characteristic
34	#14273686	C₂²	1	1	yes	characteristic
35	C₂×C₂²⋊Dic₅₉	—	2	1	no	—
36	C₂×C₂²⋊Dic₅₉	—	2	1	no	—
37	#14273661	C₂²	1	1	yes	characteristic, complemented
38	#14273660	C₂²	1	1	yes	characteristic, complemented
39	#14273659	C₂²	1	1	yes	characteristic, complemented
40	#14273658	C₂²	1	1	yes	characteristic, complemented
41	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic, complemented
42	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic, complemented
43	#14273652	C₂²	1	1	yes	characteristic, complemented
44	#14273652	C₂²	1	1	yes	characteristic, complemented
45	#14273648	C₂²	1	1	yes	characteristic
46	#14273648	C₂²	1	1	yes	characteristic
47	#14273648	C₂²	1	1	yes	characteristic, complemented
48	#14273648	C₂²	1	1	yes	characteristic, complemented
49	C₂²⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
50	#14273637	C₂²	1	1	yes	characteristic, complemented
51	C₈×Dic₅₉	C₂²	1	1	yes	characteristic, complemented
52	C₂²×D₄×C₅₉	C₂²	1	1	yes	characteristic
53	C₂×D₈×C₅₉	C₂²	1	1	yes	characteristic
54	C₂×D₈×C₅₉	C₂²	1	1	yes	characteristic
55	#121×C₅₉	C₂²	1	1	yes	characteristic
56	#102×C₅₉	C₂²	1	1	yes	characteristic
57	#102×C₅₉	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
59	C₂⁴×C₅₉	—	2	1	no	—
60	C₂×D₄×C₅₉	—	2	2	no	—
61	C₂×D₄×C₅₉	—	2	2	no	—
62	C₂×D₄×C₅₉	—	2	1	no	—
63	C₂×D₄×C₅₉	—	2	1	no	—
64	C₂×D₄×C₅₉	C₂³	1	1	yes	characteristic
65	C₂×D₄×C₅₉	C₂³	1	1	yes	characteristic
66	C₂×D₄×C₅₉	C₂³	1	1	yes	characteristic
67	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
68	C₂²×Dic₅₉	—	2	1	no	—
69	C₂²×Dic₅₉	—	2	1	no	—
70	C₂²×Dic₅₉	—	2	1	no	—
71	C₂²×Dic₅₉	—	2	1	no	—
72	C₂²×Dic₅₉	—	2	1	no	—
73	C₂×Dic₁₁₈	—	2	1	no	—
74	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
75	D₈×C₅₉	—	2	2	no	—
76	D₈×C₅₉	—	2	2	no	—
77	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
78	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
79	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
80	C₂²⋊C₄×C₅₉	—	2	1	no	—
81	C₂²⋊Dic₅₉	—	2	1	no	—
82	C₂²⋊Dic₅₉	—	2	1	no	—
83	C₂²⋊Dic₅₉	—	2	1	no	—
84	C₂²⋊Dic₅₉	—	2	1	no	—
85	C₂²⋊Dic₅₉	—	2	1	no	—
86	C₂²⋊Dic₅₉	—	2	1	no	—
87	C₂²⋊Dic₅₉	—	2	1	no	—
88	C₂²⋊Dic₅₉	—	2	1	no	—
89	C₂²⋊Dic₅₉	—	2	1	no	—
90	C₂²⋊Dic₅₉	D₄	1	1	yes	characteristic
91	C₂²⋊Dic₅₉	D₄	1	1	yes	characteristic
92	#11745670	—	2	2	no	—
93	#11745670	—	2	2	no	—
94	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
95	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
96	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
97	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
98	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
99	Dic₅₉⋊C₄	—	2	1	no	—
100	Dic₅₉⋊C₄	—	2	1	no	—
101	Dic₅₉⋊C₄	—	2	1	no	—
102	C₄×Dic₅₉	—	2	1	no	—
103	C₄×Dic₅₉	—	2	1	no	—
104	C₄×Dic₅₉	C₂³	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	C₂³×C₅₉	—	2	4	no	—
109	C₂³×C₅₉	—	2	1	no	—
110	C₂³×C₅₉	—	2	1	no	—
111	C₂³×C₅₉	—	2	1	no	—
112	C₂³×C₅₉	—	8	1	no	—
113	C₂³×C₅₉	—	8	1	no	—
114	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
115	D₄×C₅₉	—	2	4	no	—
116	D₄×C₅₉	—	2	1	no	—
117	D₄×C₅₉	—	2	1	no	—
118	D₄×C₅₉	—	2	1	no	—
119	D₄×C₅₉	—	2	1	no	—
120	D₄×C₅₉	—	8	1	no	—
121	D₄×C₅₉	#41	1	2	yes	—
122	D₄×C₅₉	#41	1	2	yes	—
123	C₂×C₄×C₅₉	—	2	1	no	—
124	C₂×C₄×C₅₉	—	2	1	no	—
125	C₂×C₄×C₅₉	—	2	1	no	—
126	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
128	C₂×Dic₅₉	—	2	1	no	—
129	C₂×Dic₅₉	—	2	1	no	—
130	C₂×Dic₅₉	—	2	1	no	—
131	C₂×Dic₅₉	—	2	1	no	—
132	C₂×Dic₅₉	—	8	1	no	—
133	C₂×Dic₅₉	—	8	1	no	—
134	C₂×Dic₅₉	—	8	1	no	—
135	C₂×Dic₅₉	—	8	1	no	—
136	C₂×Dic₅₉	—	8	1	no	—
137	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
138	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
139	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
140	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
141	Dic₁₁₈	—	2	1	no	—
142	Dic₁₁₈	—	2	1	no	—
143	Dic₁₁₈	—	8	1	no	—
144	C₈×C₅₉	—	2	1	no	—
145	C₈×C₅₉	—	2	1	no	—
146	C₅₉⋊C₈	—	2	1	no	—
147	C₅₉⋊C₈	—	2	1	no	—
148	C₂²×C₅₉	—	2	1	no	—
149	C₂²×C₅₉	—	2	2	no	—
150	C₂²×C₅₉	—	2	2	no	—
151	C₂²×C₅₉	—	2	1	no	—
152	C₂²×C₅₉	—	2	1	no	—
153	C₂²×C₅₉	—	4	1	no	—
154	C₂²×C₅₉	—	4	1	no	—
155	C₂²×C₅₉	—	4	1	no	—
156	C₂²×C₅₉	—	4	1	no	—
157	C₂²×C₅₉	—	4	1	no	—
158	C₂²×C₅₉	—	4	1	no	—
159	C₂²×C₅₉	—	4	1	no	—
160	C₂²×C₅₉	—	8	4	no	—
161	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
162	C₄×C₅₉	—	2	1	no	—
163	C₄×C₅₉	—	4	1	no	—
164	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
165	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
166	Dic₅₉	—	2	1	no	—
167	Dic₅₉	—	2	1	no	—
168	Dic₅₉	—	2	1	no	—
169	Dic₅₉	—	2	1	no	—
170	Dic₅₉	—	4	1	no	—
171	Dic₅₉	—	4	1	no	—
172	Dic₅₉	—	4	1	no	—
173	Dic₅₉	—	4	1	no	—
174	#3118	—	59	1	no	maximal, Sylow
175	C₂×C₅₉	—	4	1	no	—
176	C₂×C₅₉	—	4	2	no	—
177	C₂×C₅₉	—	4	2	no	—
178	C₂×C₅₉	—	8	1	no	—
179	C₂×C₅₉	#525	1	1	yes	characteristic
180	C₂×C₅₉	#576	1	1	yes	characteristic
181	C₂×C₅₉	#577	1	1	yes	characteristic
182	#542	—	59	1	no	—
183	#518	—	59	1	no	—
184	#485	—	59	1	no	—
185	#482	—	59	1	no	—
186	#476	—	59	1	no	—
187	#474	—	59	1	no	—
188	#463	—	59	1	no	—
189	#463	—	59	1	no	—
190	#450	—	59	1	no	—
191	#447	D₅₉	1	1	yes	characteristic
192	#442	—	59	1	no	—
193	#440	—	59	1	no	—
194	C₄×SD₁₆	—	59	1	no	—
195	C₄×D₈	—	59	1	no	—
196	#433	—	59	1	no	—
197	C₅₉	#3118	1	1	yes	characteristic, complemented, Sylow
198	C₂²×D₄	C₂×D₅₉	1	1	yes	characteristic
199	C₂×SD₁₆	—	59	1	no	—
200	C₂×SD₁₆	—	59	1	no	—
201	C₂×D₈	C₂×D₅₉	1	1	yes	characteristic
202	C₂×D₈	C₂×D₅₉	1	1	yes	characteristic
203	C₄⋊Q₈	—	59	1	no	—
204	#126	—	118	1	no	—
205	#124	—	59	1	no	—
206	#124	—	118	1	no	—
207	#123	—	118	1	no	—
208	C₂²⋊Q₈	—	59	1	no	—
209	C₂²⋊Q₈	—	118	1	no	—
210	#121	C₂×D₅₉	1	1	yes	characteristic
211	C₄×Q₈	—	59	1	no	—
212	C₄×D₄	—	59	1	no	—
213	C₄×D₄	—	59	1	no	—
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	#117	—	59	1	no	—
220	#117	—	59	1	no	—
221	C₂×C₂²⋊C₄	—	118	1	no	—
222	C₂×C₂²⋊C₄	—	118	1	no	—
223	#107	—	59	1	no	—
224	#107	—	59	1	no	—
225	#106	—	59	1	no	—
226	#106	—	59	1	no	—
227	C₄⋊C₈	—	59	1	no	—
228	C₄⋊C₈	—	59	1	no	—
229	#103	—	59	1	no	—
230	#103	—	59	1	no	—
231	#102	—	59	1	no	—
232	#102	—	59	1	no	—
233	#102	—	59	1	no	—
234	#102	—	59	1	no	—
235	#102	C₂×D₅₉	1	1	yes	characteristic
236	#102	C₂×D₅₉	1	1	yes	characteristic
237	C₂²⋊C₈	—	59	1	no	—
238	C₂²⋊C₈	C₂×D₅₉	1	1	yes	characteristic
239	#97	—	59	1	no	—
240	C₄×C₈	—	59	1	no	—
241	C₂⁴	—	2	1	no	—
242	C₂×Q₈	—	59	1	no	—
243	C₂×Q₈	—	118	1	no	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	2	2	no	—
246	C₂×D₄	—	2	1	no	—
247	C₂×D₄	—	2	1	no	—
248	C₂×D₄	C₂²×D₅₉	1	1	yes	characteristic
249	C₂×D₄	C₂²×D₅₉	1	1	yes	characteristic
250	C₂×D₄	C₂²×D₅₉	1	1	yes	characteristic
251	C₂²×C₄	—	118	1	no	—
252	C₂²×C₄	—	118	1	no	—
253	C₂²×C₄	—	118	1	no	—
254	C₂²×C₄	—	118	1	no	—
255	C₂²×C₄	—	118	1	no	—
256	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
257	SD₁₆	—	118	2	no	—
258	SD₁₆	—	118	2	no	—
259	D₈	—	2	2	no	—
260	D₈	—	2	2	no	—
261	C₂×C₈	—	59	1	no	—
262	C₂×C₈	—	59	1	no	—
263	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
264	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
265	C₄⋊C₄	—	59	1	no	—
266	C₄⋊C₄	—	59	1	no	—
267	C₄⋊C₄	—	59	1	no	—
268	C₄⋊C₄	—	59	1	no	—
269	C₄⋊C₄	—	59	1	no	—
270	C₄⋊C₄	—	118	1	no	—
271	C₄⋊C₄	—	118	1	no	—
272	C₄⋊C₄	—	118	1	no	—
273	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
274	C₂²⋊C₄	—	2	1	no	—
275	C₂²⋊C₄	—	59	1	no	—
276	C₂²⋊C₄	—	59	1	no	—
277	C₂²⋊C₄	—	118	1	no	—
278	C₂²⋊C₄	—	118	1	no	—
279	C₂²⋊C₄	—	118	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	—	59	1	no	—
287	C₄²	—	59	1	no	—
288	C₄²	—	118	1	no	—
289	C₄²	—	118	1	no	—
290	C₂³	—	2	4	no	—
291	C₂³	—	2	1	no	—
292	C₂³	—	2	1	no	—
293	C₂³	—	2	1	no	—
294	C₂³	—	8	1	no	—
295	C₂³	—	8	1	no	—
296	C₂³	D₄×D₅₉	1	1	yes	characteristic
297	Q₈	—	118	1	no	—
298	Q₈	—	118	1	no	—
299	Q₈	—	472	1	no	—
300	D₄	—	2	4	no	—
301	D₄	—	2	1	no	—
302	D₄	—	2	1	no	—
303	D₄	—	2	1	no	—
304	D₄	—	2	1	no	—
305	D₄	—	8	1	no	—
306	D₄	#11745687	1	2	yes	—
307	D₄	#11745687	1	2	yes	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	59	1	no	—
312	C₂×C₄	—	59	1	no	—
313	C₂×C₄	—	59	1	no	—
314	C₂×C₄	—	59	1	no	—
315	C₂×C₄	—	118	1	no	—
316	C₂×C₄	—	118	1	no	—
317	C₂×C₄	—	118	1	no	—
318	C₂×C₄	—	118	1	no	—
319	C₂×C₄	—	472	1	no	—
320	C₂×C₄	—	472	1	no	—
321	C₂×C₄	—	472	1	no	—
322	C₂×C₄	—	472	1	no	—
323	C₂×C₄	—	472	1	no	—
324	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
325	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
326	C₈	—	2	1	no	—
327	C₈	—	2	1	no	—
328	C₈	—	118	1	no	—
329	C₈	—	118	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	2	no	—
332	C₂²	—	2	2	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	4	1	no	—
337	C₂²	—	4	1	no	—
338	C₂²	—	4	1	no	—
339	C₂²	—	4	1	no	—
340	C₂²	—	4	1	no	—
341	C₂²	—	4	1	no	—
342	C₂²	—	8	4	no	—
343	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	4	1	no	—
346	C₄	—	118	1	no	—
347	C₄	—	118	1	no	—
348	C₄	—	118	1	no	—
349	C₄	—	118	1	no	—
350	C₄	—	236	1	no	—
351	C₄	—	236	1	no	—
352	C₄	—	236	1	no	—
353	C₄	—	236	1	no	—
354	C₄	C₂×#11745687	1	1	yes	characteristic
355	C₄	C₂×#11745687	1	1	yes	characteristic
356	C₂	—	4	1	no	—
357	C₂	—	4	2	no	—
358	C₂	—	4	2	no	—
359	C₂	—	8	1	no	—
360	C₂	#16837908	1	1	yes	characteristic
361	C₂	#16838203	1	1	yes	characteristic
362	C₂	#16838212	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#447	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#447	D₅₉
hypocenter	C₅₉	#3118
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #447

Complements

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

N	H ≅ G/N	type
1. #16838019	C₂	semidirect
2. #16837877	C₂	semidirect
3. #16837744	C₂	semidirect
4. #16837723	C₂	semidirect
5. #16837690	C₂	semidirect
6. #16837680	C₂	semidirect
7. #16837615	C₂	semidirect
8. #16837615	C₂	semidirect
9. #16837553	C₂	semidirect
10. #16837514	C₂	semidirect
11. #16837499	C₂	semidirect
12. #16837493	C₂	semidirect
13. D₈×Dic₅₉	C₂	semidirect
14. #16837460	C₂	semidirect
18. #14273726	C₂²	semidirect
21. #14273714	C₂²	semidirect
25. #14273692	C₂²	semidirect
32. D₄×Dic₅₉	C₂²	semidirect
37. #14273661	C₂²	semidirect
38. #14273660	C₂²	semidirect
39. #14273659	C₂²	semidirect
40. #14273658	C₂²	semidirect
41. Dic₅₉⋊C₈	C₂²	semidirect
42. Dic₅₉⋊C₈	C₂²	semidirect
43. #14273652	C₂²	semidirect
44. #14273652	C₂²	semidirect
47. #14273648	C₂²	semidirect
48. #14273648	C₂²	semidirect
50. #14273637	C₂²	semidirect
51. C₈×Dic₅₉	C₂²	semidirect
197. C₅₉	#3118	semidirect