Subgroups — #21434043

Order 7552.

410 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊Q₈×D₅₉	C₂	1	1	yes	maximal, characteristic
2	#117×D₅₉	C₂	1	1	yes	maximal, characteristic
3	#16837744	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837743	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837740	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837739	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837736	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837734	C₂	1	1	yes	maximal, characteristic, complemented
9	#16837731	C₂	1	1	yes	maximal, characteristic, complemented
10	#16837443	C₂	1	1	yes	maximal, characteristic, complemented
11	#16837441	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837409	C₂	1	1	yes	maximal, characteristic, complemented
13	#16837409	C₂	1	1	yes	maximal, characteristic, complemented
14	#16837364	C₂	1	1	yes	maximal, characteristic
15	C₅₉×#485	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₅₉	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₅₉	C₂²	1	1	yes	characteristic
18	C₂×#11745659	C₂²	1	1	yes	characteristic
19	C₂×#11745659	C₂²	1	1	yes	characteristic
20	#14273707	C₂²	1	1	yes	characteristic
21	#14273706	C₂²	1	1	yes	characteristic
22	#14273705	—	2	1	no	—
23	#14273704	—	2	1	no	—
24	#14273703	C₂²	1	1	yes	characteristic
25	#14273686	C₂²	1	1	yes	characteristic
26	#14273685	C₂²	1	1	yes	characteristic
27	#14273685	C₂²	1	1	yes	characteristic
28	#14273684	—	2	1	no	—
29	#14273683	—	2	1	no	—
30	C₄⋊C₄×D₅₉	—	2	1	no	—
31	C₄⋊C₄×D₅₉	C₂²	1	1	yes	characteristic
32	C₄⋊C₄×D₅₉	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×D₅₉	C₂²	1	1	yes	characteristic
34	C₂²⋊C₄×D₅₉	—	2	1	no	—
35	C₂²⋊C₄×D₅₉	—	2	1	no	—
36	C₄²×D₅₉	—	2	1	no	—
37	#14273661	C₂²	1	1	yes	characteristic, complemented
38	#14273660	C₂²	1	1	yes	characteristic, complemented
39	#14273660	C₂²	1	1	yes	characteristic, complemented
40	#14273659	C₂²	1	1	yes	characteristic, complemented
41	#14273658	C₂²	1	1	yes	characteristic, complemented
42	#14273658	C₂²	1	1	yes	characteristic, complemented
43	#14273652	C₂²	1	1	yes	characteristic, complemented
44	#14273652	C₂²	1	1	yes	characteristic, complemented
45	#14273651	C₂²	1	1	yes	characteristic, complemented
46	#14273651	C₂²	1	1	yes	characteristic, complemented
47	#14273650	C₂²	1	1	yes	characteristic, complemented
48	#14273650	C₂²	1	1	yes	characteristic, complemented
49	C₂²⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
50	(C₂×D₅₉)⋊C₈	C₂²	1	1	yes	characteristic
51	(C₂×D₅₉)⋊C₈	C₂²	1	1	yes	characteristic
52	C₂²⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
53	#117×C₅₉	C₂²	1	1	yes	characteristic
54	#107×C₅₉	C₂²	1	1	yes	characteristic, complemented
55	#106×C₅₉	C₂²	1	1	yes	characteristic, complemented
56	#103×C₅₉	C₂²	1	1	yes	characteristic, complemented
57	#103×C₅₉	C₂²	1	1	yes	characteristic, complemented
58	C₂²⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
59	C₂³×D₅₉	D₄	1	1	yes	characteristic
60	C₂×Q₈×C₅₉	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
62	C₂²×Dic₅₉	D₄	1	1	yes	characteristic
63	Q₈×D₅₉	—	2	2	no	—
64	Q₈×D₅₉	—	2	2	no	—
65	C₂×C₄×D₅₉	—	2	2	no	—
66	C₂×C₄×D₅₉	—	2	2	no	—
67	C₂×C₄×D₅₉	—	2	1	no	—
68	C₂×C₄×D₅₉	—	2	1	no	—
69	C₂×C₄×D₅₉	—	2	1	no	—
70	C₂×C₄×D₅₉	—	2	1	no	—
71	C₂×C₄×D₅₉	D₄	1	1	yes	characteristic
72	C₂×C₄×D₅₉	D₄	1	1	yes	characteristic
73	C₂×C₄×D₅₉	C₂³	1	1	yes	characteristic
74	C₂×C₄×D₅₉	C₂³	1	1	yes	characteristic
75	C₂×Dic₁₁₈	—	2	1	no	—
76	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
78	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
79	C₄⋊C₄×C₅₉	—	2	1	no	—
80	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
81	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
83	C₂²⋊C₄×C₅₉	—	2	1	no	—
84	C₂²⋊C₄×C₅₉	—	2	1	no	—
85	C₄²×C₅₉	—	2	1	no	—
86	C₂²⋊Dic₅₉	—	2	1	no	—
87	C₂²⋊Dic₅₉	—	2	1	no	—
88	(C₂×D₅₉)⋊C₄	—	2	1	no	—
89	(C₂×D₅₉)⋊C₄	—	2	1	no	—
90	(C₂×D₅₉)⋊C₄	—	2	1	no	—
91	(C₂×D₅₉)⋊C₄	—	2	1	no	—
92	C₄⋊Dic₅₉	—	2	1	no	—
93	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
94	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
95	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
96	Dic₅₉⋊C₄	—	2	1	no	—
97	Dic₅₉⋊C₄	—	2	1	no	—
98	Dic₅₉⋊C₄	—	2	1	no	—
99	Dic₅₉⋊C₄	—	2	1	no	—
100	Dic₅₉⋊C₄	—	2	1	no	—
101	C₄×Dic₅₉	—	2	1	no	—
102	C₄×Dic₅₉	—	2	1	no	—
103	C₄×Dic₅₉	—	2	1	no	—
104	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
105	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
106	#11745659	—	2	2	no	—
107	#11745659	—	2	2	no	—
108	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
109	C₂²×D₅₉	—	2	2	no	—
110	C₂²×D₅₉	—	2	2	no	—
111	C₂²×D₅₉	—	4	1	no	—
112	C₂²×D₅₉	C₂×D₄	1	1	yes	characteristic
113	C₂²×D₅₉	C₂×D₄	1	1	yes	characteristic
114	Q₈×C₅₉	—	2	1	no	—
115	Q₈×C₅₉	—	2	1	no	—
116	C₂×C₄×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	—	2	1	no	—
121	C₂×C₄×C₅₉	—	2	1	no	—
122	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
124	C₂×Dic₅₉	—	2	2	no	—
125	C₂×Dic₅₉	—	2	1	no	—
126	C₂×Dic₅₉	—	2	1	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	C₄×D₅₉	—	2	2	no	—
132	C₄×D₅₉	—	2	1	no	—
133	C₄×D₅₉	—	2	1	no	—
134	C₄×D₅₉	—	2	2	no	—
135	C₄×D₅₉	—	4	2	no	—
136	C₄×D₅₉	—	4	2	no	—
137	C₄×D₅₉	—	4	2	no	—
138	C₄×D₅₉	—	4	2	no	—
139	C₄×D₅₉	#41	1	2	yes	—
140	C₄×D₅₉	#41	1	2	yes	—
141	Dic₁₁₈	—	2	1	no	—
142	Dic₁₁₈	—	2	1	no	—
143	Dic₁₁₈	—	4	2	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	1	no	—
150	C₂²×C₅₉	—	2	1	no	—
151	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
152	C₂×D₅₉	—	2	1	no	—
153	C₂×D₅₉	—	2	1	no	—
154	C₂×D₅₉	—	2	1	no	—
155	C₂×D₅₉	—	2	1	no	—
156	C₂×D₅₉	—	4	4	no	—
157	C₂×D₅₉	#123	1	2	yes	—
158	C₂×D₅₉	#137	1	2	yes	—
159	C₄×C₅₉	—	2	1	no	—
160	C₄×C₅₉	—	4	1	no	—
161	C₄×C₅₉	—	4	1	no	—
162	C₄×C₅₉	—	4	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	—	4	1	no	—
168	Dic₅₉	—	4	1	no	—
169	Dic₅₉	—	4	1	no	—
170	Dic₅₉	—	4	1	no	—
171	Dic₅₉	#123	1	2	yes	—
172	#3095	—	59	1	no	maximal, Sylow
173	C₂×C₅₉	—	4	1	no	—
174	C₂×C₅₉	C₂×#123	1	1	yes	characteristic
175	C₂×C₅₉	C₂×#137	1	1	yes	characteristic
176	C₂×C₅₉	#575	1	1	yes	characteristic
177	D₅₉	—	2	2	no	—
178	D₅₉	—	2	2	no	—
179	C₂×C₂²⋊Q₈	—	59	1	no	—
180	C₂×#117	—	59	1	no	—
181	#485	—	59	1	no	—
182	#485	—	59	1	no	—
183	#485	—	59	1	no	—
184	#485	D₅₉	1	1	yes	characteristic, complemented
185	#484	—	59	1	no	—
186	#484	—	59	1	no	—
187	#483	—	59	1	no	—
188	#483	—	59	1	no	—
189	#428	—	59	1	no	—
190	#428	—	59	1	no	—
191	#419	—	59	1	no	—
192	#419	—	59	1	no	—
193	#407	—	59	1	no	—
194	C₅₉	#3095	1	1	yes	characteristic, complemented, Sylow
195	C₂²×Q₈	—	59	1	no	—
196	C₂³×C₄	—	59	1	no	—
197	C₂×M₁₆	—	59	1	no	—
198	C₂×M₁₆	—	59	1	no	—
199	C₂²⋊Q₈	—	59	1	no	—
200	C₂²⋊Q₈	—	59	1	no	—
201	C₂²⋊Q₈	—	59	1	no	—
202	C₂²⋊Q₈	—	118	1	no	—
203	C₂²⋊Q₈	—	118	1	no	—
204	C₂²⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
205	#117	—	59	1	no	—
206	#117	—	59	1	no	—
207	#117	—	59	1	no	—
208	#117	—	118	1	no	—
209	#117	—	118	1	no	—
210	#117	C₂×D₅₉	1	1	yes	characteristic
211	C₂×C₄⋊C₄	—	59	1	no	—
212	C₂×C₄⋊C₄	—	59	1	no	—
213	C₂×C₄⋊C₄	—	59	1	no	—
214	C₂×C₄⋊C₄	—	118	1	no	—
215	C₂×C₂²⋊C₄	—	118	1	no	—
216	C₂×C₂²⋊C₄	—	118	1	no	—
217	C₂×C₄²	—	118	1	no	—
218	#107	—	59	1	no	—
219	#107	—	59	1	no	—
220	#107	—	59	1	no	—
221	#107	C₂×D₅₉	1	1	yes	characteristic, complemented
222	#106	—	59	1	no	—
223	#106	—	59	1	no	—
224	#106	—	59	1	no	—
225	#106	C₂×D₅₉	1	1	yes	characteristic, complemented
226	#103	—	59	1	no	—
227	#103	—	59	1	no	—
228	#103	—	59	1	no	—
229	#103	—	59	1	no	—
230	#103	—	59	1	no	—
231	#103	—	59	1	no	—
232	#103	C₂×D₅₉	1	1	yes	characteristic, complemented
233	#103	C₂×D₅₉	1	1	yes	characteristic, complemented
234	C₂²⋊C₈	—	59	1	no	—
235	C₂²⋊C₈	—	59	1	no	—
236	C₂²⋊C₈	—	59	1	no	—
237	C₂²⋊C₈	C₂×D₅₉	1	1	yes	characteristic
238	C₂⁴	—	59	1	no	—
239	C₂×Q₈	—	59	1	no	—
240	C₂×Q₈	—	118	1	no	—
241	C₂×Q₈	—	118	2	no	—
242	C₂×Q₈	—	118	2	no	—
243	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
244	C₂²×C₄	—	59	1	no	—
245	C₂²×C₄	—	59	1	no	—
246	C₂²×C₄	—	59	1	no	—
247	C₂²×C₄	—	59	1	no	—
248	C₂²×C₄	—	59	1	no	—
249	C₂²×C₄	—	118	2	no	—
250	C₂²×C₄	—	118	2	no	—
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	C₂²×D₅₉	1	1	yes	characteristic
256	M₁₆	—	118	2	no	—
257	M₁₆	—	118	2	no	—
258	C₂×C₈	—	59	1	no	—
259	C₂×C₈	—	59	1	no	—
260	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
261	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	59	1	no	—
264	C₄⋊C₄	—	59	1	no	—
265	C₄⋊C₄	—	59	1	no	—
266	C₄⋊C₄	—	118	1	no	—
267	C₄⋊C₄	—	118	1	no	—
268	C₄⋊C₄	—	118	1	no	—
269	C₄⋊C₄	—	118	1	no	—
270	C₄⋊C₄	—	118	1	no	—
271	C₄⋊C₄	—	118	1	no	—
272	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
273	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
274	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
275	C₂²⋊C₄	—	2	1	no	—
276	C₂²⋊C₄	—	2	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	C₄²	—	2	1	no	—
284	C₄²	—	118	1	no	—
285	C₄²	—	118	1	no	—
286	C₄²	—	118	1	no	—
287	C₂³	—	59	1	no	—
288	C₂³	—	59	1	no	—
289	C₂³	—	118	2	no	—
290	C₂³	—	118	2	no	—
291	C₂³	—	236	1	no	—
292	C₂³	D₄×D₅₉	1	1	yes	characteristic
293	Q₈	—	2	1	no	—
294	Q₈	—	2	1	no	—
295	Q₈	—	118	1	no	—
296	Q₈	—	118	1	no	—
297	Q₈	—	236	2	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	59	1	no	—
305	C₂×C₄	—	59	2	no	—
306	C₂×C₄	—	59	2	no	—
307	C₂×C₄	—	59	1	no	—
308	C₂×C₄	—	118	2	no	—
309	C₂×C₄	—	118	2	no	—
310	C₂×C₄	—	118	2	no	—
311	C₂×C₄	—	118	1	no	—
312	C₂×C₄	—	118	1	no	—
313	C₂×C₄	—	118	1	no	—
314	C₂×C₄	—	118	1	no	—
315	C₂×C₄	—	118	1	no	—
316	C₂×C₄	—	118	1	no	—
317	C₂×C₄	—	236	2	no	—
318	C₂×C₄	—	236	2	no	—
319	C₂×C₄	—	236	2	no	—
320	C₂×C₄	—	236	2	no	—
321	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
322	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
323	C₈	—	2	1	no	—
324	C₈	—	2	1	no	—
325	C₈	—	118	1	no	—
326	C₈	—	118	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	59	2	no	—
331	C₂²	—	59	2	no	—
332	C₂²	—	118	1	no	—
333	C₂²	—	118	1	no	—
334	C₂²	—	118	1	no	—
335	C₂²	—	118	1	no	—
336	C₂²	—	236	4	no	—
337	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
338	C₄	—	2	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	4	1	no	—
341	C₄	—	4	1	no	—
342	C₄	—	4	1	no	—
343	C₄	—	59	2	no	—
344	C₄	—	118	1	no	—
345	C₄	—	236	1	no	—
346	C₄	—	236	1	no	—
347	C₄	—	236	1	no	—
348	C₄	—	236	1	no	—
349	C₄	#41×D₅₉	1	1	yes	characteristic
350	C₄	#41×D₅₉	1	1	yes	characteristic
351	C₂	—	4	1	no	—
352	C₂	—	118	2	no	—
353	C₂	—	118	2	no	—
354	C₂	#123×D₅₉	1	1	yes	characteristic
355	C₂	#137×D₅₉	1	1	yes	characteristic
356	C₂	#16838199	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#485	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#485	D₅₉
hypocenter	C₅₉	#3095
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #485

Complements

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

N	H ≅ G/N	type
3. #16837744	C₂	semidirect
4. #16837743	C₂	semidirect
5. #16837740	C₂	semidirect
6. #16837739	C₂	semidirect
7. #16837736	C₂	semidirect
8. #16837734	C₂	semidirect
9. #16837731	C₂	semidirect
10. #16837443	C₂	semidirect
11. #16837441	C₂	semidirect
12. #16837409	C₂	semidirect
13. #16837409	C₂	semidirect
15. C₅₉×#485	C₂	semidirect
37. #14273661	C₂²	semidirect
38. #14273660	C₂²	semidirect
39. #14273660	C₂²	semidirect
40. #14273659	C₂²	semidirect
41. #14273658	C₂²	semidirect
42. #14273658	C₂²	semidirect
43. #14273652	C₂²	semidirect
44. #14273652	C₂²	semidirect
45. #14273651	C₂²	semidirect
46. #14273651	C₂²	semidirect
47. #14273650	C₂²	semidirect
48. #14273650	C₂²	semidirect
54. #107×C₅₉	C₂²	semidirect
55. #106×C₅₉	C₂²	semidirect
56. #103×C₅₉	C₂²	semidirect
57. #103×C₅₉	C₂²	semidirect
184. #485	D₅₉	semidirect
194. C₅₉	#3095	semidirect
221. #107	C₂×D₅₉	semidirect
225. #106	C₂×D₅₉	semidirect
232. #103	C₂×D₅₉	semidirect
233. #103	C₂×D₅₉	semidirect