Subgroups — #21436268

Order 7552.

388 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16838096	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₂²⋊D₅₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837769	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837743	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837736	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837711	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837648	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837634	C₂	1	1	yes	maximal, characteristic, complemented
9	#16837606	C₂	1	1	yes	maximal, characteristic, complemented
10	#16837542	C₂	1	1	yes	maximal, characteristic, complemented
11	#16837541	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837505	C₂	1	1	yes	maximal, characteristic, complemented
13	#16837471	C₂	1	1	yes	maximal, characteristic, complemented
14	#103×D₅₉	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₉×#419	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₅₉	C₂²	1	1	yes	characteristic
17	C₂×#11745660	C₂²	1	1	yes	characteristic
18	C₂×#11745671	C₂²	1	1	yes	characteristic
19	C₂×#11745671	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₅₉	C₂²	1	1	yes	characteristic
21	#14273725	—	2	1	no	—
22	#14273725	C₂²	1	1	yes	characteristic
23	#14273721	—	2	1	no	—
24	#14273718	C₂²	1	1	yes	characteristic
25	#14273709	—	2	1	no	—
26	#14273706	—	2	1	no	—
27	#14273706	C₂²	1	1	yes	characteristic
28	#14273703	—	2	1	no	—
29	#14273703	C₂²	1	1	yes	characteristic
30	#14273702	C₂²	1	1	yes	characteristic
31	#14273701	C₂²	1	1	yes	characteristic
32	#14273700	—	2	1	no	—
33	Q₈×Dic₅₉	C₂²	1	1	yes	characteristic
34	C₄×D₂₃₆	—	2	1	no	—
35	#14273689	C₂²	1	1	yes	characteristic
36	C₄×C₂²⋊D₅₉	—	2	1	no	—
37	C₄×C₂²⋊D₅₉	C₂²	1	1	yes	characteristic
38	C₄⋊C₄×D₅₉	C₂²	1	1	yes	characteristic
39	#14273661	C₂²	1	1	yes	characteristic
40	#14273660	C₂²	1	1	yes	characteristic
41	#14273658	C₂²	1	1	yes	characteristic
42	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
43	#14273652	C₂²	1	1	yes	characteristic
44	#14273651	C₂²	1	1	yes	characteristic
45	#14273651	C₂²	1	1	yes	characteristic
46	#14273650	C₂²	1	1	yes	characteristic
47	#14273650	C₂²	1	1	yes	characteristic
48	#14273649	C₂²	1	1	yes	characteristic
49	C₂²⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
50	(C₂×D₅₉)⋊C₈	C₂²	1	1	yes	characteristic
51	#14273637	C₂²	1	1	yes	characteristic
52	C₂²×Q₈×C₅₉	C₂²	1	1	yes	characteristic
53	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
54	#117×C₅₉	C₂²	1	1	yes	characteristic
55	#103×C₅₉	C₂²	1	1	yes	characteristic
56	#103×C₅₉	C₂²	1	1	yes	characteristic
57	#103×C₅₉	C₂²	1	1	yes	characteristic
58	#103×C₅₉	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₅₉	—	2	1	no	—
60	C₂×Q₈×C₅₉	—	2	2	no	—
61	C₂×Q₈×C₅₉	—	4	1	no	—
62	C₂×Q₈×C₅₉	C₂³	1	1	yes	characteristic
63	C₂×Q₈×C₅₉	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₅₉	—	2	1	no	—
65	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
66	C₂×C₂²⋊D₅₉	—	2	1	no	—
67	C₂×C₂²⋊D₅₉	D₄	1	1	yes	characteristic
68	Q₈×D₅₉	—	2	2	no	—
69	Q₈×D₅₉	—	4	1	no	—
70	C₂×D₂₃₆	—	2	1	no	—
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	C₂³	1	1	yes	characteristic
76	C₂×Dic₁₁₈	—	2	1	no	—
77	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
78	M₁₆×C₅₉	—	4	1	no	—
79	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
80	C₂×C₈×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	C₄²×C₅₉	—	2	1	no	—
85	C₂²⋊Dic₅₉	—	2	1	no	—
86	C₂²⋊Dic₅₉	D₄	1	1	yes	characteristic
87	#11745671	—	2	2	no	—
88	#11745671	—	4	1	no	—
89	(C₂×D₅₉)⋊C₄	—	2	1	no	—
90	(C₂×D₅₉)⋊C₄	—	2	1	no	—
91	(C₂×D₅₉)⋊C₄	—	2	1	no	—
92	(C₂×D₅₉)⋊C₄	—	2	1	no	—
93	(C₂×D₅₉)⋊C₄	D₄	1	1	yes	characteristic
94	C₄⋊Dic₅₉	—	2	1	no	—
95	C₄⋊Dic₅₉	—	2	1	no	—
96	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
97	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
98	Dic₅₉⋊C₄	—	2	1	no	—
99	Dic₅₉⋊C₄	—	2	1	no	—
100	Dic₅₉⋊C₄	—	2	1	no	—
101	Dic₅₉⋊C₄	—	2	1	no	—
102	Dic₅₉⋊C₄	—	2	1	no	—
103	Dic₅₉⋊C₄	—	2	1	no	—
104	Dic₅₉⋊C₄	D₄	1	1	yes	characteristic
105	C₄×Dic₅₉	—	2	1	no	—
106	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
107	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
108	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
109	#11745660	—	4	1	no	—
110	C₈×D₅₉	—	4	1	no	—
111	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
112	C₂²×D₅₉	—	2	1	no	—
113	C₂²×D₅₉	C₂×D₄	1	1	yes	characteristic
114	Q₈×C₅₉	—	2	1	no	—
115	Q₈×C₅₉	—	2	1	no	—
116	Q₈×C₅₉	—	2	1	no	—
117	Q₈×C₅₉	—	4	2	no	—
118	Q₈×C₅₉	#41	1	2	yes	—
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	—	2	1	no	—
123	C₂×C₄×C₅₉	—	2	1	no	—
124	C₂×C₄×C₅₉	—	2	1	no	—
125	C₂×C₄×C₅₉	—	4	2	no	—
126	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
128	C₂²⋊D₅₉	—	2	2	no	—
129	C₂²⋊D₅₉	—	8	1	no	—
130	C₂×Dic₅₉	—	2	1	no	—
131	C₂×Dic₅₉	—	2	1	no	—
132	C₂×Dic₅₉	—	2	1	no	—
133	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
134	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
135	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
136	D₂₃₆	—	2	1	no	—
137	D₂₃₆	—	2	1	no	—
138	D₂₃₆	—	4	2	no	—
139	C₄×D₅₉	—	2	1	no	—
140	C₄×D₅₉	—	2	1	no	—
141	C₄×D₅₉	—	4	2	no	—
142	C₄×D₅₉	—	8	1	no	—
143	C₄×D₅₉	—	8	1	no	—
144	Dic₁₁₈	—	2	1	no	—
145	Dic₁₁₈	—	2	1	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	—	2	1	no	—
152	C₂²×C₅₉	—	2	1	no	—
153	C₂²×C₅₉	—	2	1	no	—
154	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
155	C₂×D₅₉	—	2	1	no	—
156	C₂×D₅₉	—	2	1	no	—
157	C₂×D₅₉	—	4	1	no	—
158	C₂×D₅₉	—	4	1	no	—
159	C₂×D₅₉	—	4	1	no	—
160	C₂×D₅₉	#137	1	2	yes	—
161	C₄×C₅₉	—	2	2	no	—
162	C₄×C₅₉	—	2	1	no	—
163	C₄×C₅₉	—	4	1	no	—
164	C₄×C₅₉	—	4	1	no	—
165	C₄×C₅₉	—	4	1	no	—
166	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
167	C₄×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
168	Dic₅₉	—	2	1	no	—
169	Dic₅₉	—	2	1	no	—
170	Dic₅₉	—	2	1	no	—
171	Dic₅₉	—	4	1	no	—
172	Dic₅₉	—	4	1	no	—
173	Dic₅₉	—	4	1	no	—
174	#3320	—	59	1	no	maximal, Sylow
175	C₂×C₅₉	—	4	1	no	—
176	C₂×C₅₉	#546	1	1	yes	characteristic
177	C₂×C₅₉	C₂×#137	1	1	yes	characteristic
178	C₂×C₅₉	#577	1	1	yes	characteristic
179	D₅₉	—	2	2	no	—
180	D₅₉	—	8	1	no	—
181	#553	—	59	1	no	—
182	D₄×Q₈	—	59	1	no	—
183	#491	—	59	1	no	—
184	#485	—	59	1	no	—
185	#484	—	59	1	no	—
186	#479	—	59	1	no	—
187	#468	—	59	1	no	—
188	#465	—	59	1	no	—
189	#461	—	59	1	no	—
190	#448	—	59	1	no	—
191	#448	—	59	1	no	—
192	#440	—	59	1	no	—
193	#435	—	59	1	no	—
194	#419	D₅₉	1	1	yes	characteristic, complemented
195	C₂×#103	—	59	1	no	—
196	C₅₉	#3320	1	1	yes	characteristic, complemented, Sylow
197	C₂²×Q₈	—	59	1	no	—
198	C₂²×Q₈	C₂×D₅₉	1	1	yes	characteristic
199	C₂×SD₁₆	—	59	1	no	—
200	C₂×SD₁₆	—	59	1	no	—
201	C₂×SD₁₆	—	59	1	no	—
202	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
203	C₂²×C₈	—	59	1	no	—
204	C₄⋊Q₈	—	59	1	no	—
205	C₄⋊Q₈	—	118	1	no	—
206	#126	—	118	1	no	—
207	#125	—	59	1	no	—
208	#123	—	118	1	no	—
209	C₂²⋊Q₈	—	59	1	no	—
210	C₂²⋊Q₈	—	59	1	no	—
211	C₂²⋊Q₈	—	118	1	no	—
212	C₂²⋊Q₈	—	118	1	no	—
213	#121	—	59	1	no	—
214	#121	—	59	1	no	—
215	#121	—	118	1	no	—
216	C₄×Q₈	—	59	1	no	—
217	C₄×D₄	—	59	1	no	—
218	C₄×D₄	—	59	1	no	—
219	C₄×D₄	—	118	1	no	—
220	C₄×D₄	—	118	1	no	—
221	#117	C₂×D₅₉	1	1	yes	characteristic
222	C₂×C₄⋊C₄	—	59	1	no	—
223	#107	—	59	1	no	—
224	#107	—	59	1	no	—
225	#106	—	59	1	no	—
226	C₄⋊C₈	—	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
233	#103	C₂×D₅₉	1	1	yes	characteristic
234	#103	C₂×D₅₉	1	1	yes	characteristic
235	#103	C₂×D₅₉	1	1	yes	characteristic
236	#102	—	59	1	no	—
237	C₂²⋊C₈	—	59	1	no	—
238	C₂²⋊C₈	—	59	1	no	—
239	#97	—	59	1	no	—
240	C₂×Q₈	—	2	1	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	4	1	no	—
243	C₂×Q₈	—	59	1	no	—
244	C₂×Q₈	—	118	1	no	—
245	C₂×Q₈	—	118	2	no	—
246	C₂×Q₈	—	236	1	no	—
247	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
248	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
249	C₂×D₄	—	59	1	no	—
250	C₂×D₄	—	59	1	no	—
251	C₂×D₄	—	118	1	no	—
252	C₂×D₄	—	118	1	no	—
253	C₂²×C₄	—	2	1	no	—
254	C₂²×C₄	—	59	1	no	—
255	C₂²×C₄	—	118	1	no	—
256	C₂²×C₄	—	118	1	no	—
257	C₂²×C₄	—	118	1	no	—
258	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
259	SD₁₆	—	118	2	no	—
260	SD₁₆	—	236	1	no	—
261	SD₁₆	—	236	1	no	—
262	M₁₆	—	4	1	no	—
263	C₂×C₈	—	59	1	no	—
264	C₂×C₈	—	59	1	no	—
265	C₂×C₈	—	236	1	no	—
266	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
267	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
268	C₄⋊C₄	—	59	1	no	—
269	C₄⋊C₄	—	59	1	no	—
270	C₄⋊C₄	—	59	1	no	—
271	C₄⋊C₄	—	118	1	no	—
272	C₄⋊C₄	—	118	1	no	—
273	C₄⋊C₄	—	118	1	no	—
274	C₄⋊C₄	—	118	1	no	—
275	C₄⋊C₄	—	118	1	no	—
276	C₄⋊C₄	—	118	1	no	—
277	C₄⋊C₄	—	118	1	no	—
278	C₄⋊C₄	—	118	1	no	—
279	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
280	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
281	C₂²⋊C₄	—	2	1	no	—
282	C₂²⋊C₄	—	59	1	no	—
283	C₂²⋊C₄	—	59	1	no	—
284	C₂²⋊C₄	—	118	1	no	—
285	C₂²⋊C₄	—	118	1	no	—
286	C₂²⋊C₄	—	118	1	no	—
287	C₂²⋊C₄	—	118	1	no	—
288	C₂²⋊C₄	—	118	1	no	—
289	C₄²	—	2	1	no	—
290	C₄²	—	59	1	no	—
291	C₄²	—	118	1	no	—
292	C₂³	—	59	1	no	—
293	C₂³	—	118	1	no	—
294	C₂³	D₄×D₅₉	1	1	yes	characteristic
295	Q₈	—	2	1	no	—
296	Q₈	—	2	1	no	—
297	Q₈	—	2	1	no	—
298	Q₈	—	4	2	no	—
299	Q₈	—	118	1	no	—
300	Q₈	—	118	1	no	—
301	Q₈	—	236	2	no	—
302	Q₈	#11745685	1	2	yes	—
303	D₄	—	118	2	no	—
304	D₄	—	118	1	no	—
305	D₄	—	118	1	no	—
306	D₄	—	472	1	no	—
307	D₄	—	236	2	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	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	1	no	—
319	C₂×C₄	—	118	1	no	—
320	C₂×C₄	—	118	1	no	—
321	C₂×C₄	—	118	1	no	—
322	C₂×C₄	—	118	1	no	—
323	C₂×C₄	—	472	1	no	—
324	C₂×C₄	—	472	1	no	—
325	C₂×C₄	—	236	2	no	—
326	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
327	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
328	C₈	—	2	1	no	—
329	C₈	—	2	1	no	—
330	C₈	—	118	1	no	—
331	C₈	—	118	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	59	2	no	—
336	C₂²	—	118	1	no	—
337	C₂²	—	118	1	no	—
338	C₂²	—	236	1	no	—
339	C₂²	—	236	1	no	—
340	C₂²	—	236	1	no	—
341	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
342	C₄	—	2	1	no	—
343	C₄	—	2	2	no	—
344	C₄	—	4	1	no	—
345	C₄	—	4	1	no	—
346	C₄	—	4	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	C₂×#11745685	1	1	yes	characteristic
354	C₄	#14273772	1	1	yes	characteristic
355	C₂	—	4	1	no	—
356	C₂	—	118	2	no	—
357	C₂	—	472	1	no	—
358	C₂	#16838042	1	1	yes	characteristic
359	C₂	#137×D₅₉	1	1	yes	characteristic
360	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×#419	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#419	D₅₉
hypocenter	C₅₉	#3320
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #419

Complements

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

N	H ≅ G/N	type
2. Q₈×C₂²⋊D₅₉	C₂	semidirect
3. #16837769	C₂	semidirect
4. #16837743	C₂	semidirect
5. #16837736	C₂	semidirect
6. #16837711	C₂	semidirect
7. #16837648	C₂	semidirect
8. #16837634	C₂	semidirect
9. #16837606	C₂	semidirect
10. #16837542	C₂	semidirect
11. #16837541	C₂	semidirect
12. #16837505	C₂	semidirect
13. #16837471	C₂	semidirect
14. #103×D₅₉	C₂	semidirect
15. C₅₉×#419	C₂	semidirect
194. #419	D₅₉	semidirect
196. C₅₉	#3320	semidirect