Subgroups — #21435968

Order 7552.

388 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16838107	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₂²⋊D₅₉	C₂	1	1	yes	maximal, characteristic
3	#16837751	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837737	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837736	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837695	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837643	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837642	C₂	1	1	yes	maximal, characteristic, complemented
9	#16837611	C₂	1	1	yes	maximal, characteristic, complemented
10	#16837559	C₂	1	1	yes	maximal, characteristic, complemented
11	#16837558	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837494	C₂	1	1	yes	maximal, characteristic, complemented
13	C₈×C₂²⋊D₅₉	C₂	1	1	yes	maximal, characteristic
14	#103×D₅₉	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#103×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₅₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₃₆	C₂²	1	1	yes	characteristic
18	C₂×#11745672	C₂²	1	1	yes	characteristic
19	C₂×#11745672	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₅₉	C₂²	1	1	yes	characteristic
21	#14273726	C₂²	1	1	yes	characteristic
22	#14273725	—	2	1	no	—
23	#14273725	C₂²	1	1	yes	characteristic
24	#14273717	—	2	1	no	—
25	#14273707	C₂²	1	1	yes	characteristic
26	#14273706	—	2	1	no	—
27	#14273706	—	2	1	no	—
28	#14273706	C₂²	1	1	yes	characteristic
29	#14273706	C₂²	1	1	yes	characteristic
30	#14273705	—	2	1	no	—
31	#14273703	—	2	1	no	—
32	#14273703	C₂²	1	1	yes	characteristic
33	Q₈×Dic₅₉	C₂²	1	1	yes	characteristic
34	#14273692	C₂²	1	1	yes	characteristic
35	C₄×C₂²⋊D₅₉	—	2	1	no	—
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	#14273660	C₂²	1	1	yes	characteristic
42	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
43	#14273652	C₂²	1	1	yes	characteristic
44	#14273652	C₂²	1	1	yes	characteristic
45	#14273651	C₂²	1	1	yes	characteristic
46	#14273651	C₂²	1	1	yes	characteristic
47	#14273650	C₂²	1	1	yes	characteristic
48	#14273650	C₂²	1	1	yes	characteristic
49	C₂²⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
50	(C₂×D₅₉)⋊C₈	C₂²	1	1	yes	characteristic
51	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
52	C₂²×Q₈×C₅₉	C₂²	1	1	yes	characteristic
53	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
54	C₂×C₄⋊C₄×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	—	2	1	no	—
66	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊D₅₉	D₄	1	1	yes	characteristic
68	Q₈×D₅₉	—	2	2	no	—
69	Q₈×D₅₉	—	4	1	no	—
70	C₂×C₄×D₅₉	—	2	1	no	—
71	C₂×C₄×D₅₉	—	2	1	no	—
72	C₂×C₄×D₅₉	C₂³	1	1	yes	characteristic
73	C₂×Dic₁₁₈	—	2	1	no	—
74	C₂×Dic₁₁₈	—	2	1	no	—
75	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
76	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₅₉	—	4	1	no	—
78	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
79	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
80	C₄⋊C₄×C₅₉	—	2	1	no	—
81	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
83	C₂²⋊Dic₅₉	—	2	1	no	—
84	C₂²⋊Dic₅₉	—	2	1	no	—
85	C₂²⋊Dic₅₉	D₄	1	1	yes	characteristic
86	#11745672	—	2	2	no	—
87	#11745672	—	4	1	no	—
88	(C₂×D₅₉)⋊C₄	—	2	1	no	—
89	(C₂×D₅₉)⋊C₄	—	2	1	no	—
90	(C₂×D₅₉)⋊C₄	D₄	1	1	yes	characteristic
91	C₄⋊Dic₅₉	—	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	Dic₅₉⋊C₄	—	2	1	no	—
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	Dic₅₉⋊C₄	—	2	1	no	—
102	Dic₅₉⋊C₄	—	2	1	no	—
103	Dic₅₉⋊C₄	D₄	1	1	yes	characteristic
104	C₄×Dic₅₉	—	2	1	no	—
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	Dic₂₃₆	—	4	1	no	—
110	C₈×D₅₉	—	4	1	no	—
111	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
112	C₂²×D₅₉	C₂×D₄	1	1	yes	characteristic
113	Q₈×C₅₉	—	2	1	no	—
114	Q₈×C₅₉	—	2	1	no	—
115	Q₈×C₅₉	—	2	1	no	—
116	Q₈×C₅₉	—	4	2	no	—
117	Q₈×C₅₉	#41	1	2	yes	—
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	—	2	1	no	—
123	C₂×C₄×C₅₉	—	2	1	no	—
124	C₂×C₄×C₅₉	—	4	2	no	—
125	C₂×C₄×C₅₉	—	8	1	no	—
126	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
128	C₂²⋊D₅₉	Q₁₆	1	4	yes	complemented
129	C₂×Dic₅₉	—	2	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	C₄×D₅₉	—	2	1	no	—
137	C₄×D₅₉	—	2	1	no	—
138	C₄×D₅₉	—	4	2	no	—
139	C₄×D₅₉	—	8	1	no	—
140	Dic₁₁₈	—	2	1	no	—
141	Dic₁₁₈	—	2	1	no	—
142	Dic₁₁₈	—	2	1	no	—
143	Dic₁₁₈	—	2	1	no	—
144	Dic₁₁₈	—	4	2	no	—
145	Dic₁₁₈	—	8	1	no	—
146	C₈×C₅₉	—	2	1	no	—
147	C₈×C₅₉	—	2	1	no	—
148	C₅₉⋊C₈	—	2	1	no	—
149	C₅₉⋊C₈	—	2	1	no	—
150	C₂²×C₅₉	—	2	1	no	—
151	C₂²×C₅₉	—	2	1	no	—
152	C₂²×C₅₉	C₂×Q₁₆	1	2	yes	—
153	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
154	C₂×D₅₉	—	2	1	no	—
155	C₂×D₅₉	—	2	1	no	—
156	C₂×D₅₉	C₂×Q₁₆	1	2	yes	—
157	C₄×C₅₉	—	2	2	no	—
158	C₄×C₅₉	—	2	1	no	—
159	C₄×C₅₉	—	4	1	no	—
160	C₄×C₅₉	—	4	1	no	—
161	C₄×C₅₉	—	4	1	no	—
162	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
163	C₄×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
164	Dic₅₉	—	2	1	no	—
165	Dic₅₉	—	2	1	no	—
166	Dic₅₉	—	4	1	no	—
167	Dic₅₉	—	4	1	no	—
168	Dic₅₉	—	4	1	no	—
169	Dic₅₉	—	4	1	no	—
170	Dic₅₉	C₂×Q₁₆	1	2	yes	—
171	#3299	—	59	1	no	maximal, Sylow
172	C₂×C₅₉	—	2	2	no	—
173	C₂×C₅₉	#546	1	1	yes	characteristic
174	C₂×C₅₉	C₂²×Q₁₆	1	1	yes	characteristic
175	C₂×C₅₉	#577	1	1	yes	characteristic
176	D₅₉	—	2	2	no	—
177	#554	—	59	1	no	—
178	D₄×Q₈	—	59	1	no	—
179	#487	—	59	1	no	—
180	#484	—	59	1	no	—
181	#484	—	59	1	no	—
182	#477	—	59	1	no	—
183	#467	—	59	1	no	—
184	#467	—	59	1	no	—
185	#462	—	59	1	no	—
186	#451	—	59	1	no	—
187	#451	—	59	1	no	—
188	C₄×Q₁₆	—	59	1	no	—
189	C₈×D₄	—	59	1	no	—
190	C₂×#103	—	59	1	no	—
191	C₂×#103	D₅₉	1	1	yes	characteristic, complemented
192	C₅₉	#3299	1	1	yes	characteristic, complemented, Sylow
193	C₂²×Q₈	—	59	1	no	—
194	C₂²×Q₈	C₂×D₅₉	1	1	yes	characteristic
195	C₂×Q₁₆	—	59	1	no	—
196	C₂×Q₁₆	—	59	1	no	—
197	C₂×Q₁₆	—	59	1	no	—
198	C₂²×C₈	—	59	1	no	—
199	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
200	C₄⋊Q₈	—	59	1	no	—
201	C₄⋊Q₈	—	59	1	no	—
202	C₄⋊Q₈	—	118	1	no	—
203	#125	—	118	1	no	—
204	C₂²⋊Q₈	—	59	1	no	—
205	C₂²⋊Q₈	—	59	1	no	—
206	C₂²⋊Q₈	—	59	1	no	—
207	C₂²⋊Q₈	—	59	1	no	—
208	C₂²⋊Q₈	—	118	1	no	—
209	C₂²⋊Q₈	—	118	1	no	—
210	C₂²⋊Q₈	—	118	1	no	—
211	C₂²⋊Q₈	—	118	1	no	—
212	C₄×Q₈	—	59	1	no	—
213	C₄×Q₈	—	59	1	no	—
214	C₄×D₄	—	59	1	no	—
215	C₄×D₄	—	118	1	no	—
216	C₄×D₄	—	118	1	no	—
217	C₂×C₄⋊C₄	—	59	1	no	—
218	C₂×C₄⋊C₄	C₂×D₅₉	1	1	yes	characteristic
219	#107	—	59	1	no	—
220	#107	—	59	1	no	—
221	#107	—	59	1	no	—
222	C₄⋊C₈	—	59	1	no	—
223	#103	—	59	1	no	—
224	#103	—	59	1	no	—
225	#103	—	59	1	no	—
226	#103	—	59	1	no	—
227	#103	—	59	1	no	—
228	#103	—	59	1	no	—
229	#103	C₂×D₅₉	1	1	yes	characteristic
230	#103	C₂×D₅₉	1	1	yes	characteristic
231	#103	C₂×D₅₉	1	1	yes	characteristic
232	#103	C₂×D₅₉	1	1	yes	characteristic
233	C₂²⋊C₈	—	59	1	no	—
234	C₂²⋊C₈	—	59	1	no	—
235	C₄×C₈	—	59	1	no	—
236	C₂×Q₈	—	2	1	no	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	—	4	1	no	—
239	C₂×Q₈	—	59	1	no	—
240	C₂×Q₈	—	59	1	no	—
241	C₂×Q₈	—	118	1	no	—
242	C₂×Q₈	—	118	1	no	—
243	C₂×Q₈	—	118	2	no	—
244	C₂×Q₈	—	236	1	no	—
245	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
246	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
247	C₂×D₄	—	59	1	no	—
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	59	1	no	—
251	C₂²×C₄	—	118	1	no	—
252	C₂²×C₄	—	118	1	no	—
253	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
254	Q₁₆	—	118	2	no	—
255	Q₁₆	—	236	1	no	—
256	Q₁₆	—	236	1	no	—
257	C₂×C₈	—	4	1	no	—
258	C₂×C₈	—	59	1	no	—
259	C₂×C₈	—	59	1	no	—
260	C₂×C₈	—	236	1	no	—
261	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
262	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	59	1	no	—
265	C₄⋊C₄	—	59	1	no	—
266	C₄⋊C₄	—	59	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	—	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	C₂²×D₅₉	1	1	yes	characteristic
278	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
279	C₂²⋊C₄	—	59	1	no	—
280	C₂²⋊C₄	—	59	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	C₄²	—	59	1	no	—
286	C₄²	—	118	1	no	—
287	C₄²	—	118	1	no	—
288	C₂³	—	59	1	no	—
289	C₂³	D₄×D₅₉	1	1	yes	characteristic
290	Q₈	—	2	1	no	—
291	Q₈	—	2	1	no	—
292	Q₈	—	2	1	no	—
293	Q₈	—	4	2	no	—
294	Q₈	—	118	1	no	—
295	Q₈	—	118	1	no	—
296	Q₈	—	118	1	no	—
297	Q₈	—	118	1	no	—
298	Q₈	—	472	1	no	—
299	Q₈	—	236	2	no	—
300	Q₈	#11745685	1	2	yes	—
301	D₄	—	59	4	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	8	1	no	—
310	C₂×C₄	—	59	1	no	—
311	C₂×C₄	—	59	1	no	—
312	C₂×C₄	—	59	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	—	118	1	no	—
318	C₂×C₄	—	118	1	no	—
319	C₂×C₄	—	472	1	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	—	59	2	no	—
330	C₂²	—	118	1	no	—
331	C₂²	—	118	1	no	—
332	C₂²	Q₁₆×D₅₉	1	2	yes	—
333	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
334	C₄	—	2	1	no	—
335	C₄	—	2	2	no	—
336	C₄	—	4	1	no	—
337	C₄	—	4	1	no	—
338	C₄	—	4	1	no	—
339	C₄	—	59	2	no	—
340	C₄	—	118	1	no	—
341	C₄	—	118	1	no	—
342	C₄	—	236	1	no	—
343	C₄	—	236	1	no	—
344	C₄	—	236	1	no	—
345	C₄	—	236	1	no	—
346	C₄	C₂×#11745685	1	1	yes	characteristic
347	C₄	#14273772	1	1	yes	characteristic
348	C₂	—	2	2	no	—
349	C₂	—	118	2	no	—
350	C₂	#16838042	1	1	yes	characteristic
351	C₂	C₂×Q₁₆×D₅₉	1	1	yes	characteristic
352	C₂	#16838208	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₂×#103×C₅₉	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	C₂×#103	D₅₉
hypocenter	C₅₉	#3299
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#103

Complements

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

N	H ≅ G/N	type
3. #16837751	C₂	semidirect
4. #16837737	C₂	semidirect
5. #16837736	C₂	semidirect
6. #16837695	C₂	semidirect
7. #16837643	C₂	semidirect
8. #16837642	C₂	semidirect
9. #16837611	C₂	semidirect
10. #16837559	C₂	semidirect
11. #16837558	C₂	semidirect
12. #16837494	C₂	semidirect
14. #103×D₅₉	C₂	semidirect
15. C₂×#103×C₅₉	C₂	semidirect
128. C₂²⋊D₅₉	Q₁₆	semidirect
191. C₂×#103	D₅₉	semidirect
192. C₅₉	#3299	semidirect