Subgroups — #21432934

Order 7552.

470 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14273759	C₂	1	1	yes	maximal, characteristic
2	#16837884	C₂	1	1	yes	maximal, characteristic
3	#16837878	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837509	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837507	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837501	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837500	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837494	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#11745660	C₂	1	1	yes	maximal, characteristic, complemented
10	#16837487	C₂	1	1	yes	maximal, characteristic, complemented
11	#16837443	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837397	C₂	1	1	yes	maximal, characteristic
13	#103×D₅₉	C₂	1	1	yes	maximal, characteristic, complemented
14	#16837356	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×Q₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11745689	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₅₉	C₂²	1	1	yes	characteristic
18	#14273759	C₄	1	8	yes	complemented
19	C₂×#11745672	C₂²	1	1	yes	characteristic
20	C₂×#11745660	C₂²	1	1	yes	characteristic
21	C₂×#11745671	C₂²	1	1	yes	characteristic
22	C₂×#11745659	C₂²	1	1	yes	characteristic
23	C₂×C₈×D₅₉	C₂²	1	1	yes	characteristic
24	Q₈×Dic₅₉	C₂²	1	1	yes	characteristic
25	Q₈×Dic₅₉	C₂²	1	1	yes	characteristic
26	C₄×Dic₁₁₈	—	2	1	no	—
27	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
28	#14273692	—	2	1	no	—
29	#14273692	C₂²	1	1	yes	characteristic
30	C₄×D₂₃₆	—	2	1	no	—
31	C₄×D₂₃₆	C₂²	1	1	yes	characteristic
32	#14273689	—	2	1	no	—
33	#14273689	C₂²	1	1	yes	characteristic
34	#14273685	—	2	1	no	—
35	#14273685	C₂²	1	1	yes	characteristic
36	#14273684	—	2	1	no	—
37	#14273684	—	2	1	no	—
38	#14273684	C₂²	1	1	yes	characteristic
39	C₄⋊C₄×D₅₉	—	2	1	no	—
40	C₄⋊C₄×D₅₉	C₂²	1	1	yes	characteristic
41	#14273660	C₂²	1	1	yes	characteristic
42	#14273659	C₂²	1	1	yes	characteristic
43	#14273658	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	#14273649	C₂²	1	1	yes	characteristic
49	#14273647	C₂²	1	1	yes	characteristic
50	#14273638	C₂²	1	1	yes	characteristic
51	#14273637	C₂²	1	1	yes	characteristic
52	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
53	C₂×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
55	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
56	#107×C₅₉	C₂²	1	1	yes	characteristic
57	#103×C₅₉	C₂²	1	1	yes	characteristic
58	#103×C₅₉	C₂²	1	1	yes	characteristic
59	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
60	C₂×Q₈×C₅₉	C₂³	1	1	yes	characteristic
61	C₂×Q₈×C₅₉	C₂³	1	1	yes	characteristic
62	#11745689	—	2	2	no	—
63	#11745689	C₂×C₄	1	4	yes	—
64	Q₈×D₅₉	—	2	2	no	—
65	Q₈×D₅₉	C₂×C₄	1	4	yes	complemented
66	C₂×D₂₃₆	—	2	1	no	—
67	C₂×D₂₃₆	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
72	C₂×C₄×D₅₉	—	2	1	no	—
73	C₂×C₄×D₅₉	C₂³	1	1	yes	characteristic
74	C₂×Dic₁₁₈	—	2	1	no	—
75	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
76	Q₁₆×C₅₉	C₂×C₄	1	4	yes	complemented
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	—	2	1	no	—
81	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
83	C₄²×C₅₉	—	2	1	no	—
84	C₄²×C₅₉	—	2	1	no	—
85	C₄²×C₅₉	C₂³	1	1	yes	characteristic
86	#11745672	C₂×C₄	1	4	yes	complemented
87	#11745671	C₂×C₄	1	4	yes	complemented
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	D₄	1	1	yes	characteristic
92	(C₂×D₅₉)⋊C₄	D₄	1	1	yes	characteristic
93	C₄⋊Dic₅₉	—	2	1	no	—
94	C₄⋊Dic₅₉	—	2	1	no	—
95	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
96	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
97	Dic₅₉⋊C₄	—	2	1	no	—
98	Dic₅₉⋊C₄	—	2	1	no	—
99	Dic₅₉⋊C₄	—	2	1	no	—
100	Dic₅₉⋊C₄	D₄	1	1	yes	characteristic
101	Dic₅₉⋊C₄	D₄	1	1	yes	characteristic
102	C₄×Dic₅₉	—	2	1	no	—
103	C₄×Dic₅₉	—	2	1	no	—
104	C₄×Dic₅₉	—	2	1	no	—
105	C₄×Dic₅₉	—	2	1	no	—
106	C₄×Dic₅₉	—	2	1	no	—
107	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
108	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
109	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
110	#11745660	C₂×C₄	1	4	yes	complemented
111	#11745659	C₂×C₄	1	4	yes	complemented
112	C₈×D₅₉	—	2	2	no	—
113	C₂²×D₅₉	—	2	1	no	—
114	C₂²×D₅₉	C₂×D₄	1	1	yes	characteristic
115	Q₈×C₅₉	—	2	1	no	—
116	Q₈×C₅₉	—	2	1	no	—
117	Q₈×C₅₉	C₂²×C₄	1	2	yes	—
118	Q₈×C₅₉	C₂²×C₄	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	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
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	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
132	D₂₃₆	—	2	4	no	—
133	D₂₃₆	—	2	1	no	—
134	D₂₃₆	C₂²×C₄	1	2	yes	—
135	C₄×D₅₉	—	2	4	no	—
136	C₄×D₅₉	—	2	4	no	—
137	C₄×D₅₉	—	8	1	no	—
138	C₄×D₅₉	—	8	1	no	—
139	C₄×D₅₉	—	8	1	no	—
140	C₄×D₅₉	C₂²×C₄	1	2	yes	—
141	C₄×D₅₉	#41	1	2	yes	—
142	Dic₁₁₈	—	2	1	no	—
143	Dic₁₁₈	—	2	4	no	—
144	Dic₁₁₈	C₂²×C₄	1	2	yes	—
145	C₈×C₅₉	—	2	1	no	—
146	C₈×C₅₉	C₂²×C₄	1	2	yes	—
147	C₅₉⋊C₈	—	2	1	no	—
148	C₅₉⋊C₈	C₂²×C₄	1	2	yes	—
149	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
150	C₂×D₅₉	—	2	1	no	—
151	C₂×D₅₉	—	2	1	no	—
152	C₂×D₅₉	—	2	2	no	—
153	C₂×D₅₉	—	4	1	no	—
154	C₂×D₅₉	—	4	1	no	—
155	C₂×D₅₉	C₄×D₄	1	2	yes	—
156	C₄×C₅₉	—	2	2	no	—
157	C₄×C₅₉	—	2	2	no	—
158	C₄×C₅₉	—	2	1	no	—
159	C₄×C₅₉	—	2	1	no	—
160	C₄×C₅₉	—	4	1	no	—
161	C₄×C₅₉	—	4	1	no	—
162	C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
163	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
164	Dic₅₉	—	2	2	no	—
165	Dic₅₉	—	2	1	no	—
166	Dic₅₉	—	2	1	no	—
167	Dic₅₉	—	4	1	no	—
168	Dic₅₉	—	4	1	no	—
169	Dic₅₉	C₄×D₄	1	2	yes	—
170	#2947	—	59	1	no	maximal, Sylow
171	C₂×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
172	C₂×C₅₉	#577	1	1	yes	characteristic
173	C₂×C₅₉	#578	1	1	yes	characteristic
174	D₅₉	—	2	2	no	—
175	D₅₉	—	4	2	no	—
176	C₂×#137	—	59	1	no	—
177	#520	—	59	1	no	—
178	#519	—	59	1	no	—
179	#441	—	59	1	no	—
180	#441	—	59	1	no	—
181	#440	—	59	1	no	—
182	#440	—	59	1	no	—
183	C₄×Q₁₆	—	59	1	no	—
184	C₄×Q₁₆	D₅₉	1	1	yes	characteristic, complemented
185	C₄×SD₁₆	—	59	1	no	—
186	C₄×SD₁₆	—	59	1	no	—
187	#428	—	59	1	no	—
188	#416	—	59	1	no	—
189	C₂×#103	—	59	1	no	—
190	#405	—	59	1	no	—
191	C₅₉	#2947	1	1	yes	characteristic, complemented, Sylow
192	C₂×#41	—	59	1	no	—
193	C₂²×Q₈	—	59	1	no	—
194	#137	—	59	8	no	—
195	C₂×Q₁₆	—	59	1	no	—
196	C₂×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
197	C₂×SD₁₆	—	59	1	no	—
198	C₂×SD₁₆	—	59	1	no	—
199	C₂×M₁₆	—	59	1	no	—
200	C₂²×C₈	—	59	1	no	—
201	C₄×Q₈	—	59	1	no	—
202	C₄×Q₈	—	59	1	no	—
203	C₄×Q₈	—	59	1	no	—
204	C₄×Q₈	—	59	1	no	—
205	C₄×Q₈	—	118	1	no	—
206	C₄×Q₈	—	118	1	no	—
207	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
208	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
209	C₄×D₄	—	59	1	no	—
210	C₄×D₄	—	59	1	no	—
211	C₄×D₄	—	118	1	no	—
212	C₄×D₄	—	118	1	no	—
213	#117	—	59	1	no	—
214	#117	—	59	1	no	—
215	#117	—	118	1	no	—
216	#117	—	118	1	no	—
217	#117	—	118	1	no	—
218	C₂×C₄⋊C₄	—	59	1	no	—
219	C₂×C₄⋊C₄	—	118	1	no	—
220	#107	—	59	1	no	—
221	#107	C₂×D₅₉	1	1	yes	characteristic
222	#106	—	59	1	no	—
223	#106	—	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	C₂×D₅₉	1	1	yes	characteristic
229	#103	C₂×D₅₉	1	1	yes	characteristic
230	#102	—	59	1	no	—
231	#102	—	59	1	no	—
232	#97	—	59	1	no	—
233	#97	—	59	1	no	—
234	C₄×C₈	—	59	1	no	—
235	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
236	#41	—	59	4	no	—
237	#41	—	118	2	no	—
238	C₂×Q₈	—	59	4	no	—
239	C₂×Q₈	—	59	1	no	—
240	C₂×Q₈	—	118	1	no	—
241	C₂×Q₈	—	118	2	no	—
242	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
243	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
244	C₂×D₄	—	59	1	no	—
245	C₂×D₄	—	118	1	no	—
246	C₂²×C₄	—	59	1	no	—
247	C₂²×C₄	—	118	1	no	—
248	C₂²×C₄	—	118	1	no	—
249	C₂²×C₄	—	118	1	no	—
250	C₂²×C₄	—	118	1	no	—
251	C₂²×C₄	—	118	1	no	—
252	Q₁₆	—	59	4	no	—
253	Q₁₆	C₄×D₅₉	1	4	yes	complemented
254	SD₁₆	—	59	4	no	—
255	SD₁₆	—	59	4	no	—
256	M₁₆	—	59	4	no	—
257	C₂×C₈	—	59	1	no	—
258	C₂×C₈	—	59	1	no	—
259	C₂×C₈	—	118	2	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	—	2	1	no	—
264	C₄⋊C₄	—	59	1	no	—
265	C₄⋊C₄	—	59	1	no	—
266	C₄⋊C₄	—	59	1	no	—
267	C₄⋊C₄	—	59	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	C₂²×D₅₉	1	1	yes	characteristic
274	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
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	C₄²	—	2	1	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	59	1	no	—
283	C₄²	—	118	1	no	—
284	C₄²	—	118	1	no	—
285	C₄²	—	118	1	no	—
286	C₄²	—	118	1	no	—
287	C₄²	—	118	1	no	—
288	C₄²	C₂²×D₅₉	1	1	yes	characteristic
289	C₂³	—	59	1	no	—
290	C₂³	—	118	1	no	—
291	Q₈	—	2	1	no	—
292	Q₈	—	2	1	no	—
293	Q₈	—	59	2	no	—
294	Q₈	—	118	4	no	—
295	Q₈	—	118	1	no	—
296	Q₈	C₂×C₄×D₅₉	1	2	yes	—
297	Q₈	C₂×C₄×D₅₉	1	2	yes	—
298	D₄	—	59	2	no	—
299	D₄	—	118	4	no	—
300	D₄	—	118	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	59	1	no	—
306	C₂×C₄	—	59	2	no	—
307	C₂×C₄	—	59	2	no	—
308	C₂×C₄	—	59	1	no	—
309	C₂×C₄	—	59	1	no	—
310	C₂×C₄	—	118	4	no	—
311	C₂×C₄	—	118	4	no	—
312	C₂×C₄	—	118	1	no	—
313	C₂×C₄	—	118	1	no	—
314	C₂×C₄	—	118	1	no	—
315	C₂×C₄	—	472	1	no	—
316	C₂×C₄	—	472	1	no	—
317	C₂×C₄	—	472	1	no	—
318	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
319	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
320	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
321	C₈	—	2	1	no	—
322	C₈	—	59	2	no	—
323	C₈	—	118	1	no	—
324	C₈	C₂×C₄×D₅₉	1	2	yes	—
325	C₂²	—	59	2	no	—
326	C₂²	—	118	1	no	—
327	C₂²	—	118	1	no	—
328	C₂²	—	118	2	no	—
329	C₂²	—	236	1	no	—
330	C₂²	—	236	1	no	—
331	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	2	2	no	—
335	C₄	—	2	2	no	—
336	C₄	—	4	1	no	—
337	C₄	—	4	1	no	—
338	C₄	—	59	2	no	—
339	C₄	—	118	1	no	—
340	C₄	—	118	1	no	—
341	C₄	—	118	2	no	—
342	C₄	—	236	1	no	—
343	C₄	—	236	1	no	—
344	C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
345	C₄	#41×D₅₉	1	1	yes	characteristic
346	C₂	—	118	2	no	—
347	C₂	—	236	2	no	—
348	C₂	C₄×D₄×D₅₉	1	1	yes	characteristic
349	C₂	#16838209	1	1	yes	characteristic
350	C₂	#16838215	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₉
commutator	C₄×C₅₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₄×Q₁₆×C₅₉	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	C₄×Q₁₆	D₅₉
hypocenter	C₅₉	#2947
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
3. #16837878	C₂	semidirect
4. #16837509	C₂	semidirect
5. #16837507	C₂	semidirect
6. #16837501	C₂	semidirect
7. #16837500	C₂	semidirect
8. #16837494	C₂	semidirect
9. C₄×#11745660	C₂	semidirect
10. #16837487	C₂	semidirect
11. #16837443	C₂	semidirect
13. #103×D₅₉	C₂	semidirect
14. #16837356	C₂	semidirect
15. C₄×Q₁₆×C₅₉	C₂	semidirect
18. #14273759	C₄	semidirect
65. Q₈×D₅₉	C₂×C₄	semidirect
76. Q₁₆×C₅₉	C₂×C₄	semidirect
86. #11745672	C₂×C₄	semidirect
87. #11745671	C₂×C₄	semidirect
110. #11745660	C₂×C₄	semidirect
111. #11745659	C₂×C₄	semidirect
184. C₄×Q₁₆	D₅₉	semidirect
191. C₅₉	#2947	semidirect
253. Q₁₆	C₄×D₅₉	semidirect