Subgroups — #21436686

Order 7552.

386 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	#16838116	C₂	1	1	yes	maximal, characteristic, complemented
2	#16838052	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837799	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837724	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837723	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837719	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837718	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837690	C₂	1	1	yes	maximal, characteristic, complemented
9	#16837680	C₂	1	1	yes	maximal, characteristic, complemented
10	#16837652	C₂	1	1	yes	maximal, characteristic, complemented
11	D₈×Dic₅₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837478	C₂	1	1	yes	maximal, characteristic, complemented
13	#16837444	C₂	1	1	yes	maximal, characteristic, complemented
14	#16837441	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₉×#469	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11745687	—	2	1	no	—
17	C₂×#11745659	C₂²	1	1	yes	characteristic
18	#14273726	C₂²	1	1	yes	characteristic
19	#14273726	C₂²	1	1	yes	characteristic
20	#14273714	—	2	1	no	—
21	#14273713	—	2	1	no	—
22	#14273709	—	2	1	no	—
23	#14273708	—	2	1	no	—
24	#14273704	—	2	1	no	—
25	#14273699	—	2	1	no	—
26	#14273698	C₂²	1	1	yes	characteristic
27	#14273698	C₂²	1	1	yes	characteristic
28	D₄×Dic₅₉	C₂²	1	1	yes	characteristic
29	D₄×Dic₅₉	C₂²	1	1	yes	characteristic
30	C₄×C₂²⋊D₅₉	C₂²	1	1	yes	characteristic
31	#14273686	C₂²	1	1	yes	characteristic
32	#14273685	C₂²	1	1	yes	characteristic
33	#14273683	—	2	1	no	—
34	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
35	C₄⋊C₄×D₅₉	C₂²	1	1	yes	characteristic
36	#14273661	C₂²	1	1	yes	characteristic
37	#14273661	C₂²	1	1	yes	characteristic
38	#14273660	C₂²	1	1	yes	characteristic
39	#14273659	C₂²	1	1	yes	characteristic
40	#14273659	C₂²	1	1	yes	characteristic
41	#14273658	C₂²	1	1	yes	characteristic
42	#14273658	C₂²	1	1	yes	characteristic
43	#14273658	C₂²	1	1	yes	characteristic
44	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
45	#14273648	C₂²	1	1	yes	characteristic
46	#14273648	C₂²	1	1	yes	characteristic
47	#14273648	C₂²	1	1	yes	characteristic
48	#14273648	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	C₂×D₈×C₅₉	C₂²	1	1	yes	characteristic
53	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
54	#121×C₅₉	C₂²	1	1	yes	characteristic
55	#121×C₅₉	C₂²	1	1	yes	characteristic
56	#106×C₅₉	C₂²	1	1	yes	characteristic
57	#102×C₅₉	C₂²	1	1	yes	characteristic
58	#102×C₅₉	C₂²	1	1	yes	characteristic
59	C₂×D₄×C₅₉	—	2	1	no	—
60	C₂×D₄×C₅₉	—	2	1	no	—
61	C₂×D₄×C₅₉	C₂³	1	1	yes	characteristic
62	C₂×D₄×C₅₉	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊D₅₉	—	2	1	no	—
65	C₂×C₂²⋊D₅₉	—	2	1	no	—
66	C₂×C₂²⋊D₅₉	D₄	1	1	yes	characteristic
67	C₂²×Dic₅₉	—	2	1	no	—
68	C₂²×Dic₅₉	—	2	1	no	—
69	C₂²×Dic₅₉	—	2	1	no	—
70	#11745687	—	4	4	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	D₈×C₅₉	—	2	2	no	—
76	M₁₆×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	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
81	C₂²⋊C₄×C₅₉	—	2	1	no	—
82	C₂²⋊C₄×C₅₉	—	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	D₄	1	1	yes	characteristic
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	C₂³	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
100	C₄×Dic₅₉	—	2	1	no	—
101	C₄×Dic₅₉	—	2	1	no	—
102	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
103	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
104	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
105	#11745659	—	2	2	no	—
106	C₂³×C₅₉	—	2	1	no	—
107	C₂³×C₅₉	—	2	1	no	—
108	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
109	C₂²×D₅₉	C₂×D₄	1	1	yes	characteristic
110	D₄×C₅₉	—	2	1	no	—
111	D₄×C₅₉	—	2	1	no	—
112	D₄×C₅₉	—	2	1	no	—
113	D₄×C₅₉	—	2	1	no	—
114	D₄×C₅₉	—	4	2	no	—
115	D₄×C₅₉	—	8	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	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
122	C₂²⋊D₅₉	—	2	2	no	—
123	C₂²⋊D₅₉	—	4	2	no	—
124	C₂²⋊D₅₉	—	8	1	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	—	4	2	no	—
130	C₂×Dic₅₉	—	4	2	no	—
131	C₂×Dic₅₉	—	4	2	no	—
132	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
133	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
134	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
135	C₄×D₅₉	—	2	1	no	—
136	C₄×D₅₉	—	2	1	no	—
137	C₄×D₅₉	—	4	2	no	—
138	Dic₁₁₈	—	4	2	no	—
139	Dic₁₁₈	—	8	1	no	—
140	C₈×C₅₉	—	2	1	no	—
141	C₈×C₅₉	#41	1	2	yes	complemented
142	C₅₉⋊C₈	—	2	1	no	—
143	C₅₉⋊C₈	—	2	1	no	—
144	C₂²×C₅₉	—	2	1	no	—
145	C₂²×C₅₉	—	2	1	no	—
146	C₂²×C₅₉	—	2	1	no	—
147	C₂²×C₅₉	—	4	1	no	—
148	C₂²×C₅₉	—	4	1	no	—
149	C₂²×C₅₉	—	4	1	no	—
150	C₂²×C₅₉	—	4	1	no	—
151	C₂²×C₅₉	—	4	1	no	—
152	C₂²×C₅₉	—	4	1	no	—
153	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
154	C₂×D₅₉	—	2	1	no	—
155	C₂×D₅₉	—	2	1	no	—
156	C₂×D₅₉	—	2	1	no	—
157	C₄×C₅₉	—	2	1	no	—
158	C₄×C₅₉	—	4	1	no	—
159	C₄×C₅₉	—	4	1	no	—
160	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
161	C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
162	Dic₅₉	—	2	1	no	—
163	Dic₅₉	—	2	1	no	—
164	Dic₅₉	—	4	1	no	—
165	Dic₅₉	—	4	1	no	—
166	Dic₅₉	—	4	1	no	—
167	Dic₅₉	—	4	1	no	—
168	Dic₅₉	AGL(1,ℤ/8)	1	2	yes	—
169	#3354	—	59	1	no	maximal, Sylow
170	C₂×C₅₉	—	4	1	no	—
171	C₂×C₅₉	—	8	1	no	—
172	C₂×C₅₉	—	8	1	no	—
173	C₂×C₅₉	#547	1	1	yes	characteristic
174	C₂×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
175	C₂×C₅₉	#577	1	1	yes	characteristic
176	D₅₉	—	4	1	no	—
177	#555	—	59	1	no	—
178	#547	—	59	1	no	—
179	#498	—	59	1	no	—
180	#482	—	59	1	no	—
181	#482	—	59	1	no	—
182	#481	—	59	1	no	—
183	#481	—	59	1	no	—
184	#476	—	59	1	no	—
185	#474	—	59	1	no	—
186	#469	—	59	1	no	—
187	#469	D₅₉	1	1	yes	characteristic, complemented
188	C₄×D₈	—	59	1	no	—
189	#436	—	59	1	no	—
190	#428	—	59	1	no	—
191	#428	—	59	1	no	—
192	C₅₉	#3354	1	1	yes	characteristic, complemented, Sylow
193	C₂×#41	—	118	1	no	—
194	C₂×D₈	C₂×D₅₉	1	1	yes	characteristic
195	C₂×M₁₆	—	59	1	no	—
196	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
197	C₄⋊Q₈	—	59	1	no	—
198	C₄⋊Q₈	—	59	1	no	—
199	#124	—	118	1	no	—
200	#124	—	118	1	no	—
201	#123	—	118	1	no	—
202	#123	—	118	1	no	—
203	C₂²⋊Q₈	—	118	1	no	—
204	#121	—	59	1	no	—
205	#121	—	59	1	no	—
206	#121	—	118	1	no	—
207	#121	C₂×D₅₉	1	1	yes	characteristic
208	#121	C₂×D₅₉	1	1	yes	characteristic
209	C₄×D₄	—	59	1	no	—
210	C₄×D₄	—	59	1	no	—
211	C₄×D₄	—	59	1	no	—
212	#117	—	59	1	no	—
213	#117	—	59	1	no	—
214	#117	—	118	1	no	—
215	C₂×C₄⋊C₄	—	59	1	no	—
216	C₂×C₄⋊C₄	—	59	1	no	—
217	#107	—	59	1	no	—
218	#107	—	59	1	no	—
219	#107	—	59	1	no	—
220	#106	—	59	1	no	—
221	#106	—	59	1	no	—
222	#106	—	59	1	no	—
223	#106	—	59	1	no	—
224	#106	—	59	1	no	—
225	#106	C₂×D₅₉	1	1	yes	characteristic
226	C₄⋊C₈	—	59	1	no	—
227	#102	—	59	1	no	—
228	#102	—	59	1	no	—
229	#102	—	59	1	no	—
230	#102	—	59	1	no	—
231	#102	C₂×D₅₉	1	1	yes	characteristic
232	#102	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	#41	—	236	4	no	—
237	C₂×Q₈	—	118	1	no	—
238	C₂×Q₈	—	118	1	no	—
239	C₂×D₄	—	2	1	no	—
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	—	59	1	no	—
242	C₂×D₄	—	118	1	no	—
243	C₂×D₄	—	118	1	no	—
244	C₂×D₄	C₂²×D₅₉	1	1	yes	characteristic
245	C₂×D₄	C₂²×D₅₉	1	1	yes	characteristic
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	C₂²×D₅₉	1	1	yes	characteristic
252	D₈	—	2	2	no	—
253	M₁₆	—	2	2	no	—
254	M₁₆	—	118	2	no	—
255	C₂×C₈	—	59	1	no	—
256	C₂×C₈	—	59	1	no	—
257	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
258	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
259	C₄⋊C₄	—	59	1	no	—
260	C₄⋊C₄	—	59	1	no	—
261	C₄⋊C₄	—	59	1	no	—
262	C₄⋊C₄	—	59	1	no	—
263	C₄⋊C₄	—	59	1	no	—
264	C₄⋊C₄	—	118	1	no	—
265	C₄⋊C₄	—	118	1	no	—
266	C₄⋊C₄	—	118	1	no	—
267	C₄⋊C₄	—	118	1	no	—
268	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
269	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
270	C₂²⋊C₄	—	2	1	no	—
271	C₂²⋊C₄	—	2	1	no	—
272	C₂²⋊C₄	—	59	1	no	—
273	C₂²⋊C₄	—	59	1	no	—
274	C₂²⋊C₄	—	118	1	no	—
275	C₂²⋊C₄	—	118	1	no	—
276	C₂²⋊C₄	—	118	1	no	—
277	C₂²⋊C₄	—	118	1	no	—
278	C₂²⋊C₄	—	118	1	no	—
279	C₂²⋊C₄	—	118	1	no	—
280	C₄²	—	59	1	no	—
281	C₄²	—	118	1	no	—
282	C₄²	—	118	1	no	—
283	C₂³	—	2	1	no	—
284	C₂³	—	2	1	no	—
285	C₂³	—	59	1	no	—
286	C₂³	D₄×D₅₉	1	1	yes	characteristic
287	Q₈	—	472	1	no	—
288	Q₈	—	236	2	no	—
289	D₄	—	2	1	no	—
290	D₄	—	2	1	no	—
291	D₄	—	2	1	no	—
292	D₄	—	2	1	no	—
293	D₄	—	4	2	no	—
294	D₄	—	8	1	no	—
295	D₄	—	118	2	no	—
296	D₄	—	472	1	no	—
297	D₄	—	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	—	59	1	no	—
303	C₂×C₄	—	59	1	no	—
304	C₂×C₄	—	59	1	no	—
305	C₂×C₄	—	118	1	no	—
306	C₂×C₄	—	118	1	no	—
307	C₂×C₄	—	118	1	no	—
308	C₂×C₄	—	118	1	no	—
309	C₂×C₄	—	118	1	no	—
310	C₂×C₄	—	118	1	no	—
311	C₂×C₄	—	236	2	no	—
312	C₂×C₄	—	236	2	no	—
313	C₂×C₄	—	236	2	no	—
314	C₂×C₄	—	236	2	no	—
315	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
316	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
317	C₈	—	2	1	no	—
318	C₈	—	118	1	no	—
319	C₈	—	118	1	no	—
320	C₈	#11745687	1	2	yes	complemented
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	—	4	1	no	—
326	C₂²	—	4	1	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	—	118	1	no	—
331	C₂²	—	118	1	no	—
332	C₂²	—	118	1	no	—
333	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
334	C₄	—	2	1	no	—
335	C₄	—	4	1	no	—
336	C₄	—	4	1	no	—
337	C₄	—	59	2	no	—
338	C₄	—	118	1	no	—
339	C₄	—	118	1	no	—
340	C₄	—	236	1	no	—
341	C₄	—	236	1	no	—
342	C₄	—	236	1	no	—
343	C₄	—	236	1	no	—
344	C₄	C₂×#11745687	1	1	yes	characteristic
345	C₄	#14273774	1	1	yes	characteristic
346	C₂	—	4	1	no	—
347	C₂	—	8	1	no	—
348	C₂	—	8	1	no	—
349	C₂	—	236	1	no	—
350	C₂	#16838052	1	1	yes	characteristic
351	C₂	AGL(1,ℤ/8)×D₅₉	1	1	yes	characteristic
352	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×#469	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#469	D₅₉
hypocenter	C₅₉	#3354
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #469

Complements

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

N	H ≅ G/N	type
1. #16838116	C₂	semidirect
2. #16838052	C₂	semidirect
3. #16837799	C₂	semidirect
4. #16837724	C₂	semidirect
5. #16837723	C₂	semidirect
6. #16837719	C₂	semidirect
7. #16837718	C₂	semidirect
8. #16837690	C₂	semidirect
9. #16837680	C₂	semidirect
10. #16837652	C₂	semidirect
11. D₈×Dic₅₉	C₂	semidirect
12. #16837478	C₂	semidirect
13. #16837444	C₂	semidirect
14. #16837441	C₂	semidirect
15. C₅₉×#469	C₂	semidirect
141. C₈×C₅₉	#41	semidirect
187. #469	D₅₉	semidirect
192. C₅₉	#3354	semidirect
320. C₈	#11745687	semidirect