Subgroups — #20291473

Order 6784.

470 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811692	C₂	1	1	yes	maximal, characteristic
2	#16811691	C₂	1	1	yes	maximal, characteristic, complemented
3	#16811663	C₂	1	2	yes	maximal, complemented
4	#16811662	C₂	1	2	yes	maximal, complemented
5	Q₁₆×#6157	C₂	1	2	yes	maximal, complemented
6	SD₁₆×#6157	C₂	1	2	yes	maximal, complemented
7	#16811634	C₂	1	1	yes	maximal, characteristic, complemented
8	#16811613	C₂	1	1	yes	maximal, characteristic
9	C₂×#13133280	C₂	1	1	yes	maximal, characteristic, complemented
10	#16811589	C₂	1	1	yes	maximal, characteristic, complemented
11	#137×D₅₃	C₂	1	1	yes	maximal, characteristic
12	C₂×D₅₃⋊C₈	C₂²	1	1	yes	characteristic
13	Q₈×#6157	—	2	1	no	—
14	Q₈×#6157	—	2	1	no	—
15	Q₈×#6157	C₂²	1	2	yes	—
16	Q₈×#6157	C₂²	1	2	yes	—
17	Q₈×#6157	C₂²	1	2	yes	—
18	D₄×#6157	—	2	2	no	—
19	D₄×#6157	C₂²	1	2	yes	—
20	#13133296	—	2	2	no	—
21	#13133296	—	2	1	no	—
22	#13133296	C₂²	1	1	yes	characteristic
23	#13133296	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊#6157	—	2	1	no	—
25	C₂×C₄⋊#6157	C₂²	1	1	yes	characteristic
26	#13133286	C₂²	1	2	yes	—
27	#13133285	C₂²	1	2	yes	—
28	#13133280	C₂²	1	2	yes	—
29	#13133280	C₂²	1	2	yes	—
30	#13133280	C₂²	1	1	yes	characteristic
31	#13133280	C₂²	1	1	yes	characteristic
32	#13133279	C₂²	1	1	yes	characteristic
33	#13133279	C₂²	1	1	yes	characteristic
34	#13133271	C₂²	1	2	yes	—
35	C₈×#6157	C₂²	1	2	yes	—
36	#41×D₅₃	C₂²	1	1	yes	characteristic
37	C₂×Q₈×D₅₃	C₂²	1	1	yes	characteristic
38	#13133250	C₄	1	1	yes	characteristic, complemented
39	#13133249	C₄	1	2	yes	complemented
40	#13133248	C₄	1	1	yes	characteristic, complemented
41	#13133247	C₄	1	1	yes	characteristic, complemented
42	#13133246	C₄	1	2	yes	complemented
43	Q₁₆×D₅₃	C₂²	1	2	yes	—
44	SD₁₆×D₅₃	C₂²	1	2	yes	—
45	M₁₆×D₅₃	C₂²	1	1	yes	characteristic
46	#137×C₅₃	C₄	1	1	yes	characteristic, complemented
47	C₂²×#6157	—	2	1	no	—
48	C₂²×#6157	—	2	2	no	—
49	#41×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
51	#11719949	C₂×C₄	1	1	yes	characteristic
52	Q₈×D₅₃	—	2	1	no	—
53	Q₈×D₅₃	—	2	1	no	—
54	Q₈×D₅₃	C₂×C₄	1	2	yes	complemented
55	Q₈×D₅₃	C₂³	1	2	yes	—
56	Q₈×D₅₃	C₂³	1	1	yes	characteristic
57	#11719947	—	2	1	no	—
58	#11719947	C₂×C₄	1	1	yes	characteristic
59	D₄×D₅₃	—	2	1	no	—
60	D₄×D₅₃	C₂³	1	1	yes	characteristic
61	#11719945	—	2	1	no	—
62	#11719945	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×D₅₃	—	2	1	no	—
64	C₂×C₄×D₅₃	—	2	1	no	—
65	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
66	C₂×Dic₁₀₆	—	2	1	no	—
67	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic, complemented
68	C₂²⋊#6157	—	2	1	no	—
69	C₂²⋊#6157	—	2	2	no	—
70	C₂²⋊#6157	D₄	1	2	yes	—
71	C₂×#88324	—	2	1	no	—
72	C₄⋊#6157	—	2	2	no	—
73	C₄⋊#6157	—	2	2	no	—
74	C₄⋊#6157	—	2	2	no	—
75	C₄⋊#6157	—	2	1	no	—
76	C₄⋊#6157	D₄	1	1	yes	characteristic
77	C₄⋊#6157	D₄	1	1	yes	characteristic
78	C₄⋊#6157	C₂³	1	1	yes	characteristic
79	C₄⋊#6157	C₂³	1	1	yes	characteristic
80	C₄⋊#6157	C₂³	1	2	yes	—
81	C₄×#6157	—	2	2	no	—
82	C₄×#6157	—	2	2	no	—
83	C₄×#6157	—	2	2	no	—
84	C₄×#6157	—	2	1	no	—
85	C₄×#6157	C₂³	1	2	yes	—
86	D₅₃⋊C₈	—	2	1	no	—
87	D₅₃⋊C₈	C₂³	1	1	yes	characteristic
88	D₅₃⋊C₈	C₂³	1	1	yes	characteristic
89	Q₁₆×C₅₃	C₂×C₄	1	2	yes	complemented
90	SD₁₆×C₅₃	C₂×C₄	1	2	yes	complemented
91	M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
92	#11719925	C₂×C₄	1	2	yes	complemented
93	#11719925	C₂×C₄	1	2	yes	complemented
94	#11719924	C₂×C₄	1	2	yes	complemented
95	#11719923	C₂×C₄	1	2	yes	complemented
96	#11719917	C₂×C₄	1	1	yes	characteristic, complemented
97	Dic₂₁₂	C₂×C₄	1	2	yes	complemented
98	#11719913	C₂×C₄	1	2	yes	complemented
99	#11719912	C₂×C₄	1	2	yes	complemented
100	C₈×D₅₃	C₂³	1	2	yes	—
101	C₂²×D₅₃	—	2	1	no	—
102	C₂²×D₅₃	C₂×D₄	1	1	yes	characteristic
103	C₂×#6157	—	2	1	no	—
104	C₂×#6157	—	2	1	no	—
105	C₂×#6157	—	2	2	no	—
106	C₂×#6157	—	8	2	no	—
107	C₂×#6157	—	8	1	no	—
108	C₂×#6157	C₂×D₄	1	2	yes	—
109	C₂×#6157	C₂×D₄	1	2	yes	—
110	Q₈×C₅₃	—	2	1	no	—
111	Q₈×C₅₃	C₂²×C₄	1	2	yes	—
112	Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
113	D₄×C₅₃	—	2	1	no	—
114	D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₄×C₅₃	—	2	1	no	—
116	C₂×C₄×C₅₃	—	2	1	no	—
117	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
118	C₂²⋊D₅₃	—	2	1	no	—
119	C₂²⋊D₅₃	—	2	1	no	—
120	C₂²⋊D₅₃	—	2	1	no	—
121	C₂×Dic₅₃	—	2	1	no	—
122	C₂×Dic₅₃	—	2	1	no	—
123	C₂×Dic₅₃	#41	1	1	yes	characteristic
124	D₂₁₂	—	2	1	no	—
125	D₂₁₂	C₂²×C₄	1	1	yes	characteristic
126	C₄×D₅₃	—	2	2	no	—
127	C₄×D₅₃	—	2	1	no	—
128	C₄×D₅₃	—	2	2	no	—
129	C₄×D₅₃	—	2	1	no	—
130	C₄×D₅₃	—	2	1	no	—
131	C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
132	C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
133	C₄×D₅₃	#41	1	1	yes	characteristic
134	C₄×D₅₃	C₂⁴	1	1	yes	characteristic
135	Dic₁₀₆	—	2	1	no	—
136	Dic₁₀₆	—	2	2	no	—
137	Dic₁₀₆	—	2	2	no	—
138	Dic₁₀₆	—	2	1	no	—
139	Dic₁₀₆	C₂²×C₄	1	2	yes	—
140	Dic₁₀₆	C₂²×C₄	1	1	yes	characteristic
141	#88324	—	2	1	no	—
142	#88324	—	2	1	no	—
143	C₈×C₅₃	C₂²×C₄	1	2	yes	—
144	#88322	C₂²×C₄	1	2	yes	—
145	C₂²×C₅₃	—	2	1	no	—
146	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
147	C₂×D₅₃	—	2	1	no	—
148	C₂×D₅₃	—	2	1	no	—
149	C₂×D₅₃	—	2	1	no	—
150	C₂×D₅₃	—	4	1	no	—
151	C₂×D₅₃	—	4	1	no	—
152	C₂×D₅₃	C₄×D₄	1	1	yes	characteristic
153	C₂×D₅₃	C₂²×D₄	1	1	yes	characteristic
154	#6157	—	2	2	no	—
155	#6157	—	2	2	no	—
156	#6157	—	4	1	no	—
157	#6157	—	4	2	no	—
158	#6157	—	4	1	no	—
159	C₄×C₅₃	—	2	2	no	—
160	C₄×C₅₃	—	2	1	no	—
161	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
162	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
163	Dic₅₃	—	2	2	no	—
164	Dic₅₃	—	2	1	no	—
165	Dic₅₃	C₄×D₄	1	1	yes	characteristic
166	Dic₅₃	C₂×#41	1	1	yes	characteristic
167	#2947	—	53	1	no	maximal, Sylow
168	C₂×C₅₃	—	2	1	no	—
169	C₂×C₅₃	—	4	1	no	—
170	C₂×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
171	D₅₃	—	2	1	no	—
172	D₅₃	—	4	1	no	—
173	D₅₃	#577	1	1	yes	characteristic
174	D₅₃	#578	1	1	yes	characteristic
175	C₂×#137	—	53	1	no	—
176	#520	—	53	1	no	—
177	#519	—	53	1	no	—
178	#441	—	53	2	no	—
179	#440	—	53	2	no	—
180	C₄×Q₁₆	—	53	2	no	—
181	C₄×SD₁₆	—	53	2	no	—
182	#428	—	53	1	no	—
183	#416	—	53	1	no	—
184	C₂×#103	—	53	1	no	—
185	#405	—	53	1	no	—
186	C₅₃	#2947	1	1	yes	characteristic, complemented, Sylow
187	C₂×#41	—	53	1	no	—
188	C₂²×Q₈	—	53	1	no	—
189	#137	—	53	1	no	—
190	#137	—	53	1	no	—
191	#137	—	53	2	no	—
192	#137	—	53	2	no	—
193	#137	—	53	1	no	—
194	#137	#6157	1	1	yes	characteristic, complemented
195	C₂×Q₁₆	—	53	2	no	—
196	C₂×SD₁₆	—	53	2	no	—
197	C₂×M₁₆	—	53	1	no	—
198	C₂²×C₈	—	53	1	no	—
199	C₄×Q₈	—	53	2	no	—
200	C₄×Q₈	—	53	2	no	—
201	C₄×Q₈	—	53	2	no	—
202	C₄×Q₈	—	106	1	no	—
203	C₄×Q₈	—	106	1	no	—
204	C₄×D₄	—	53	2	no	—
205	C₄×D₄	—	106	2	no	—
206	#117	—	53	1	no	—
207	#117	—	53	1	no	—
208	#117	—	106	2	no	—
209	#117	—	106	1	no	—
210	C₂×C₄⋊C₄	—	53	1	no	—
211	C₂×C₄⋊C₄	—	106	1	no	—
212	#107	—	53	2	no	—
213	#106	—	53	2	no	—
214	#103	—	53	2	no	—
215	#103	—	53	2	no	—
216	#103	—	53	1	no	—
217	#103	—	53	1	no	—
218	#102	—	53	1	no	—
219	#102	—	53	1	no	—
220	#97	—	53	2	no	—
221	C₄×C₈	—	53	2	no	—
222	#41	—	53	1	no	—
223	#41	—	53	1	no	—
224	#41	—	53	1	no	—
225	#41	—	106	1	no	—
226	#41	—	106	1	no	—
227	#41	C₂×#6157	1	1	yes	characteristic
228	C₂×Q₈	—	53	2	no	—
229	C₂×Q₈	—	53	2	no	—
230	C₂×Q₈	—	53	1	no	—
231	C₂×Q₈	—	53	1	no	—
232	C₂×Q₈	—	106	1	no	—
233	C₂×Q₈	—	106	1	no	—
234	C₂×Q₈	—	106	1	no	—
235	C₂×Q₈	C₂×#6157	1	1	yes	characteristic, complemented
236	C₂×D₄	—	53	1	no	—
237	C₂×D₄	—	106	1	no	—
238	C₂²×C₄	—	53	1	no	—
239	C₂²×C₄	—	106	1	no	—
240	C₂²×C₄	—	106	1	no	—
241	C₂²×C₄	—	106	2	no	—
242	C₂²×C₄	—	106	1	no	—
243	Q₁₆	—	53	2	no	—
244	Q₁₆	—	53	2	no	—
245	Q₁₆	—	53	2	no	—
246	Q₁₆	C₂×#6157	1	2	yes	complemented
247	SD₁₆	—	53	2	no	—
248	SD₁₆	—	53	2	no	—
249	SD₁₆	—	53	2	no	—
250	SD₁₆	C₂×#6157	1	2	yes	complemented
251	M₁₆	—	53	1	no	—
252	M₁₆	—	53	2	no	—
253	M₁₆	C₂×#6157	1	1	yes	characteristic, complemented
254	C₂×C₈	—	53	2	no	—
255	C₂×C₈	—	53	1	no	—
256	C₂×C₈	—	53	1	no	—
257	C₂×C₈	—	106	1	no	—
258	C₂×C₈	—	106	1	no	—
259	C₄⋊C₄	—	53	1	no	—
260	C₄⋊C₄	—	53	1	no	—
261	C₄⋊C₄	—	53	1	no	—
262	C₄⋊C₄	—	53	1	no	—
263	C₄⋊C₄	—	53	2	no	—
264	C₄⋊C₄	—	106	1	no	—
265	C₄⋊C₄	—	106	2	no	—
266	C₄⋊C₄	—	106	2	no	—
267	C₄⋊C₄	—	106	2	no	—
268	C₂²⋊C₄	—	53	2	no	—
269	C₂²⋊C₄	—	106	2	no	—
270	C₂²⋊C₄	—	106	1	no	—
271	C₄²	—	53	2	no	—
272	C₄²	—	106	1	no	—
273	C₄²	—	106	2	no	—
274	C₄²	—	106	2	no	—
275	C₄²	—	106	2	no	—
276	C₂³	—	53	1	no	—
277	C₂³	—	106	1	no	—
278	Q₈	—	2	1	no	—
279	Q₈	—	53	2	no	—
280	Q₈	—	53	1	no	—
281	Q₈	—	106	2	no	—
282	Q₈	—	106	2	no	—
283	Q₈	—	106	1	no	—
284	Q₈	—	106	1	no	—
285	Q₈	C₂²×#6157	1	2	yes	—
286	Q₈	C₂²×#6157	1	1	yes	characteristic
287	D₄	—	2	1	no	—
288	D₄	—	53	1	no	—
289	D₄	—	106	1	no	—
290	D₄	—	106	1	no	—
291	D₄	—	106	1	no	—
292	D₄	—	106	1	no	—
293	D₄	C₂²×#6157	1	1	yes	characteristic
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	53	1	no	—
297	C₂×C₄	—	53	1	no	—
298	C₂×C₄	—	53	1	no	—
299	C₂×C₄	—	53	1	no	—
300	C₂×C₄	—	53	1	no	—
301	C₂×C₄	—	53	2	no	—
302	C₂×C₄	—	53	2	no	—
303	C₂×C₄	—	106	1	no	—
304	C₂×C₄	—	106	1	no	—
305	C₂×C₄	—	106	2	no	—
306	C₂×C₄	—	106	2	no	—
307	C₂×C₄	—	106	1	no	—
308	C₂×C₄	—	106	1	no	—
309	C₂×C₄	—	106	1	no	—
310	C₂×C₄	—	106	1	no	—
311	C₂×C₄	—	106	1	no	—
312	C₂×C₄	—	106	2	no	—
313	C₂×C₄	—	424	2	no	—
314	C₂×C₄	—	424	1	no	—
315	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
316	C₈	—	53	2	no	—
317	C₈	—	106	1	no	—
318	C₈	—	106	1	no	—
319	C₈	C₂²×#6157	1	2	yes	—
320	C₂²	—	2	1	no	—
321	C₂²	—	53	1	no	—
322	C₂²	—	53	1	no	—
323	C₂²	—	106	1	no	—
324	C₂²	—	106	1	no	—
325	C₂²	—	106	1	no	—
326	C₂²	—	212	1	no	—
327	C₂²	—	212	1	no	—
328	C₂²	D₄×#6157	1	1	yes	characteristic
329	C₄	—	2	1	no	—
330	C₄	—	2	2	no	—
331	C₄	—	53	1	no	—
332	C₄	—	53	1	no	—
333	C₄	—	106	2	no	—
334	C₄	—	106	2	no	—
335	C₄	—	106	1	no	—
336	C₄	—	106	2	no	—
337	C₄	—	212	2	no	—
338	C₄	—	212	1	no	—
339	C₄	—	212	1	no	—
340	C₄	D₄×#6157	1	1	yes	characteristic
341	C₄	C₂³×#6157	1	1	yes	characteristic
342	C₂	—	2	1	no	—
343	C₂	—	4	1	no	—
344	C₂	—	53	1	no	—
345	C₂	—	53	1	no	—
346	C₂	—	106	1	no	—
347	C₂	—	212	1	no	—
348	C₂	C₂×D₄×#6157	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×#6157
commutator	C₄×C₅₃	C₂³×C₄
Frattini	C₄	C₂³×#6157
Fitting	#137×C₅₃	C₄
socle	C₂×C₅₃	C₂×C₄×D₄
hypercenter	#137	#6157
hypocenter	C₅₃	#2947
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #137

Complements

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

N	H ≅ G/N	type
2. #16811691	C₂	semidirect
3. #16811663	C₂	semidirect
4. #16811662	C₂	semidirect
5. Q₁₆×#6157	C₂	semidirect
6. SD₁₆×#6157	C₂	semidirect
7. #16811634	C₂	semidirect
9. C₂×#13133280	C₂	semidirect
10. #16811589	C₂	semidirect
38. #13133250	C₄	semidirect
39. #13133249	C₄	semidirect
40. #13133248	C₄	semidirect
41. #13133247	C₄	semidirect
42. #13133246	C₄	semidirect
46. #137×C₅₃	C₄	semidirect
50. C₂×Q₈×C₅₃	C₂×C₄	semidirect
54. Q₈×D₅₃	C₂×C₄	semidirect
67. C₂×Dic₁₀₆	C₂×C₄	semidirect
89. Q₁₆×C₅₃	C₂×C₄	semidirect
90. SD₁₆×C₅₃	C₂×C₄	semidirect
91. M₁₆×C₅₃	C₂×C₄	semidirect
92. #11719925	C₂×C₄	semidirect
93. #11719925	C₂×C₄	semidirect
94. #11719924	C₂×C₄	semidirect
95. #11719923	C₂×C₄	semidirect
96. #11719917	C₂×C₄	semidirect
97. Dic₂₁₂	C₂×C₄	semidirect
98. #11719913	C₂×C₄	semidirect
99. #11719912	C₂×C₄	semidirect
186. C₅₃	#2947	semidirect
194. #137	#6157	semidirect
235. C₂×Q₈	C₂×#6157	semidirect
246. Q₁₆	C₂×#6157	semidirect
250. SD₁₆	C₂×#6157	semidirect
253. M₁₆	C₂×#6157	semidirect