Subgroups — #27538666

Order 13568.

642 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811039	C₂	1	1	yes	maximal, characteristic, complemented
2	#20284431	C₂	1	2	yes	maximal, complemented
3	#20276842	C₂	1	4	yes	maximal, complemented
4	#20276837	C₂	1	2	yes	maximal, complemented
5	#20276832	C₂	1	2	yes	maximal, complemented
6	C₂×#16810453	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276464	C₂	1	2	yes	maximal, complemented
8	C₂×C₅₃×#508	C₂	1	1	yes	maximal, characteristic, complemented
9	#16811421	—	2	2	no	—
10	C₂×#13133223	C₂²	1	1	yes	characteristic
11	#16811039	—	2	1	no	—
12	#16811039	—	2	1	no	—
13	#16811039	C₂²	1	2	yes	—
14	#16811039	C₂²	1	2	yes	—
15	C₂×#13133157	C₂²	1	1	yes	characteristic
16	C₂×#13133156	—	2	1	no	—
17	C₂×C₁₆×D₅₃	—	2	1	no	—
18	#16810895	C₂²	1	4	yes	—
19	#16810890	C₂²	1	4	yes	—
20	#16810883	—	2	4	no	—
21	C₂×#13133153	C₂²	1	2	yes	—
22	#16810607	—	2	2	no	—
23	#16810453	C₂²	1	2	yes	complemented
24	#16810453	C₂²	1	2	yes	complemented
25	#16810439	C₂²	1	4	yes	complemented
26	#16810436	C₂²	1	4	yes	complemented
27	C₂×#135×C₅₃	C₂²	1	2	yes	—
28	C₅₃×#508	—	2	1	no	—
29	C₅₃×#508	—	2	1	no	—
30	C₅₃×#508	—	2	1	no	—
31	C₅₃×#508	—	2	1	no	—
32	C₂×Q₃₂×C₅₃	C₂²	1	1	yes	characteristic, complemented
33	C₂×SD₃₂×C₅₃	C₂²	1	2	yes	complemented
34	C₂×D₁₆×C₅₃	C₂²	1	1	yes	characteristic, complemented
35	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
36	C₂×#11719945	D₄	1	1	yes	characteristic
37	#13133248	—	2	4	no	—
38	#13133247	—	4	2	no	—
39	#13133243	—	2	4	no	—
40	#13133242	—	4	2	no	—
41	#13133237	—	4	4	no	—
42	#13133223	—	2	1	no	—
43	#13133223	—	2	1	no	—
44	#13133223	D₄	1	2	yes	—
45	#13133223	C₂³	1	2	yes	—
46	C₂×#11719917	—	2	2	no	—
47	C₂×#11719917	D₄	1	1	yes	characteristic
48	C₂×#11719912	—	2	1	no	—
49	C₂×C₈×D₅₃	—	2	1	no	—
50	#13133157	—	2	1	no	—
51	#13133157	C₂³	1	2	yes	—
52	#13133156	—	2	2	no	—
53	#13133156	—	2	2	no	—
54	C₂×#11719908	—	2	1	no	—
55	C₁₆×D₅₃	—	2	2	no	—
56	C₁₆×D₅₃	—	2	2	no	—
57	#13133153	—	2	2	no	—
58	#13133153	C₂³	1	4	yes	—
59	#13133136	—	2	4	no	—
60	#13133134	—	2	4	no	—
61	C₂×#41×C₅₃	—	2	2	no	—
62	#135×C₅₃	—	2	2	no	—
63	#135×C₅₃	—	2	2	no	—
64	#135×C₅₃	—	2	2	no	—
65	#135×C₅₃	—	2	2	no	—
66	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
67	C₂×SD₁₆×C₅₃	—	2	2	no	—
68	C₂×D₈×C₅₃	C₂³	1	2	yes	—
69	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
70	Q₃₂×C₅₃	—	2	1	no	—
71	Q₃₂×C₅₃	—	2	1	no	—
72	SD₃₂×C₅₃	—	2	2	no	—
73	SD₃₂×C₅₃	—	2	2	no	—
74	D₁₆×C₅₃	—	2	1	no	—
75	D₁₆×C₅₃	—	2	1	no	—
76	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
77	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
78	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
79	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
80	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
81	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
82	#41×C₅₃	—	4	2	no	—
83	#41×C₅₃	—	4	2	no	—
84	#41×C₅₃	—	4	2	no	—
85	#41×C₅₃	—	4	2	no	—
86	C₂×Q₈×C₅₃	—	2	2	no	—
87	C₂×D₄×C₅₃	—	2	2	no	—
88	C₂×D₄×C₅₃	—	4	2	no	—
89	C₂²×C₄×C₅₃	—	4	2	no	—
90	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₂²⋊D₅₃	—	2	1	no	—
92	#11719945	—	2	1	no	—
93	#11719945	—	2	1	no	—
94	#11719945	D₈	1	2	yes	—
95	#11719945	C₂×D₄	1	2	yes	—
96	C₂×D₂₁₂	D₈	1	1	yes	characteristic
97	C₂×C₄×D₅₃	—	2	1	no	—
98	C₂×Dic₁₀₆	D₈	1	1	yes	characteristic
99	Q₁₆×C₅₃	—	2	2	no	—
100	Q₁₆×C₅₃	—	2	2	no	—
101	SD₁₆×C₅₃	—	4	2	no	—
102	SD₁₆×C₅₃	—	4	2	no	—
103	D₈×C₅₃	—	2	2	no	—
104	D₈×C₅₃	—	2	2	no	—
105	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
109	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
110	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
111	#11719925	—	4	4	no	—
112	#11719924	—	4	4	no	—
113	#11719923	—	4	4	no	—
114	#11719922	—	4	4	no	—
115	#11719917	—	2	1	no	—
116	#11719917	—	2	4	no	—
117	#11719917	—	4	2	no	—
118	#11719917	C₂×D₄	1	2	yes	—
119	C₂×#88322	—	2	1	no	—
120	C₂×#88322	—	4	2	no	—
121	#11719912	—	2	2	no	—
122	#11719912	—	2	2	no	—
123	C₈×D₅₃	—	2	2	no	—
124	C₈×D₅₃	—	2	2	no	—
125	C₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
126	C₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
127	C₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
128	C₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
129	#11719908	—	2	2	no	—
130	C₂³×C₅₃	—	4	2	no	—
131	C₂³×C₅₃	#135	1	1	yes	characteristic
132	C₂²×D₅₃	—	2	1	no	—
133	Q₈×C₅₃	—	4	2	no	—
134	Q₈×C₅₃	—	4	2	no	—
135	D₄×C₅₃	—	4	2	no	—
136	D₄×C₅₃	—	4	2	no	—
137	D₄×C₅₃	—	8	2	no	—
138	D₄×C₅₃	—	8	2	no	—
139	C₂×C₄×C₅₃	—	4	2	no	—
140	C₂×C₄×C₅₃	—	8	2	no	—
141	C₂×C₄×C₅₃	—	8	2	no	—
142	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
143	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
144	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
147	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
148	C₂²⋊D₅₃	—	2	2	no	—
149	C₂²⋊D₅₃	—	2	2	no	—
150	C₂×Dic₅₃	—	2	1	no	—
151	D₂₁₂	—	2	1	no	—
152	D₂₁₂	C₂×D₈	1	2	yes	—
153	C₄×D₅₃	—	2	2	no	—
154	C₄×D₅₃	—	2	2	no	—
155	Dic₁₀₆	—	2	1	no	—
156	Dic₁₀₆	C₂×D₈	1	2	yes	—
157	C₈×C₅₃	#121	1	1	yes	characteristic
158	C₈×C₅₃	#121	1	1	yes	characteristic
159	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
160	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
161	#88322	—	2	2	no	—
162	#88322	—	4	4	no	—
163	#33617	—	53	1	no	maximal, Sylow
164	C₂²×C₅₃	—	4	1	no	—
165	C₂²×C₅₃	—	8	2	no	—
166	C₂²×C₅₃	—	16	2	no	—
167	C₂²×C₅₃	#466	1	1	yes	characteristic
168	C₂²×C₅₃	#466	1	1	yes	characteristic
169	C₂²×C₅₃	C₂×#135	1	1	yes	characteristic
170	C₂×D₅₃	—	2	2	no	—
171	C₂×D₅₃	—	4	2	no	—
172	C₄×C₅₃	—	8	2	no	—
173	C₄×C₅₃	#466	1	1	yes	characteristic
174	C₄×C₅₃	#466	1	1	yes	characteristic
175	C₄×C₅₃	C₂×#121	1	1	yes	characteristic
176	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
177	Dic₅₃	—	2	2	no	—
178	C₂×#508	D₅₃	1	1	yes	characteristic, complemented
179	C₂×#504	—	53	1	no	—
180	#3067	—	53	2	no	—
181	#2223	—	53	4	no	—
182	#2222	—	53	2	no	—
183	#2221	—	53	2	no	—
184	C₂×#368	—	53	1	no	—
185	#2148	—	53	2	no	—
186	C₂×C₅₃	—	2	1	no	—
187	C₂×C₅₃	—	2	1	no	—
188	C₂×C₅₃	—	2	1	no	—
189	C₂×C₅₃	—	16	2	no	—
190	C₂×C₅₃	C₂×#466	1	1	yes	characteristic
191	D₅₃	—	4	2	no	—
192	#575	—	106	2	no	—
193	C₂×#135	C₂×D₅₃	1	2	yes	—
194	C₂×#131	—	53	1	no	—
195	#508	—	2	1	no	—
196	#508	—	2	1	no	—
197	#508	—	2	1	no	—
198	#508	—	2	1	no	—
199	C₂×Q₃₂	C₂×D₅₃	1	1	yes	characteristic, complemented
200	C₂×SD₃₂	C₂×D₅₃	1	2	yes	complemented
201	C₂×D₁₆	C₂×D₅₃	1	1	yes	characteristic, complemented
202	#504	—	53	2	no	—
203	#504	—	53	2	no	—
204	#504	—	106	1	no	—
205	#504	—	106	1	no	—
206	C₂×M₃₂	—	53	1	no	—
207	C₂×M₃₂	—	106	1	no	—
208	C₂²×C₁₆	—	106	1	no	—
209	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
210	#473	—	53	4	no	—
211	#472	—	53	4	no	—
212	#471	—	106	4	no	—
213	C₂×#108	—	53	2	no	—
214	#413	—	106	2	no	—
215	#368	—	53	2	no	—
216	#368	—	53	2	no	—
217	#362	—	53	4	no	—
218	#361	—	53	4	no	—
219	C₅₃	#33617	1	1	yes	characteristic, complemented, Sylow
220	C₂×#41	—	2	2	no	—
221	C₂×#41	—	53	1	no	—
222	#137	—	106	4	no	—
223	#137	—	212	2	no	—
224	AGL(1,ℤ/8)	—	106	4	no	—
225	AGL(1,ℤ/8)	—	212	2	no	—
226	#135	—	2	2	no	—
227	#135	—	2	2	no	—
228	#135	—	2	2	no	—
229	#135	—	2	2	no	—
230	#135	—	212	4	no	—
231	C₂×Q₁₆	C₂²×D₅₃	1	2	yes	—
232	C₂×SD₁₆	—	2	2	no	—
233	C₂×D₈	C₂²×D₅₃	1	2	yes	—
234	#131	—	53	2	no	—
235	#131	—	53	2	no	—
236	#131	—	106	1	no	—
237	#131	—	106	1	no	—
238	C₂×M₁₆	—	53	1	no	—
239	C₂×M₁₆	—	106	1	no	—
240	C₂×M₁₆	—	106	2	no	—
241	C₂²×C₈	—	106	1	no	—
242	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
243	Q₃₂	—	2	1	no	—
244	Q₃₂	—	2	1	no	—
245	SD₃₂	—	2	2	no	—
246	SD₃₂	—	2	2	no	—
247	D₁₆	—	2	1	no	—
248	D₁₆	—	2	1	no	—
249	M₃₂	—	53	2	no	—
250	M₃₂	—	106	1	no	—
251	M₃₂	—	106	2	no	—
252	M₃₂	—	106	2	no	—
253	C₂×C₁₆	—	106	1	no	—
254	C₂×C₁₆	—	106	2	no	—
255	C₂×C₁₆	—	106	2	no	—
256	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
257	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
258	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
259	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
260	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
261	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
262	#108	—	53	4	no	—
263	#108	—	106	2	no	—
264	#101	—	106	4	no	—
265	#100	—	106	4	no	—
266	#41	—	4	2	no	—
267	#41	—	4	2	no	—
268	#41	—	4	2	no	—
269	#41	—	4	2	no	—
270	#41	—	53	2	no	—
271	#41	—	53	2	no	—
272	#41	—	106	1	no	—
273	#41	—	106	1	no	—
274	C₂×Q₈	—	2	2	no	—
275	C₂×Q₈	—	53	1	no	—
276	C₂×D₄	—	2	2	no	—
277	C₂×D₄	—	4	2	no	—
278	C₂×D₄	—	53	1	no	—
279	C₂×D₄	—	106	1	no	—
280	C₂²×C₄	—	4	2	no	—
281	C₂²×C₄	—	106	1	no	—
282	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
283	Q₁₆	—	2	2	no	—
284	Q₁₆	—	2	2	no	—
285	Q₁₆	—	212	4	no	—
286	SD₁₆	—	4	2	no	—
287	SD₁₆	—	4	2	no	—
288	SD₁₆	—	212	4	no	—
289	SD₁₆	—	212	4	no	—
290	D₈	—	2	2	no	—
291	D₈	—	2	2	no	—
292	D₈	—	212	4	no	—
293	M₁₆	—	53	2	no	—
294	M₁₆	—	106	1	no	—
295	M₁₆	—	106	2	no	—
296	M₁₆	—	106	2	no	—
297	M₁₆	—	106	4	no	—
298	M₁₆	—	212	2	no	—
299	C₂×C₈	—	106	1	no	—
300	C₂×C₈	—	106	2	no	—
301	C₂×C₈	—	106	2	no	—
302	C₂×C₈	—	212	2	no	—
303	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
304	C₂×C₈	#11719947	1	1	yes	characteristic
305	C₂×C₈	#11719947	1	1	yes	characteristic
306	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
307	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
308	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
309	C₁₆	—	106	2	no	—
310	C₁₆	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
311	C₁₆	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
312	C₁₆	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
313	C₁₆	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
314	C₂³	—	4	2	no	—
315	C₂³	—	106	1	no	—
316	C₂³	#13133236	1	1	yes	characteristic
317	Q₈	—	4	2	no	—
318	Q₈	—	4	2	no	—
319	Q₈	—	53	2	no	—
320	Q₈	—	106	1	no	—
321	D₄	—	4	2	no	—
322	D₄	—	4	2	no	—
323	D₄	—	8	2	no	—
324	D₄	—	8	2	no	—
325	D₄	—	53	2	no	—
326	D₄	—	106	2	no	—
327	D₄	—	106	2	no	—
328	D₄	—	106	1	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	8	2	no	—
331	C₂×C₄	—	8	2	no	—
332	C₂×C₄	—	106	1	no	—
333	C₂×C₄	—	106	2	no	—
334	C₂×C₄	—	106	2	no	—
335	C₂×C₄	#13133188	1	1	yes	characteristic
336	C₂×C₄	#13133188	1	1	yes	characteristic
337	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
338	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
339	C₂×C₄	#13133236	1	1	yes	characteristic
340	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
341	C₈	—	106	2	no	—
342	C₈	—	212	4	no	—
343	C₈	#13133188	1	1	yes	characteristic
344	C₈	#13133188	1	1	yes	characteristic
345	C₈	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
346	C₈	C₂×#11719947	1	1	yes	characteristic
347	C₂²	—	4	1	no	—
348	C₂²	—	8	2	no	—
349	C₂²	—	16	2	no	—
350	C₂²	—	106	2	no	—
351	C₂²	—	212	2	no	—
352	C₂²	#16810856	1	1	yes	characteristic
353	C₂²	#16810856	1	1	yes	characteristic
354	C₂²	C₂×#13133236	1	1	yes	characteristic
355	C₄	—	8	2	no	—
356	C₄	—	106	2	no	—
357	C₄	#16810856	1	1	yes	characteristic
358	C₄	#16810856	1	1	yes	characteristic
359	C₄	C₂×#13133188	1	1	yes	characteristic
360	C₄	C₂×D₈×D₅₃	1	1	yes	characteristic
361	C₂	—	2	1	no	—
362	C₂	—	2	1	no	—
363	C₂	—	2	1	no	—
364	C₂	—	16	2	no	—
365	C₂	—	212	2	no	—
366	C₂	C₂×#16810856	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16810856
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#508	C₂
socle	C₂×C₅₃	C₂×#466
hypercenter	C₂×#508	D₅₃
hypocenter	C₅₃	#33617
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#508

Complements

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

N	H ≅ G/N	type
1. C₂×#16811039	C₂	semidirect
2. #20284431	C₂	semidirect
3. #20276842	C₂	semidirect
4. #20276837	C₂	semidirect
5. #20276832	C₂	semidirect
6. C₂×#16810453	C₂	semidirect
7. #20276464	C₂	semidirect
8. C₂×C₅₃×#508	C₂	semidirect
23. #16810453	C₂²	semidirect
24. #16810453	C₂²	semidirect
25. #16810439	C₂²	semidirect
26. #16810436	C₂²	semidirect
32. C₂×Q₃₂×C₅₃	C₂²	semidirect
33. C₂×SD₃₂×C₅₃	C₂²	semidirect
34. C₂×D₁₆×C₅₃	C₂²	semidirect
178. C₂×#508	D₅₃	semidirect
199. C₂×Q₃₂	C₂×D₅₃	semidirect
200. C₂×SD₃₂	C₂×D₅₃	semidirect
201. C₂×D₁₆	C₂×D₅₃	semidirect
219. C₅₃	#33617	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#16811039	2
#20284431	2
#20276842	2
#20276837	2
#20276832	2
C₂×#16810453	2
#20276464	2
C₂×C₅₃×#508	2
#33617	53