Subgroups — #28328543

Order 13568.

494 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291132	C₂	1	4	yes	maximal, complemented
2	#20291131	C₂	1	2	yes	maximal, complemented
3	#20291129	C₂	1	2	yes	maximal, complemented
4	#20291077	C₂	1	2	yes	maximal
5	#20291076	C₂	1	1	yes	maximal, characteristic
6	#20291073	C₂	1	2	yes	maximal
7	#20291072	C₂	1	1	yes	maximal, characteristic
8	C₂×#16810747	C₂	1	1	yes	maximal, characteristic
9	#16811544	C₂²	1	4	yes	—
10	#16811544	C₂²	1	4	yes	—
11	C₄⋊#11719910	C₂²	1	2	yes	—
12	C₄⋊#11719910	C₂²	1	2	yes	—
13	C₂²⋊#11719910	C₂²	1	1	yes	characteristic
14	C₂²⋊#11719910	C₂²	1	1	yes	characteristic
15	C₂²⋊#11719910	C₂²	1	2	yes	—
16	#16811512	C₂²	1	4	yes	—
17	#16811511	C₂²	1	2	yes	—
18	#16811509	C₂²	1	4	yes	—
19	C₄×#11719910	C₂²	1	2	yes	—
20	C₂×C₄×Dic₁₀₆	C₄	1	1	yes	characteristic
21	#16810747	C₄	1	1	yes	characteristic
22	#16810747	C₄	1	1	yes	characteristic
23	#16810747	C₄	1	2	yes	—
24	#16810747	C₂²	1	1	yes	characteristic
25	#16810747	C₂²	1	1	yes	characteristic
26	#16810747	C₂²	1	2	yes	—
27	C₂×#13133146	C₄	1	2	yes	—
28	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
29	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
30	C₂×C₄⋊C₈×C₅₃	C₄	1	1	yes	characteristic
31	C₂×#11719910	C₂³	1	2	yes	—
32	C₂×#11719910	C₂³	1	2	yes	—
33	C₂×#11719910	C₂³	1	4	yes	—
34	C₂²×Dic₁₀₆	C₈	1	1	yes	characteristic
35	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
36	C₂²×#88322	C₂³	1	1	yes	characteristic
37	C₄×Dic₁₀₆	C₈	1	1	yes	characteristic
38	C₄×Dic₁₀₆	C₈	1	2	yes	—
39	C₄×Dic₁₀₆	C₈	1	1	yes	characteristic
40	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
41	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
42	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄⋊Dic₅₃	C₈	1	1	yes	characteristic
44	C₂×#11719919	C₈	1	2	yes	—
45	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
46	#13133146	C₂×C₄	1	2	yes	—
47	#13133146	C₂×C₄	1	2	yes	—
48	#13133146	C₂×C₄	1	2	yes	—
49	#13133146	C₂×C₄	1	2	yes	—
50	C₄×#88322	C₂×C₄	1	1	yes	characteristic
51	C₄×#88322	C₂×C₄	1	1	yes	characteristic
52	C₄×#88322	C₂³	1	1	yes	characteristic
53	C₄×#88322	C₂³	1	1	yes	characteristic
54	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
55	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
56	C₈×Dic₅₃	C₂³	1	2	yes	—
57	C₈×Dic₅₃	C₂³	1	2	yes	—
58	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
59	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
61	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
62	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
63	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
67	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
68	C₂×Dic₁₀₆	—	2	4	no	—
69	C₂×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
70	C₂×Dic₁₀₆	C₂×C₈	1	2	yes	—
71	C₂×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
72	C₂×C₈×C₅₃	—	4	2	no	—
73	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
75	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
76	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
77	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
80	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
81	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
82	#11719919	C₂×C₈	1	2	yes	—
83	#11719919	C₂×C₈	1	2	yes	—
84	#11719919	C₂×C₈	1	2	yes	—
85	#11719919	C₂×C₈	1	2	yes	—
86	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
87	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
88	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
89	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
90	C₂×#88322	—	2	2	no	—
91	C₂×#88322	—	4	1	no	—
92	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
93	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
94	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
95	C₂×#88322	C₂⁴	1	1	yes	characteristic
96	#11719910	—	2	2	no	—
97	#11719910	—	2	2	no	—
98	#11719910	—	2	4	no	—
99	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₅₃	—	2	2	no	—
101	C₂×C₄×C₅₃	—	2	1	no	—
102	C₂×C₄×C₅₃	—	2	1	no	—
103	C₂×C₄×C₅₃	—	2	2	no	—
104	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
110	C₂×Dic₅₃	—	2	4	no	—
111	C₂×Dic₅₃	—	2	4	no	—
112	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
113	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
114	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
115	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
116	Dic₁₀₆	—	2	2	no	—
117	Dic₁₀₆	—	2	4	no	—
118	Dic₁₀₆	M₃₂	1	4	yes	—
119	C₈×C₅₃	—	2	2	no	—
120	C₈×C₅₃	—	2	2	no	—
121	#88322	—	2	1	no	—
122	#88322	—	2	1	no	—
123	#88322	—	2	1	no	—
124	#88322	2₋¹⁺⁴	1	2	yes	—
125	#33498	—	53	1	no	maximal, Sylow
126	C₂²×C₅₃	—	2	1	no	—
127	C₂²×C₅₃	—	2	1	no	—
128	C₂²×C₅₃	#504	1	2	yes	—
129	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
130	C₄×C₅₃	—	2	1	no	—
131	C₄×C₅₃	—	2	1	no	—
132	C₄×C₅₃	—	2	1	no	—
133	C₄×C₅₃	—	2	1	no	—
134	C₄×C₅₃	C₂×M₃₂	1	2	yes	—
135	C₄×C₅₃	#519	1	1	yes	characteristic
136	C₄×C₅₃	#568	1	1	yes	characteristic
137	Dic₅₃	—	2	2	no	—
138	Dic₅₃	—	2	2	no	—
139	Dic₅₃	—	2	2	no	—
140	Dic₅₃	C₂×M₃₂	1	4	yes	—
141	C₂×C₈×Q₈	—	53	1	no	—
142	#2183	—	53	2	no	—
143	#2183	—	53	4	no	—
144	Q₈×C₁₆	—	53	2	no	—
145	#2163	—	53	1	no	—
146	#2163	—	53	2	no	—
147	#2162	—	53	1	no	—
148	#2162	—	53	2	no	—
149	C₂×C₅₃	—	2	2	no	—
150	C₂×C₅₃	#2982	1	1	yes	characteristic
151	C₂×C₅₃	C₂²×M₃₂	1	1	yes	characteristic
152	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
153	C₂×C₄×Q₈	—	53	1	no	—
154	C₈×Q₈	—	53	1	no	—
155	C₈×Q₈	—	53	2	no	—
156	C₈×Q₈	—	53	1	no	—
157	C₈×Q₈	—	53	2	no	—
158	C₈×Q₈	—	53	1	no	—
159	C₈×Q₈	—	53	1	no	—
160	C₂×C₄⋊C₈	—	53	2	no	—
161	C₂×C₄⋊C₈	#6157	1	1	yes	characteristic
162	C₂×C₄×C₈	—	53	2	no	—
163	C₂×C₄×C₈	—	53	1	no	—
164	C₄⋊C₁₆	—	53	2	no	—
165	C₄⋊C₁₆	—	53	2	no	—
166	C₄⋊C₁₆	—	53	4	no	—
167	C₄⋊C₁₆	—	53	4	no	—
168	C₂²⋊C₁₆	—	53	1	no	—
169	C₂²⋊C₁₆	—	53	1	no	—
170	C₂²⋊C₁₆	—	53	2	no	—
171	#346	—	53	2	no	—
172	#346	—	53	4	no	—
173	C₄×C₁₆	—	53	2	no	—
174	C₄×C₁₆	—	53	4	no	—
175	C₅₃	#33498	1	1	yes	characteristic, complemented, Sylow
176	C₂²×Q₈	—	53	1	no	—
177	C₂²×C₈	—	53	1	no	—
178	C₂²×C₈	—	53	1	no	—
179	C₂²×C₈	C₂×#6157	1	2	yes	—
180	C₄×Q₈	—	53	1	no	—
181	C₄×Q₈	—	53	2	no	—
182	C₄×Q₈	—	53	1	no	—
183	C₄×Q₈	—	53	1	no	—
184	C₄×Q₈	—	53	2	no	—
185	C₄×Q₈	—	53	1	no	—
186	C₂×C₄⋊C₄	—	53	2	no	—
187	C₂×C₄⋊C₄	—	53	1	no	—
188	C₂×C₄²	—	53	2	no	—
189	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
190	C₂×C₁₆	—	53	2	no	—
191	C₂×C₁₆	—	53	2	no	—
192	C₂×C₁₆	—	53	4	no	—
193	C₄⋊C₈	—	53	2	no	—
194	C₄⋊C₈	—	53	2	no	—
195	C₄⋊C₈	—	53	2	no	—
196	C₄⋊C₈	—	53	2	no	—
197	C₄⋊C₈	C₂×#6157	1	1	yes	characteristic
198	C₄⋊C₈	C₂×#6157	1	2	yes	—
199	C₄⋊C₈	C₂×#6157	1	1	yes	characteristic
200	C₄×C₈	—	53	2	no	—
201	C₄×C₈	—	53	2	no	—
202	C₄×C₈	—	53	2	no	—
203	C₄×C₈	—	53	2	no	—
204	C₄×C₈	—	53	1	no	—
205	C₄×C₈	—	53	1	no	—
206	C₄×C₈	—	53	1	no	—
207	C₄×C₈	—	53	1	no	—
208	C₂×Q₈	—	53	1	no	—
209	C₂×Q₈	—	53	2	no	—
210	C₂×Q₈	—	53	1	no	—
211	C₂×Q₈	—	106	4	no	—
212	C₂²×C₄	—	53	2	no	—
213	C₂²×C₄	—	53	2	no	—
214	C₂²×C₄	D₅₃⋊C₈	1	1	yes	characteristic
215	C₂²×C₄	D₅₃⋊C₈	1	1	yes	characteristic
216	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
217	C₂×C₈	—	4	2	no	—
218	C₂×C₈	—	53	1	no	—
219	C₂×C₈	—	53	1	no	—
220	C₂×C₈	—	53	1	no	—
221	C₂×C₈	—	53	1	no	—
222	C₂×C₈	—	106	2	no	—
223	C₂×C₈	—	212	1	no	—
224	C₂×C₈	C₂²×#6157	1	2	yes	—
225	C₂×C₈	C₂²×#6157	1	2	yes	—
226	C₄⋊C₄	—	53	2	no	—
227	C₄⋊C₄	—	53	2	no	—
228	C₄⋊C₄	—	53	2	no	—
229	C₄⋊C₄	—	53	2	no	—
230	C₄⋊C₄	—	53	2	no	—
231	C₄⋊C₄	—	53	1	no	—
232	C₄⋊C₄	—	53	1	no	—
233	C₄²	—	53	2	no	—
234	C₄²	—	53	2	no	—
235	C₄²	—	53	2	no	—
236	C₄²	—	53	2	no	—
237	C₄²	D₅₃⋊C₈	1	1	yes	characteristic
238	C₄²	D₅₃⋊C₈	1	1	yes	characteristic
239	C₄²	C₂²×#6157	1	1	yes	characteristic
240	C₄²	C₂²×#6157	1	1	yes	characteristic
241	C₁₆	—	106	2	no	—
242	C₁₆	—	106	2	no	—
243	C₁₆	—	106	4	no	—
244	C₂³	C₂×D₅₃⋊C₈	1	1	yes	characteristic
245	Q₈	—	53	4	no	—
246	Q₈	—	106	4	no	—
247	Q₈	—	106	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	53	2	no	—
253	C₂×C₄	—	53	2	no	—
254	C₂×C₄	—	53	2	no	—
255	C₂×C₄	—	53	2	no	—
256	C₂×C₄	—	106	4	no	—
257	C₂×C₄	—	106	4	no	—
258	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
259	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
260	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
261	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
262	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
263	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	2	2	no	—
266	C₈	—	53	2	no	—
267	C₈	—	106	1	no	—
268	C₈	—	106	1	no	—
269	C₈	—	106	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	#16811682	1	2	yes	—
273	C₂²	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	53	4	no	—
279	C₄	—	106	2	no	—
280	C₄	—	106	2	no	—
281	C₄	—	106	2	no	—
282	C₄	D₅₃⋊M₃₂	1	2	yes	—
283	C₄	#16811691	1	1	yes	characteristic
284	C₄	#16811701	1	1	yes	characteristic
285	C₂	—	2	2	no	—
286	C₂	#20291610	1	1	yes	characteristic
287	C₂	C₂×D₅₃⋊M₃₂	1	1	yes	characteristic
288	C₂	C₂×#16811682	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6157
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₂×C₄⋊C₈×C₅₃	C₄
prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂×#88322	C₂⁴
hypercenter	C₂×C₄⋊C₈	#6157
hypocenter	C₅₃	#33498
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. #20291132	C₂	semidirect
2. #20291131	C₂	semidirect
3. #20291129	C₂	semidirect
175. C₅₃	#33498	semidirect

Maximal subgroups

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

subgroup	index
#20291132	2
#20291131	2
#20291129	2
#20291077	2
#20291076	2
#20291073	2
#20291072	2
C₂×#16810747	2
#33498	53