Subgroups — #28298367

Order 13568.

282 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811509	C₂	1	1	yes	maximal, characteristic
2	C₂×#16811486	C₂	1	1	yes	maximal, characteristic
3	#107×Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11719910	—	2	1	no	—
5	C₂×C₄⋊#88324	—	2	1	no	—
6	#16811509	—	2	1	no	—
7	#16811509	C₄	1	2	yes	complemented
8	#16811486	—	2	1	no	—
9	#16811486	C₄	1	2	yes	complemented
10	C₂×#13133151	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×Dic₅₃	—	2	1	no	—
13	#16810385	—	2	1	no	—
14	C₂×#107×C₅₃	C₄	1	1	yes	characteristic
15	C₂²×#88324	—	4	1	no	—
16	C₂×#11719910	—	2	2	no	—
17	C₂×#11719910	—	2	2	no	—
18	C₂×#11719910	—	2	2	no	—
19	C₄⋊#88324	—	2	2	no	—
20	C₄⋊#88324	—	4	1	no	—
21	C₂²×#88322	D₄	1	1	yes	characteristic
22	C₂×C₄⋊Dic₅₃	—	2	1	no	—
23	C₂×C₄×Dic₅₃	—	4	1	no	—
24	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
25	#13133151	—	2	1	no	—
26	#13133151	C₈	1	2	yes	complemented
27	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
28	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₅₃	D₄	1	1	yes	characteristic
30	C₈×Dic₅₃	Q₈	1	1	yes	characteristic
31	#13133124	—	4	1	no	—
32	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
34	#107×C₅₃	—	2	1	no	—
35	#107×C₅₃	C₈	1	2	yes	complemented
36	C₂²×C₄×C₅₃	—	4	1	no	—
37	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×Dic₅₃	—	4	1	no	—
39	C₂²×Dic₅₃	D₈	1	1	yes	characteristic
40	C₂²×Dic₅₃	SD₁₆	1	1	yes	characteristic
41	C₂×#88324	—	4	2	no	—
42	C₂×#88324	—	8	1	no	—
43	C₂×#88324	—	8	1	no	—
44	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
45	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
47	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
48	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
49	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
50	C₄⋊C₄×C₅₃	—	2	2	no	—
51	C₄⋊C₄×C₅₃	—	4	1	no	—
52	C₄⋊Dic₅₃	—	2	2	no	—
53	C₄⋊Dic₅₃	—	4	1	no	—
54	C₄×Dic₅₃	—	8	1	no	—
55	C₄×Dic₅₃	—	8	1	no	—
56	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
58	C₄×Dic₅₃	SD₁₆	1	1	yes	characteristic
59	C₄×Dic₅₃	Q₁₆	1	1	yes	characteristic
60	C₂×#88322	—	2	1	no	—
61	C₂×#88322	—	2	1	no	—
62	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
64	#11719910	—	2	4	no	—
65	C₂³×C₅₃	#102	1	1	yes	characteristic
66	C₂×C₄×C₅₃	—	4	2	no	—
67	C₂×C₄×C₅₃	—	8	1	no	—
68	C₂×C₄×C₅₃	—	8	1	no	—
69	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
72	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
73	C₂×C₄×C₅₃	#104	1	1	yes	characteristic
74	C₂×C₄×C₅₃	#104	1	1	yes	characteristic
75	C₂×Dic₅₃	—	2	1	no	—
76	C₂×Dic₅₃	—	2	1	no	—
77	C₂×Dic₅₃	—	4	2	no	—
78	C₂×Dic₅₃	—	8	1	no	—
79	C₂×Dic₅₃	—	8	1	no	—
80	C₂×Dic₅₃	#102	1	1	yes	characteristic
81	C₂×Dic₅₃	#103	1	1	yes	characteristic
82	C₂×Dic₅₃	#106	1	1	yes	characteristic
83	C₂×Dic₅₃	#107	1	1	yes	characteristic
84	C₂×Dic₅₃	D₁₆	1	1	yes	characteristic
85	C₂×Dic₅₃	SD₃₂	1	1	yes	characteristic
86	C₂×Dic₅₃	SD₃₂	1	1	yes	characteristic
87	C₂×Dic₅₃	Q₃₂	1	1	yes	characteristic
88	#88324	—	8	2	no	—
89	C₈×C₅₃	C₄×C₈	1	1	yes	characteristic
90	C₈×C₅₃	#97	1	1	yes	characteristic
91	C₈×C₅₃	C₄⋊C₈	1	1	yes	characteristic
92	C₈×C₅₃	C₄⋊C₈	1	1	yes	characteristic
93	#88322	—	2	1	no	—
94	#88322	—	2	1	no	—
95	#7171	—	53	1	no	maximal, Sylow
96	C₂²×C₅₃	D₄⋊C₈	1	1	yes	characteristic
97	C₂²×C₅₃	D₄⋊C₈	1	1	yes	characteristic
98	C₂²×C₅₃	#340	1	1	yes	characteristic
99	C₂²×C₅₃	#357	1	1	yes	characteristic
100	C₂²×C₅₃	#358	1	1	yes	characteristic
101	C₂²×C₅₃	#359	1	1	yes	characteristic
102	C₂²×C₅₃	#359	1	1	yes	characteristic
103	C₄×C₅₃	—	8	2	no	—
104	C₄×C₅₃	Q₈⋊C₈	1	1	yes	characteristic
105	C₄×C₅₃	Q₈⋊C₈	1	1	yes	characteristic
106	C₄×C₅₃	#336	1	1	yes	characteristic
107	C₄×C₅₃	#339	1	1	yes	characteristic
108	Dic₅₃	—	2	1	no	—
109	Dic₅₃	—	2	1	no	—
110	Dic₅₃	—	8	2	no	—
111	Dic₅₃	#357	1	1	yes	characteristic
112	Dic₅₃	#358	1	1	yes	characteristic
113	Dic₅₃	#366	1	1	yes	characteristic
114	Dic₅₃	#367	1	1	yes	characteristic
115	C₂×C₄×C₁₆	—	53	1	no	—
116	C₄×#107	—	53	1	no	—
117	C₂×#335	—	53	1	no	—
118	C₂×C₅₃	#1279	1	1	yes	characteristic
119	C₂×C₅₃	#1331	1	1	yes	characteristic
120	C₂×C₅₃	#1331	1	1	yes	characteristic
121	C₂×C₅₃	#1332	1	1	yes	characteristic
122	C₂×C₅₃	#1332	1	1	yes	characteristic
123	C₂×C₅₃	#1380	1	1	yes	characteristic
124	C₂×C₅₃	#1382	1	1	yes	characteristic
125	C₂²×C₁₆	—	106	1	no	—
126	C₂×#107	—	53	1	no	—
127	C₂×#107	#6157	1	1	yes	characteristic
128	C₂×C₄⋊C₈	—	106	1	no	—
129	C₂×C₄×C₈	—	53	1	no	—
130	C₄×C₄⋊C₄	—	106	1	no	—
131	C₄×C₁₆	—	53	2	no	—
132	C₄×C₁₆	—	106	1	no	—
133	#340	—	106	1	no	—
134	#335	—	53	2	no	—
135	#335	—	106	1	no	—
136	C₅₃	#7171	1	1	yes	characteristic, complemented, Sylow
137	C₂²×C₈	—	53	1	no	—
138	C₂²×C₈	—	212	1	no	—
139	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
140	C₂×C₄⋊C₄	—	2	1	no	—
141	C₂×C₄⋊C₄	—	106	1	no	—
142	C₂×C₄²	—	53	1	no	—
143	C₂×C₄²	—	212	1	no	—
144	C₂×C₁₆	—	106	2	no	—
145	C₂×C₁₆	—	106	2	no	—
146	C₂×C₁₆	—	106	2	no	—
147	#107	—	2	1	no	—
148	#107	—	53	2	no	—
149	#107	—	106	1	no	—
150	#107	#88324	1	2	yes	complemented
151	C₄⋊C₈	—	106	2	no	—
152	C₄⋊C₈	—	212	1	no	—
153	C₄×C₈	—	53	1	no	—
154	C₄×C₈	—	53	1	no	—
155	C₄×C₈	—	53	1	no	—
156	C₄×C₈	—	53	1	no	—
157	#95	—	212	1	no	—
158	C₂²×C₄	—	4	1	no	—
159	C₂²×C₄	—	53	1	no	—
160	C₂²×C₄	—	53	1	no	—
161	C₂²×C₄	—	212	1	no	—
162	C₂²×C₄	C₂²⋊#6157	1	1	yes	characteristic
163	C₂×C₈	—	53	1	no	—
164	C₂×C₈	—	53	1	no	—
165	C₂×C₈	—	106	1	no	—
166	C₂×C₈	—	106	1	no	—
167	C₂×C₈	—	424	1	no	—
168	C₂×C₈	—	424	1	no	—
169	C₂×C₈	—	212	2	no	—
170	C₂×C₈	D₅₃⋊C₈	1	1	yes	characteristic
171	C₂×C₈	#11719936	1	1	yes	characteristic
172	C₂×C₈	C₄×#6157	1	1	yes	characteristic
173	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
174	C₂×C₈	C₂×#88324	1	1	yes	characteristic
175	C₂×C₈	#11719940	1	1	yes	characteristic
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	4	1	no	—
178	C₄⋊C₄	—	106	2	no	—
179	C₄⋊C₄	—	212	1	no	—
180	C₄²	—	53	1	no	—
181	C₄²	—	53	1	no	—
182	C₄²	—	53	1	no	—
183	C₄²	—	53	1	no	—
184	C₄²	—	424	1	no	—
185	C₄²	—	424	1	no	—
186	C₁₆	—	106	4	no	—
187	C₂³	#13133279	1	1	yes	characteristic
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	8	1	no	—
190	C₂×C₄	—	8	1	no	—
191	C₂×C₄	—	53	1	no	—
192	C₂×C₄	—	53	1	no	—
193	C₂×C₄	—	53	1	no	—
194	C₂×C₄	—	53	1	no	—
195	C₂×C₄	—	53	1	no	—
196	C₂×C₄	—	53	1	no	—
197	C₂×C₄	—	53	1	no	—
198	C₂×C₄	—	53	1	no	—
199	C₂×C₄	—	106	1	no	—
200	C₂×C₄	—	106	1	no	—
201	C₂×C₄	—	424	1	no	—
202	C₂×C₄	—	424	1	no	—
203	C₂×C₄	—	212	2	no	—
204	C₂×C₄	(C₂×#6157)⋊C₄	1	1	yes	characteristic
205	C₂×C₄	C₂²⋊#88324	1	1	yes	characteristic
206	C₂×C₄	#13133274	1	1	yes	characteristic
207	C₂×C₄	#13133280	1	1	yes	characteristic
208	C₂×C₄	#13133281	1	1	yes	characteristic
209	C₂×C₄	#13133282	1	1	yes	characteristic
210	C₈	—	106	1	no	—
211	C₈	—	106	1	no	—
212	C₈	—	424	2	no	—
213	C₈	C₄×#88324	1	1	yes	characteristic
214	C₈	#13133272	1	1	yes	characteristic
215	C₈	C₄⋊#88324	1	1	yes	characteristic
216	C₈	#13133284	1	1	yes	characteristic
217	C₂²	D₄⋊#88324	1	1	yes	characteristic
218	C₂²	#16811471	1	1	yes	characteristic
219	C₂²	#16811496	1	1	yes	characteristic
220	C₂²	#16811535	1	1	yes	characteristic
221	C₂²	#16811536	1	1	yes	characteristic
222	C₂²	#16811537	1	1	yes	characteristic
223	C₂²	#16811538	1	1	yes	characteristic
224	C₄	—	8	2	no	—
225	C₄	—	53	1	no	—
226	C₄	—	53	1	no	—
227	C₄	—	53	1	no	—
228	C₄	—	53	1	no	—
229	C₄	—	106	1	no	—
230	C₄	—	106	1	no	—
231	C₄	—	424	2	no	—
232	C₄	Q₈⋊#88324	1	1	yes	characteristic
233	C₄	#16811473	1	1	yes	characteristic
234	C₄	#16811489	1	1	yes	characteristic
235	C₄	#16811493	1	1	yes	characteristic
236	C₂	#20289551	1	1	yes	characteristic
237	C₂	#20289669	1	1	yes	characteristic
238	C₂	#20289670	1	1	yes	characteristic
239	C₂	#20289671	1	1	yes	characteristic
240	C₂	#20289672	1	1	yes	characteristic
241	C₂	#20289772	1	1	yes	characteristic
242	C₂	#20289776	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133279
commutator	C₈×C₅₃	C₄×C₈
Frattini	C₂²×C₈	C₂×#6157
Fitting	C₂×#107×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	#102
cosocle	C₂×C₈×Dic₅₃	C₂²
hypercenter	C₂×#107	#6157
hypocenter	C₅₃	#7171
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#107

Complements

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

N	H ≅ G/N	type
7. #16811509	C₄	semidirect
9. #16811486	C₄	semidirect
26. #13133151	C₈	semidirect
35. #107×C₅₃	C₈	semidirect
136. C₅₃	#7171	semidirect
150. #107	#88324	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811509	2
C₂×#16811486	2
#107×Dic₅₃	2
#7171	53