Subgroups — #28298701

Order 13568.

262 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811544	C₂	1	1	yes	maximal, characteristic
2	C₂×#16811487	C₂	1	1	yes	maximal, characteristic
3	#20274873	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11719910	—	2	1	no	—
5	C₂×C₄⋊#88324	—	2	1	no	—
6	#16811544	—	2	1	no	—
7	#16811544	C₄	1	2	yes	complemented
8	#16811487	—	2	1	no	—
9	#16811487	C₄	1	2	yes	complemented
10	C₂×#13133151	C₄	1	1	yes	characteristic
11	C₂×#13133150	—	2	1	no	—
12	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
13	#16810492	—	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	—	4	1	no	—
18	C₂×#11719910	—	4	1	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×#11719919	—	4	1	no	—
24	C₂×#11719919	—	4	1	no	—
25	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
26	#13133151	—	4	1	no	—
27	#13133150	—	4	1	no	—
28	#13133150	—	4	1	no	—
29	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
30	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
31	C₈×Dic₅₃	D₄	1	1	yes	characteristic
32	C₈×Dic₅₃	Q₈	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
35	#107×C₅₃	—	4	1	no	—
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	—	2	1	no	—
45	C₂×C₈×C₅₃	—	2	1	no	—
46	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
47	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
48	C₄⋊C₄×C₅₃	—	4	1	no	—
49	C₄⋊C₄×C₅₃	—	4	1	no	—
50	C₄⋊Dic₅₃	—	4	1	no	—
51	C₄⋊Dic₅₃	—	4	1	no	—
52	#11719919	—	8	2	no	—
53	#11719919	—	8	1	no	—
54	#11719919	—	8	1	no	—
55	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
57	C₄×Dic₅₃	SD₁₆	1	1	yes	characteristic
58	C₄×Dic₅₃	Q₁₆	1	1	yes	characteristic
59	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
61	C₂×#88322	D₈	1	1	yes	characteristic
62	C₂×#88322	SD₁₆	1	1	yes	characteristic
63	C₂×#88322	SD₁₆	1	1	yes	characteristic
64	C₂×#88322	Q₁₆	1	1	yes	characteristic
65	#11719910	—	4	2	no	—
66	C₂³×C₅₃	#102	1	1	yes	characteristic
67	C₂×C₄×C₅₃	—	8	1	no	—
68	C₂×C₄×C₅₃	—	8	1	no	—
69	C₂×C₄×C₅₃	—	8	1	no	—
70	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
71	C₂×C₄×C₅₃	#100	1	1	yes	characteristic
72	C₂×C₄×C₅₃	#101	1	1	yes	characteristic
73	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
74	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
75	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
76	C₂×Dic₅₃	—	2	1	no	—
77	C₂×Dic₅₃	—	2	1	no	—
78	C₂×Dic₅₃	—	8	1	no	—
79	C₂×Dic₅₃	—	8	1	no	—
80	C₂×Dic₅₃	—	8	1	no	—
81	C₂×Dic₅₃	#102	1	1	yes	characteristic
82	C₂×Dic₅₃	#103	1	1	yes	characteristic
83	C₂×Dic₅₃	#106	1	1	yes	characteristic
84	C₂×Dic₅₃	#107	1	1	yes	characteristic
85	C₂×Dic₅₃	D₁₆	1	1	yes	characteristic
86	C₂×Dic₅₃	SD₃₂	1	1	yes	characteristic
87	C₂×Dic₅₃	SD₃₂	1	1	yes	characteristic
88	C₂×Dic₅₃	Q₃₂	1	1	yes	characteristic
89	#88324	—	8	2	no	—
90	C₈×C₅₃	—	2	1	no	—
91	C₈×C₅₃	—	2	1	no	—
92	#88322	#102	1	1	yes	characteristic
93	#88322	#103	1	1	yes	characteristic
94	#88322	#106	1	1	yes	characteristic
95	#88322	#107	1	1	yes	characteristic
96	#7274	—	53	1	no	maximal, Sylow
97	C₂²×C₅₃	#331	1	1	yes	characteristic
98	C₂²×C₅₃	#332	1	1	yes	characteristic
99	C₂²×C₅₃	#340	1	1	yes	characteristic
100	C₂²×C₅₃	#357	1	1	yes	characteristic
101	C₂²×C₅₃	#358	1	1	yes	characteristic
102	C₂²×C₅₃	#361	1	1	yes	characteristic
103	C₂²×C₅₃	#362	1	1	yes	characteristic
104	C₄×C₅₃	—	16	1	no	—
105	C₄×C₅₃	#332	1	1	yes	characteristic
106	C₄×C₅₃	#333	1	1	yes	characteristic
107	C₄×C₅₃	#340	1	1	yes	characteristic
108	C₄×C₅₃	#343	1	1	yes	characteristic
109	Dic₅₃	—	2	1	no	—
110	Dic₅₃	—	2	1	no	—
111	Dic₅₃	—	16	1	no	—
112	Dic₅₃	#357	1	1	yes	characteristic
113	Dic₅₃	#358	1	1	yes	characteristic
114	Dic₅₃	#366	1	1	yes	characteristic
115	Dic₅₃	#367	1	1	yes	characteristic
116	C₂×C₄⋊C₁₆	—	53	1	no	—
117	#1845	—	53	1	no	—
118	C₂×#335	—	53	1	no	—
119	C₂×C₅₃	#1296	1	1	yes	characteristic
120	C₂×C₅₃	#1361	1	1	yes	characteristic
121	C₂×C₅₃	#1362	1	1	yes	characteristic
122	C₂×C₅₃	#1364	1	1	yes	characteristic
123	C₂×C₅₃	#1365	1	1	yes	characteristic
124	C₂×C₅₃	#1380	1	1	yes	characteristic
125	C₂×C₅₃	#1387	1	1	yes	characteristic
126	C₂²×C₁₆	—	106	1	no	—
127	C₂×#107	—	53	1	no	—
128	C₂×#107	—	106	1	no	—
129	C₂×#107	#6157	1	1	yes	characteristic
130	C₂×C₄⋊C₈	—	106	1	no	—
131	C₂×C₄×C₈	—	53	1	no	—
132	#384	—	106	1	no	—
133	C₄⋊C₁₆	—	53	2	no	—
134	C₄⋊C₁₆	—	106	1	no	—
135	#335	—	53	2	no	—
136	#335	—	106	1	no	—
137	C₅₃	#7274	1	1	yes	characteristic, complemented, Sylow
138	C₂²×C₈	—	53	1	no	—
139	C₂²×C₈	—	212	1	no	—
140	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
141	C₂×C₄⋊C₄	—	2	1	no	—
142	C₂×C₄⋊C₄	—	106	1	no	—
143	C₂×C₄⋊C₄	—	212	1	no	—
144	C₂×C₄⋊C₄	—	212	1	no	—
145	C₂×C₄²	—	53	1	no	—
146	C₂×C₁₆	—	106	2	no	—
147	C₂×C₁₆	—	212	1	no	—
148	C₂×C₁₆	—	212	1	no	—
149	#107	—	4	1	no	—
150	#107	—	212	1	no	—
151	#107	—	212	1	no	—
152	#107	—	212	1	no	—
153	C₄⋊C₈	—	106	2	no	—
154	C₄⋊C₈	—	212	1	no	—
155	C₄×C₈	—	53	1	no	—
156	C₄×C₈	—	53	1	no	—
157	C₄×C₈	—	53	1	no	—
158	C₄×C₈	—	53	1	no	—
159	C₂²×C₄	—	4	1	no	—
160	C₂²×C₄	—	53	1	no	—
161	C₂²×C₄	—	53	1	no	—
162	C₂²×C₄	—	212	1	no	—
163	C₂²×C₄	C₂²⋊#6157	1	1	yes	characteristic
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	53	1	no	—
167	C₂×C₈	—	53	1	no	—
168	C₂×C₈	—	53	1	no	—
169	C₂×C₈	—	53	1	no	—
170	C₂×C₈	—	53	1	no	—
171	C₂×C₈	—	53	1	no	—
172	C₂×C₈	—	424	1	no	—
173	C₂×C₈	—	424	1	no	—
174	C₂×C₈	—	212	2	no	—
175	C₂×C₈	C₄×#6157	1	1	yes	characteristic
176	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
177	C₄⋊C₄	—	4	1	no	—
178	C₄⋊C₄	—	4	1	no	—
179	C₄⋊C₄	—	424	1	no	—
180	C₄⋊C₄	—	424	1	no	—
181	C₄⋊C₄	—	424	2	no	—
182	C₄⋊C₄	—	212	1	no	—
183	C₄⋊C₄	—	212	1	no	—
184	C₄²	—	53	1	no	—
185	C₄²	—	53	1	no	—
186	C₄²	—	53	1	no	—
187	C₄²	—	53	1	no	—
188	C₁₆	—	212	2	no	—
189	C₂³	#13133279	1	1	yes	characteristic
190	C₂×C₄	—	8	1	no	—
191	C₂×C₄	—	8	1	no	—
192	C₂×C₄	—	8	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	—	53	1	no	—
200	C₂×C₄	—	53	1	no	—
201	C₂×C₄	—	106	1	no	—
202	C₂×C₄	—	106	1	no	—
203	C₂×C₄	—	424	1	no	—
204	C₂×C₄	—	424	1	no	—
205	C₂×C₄	—	424	1	no	—
206	C₂×C₄	(C₂×#6157)⋊C₄	1	1	yes	characteristic
207	C₂×C₄	#13133277	1	1	yes	characteristic
208	C₂×C₄	#13133278	1	1	yes	characteristic
209	C₂×C₄	#13133279	1	1	yes	characteristic
210	C₂×C₄	#13133280	1	1	yes	characteristic
211	C₂×C₄	#13133280	1	1	yes	characteristic
212	C₈	—	2	1	no	—
213	C₈	—	2	1	no	—
214	C₈	—	53	1	no	—
215	C₈	—	53	1	no	—
216	C₈	—	53	1	no	—
217	C₈	—	53	1	no	—
218	C₈	—	424	2	no	—
219	C₂²	#16811481	1	1	yes	characteristic
220	C₂²	#16811482	1	1	yes	characteristic
221	C₂²	#16811496	1	1	yes	characteristic
222	C₂²	#16811535	1	1	yes	characteristic
223	C₂²	#16811536	1	1	yes	characteristic
224	C₂²	#16811541	1	1	yes	characteristic
225	C₂²	#16811542	1	1	yes	characteristic
226	C₄	—	16	1	no	—
227	C₄	—	53	1	no	—
228	C₄	—	53	1	no	—
229	C₄	—	53	1	no	—
230	C₄	—	53	1	no	—
231	C₄	—	848	1	no	—
232	C₄	—	106	1	no	—
233	C₄	—	106	1	no	—
234	C₄	#16811482	1	1	yes	characteristic
235	C₄	#16811483	1	1	yes	characteristic
236	C₄	#16811496	1	1	yes	characteristic
237	C₄	#16811503	1	1	yes	characteristic
238	C₂	#20289593	1	1	yes	characteristic
239	C₂	#20289729	1	1	yes	characteristic
240	C₂	#20289730	1	1	yes	characteristic
241	C₂	#20289733	1	1	yes	characteristic
242	C₂	#20289734	1	1	yes	characteristic
243	C₂	#20289772	1	1	yes	characteristic
244	C₂	#20289789	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133279
commutator	C₂×C₈×C₅₃	C₄²
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	#7274
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×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. #16811544	C₄	semidirect
9. #16811487	C₄	semidirect
137. C₅₃	#7274	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811544	2
C₂×#16811487	2
#20274873	2
#7274	53