Subgroups — #28310780

Order 13568.

374 conjugacy classes of subgroups.

The subgroup lattice has 250 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×#16811512	C₂	1	1	yes	maximal, characteristic
3	C₂×#16811511	C₂	1	1	yes	maximal, characteristic
4	C₂×#16811509	C₂	1	1	yes	maximal, characteristic
5	#20289754	C₂	1	1	yes	maximal, characteristic
6	#20289753	C₂	1	1	yes	maximal, characteristic
7	#20274576	C₂	1	1	yes	maximal, characteristic
8	C₂²×#11719910	C₂²	1	1	yes	characteristic
9	C₂²×#11719910	C₂²	1	1	yes	characteristic
10	C₂²×#11719910	C₂²	1	1	yes	characteristic
11	C₂²×#11719910	C₂²	1	1	yes	characteristic
12	#16811544	C₄	1	4	yes	complemented
13	#16811512	—	2	2	no	—
14	#16811511	—	2	2	no	—
15	#16811509	C₄	1	4	yes	complemented
16	C₂×#13133146	C₄	1	1	yes	characteristic
17	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
18	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
20	C₄×#11719919	C₄	1	1	yes	characteristic
21	#16810377	C₄	1	1	yes	characteristic
22	C₅₃×#336	C₄	1	1	yes	characteristic
23	C₂×#11719910	—	2	1	no	—
24	C₂×#11719910	—	2	1	no	—
25	C₂×#11719910	—	2	1	no	—
26	C₂×#11719910	—	2	1	no	—
27	C₂×#11719910	—	2	1	no	—
28	C₂×#11719910	—	2	1	no	—
29	C₂×#11719910	C₂×C₄	1	2	yes	—
30	C₂×#11719910	C₂×C₄	1	2	yes	—
31	C₂×#11719910	C₂×C₄	1	2	yes	—
32	C₂×#11719910	C₂×C₄	1	2	yes	—
33	C₂×#11719910	D₄	1	2	yes	—
34	C₂×#11719910	Q₈	1	2	yes	—
35	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
36	C₂²×#88322	C₂³	1	1	yes	characteristic
37	C₂×#11719919	C₈	1	2	yes	—
38	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
40	#13133146	C₈	1	4	yes	—
41	C₄×#88322	—	2	1	no	—
42	C₄×#88322	—	2	1	no	—
43	C₈×Dic₅₃	—	2	1	no	—
44	C₈×Dic₅₃	—	2	1	no	—
45	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
46	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
47	#13133124	C₈	1	2	yes	—
48	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
52	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
54	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
55	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
56	C₂×C₈×C₅₃	—	2	1	no	—
57	C₂×C₈×C₅₃	—	2	1	no	—
58	C₂×C₈×C₅₃	—	2	1	no	—
59	C₂×C₈×C₅₃	—	2	1	no	—
60	C₂×C₈×C₅₃	C₄²	1	2	yes	—
61	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
62	C₄²×C₅₃	—	2	1	no	—
63	C₄²×C₅₃	—	2	1	no	—
64	#11719919	—	2	4	no	—
65	C₄×Dic₅₃	—	2	1	no	—
66	C₄×Dic₅₃	—	2	1	no	—
67	C₄×Dic₅₃	C₄²	1	2	yes	—
68	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
69	C₂×#88322	—	2	1	no	—
70	C₂×#88322	—	2	1	no	—
71	C₂×#88322	C₂×C₈	1	2	yes	—
72	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
73	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
74	C₂×#88322	C₂×D₄	1	1	yes	characteristic
75	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
76	C₂×#88322	#41	1	1	yes	characteristic
77	C₂×#88322	#41	1	1	yes	characteristic
78	#11719910	—	2	2	no	—
79	#11719910	—	2	2	no	—
80	#11719910	—	2	2	no	—
81	#11719910	C₄⋊C₄	1	4	yes	complemented
82	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
83	C₂×C₄×C₅₃	—	2	2	no	—
84	C₂×C₄×C₅₃	—	2	2	no	—
85	C₂×C₄×C₅₃	—	2	2	no	—
86	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
91	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
92	C₂×Dic₅₃	—	2	2	no	—
93	C₂×Dic₅₃	—	2	2	no	—
94	C₂×Dic₅₃	—	2	2	no	—
95	C₂×Dic₅₃	—	2	2	no	—
96	C₂×Dic₅₃	C₄×C₈	1	2	yes	—
97	C₂×Dic₅₃	C₄×C₈	1	2	yes	—
98	C₂×Dic₅₃	M₃₂	1	4	yes	—
99	C₈×C₅₃	—	2	2	no	—
100	C₈×C₅₃	—	2	2	no	—
101	#88322	—	2	2	no	—
102	#88322	C₂×C₄⋊C₄	1	1	yes	characteristic
103	#88322	#117	1	1	yes	characteristic
104	#88322	C₄×D₄	1	1	yes	characteristic
105	#88322	C₄×Q₈	1	1	yes	characteristic
106	#12662	—	53	1	no	maximal, Sylow
107	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
108	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
109	C₂²×C₅₃	C₈×Q₈	1	1	yes	characteristic
110	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
111	C₂²×C₅₃	#504	1	2	yes	—
112	C₄×C₅₃	—	4	2	no	—
113	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
114	C₄×C₅₃	#405	1	1	yes	characteristic
115	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
116	C₄×C₅₃	C₈×Q₈	1	1	yes	characteristic
117	Dic₅₃	—	2	2	no	—
118	Dic₅₃	—	4	2	no	—
119	Dic₅₃	#346	1	4	yes	complemented
120	C₂×C₄⋊C₁₆	—	53	1	no	—
121	C₂×#346	—	53	1	no	—
122	C₂×#346	—	53	1	no	—
123	C₂×C₄×C₁₆	—	53	1	no	—
124	C₈×C₄⋊C₄	—	53	1	no	—
125	#1374	—	53	1	no	—
126	#1374	—	53	1	no	—
127	C₂×C₅₃	C₈×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₅₃	C₂×#346	1	1	yes	characteristic
129	C₂×C₅₃	#2108	1	1	yes	characteristic
130	C₂×C₅₃	#2169	1	2	yes	—
131	C₂×C₅₃	#2183	1	2	yes	—
132	C₂²×C₁₆	—	53	1	no	—
133	C₂²×C₁₆	—	53	1	no	—
134	C₂²×C₁₆	—	53	1	no	—
135	C₂²×C₁₆	—	53	1	no	—
136	C₂×C₄⋊C₈	—	53	1	no	—
137	C₂×C₄×C₈	—	53	1	no	—
138	C₂×C₄×C₈	—	53	1	no	—
139	C₂×C₄×C₈	—	53	1	no	—
140	C₄×C₄⋊C₄	—	53	1	no	—
141	C₄⋊C₁₆	—	53	4	no	—
142	#346	—	106	2	no	—
143	#346	—	106	2	no	—
144	C₄×C₁₆	—	53	4	no	—
145	#336	—	53	1	no	—
146	#336	#6157	1	1	yes	characteristic
147	C₅₃	#12662	1	1	yes	characteristic, complemented, Sylow
148	C₂²×C₈	—	53	1	no	—
149	C₂²×C₈	—	53	1	no	—
150	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
151	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
152	C₂×C₄⋊C₄	—	53	2	no	—
153	C₂×C₄²	—	53	1	no	—
154	C₂×C₄²	—	53	1	no	—
155	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
156	C₂×C₁₆	—	53	2	no	—
157	C₂×C₁₆	—	53	2	no	—
158	C₂×C₁₆	—	53	2	no	—
159	C₂×C₁₆	—	53	2	no	—
160	C₂×C₁₆	—	53	2	no	—
161	C₂×C₁₆	—	53	2	no	—
162	C₂×C₁₆	—	106	1	no	—
163	C₂×C₁₆	—	106	1	no	—
164	C₂×C₁₆	—	106	1	no	—
165	C₂×C₁₆	—	106	1	no	—
166	C₂×C₁₆	—	106	1	no	—
167	C₂×C₁₆	—	106	1	no	—
168	C₄⋊C₈	—	53	4	no	—
169	C₄×C₈	—	53	2	no	—
170	C₄×C₈	—	53	2	no	—
171	C₄×C₈	—	106	1	no	—
172	C₄×C₈	—	106	1	no	—
173	C₄×C₈	—	106	1	no	—
174	C₄×C₈	—	106	1	no	—
175	#95	—	53	2	no	—
176	C₂²×C₄	—	53	1	no	—
177	C₂²×C₄	—	53	1	no	—
178	C₂²×C₄	—	53	2	no	—
179	C₂²×C₄	D₅₃⋊C₈	1	2	yes	—
180	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	1	no	—
183	C₂×C₈	—	2	1	no	—
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	53	1	no	—
186	C₂×C₈	—	53	1	no	—
187	C₂×C₈	—	53	1	no	—
188	C₂×C₈	—	53	1	no	—
189	C₂×C₈	—	53	1	no	—
190	C₂×C₈	—	53	1	no	—
191	C₂×C₈	—	53	2	no	—
192	C₂×C₈	—	106	1	no	—
193	C₂×C₈	—	106	1	no	—
194	C₂×C₈	D₅₃⋊C₈	1	2	yes	—
195	C₂×C₈	C₄×#6157	1	2	yes	—
196	C₄⋊C₄	—	106	4	no	—
197	C₄²	—	2	1	no	—
198	C₄²	—	2	1	no	—
199	C₄²	—	53	2	no	—
200	C₄²	—	53	2	no	—
201	C₄²	—	106	1	no	—
202	C₄²	—	106	1	no	—
203	C₁₆	—	53	4	no	—
204	C₁₆	—	106	2	no	—
205	C₁₆	—	106	2	no	—
206	C₁₆	—	106	2	no	—
207	C₂³	C₂×D₅₃⋊C₈	1	1	yes	characteristic
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	53	2	no	—
212	C₂×C₄	—	53	4	no	—
213	C₂×C₄	—	53	2	no	—
214	C₂×C₄	—	106	2	no	—
215	C₂×C₄	—	106	2	no	—
216	C₂×C₄	—	106	2	no	—
217	C₂×C₄	—	106	2	no	—
218	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
219	C₂×C₄	D₄×#6157	1	1	yes	characteristic
220	C₂×C₄	Q₈×#6157	1	1	yes	characteristic
221	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
222	C₂×C₄	#13133304	1	1	yes	characteristic
223	C₂×C₄	#13133304	1	1	yes	characteristic
224	C₈	—	2	2	no	—
225	C₈	—	2	2	no	—
226	C₈	—	53	1	no	—
227	C₈	—	53	1	no	—
228	C₈	—	53	1	no	—
229	C₈	—	53	1	no	—
230	C₈	—	106	2	no	—
231	C₂²	C₄×D₅₃⋊C₈	1	1	yes	characteristic
232	C₂²	#16811650	1	1	yes	characteristic
233	C₂²	#16811672	1	1	yes	characteristic
234	C₂²	D₅₃⋊M₃₂	1	2	yes	—
235	C₂²	#16811682	1	2	yes	—
236	C₄	—	4	2	no	—
237	C₄	—	53	4	no	—
238	C₄	—	106	2	no	—
239	C₄	—	212	2	no	—
240	C₄	C₄⋊C₄×#6157	1	1	yes	characteristic
241	C₄	#16811587	1	1	yes	characteristic
242	C₄	#16811650	1	1	yes	characteristic
243	C₄	#16811672	1	1	yes	characteristic
244	C₂	#20290507	1	1	yes	characteristic
245	C₂	#20290916	1	1	yes	characteristic
246	C₂	#20290925	1	1	yes	characteristic
247	C₂	#20291100	1	2	yes	—
248	C₂	#20291132	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#6157
commutator	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₅₃×#336	C₄
socle, prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂²×#88322	C₂³
hypercenter	#336	#6157
hypocenter	C₅₃	#12662
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ #336

Complements

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

N	H ≅ G/N	type
12. #16811544	C₄	semidirect
15. #16811509	C₄	semidirect
81. #11719910	C₄⋊C₄	semidirect
119. Dic₅₃	#346	semidirect
147. C₅₃	#12662	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×#16811512	2
C₂×#16811511	2
C₂×#16811509	2
#20289754	2
#20289753	2
#20274576	2
#12662	53