Subgroups — #27282985

Order 13568.

482 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20275652	C₂	1	2	yes	maximal
2	C₂×#16810376	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊#88322	C₂	1	2	yes	maximal
4	C₂×#16810345	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	C₂²	1	2	yes	—
7	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
8	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
9	#16810378	—	2	4	no	—
10	#16810376	—	2	2	no	—
11	Q₈⋊#88322	—	2	4	no	—
12	#16810345	—	2	2	no	—
13	C₂²×Q₁₆×C₅₃	C₄	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
15	C₄×Q₁₆×C₅₃	—	2	2	no	—
16	C₄×Q₁₆×C₅₃	—	2	2	no	—
17	C₂×#107×C₅₃	C₄	1	1	yes	characteristic
18	C₂×#103×C₅₃	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×#88322	—	2	2	no	—
21	C₂²×#88322	—	2	1	no	—
22	C₂²×#88322	D₄	1	2	yes	—
23	C₄⋊#88322	—	2	2	no	—
24	C₄⋊#88322	—	2	2	no	—
25	C₄×#88322	—	2	1	no	—
26	C₄×#88322	—	2	1	no	—
27	C₄×#88322	—	2	1	no	—
28	C₄×#88322	D₄	1	2	yes	—
29	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
30	C₂×Q₁₆×C₅₃	—	2	2	no	—
31	C₂×Q₁₆×C₅₃	—	2	4	no	—
32	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₄×Q₈×C₅₃	—	2	2	no	—
35	C₄×Q₈×C₅₃	—	2	2	no	—
36	C₄×Q₈×C₅₃	—	2	2	no	—
37	C₄×Q₈×C₅₃	—	2	2	no	—
38	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
39	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₅₃	—	2	2	no	—
41	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
42	#107×C₅₃	—	2	2	no	—
43	#103×C₅₃	—	2	4	no	—
44	C₄×C₈×C₅₃	D₄	1	2	yes	—
45	C₄×C₈×C₅₃	D₄	1	2	yes	—
46	C₂×Q₈×C₅₃	—	2	2	no	—
47	C₂×Q₈×C₅₃	—	2	2	no	—
48	C₂×Q₈×C₅₃	—	2	2	no	—
49	C₂×Q₈×C₅₃	—	2	2	no	—
50	C₂×Q₈×C₅₃	—	2	2	no	—
51	C₂×Q₈×C₅₃	M₁₆	1	4	yes	—
52	C₂²×C₄×C₅₃	—	2	2	no	—
53	C₂²×C₄×C₅₃	—	2	2	no	—
54	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
57	Q₁₆×C₅₃	—	2	8	no	—
58	C₂×C₈×C₅₃	—	2	2	no	—
59	C₂×C₈×C₅₃	—	2	2	no	—
60	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
61	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
62	C₄⋊C₄×C₅₃	—	2	2	no	—
63	C₄⋊C₄×C₅₃	—	2	2	no	—
64	C₄⋊C₄×C₅₃	—	4	2	no	—
65	C₄⋊C₄×C₅₃	—	4	2	no	—
66	C₄²×C₅₃	—	4	2	no	—
67	C₄²×C₅₃	—	4	2	no	—
68	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₅₃	#41	1	1	yes	characteristic
72	C₂×#88322	—	2	2	no	—
73	C₂×#88322	—	2	2	no	—
74	C₂×#88322	—	2	2	no	—
75	C₂×#88322	—	4	2	no	—
76	C₂×#88322	—	4	2	no	—
77	C₂×#88322	—	4	4	no	—
78	C₂×#88322	Q₁₆	1	4	yes	complemented
79	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
80	Q₈×C₅₃	—	2	4	no	—
81	Q₈×C₅₃	—	2	4	no	—
82	Q₈×C₅₃	—	4	2	no	—
83	Q₈×C₅₃	—	4	2	no	—
84	C₂×C₄×C₅₃	—	2	2	no	—
85	C₂×C₄×C₅₃	—	2	4	no	—
86	C₂×C₄×C₅₃	—	4	2	no	—
87	C₂×C₄×C₅₃	—	4	2	no	—
88	C₂×C₄×C₅₃	—	4	2	no	—
89	C₂×C₄×C₅₃	—	4	4	no	—
90	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
93	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
94	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
95	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
97	C₂×C₄×C₅₃	#131	1	2	yes	—
98	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
99	C₂×C₄×C₅₃	#135	1	2	yes	—
100	C₈×C₅₃	—	2	2	no	—
101	C₈×C₅₃	—	2	2	no	—
102	#88322	—	2	4	no	—
103	#88322	—	4	2	no	—
104	#88322	—	8	2	no	—
105	#9662	—	53	1	no	maximal, Sylow
106	C₂²×C₅₃	#390	1	1	yes	characteristic
107	C₂²×C₅₃	#435	1	2	yes	—
108	C₂²×C₅₃	C₄×Q₁₆	1	2	yes	—
109	C₂²×C₅₃	#444	1	2	yes	—
110	C₄×C₅₃	—	2	1	no	—
111	C₄×C₅₃	—	2	1	no	—
112	C₄×C₅₃	—	4	4	no	—
113	C₄×C₅₃	—	8	2	no	—
114	C₄×C₅₃	#407	1	1	yes	characteristic
115	C₄×C₅₃	#408	1	1	yes	characteristic
116	C₄×C₅₃	#435	1	2	yes	—
117	C₄×C₅₃	#467	1	2	yes	—
118	C₄×C₅₃	#494	1	1	yes	characteristic
119	C₄×C₅₃	#495	1	1	yes	characteristic
120	C₂×C₄×Q₁₆	D₅₃	1	1	yes	characteristic
121	#1984	—	53	2	no	—
122	C₂×#335	—	53	1	no	—
123	C₂×Q₈⋊C₈	—	53	2	no	—
124	C₂×#322	—	53	1	no	—
125	C₂×C₅₃	#1584	1	4	yes	—
126	C₂×C₅₃	#1955	1	1	yes	characteristic
127	C₂×C₅₃	#1969	1	1	yes	characteristic
128	C₂×C₅₃	#1975	1	1	yes	characteristic
129	C₂²×Q₁₆	Dic₅₃	1	1	yes	characteristic
130	C₂×C₄×Q₈	C₂×D₅₃	1	2	yes	—
131	C₄×Q₁₆	—	2	2	no	—
132	C₄×Q₁₆	—	2	2	no	—
133	C₂×#107	Dic₅₃	1	1	yes	characteristic
134	C₂×C₄⋊C₈	—	53	2	no	—
135	C₂×#103	Dic₅₃	1	2	yes	—
136	C₂×C₄×C₈	—	53	1	no	—
137	C₂×C₄×C₈	—	53	1	no	—
138	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
139	#336	—	106	4	no	—
140	#335	—	106	2	no	—
141	Q₈⋊C₈	—	106	4	no	—
142	#322	—	106	2	no	—
143	C₅₃	#9662	1	1	yes	characteristic, complemented, Sylow
144	C₂²×Q₈	C₂×Dic₅₃	1	2	yes	—
145	C₂×Q₁₆	—	2	4	no	—
146	C₂×Q₁₆	—	2	2	no	—
147	C₂²×C₈	—	53	2	no	—
148	C₂²×C₈	—	106	1	no	—
149	C₂²×C₈	—	106	2	no	—
150	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
151	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
152	C₄×Q₈	—	2	2	no	—
153	C₄×Q₈	—	2	2	no	—
154	C₄×Q₈	—	2	2	no	—
155	C₄×Q₈	—	2	2	no	—
156	C₂×C₄⋊C₄	—	2	2	no	—
157	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
158	C₂×C₄²	—	2	2	no	—
159	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
160	#107	—	2	2	no	—
161	C₄⋊C₈	—	106	2	no	—
162	C₄⋊C₈	—	106	2	no	—
163	#103	—	2	4	no	—
164	C₄×C₈	—	53	2	no	—
165	C₄×C₈	—	106	1	no	—
166	C₄×C₈	—	106	1	no	—
167	C₄×C₈	—	106	1	no	—
168	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
169	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
170	C₂×Q₈	—	2	2	no	—
171	C₂×Q₈	—	2	2	no	—
172	C₂×Q₈	—	2	2	no	—
173	C₂×Q₈	—	2	2	no	—
174	C₂×Q₈	—	2	2	no	—
175	C₂×Q₈	#11719917	1	4	yes	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
179	C₂²×C₄	#11719947	1	1	yes	characteristic
180	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
181	Q₁₆	—	2	8	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	53	4	no	—
185	C₂×C₈	—	106	2	no	—
186	C₂×C₈	—	106	2	no	—
187	C₂×C₈	—	106	2	no	—
188	C₂×C₈	—	212	2	no	—
189	C₂×C₈	—	212	4	no	—
190	C₂×C₈	—	212	2	no	—
191	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
192	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄²	—	4	2	no	—
198	C₄²	—	4	2	no	—
199	C₄²	D₄×D₅₃	1	1	yes	characteristic
200	C₄²	#11719947	1	1	yes	characteristic
201	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
202	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
203	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
204	Q₈	—	2	4	no	—
205	Q₈	—	2	4	no	—
206	Q₈	—	4	2	no	—
207	Q₈	—	4	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
215	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
216	C₂×C₄	#13133188	1	1	yes	characteristic
217	C₂×C₄	#13133188	1	1	yes	characteristic
218	C₂×C₄	#13133204	1	1	yes	characteristic
219	C₂×C₄	#13133213	1	1	yes	characteristic
220	C₂×C₄	C₂×#11719917	1	2	yes	—
221	C₂×C₄	#13133224	1	2	yes	—
222	C₂×C₄	Q₁₆×D₅₃	1	2	yes	—
223	C₂×C₄	#13133240	1	2	yes	—
224	C₈	—	2	2	no	—
225	C₈	—	2	2	no	—
226	C₈	—	106	4	no	—
227	C₈	—	424	2	no	—
228	C₈	—	212	2	no	—
229	C₂²	#16810526	1	1	yes	characteristic
230	C₂²	#16810695	1	2	yes	—
231	C₂²	Q₁₆×Dic₅₃	1	2	yes	—
232	C₂²	#16810745	1	2	yes	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	4	4	no	—
236	C₄	—	8	2	no	—
237	C₄	#16810587	1	1	yes	characteristic
238	C₄	#16810590	1	1	yes	characteristic
239	C₄	#16810695	1	2	yes	—
240	C₄	#16810861	1	2	yes	—
241	C₄	#16811001	1	1	yes	characteristic
242	C₄	#16811008	1	1	yes	characteristic
243	C₂	#20273566	1	4	yes	—
244	C₂	#20275514	1	1	yes	characteristic
245	C₂	#20275577	1	1	yes	characteristic
246	C₂	#20275609	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅₃
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₄×Q₁₆×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×C₄×Q₁₆	D₅₃
hypocenter	C₅₃	#9662
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
78. C₂×#88322	Q₁₆	semidirect
143. C₅₃	#9662	semidirect

Maximal subgroups

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

subgroup	index
#20275652	2
C₂×#16810376	2
C₂×Q₈⋊#88322	2
C₂×#16810345	2
C₂×C₄×Q₁₆×C₅₃	2
#9662	53