Subgroups — #28305580

Order 13568.

398 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20290812	C₂	1	2	yes	maximal
2	C₂×#16811484	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×#88324	C₂	1	1	yes	maximal, characteristic
4	#20289551	C₂	1	2	yes	maximal
5	#106×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88324	—	2	2	no	—
7	C₂×C₄⋊#88324	C₂²	1	2	yes	—
8	C₂×C₄×#88324	C₂²	1	1	yes	characteristic
9	C₂×C₄×#88324	C₂²	1	1	yes	characteristic
10	#16811489	—	2	2	no	—
11	#16811484	—	2	2	no	—
12	C₈×#88324	—	2	2	no	—
13	C₂×#13133149	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×Dic₅₃	C₂²	1	2	yes	—
16	#16810385	C₄	1	2	yes	—
17	C₂×#106×C₅₃	C₄	1	1	yes	characteristic
18	C₂²×#88324	—	2	1	no	—
19	C₂²×#88324	—	2	2	no	—
20	C₂²×#88324	D₄	1	2	yes	—
21	C₄⋊#88324	—	2	2	no	—
22	C₄⋊#88324	—	2	2	no	—
23	C₄⋊#88324	—	4	4	no	—
24	C₄×#88324	—	2	1	no	—
25	C₄×#88324	—	2	1	no	—
26	C₄×#88324	—	2	1	no	—
27	C₄×#88324	Q₈	1	2	yes	—
28	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
30	C₂×C₄×Dic₅₃	—	2	2	no	—
31	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
32	#13133149	—	2	2	no	—
33	C₈×Dic₅₃	D₄	1	2	yes	—
34	C₈×Dic₅₃	Q₈	1	2	yes	—
35	#13133124	—	2	2	no	—
36	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
38	#106×C₅₃	C₈	1	4	yes	complemented
39	C₂²×C₄×C₅₃	—	2	2	no	—
40	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×Dic₅₃	—	2	2	no	—
42	C₂²×Dic₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₅₃	#41	1	1	yes	characteristic
44	C₂×#88324	—	2	4	no	—
45	C₂×#88324	—	2	2	no	—
46	C₂×#88324	—	2	2	no	—
47	C₂×#88324	—	2	2	no	—
48	C₂×#88324	—	4	2	no	—
49	C₂×#88324	—	4	2	no	—
50	C₂×#88324	—	4	2	no	—
51	C₂×#88324	SD₁₆	1	4	yes	—
52	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
53	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
54	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
55	C₄⋊C₄×C₅₃	—	2	2	no	—
56	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
57	C₄⋊Dic₅₃	—	2	2	no	—
58	C₄⋊Dic₅₃	—	2	2	no	—
59	C₄×Dic₅₃	—	4	2	no	—
60	C₄×Dic₅₃	—	4	2	no	—
61	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
62	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
63	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
64	C₄×Dic₅₃	#41	1	1	yes	characteristic
65	C₂×#88322	—	2	1	no	—
66	C₂×#88322	—	2	1	no	—
67	C₂×#88322	C₄⋊C₄	1	2	yes	—
68	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₅₃	—	2	4	no	—
70	C₂×C₄×C₅₃	—	4	2	no	—
71	C₂×C₄×C₅₃	—	4	2	no	—
72	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
76	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
77	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
78	C₂×Dic₅₃	—	2	2	no	—
79	C₂×Dic₅₃	—	4	2	no	—
80	C₂×Dic₅₃	—	4	2	no	—
81	C₂×Dic₅₃	—	4	2	no	—
82	C₂×Dic₅₃	#121	1	1	yes	characteristic
83	C₂×Dic₅₃	C₂²⋊Q₈	1	1	yes	characteristic
84	C₂×Dic₅₃	#125	1	1	yes	characteristic
85	C₂×Dic₅₃	C₄⋊Q₈	1	1	yes	characteristic
86	C₂×Dic₅₃	C₂×SD₁₆	1	2	yes	—
87	C₂×Dic₅₃	#135	1	2	yes	—
88	#88324	—	2	4	no	—
89	#88324	—	2	4	no	—
90	#88324	—	8	2	no	—
91	C₈×C₅₃	C₄⋊C₈	1	4	yes	complemented
92	#88322	—	2	2	no	—
93	#9593	—	53	1	no	maximal, Sylow
94	C₂²×C₅₃	#389	1	1	yes	characteristic
95	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
96	C₂²×C₅₃	#436	1	1	yes	characteristic
97	C₂²×C₅₃	C₄×SD₁₆	1	2	yes	—
98	C₂²×C₅₃	#443	1	2	yes	—
99	C₄×C₅₃	—	4	4	no	—
100	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
101	C₄×C₅₃	#424	1	1	yes	characteristic
102	C₄×C₅₃	C₈×Q₈	1	1	yes	characteristic
103	C₄×C₅₃	#446	1	1	yes	characteristic
104	Dic₅₃	—	2	1	no	—
105	Dic₅₃	—	2	1	no	—
106	Dic₅₃	—	8	2	no	—
107	Dic₅₃	#465	1	2	yes	—
108	Dic₅₃	#498	1	1	yes	characteristic
109	Dic₅₃	#499	1	1	yes	characteristic
110	#1939	—	53	2	no	—
111	C₄×#106	—	53	1	no	—
112	C₂×#334	—	53	1	no	—
113	C₂×C₈²	—	53	1	no	—
114	#1279	—	53	2	no	—
115	C₂×C₅₃	C₈×SD₁₆	1	2	yes	—
116	C₂×C₅₃	#1590	1	2	yes	—
117	C₂×C₅₃	#1939	1	1	yes	characteristic
118	C₂×C₅₃	#1941	1	1	yes	characteristic
119	C₂×C₅₃	#1945	1	1	yes	characteristic
120	C₂×#106	—	53	1	no	—
121	C₂×#106	#6157	1	1	yes	characteristic
122	C₂×C₄⋊C₈	—	53	2	no	—
123	C₂×C₄⋊C₈	—	106	2	no	—
124	C₂×C₄×C₈	—	53	1	no	—
125	C₂×C₄×C₈	—	53	1	no	—
126	C₂×C₄×C₈	—	53	1	no	—
127	C₄×C₄⋊C₄	—	53	2	no	—
128	#340	—	53	2	no	—
129	#336	—	106	2	no	—
130	#334	—	106	2	no	—
131	C₈²	—	106	2	no	—
132	C₅₃	#9593	1	1	yes	characteristic, complemented, Sylow
133	C₂²×C₈	—	53	2	no	—
134	C₂²×C₈	—	53	1	no	—
135	C₂²×C₈	—	106	1	no	—
136	C₂²×C₈	—	106	2	no	—
137	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
138	C₂×C₄⋊C₄	—	53	2	no	—
139	C₂×C₄⋊C₄	C₂×#6157	1	2	yes	—
140	C₂×C₄²	—	53	1	no	—
141	C₂×C₄²	—	106	2	no	—
142	#106	—	106	2	no	—
143	#106	#88324	1	4	yes	complemented
144	C₄⋊C₈	—	106	2	no	—
145	C₄⋊C₈	—	106	2	no	—
146	C₄⋊C₈	—	212	4	no	—
147	C₄×C₈	—	53	2	no	—
148	C₄×C₈	—	53	2	no	—
149	C₄×C₈	—	53	2	no	—
150	C₄×C₈	—	106	1	no	—
151	C₄×C₈	—	106	1	no	—
152	C₄×C₈	—	106	1	no	—
153	#95	—	106	2	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	53	1	no	—
156	C₂²×C₄	—	53	1	no	—
157	C₂²×C₄	—	106	2	no	—
158	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
159	C₂×C₈	—	53	4	no	—
160	C₂×C₈	—	53	2	no	—
161	C₂×C₈	—	106	2	no	—
162	C₂×C₈	—	106	2	no	—
163	C₂×C₈	—	106	2	no	—
164	C₂×C₈	—	106	4	no	—
165	C₂×C₈	—	106	1	no	—
166	C₂×C₈	—	106	1	no	—
167	C₂×C₈	—	212	2	no	—
168	C₂×C₈	—	212	2	no	—
169	C₂×C₈	—	212	2	no	—
170	C₂×C₈	#11719936	1	2	yes	—
171	C₂×C₈	C₄⋊#6157	1	2	yes	—
172	C₂×C₈	C₂×#88324	1	2	yes	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	106	2	no	—
175	C₄⋊C₄	—	106	2	no	—
176	C₄⋊C₄	C₂×#88324	1	4	yes	—
177	C₄²	—	53	1	no	—
178	C₄²	—	53	1	no	—
179	C₄²	—	53	1	no	—
180	C₄²	—	53	1	no	—
181	C₄²	—	212	2	no	—
182	C₄²	—	212	2	no	—
183	C₂³	D₄×#6157	1	1	yes	characteristic
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	4	2	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	53	1	no	—
188	C₂×C₄	—	53	1	no	—
189	C₂×C₄	—	53	2	no	—
190	C₂×C₄	—	53	2	no	—
191	C₂×C₄	—	53	1	no	—
192	C₂×C₄	—	53	1	no	—
193	C₂×C₄	—	106	2	no	—
194	C₂×C₄	—	212	2	no	—
195	C₂×C₄	—	212	2	no	—
196	C₂×C₄	—	212	2	no	—
197	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
198	C₂×C₄	Q₈×#6157	1	1	yes	characteristic
199	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
200	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
201	C₂×C₄	#13133304	1	1	yes	characteristic
202	C₂×C₄	#13133305	1	1	yes	characteristic
203	C₈	—	106	4	no	—
204	C₈	—	106	4	no	—
205	C₈	—	106	2	no	—
206	C₈	—	424	2	no	—
207	C₈	C₄⋊#88324	1	4	yes	complemented
208	C₂²	#16811574	1	1	yes	characteristic
209	C₂²	D₄×#88324	1	1	yes	characteristic
210	C₂²	#16811657	1	1	yes	characteristic
211	C₂²	SD₁₆×#6157	1	2	yes	—
212	C₂²	#16811666	1	2	yes	—
213	C₄	—	4	4	no	—
214	C₄	—	53	2	no	—
215	C₄	—	53	1	no	—
216	C₄	—	53	1	no	—
217	C₄	—	106	1	no	—
218	C₄	—	106	1	no	—
219	C₄	—	424	2	no	—
220	C₄	C₂×C₄⋊#88324	1	1	yes	characteristic
221	C₄	#16811630	1	1	yes	characteristic
222	C₄	Q₈×#88324	1	1	yes	characteristic
223	C₄	#16811673	1	1	yes	characteristic
224	C₂	SD₁₆×#88324	1	2	yes	—
225	C₂	#20290350	1	2	yes	—
226	C₂	#20290812	1	1	yes	characteristic
227	C₂	#20290818	1	1	yes	characteristic
228	C₂	#20290822	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄×#6157
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₂×#106×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₂×C₄×Dic₅₃	C₂³
hypercenter	C₂×#106	#6157
hypocenter	C₅₃	#9593
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

Complements

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

N	H ≅ G/N	type
38. #106×C₅₃	C₈	semidirect
91. C₈×C₅₃	C₄⋊C₈	semidirect
132. C₅₃	#9593	semidirect
143. #106	#88324	semidirect
207. C₈	C₄⋊#88324	semidirect

Maximal subgroups

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

subgroup	index
#20290812	2
C₂×#16811484	2
C₂×C₈×#88324	2
#20289551	2
#106×Dic₅₃	2
#9593	53