Subgroups — #27302467

Order 13568.

438 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₄₂₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810449	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810428	C₂	1	2	yes	maximal
4	#20275576	C₂	1	2	yes	maximal
5	C₂×C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₁₂	C₂²	1	2	yes	—
7	C₂×Dic₄₂₄	—	2	2	no	—
8	C₂×Dic₄₂₄	—	2	2	no	—
9	C₂×Dic₄₂₄	C₄	1	4	yes	complemented
10	C₂×#13133151	C₂²	1	2	yes	—
11	C₂×#13133142	—	2	2	no	—
12	#16810529	—	2	2	no	—
13	#16810449	—	2	2	no	—
14	#16810428	—	2	4	no	—
15	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₄×C₁₆×C₅₃	—	2	2	no	—
19	C₂²×Dic₁₀₆	—	2	2	no	—
20	C₂×Dic₂₁₂	—	2	2	no	—
21	C₂×Dic₂₁₂	—	2	2	no	—
22	C₂×Dic₂₁₂	—	2	2	no	—
23	C₂×Dic₂₁₂	—	2	2	no	—
24	C₂×Dic₂₁₂	—	2	2	no	—
25	C₂×Dic₂₁₂	C₂×C₄	1	4	yes	—
26	C₂×C₄⋊Dic₅₃	—	2	2	no	—
27	Dic₄₂₄	—	2	8	no	—
28	#13133151	—	2	2	no	—
29	#13133151	—	2	2	no	—
30	#13133142	—	4	4	no	—
31	#13133124	—	4	4	no	—
32	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
35	C₂×C₁₆×C₅₃	—	2	1	no	—
36	C₂×C₁₆×C₅₃	—	2	1	no	—
37	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
38	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
39	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
40	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
41	C₄×C₈×C₅₃	—	2	1	no	—
42	C₄×C₈×C₅₃	—	2	1	no	—
43	C₂²×C₄×C₅₃	D₈	1	2	yes	—
44	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×Dic₅₃	—	4	2	no	—
46	C₂²×Dic₅₃	—	4	2	no	—
47	C₂×Dic₁₀₆	—	2	4	no	—
48	C₂×Dic₁₀₆	—	4	2	no	—
49	C₂×Dic₁₀₆	—	4	2	no	—
50	C₂×Dic₁₀₆	—	4	2	no	—
51	C₂×Dic₁₀₆	—	4	2	no	—
52	C₂×Dic₁₀₆	—	4	2	no	—
53	C₂×C₈×C₅₃	—	2	1	no	—
54	C₂×C₈×C₅₃	—	2	1	no	—
55	C₂×C₈×C₅₃	D₈	1	2	yes	—
56	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
61	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
62	C₄²×C₅₃	—	2	1	no	—
63	C₄²×C₅₃	—	2	1	no	—
64	C₄⋊Dic₅₃	—	4	2	no	—
65	C₄⋊Dic₅₃	—	4	2	no	—
66	Dic₂₁₂	—	2	4	no	—
67	Dic₂₁₂	—	2	4	no	—
68	Dic₂₁₂	—	4	2	no	—
69	Dic₂₁₂	—	4	2	no	—
70	C₁₆×C₅₃	—	2	2	no	—
71	C₁₆×C₅₃	C₂²⋊C₄	1	4	yes	—
72	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₅₃	—	2	2	no	—
74	C₂×C₄×C₅₃	—	2	2	no	—
75	C₂×C₄×C₅₃	Q₃₂	1	4	yes	complemented
76	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
77	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
78	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
81	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
82	C₂×Dic₅₃	—	4	4	no	—
83	C₂×Dic₅₃	—	8	2	no	—
84	C₂×Dic₅₃	—	8	2	no	—
85	C₂×Dic₅₃	—	8	2	no	—
86	C₂×Dic₅₃	—	8	4	no	—
87	Dic₁₀₆	—	4	4	no	—
88	Dic₁₀₆	—	4	4	no	—
89	Dic₁₀₆	—	8	2	no	—
90	Dic₁₀₆	—	8	2	no	—
91	C₈×C₅₃	—	2	2	no	—
92	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
94	C₈×C₅₃	#121	1	1	yes	characteristic
95	C₈×C₅₃	#124	1	1	yes	characteristic
96	#13236	—	53	1	no	maximal, Sylow
97	C₂²×C₅₃	C₄×D₈	1	1	yes	characteristic
98	C₂²×C₅₃	#466	1	1	yes	characteristic
99	C₂²×C₅₃	#493	1	1	yes	characteristic
100	C₂²×C₅₃	C₂×Q₃₂	1	2	yes	—
101	C₂²×C₅₃	#508	1	2	yes	—
102	C₄×C₅₃	—	2	4	no	—
103	C₄×C₅₃	#390	1	1	yes	characteristic
104	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
105	C₄×C₅₃	#466	1	1	yes	characteristic
106	C₄×C₅₃	#495	1	1	yes	characteristic
107	Dic₅₃	—	8	4	no	—
108	Dic₅₃	—	16	2	no	—
109	C₂²×Q₃₂	—	53	1	no	—
110	C₂×#366	—	53	1	no	—
111	C₂×#358	—	53	2	no	—
112	C₂×C₄×C₁₆	D₅₃	1	1	yes	characteristic
113	#1969	—	53	2	no	—
114	C₂×C₅₃	#1970	1	1	yes	characteristic
115	C₂×C₅₃	C₄×Q₃₂	1	2	yes	—
116	C₂×C₅₃	#2216	1	2	yes	—
117	C₂×C₅₃	#2247	1	1	yes	characteristic
118	C₂×C₅₃	#2249	1	1	yes	characteristic
119	C₂²×Q₁₆	—	53	2	no	—
120	C₂×Q₃₂	—	53	4	no	—
121	C₂×Q₃₂	—	106	2	no	—
122	C₂×Q₃₂	—	106	2	no	—
123	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
124	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
125	C₂×#107	—	53	2	no	—
126	C₂×#103	—	106	2	no	—
127	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
128	#391	—	106	2	no	—
129	#366	—	106	2	no	—
130	#358	—	106	4	no	—
131	C₄×C₁₆	—	2	2	no	—
132	C₅₃	#13236	1	1	yes	characteristic, complemented, Sylow
133	C₂²×Q₈	—	106	2	no	—
134	C₂×Q₁₆	—	53	4	no	—
135	C₂×Q₁₆	—	106	2	no	—
136	C₂×Q₁₆	—	106	2	no	—
137	C₂×Q₁₆	—	106	2	no	—
138	C₂×Q₁₆	—	106	2	no	—
139	C₂×Q₁₆	—	106	2	no	—
140	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
141	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
142	C₂×C₄⋊C₄	—	106	2	no	—
143	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
144	Q₃₂	—	106	8	no	—
145	C₂×C₁₆	—	2	1	no	—
146	C₂×C₁₆	—	2	1	no	—
147	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
148	C₂×C₁₆	D₂₁₂	1	2	yes	—
149	C₂×C₁₆	D₂₁₂	1	2	yes	—
150	C₂×C₁₆	C₂²⋊D₅₃	1	2	yes	—
151	#107	—	106	2	no	—
152	#107	—	106	2	no	—
153	#103	—	212	4	no	—
154	C₄×C₈	—	2	1	no	—
155	C₄×C₈	—	2	1	no	—
156	#95	—	212	4	no	—
157	C₂×Q₈	—	106	4	no	—
158	C₂×Q₈	—	212	2	no	—
159	C₂×Q₈	—	212	2	no	—
160	C₂×Q₈	—	212	2	no	—
161	C₂×Q₈	—	212	2	no	—
162	C₂×Q₈	—	212	2	no	—
163	C₂²×C₄	—	212	2	no	—
164	C₂²×C₄	—	212	2	no	—
165	C₂²×C₄	D₄₂₄	1	2	yes	—
166	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
167	Q₁₆	—	106	4	no	—
168	Q₁₆	—	106	4	no	—
169	Q₁₆	—	212	2	no	—
170	Q₁₆	—	212	2	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	D₄₂₄	1	2	yes	—
174	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
175	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
176	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
177	C₂×C₈	#11719945	1	1	yes	characteristic
178	C₂×C₈	#11719945	1	1	yes	characteristic
179	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
180	C₄⋊C₄	—	212	2	no	—
181	C₄⋊C₄	—	212	2	no	—
182	C₄²	—	2	1	no	—
183	C₄²	—	2	1	no	—
184	C₁₆	—	2	2	no	—
185	C₁₆	#11719921	1	4	yes	—
186	C₂³	C₂×D₄₂₄	1	1	yes	characteristic
187	Q₈	—	424	2	no	—
188	Q₈	—	424	2	no	—
189	Q₈	—	212	4	no	—
190	Q₈	—	212	4	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	424	4	no	—
194	C₂×C₄	—	424	2	no	—
195	C₂×C₄	—	424	2	no	—
196	C₂×C₄	—	424	2	no	—
197	C₂×C₄	—	212	4	no	—
198	C₂×C₄	Dic₄₂₄	1	4	yes	complemented
199	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
200	C₂×C₄	#13133191	1	1	yes	characteristic
201	C₂×C₄	#13133214	1	1	yes	characteristic
202	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
203	C₂×C₄	#13133239	1	1	yes	characteristic
204	C₂×C₄	#13133239	1	1	yes	characteristic
205	C₈	—	2	2	no	—
206	C₈	C₂×#11719921	1	1	yes	characteristic
207	C₈	C₄×D₂₁₂	1	1	yes	characteristic
208	C₈	#13133191	1	1	yes	characteristic
209	C₈	#13133205	1	1	yes	characteristic
210	C₂²	C₄×D₄₂₄	1	1	yes	characteristic
211	C₂²	#16810859	1	1	yes	characteristic
212	C₂²	#16811000	1	1	yes	characteristic
213	C₂²	C₂×Dic₄₂₄	1	2	yes	—
214	C₂²	#16811055	1	2	yes	—
215	C₄	—	2	4	no	—
216	C₄	—	848	2	no	—
217	C₄	—	424	4	no	—
218	C₄	#16810525	1	1	yes	characteristic
219	C₄	C₄×D₄₂₄	1	1	yes	characteristic
220	C₄	#16810859	1	1	yes	characteristic
221	C₄	#16811007	1	1	yes	characteristic
222	C₂	#20275581	1	1	yes	characteristic
223	C₂	C₄×Dic₄₂₄	1	2	yes	—
224	C₂	#20276806	1	2	yes	—
225	C₂	#20276968	1	1	yes	characteristic
226	C₂	#20276976	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄₂₄
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₄×C₁₆×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×C₄×C₁₆	D₅₃
hypocenter	C₅₃	#13236
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
9. C₂×Dic₄₂₄	C₄	semidirect
75. C₂×C₄×C₅₃	Q₃₂	semidirect
132. C₅₃	#13236	semidirect
198. C₂×C₄	Dic₄₂₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₄₂₄	2
C₂×#16810449	2
C₂×#16810428	2
#20275576	2
C₂×C₄×C₁₆×C₅₃	2
#13236	53