Subgroups — #27539609

Order 13568.

474 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285254	C₂	1	2	yes	maximal
2	#20277001	C₂	1	4	yes	maximal, complemented
3	#20276997	C₂	1	2	yes	maximal, complemented
4	#20276994	C₂	1	2	yes	maximal, complemented
5	#20276503	C₂	1	2	yes	maximal
6	#20276499	C₂	1	1	yes	maximal, characteristic
7	C₂×#16810451	C₂	1	1	yes	maximal, characteristic
8	C₅₃×#2108	C₂	1	1	yes	maximal, characteristic
9	#16811251	—	2	2	no	—
10	#16811032	—	2	4	no	—
11	#16811028	C₂²	1	4	yes	—
12	#16811025	C₂²	1	4	yes	—
13	#16810665	—	2	2	no	—
14	#16810661	C₂²	1	2	yes	—
15	C₂×#13133151	C₂²	1	2	yes	—
16	#16810451	C₂²	1	1	yes	characteristic
17	#16810451	C₂²	1	2	yes	—
18	#16810451	C₂²	1	1	yes	characteristic
19	#16810451	C₂²	1	4	yes	—
20	#16810451	C₂²	1	2	yes	—
21	#16810449	C₂²	1	4	yes	—
22	#16810449	C₂²	1	2	yes	—
23	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
24	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
25	C₅₃×#405	C₂²	1	1	yes	characteristic
26	C₅₃×#346	C₂²	1	2	yes	—
27	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
28	#13133217	—	2	4	no	—
29	#13133216	—	4	2	no	—
30	#13133209	—	2	4	no	—
31	#13133208	—	4	2	no	—
32	#13133194	—	4	4	no	—
33	#13133176	—	2	2	no	—
34	C₂×C₄⋊Dic₅₃	—	2	2	no	—
35	#13133151	—	2	4	no	—
36	#13133151	—	2	2	no	—
37	#13133151	C₂³	1	2	yes	—
38	#13133151	C₂³	1	4	yes	—
39	#13133151	C₂³	1	2	yes	—
40	#13133149	—	2	4	no	—
41	#13133149	—	2	2	no	—
42	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
44	#117×C₅₃	D₄	1	1	yes	characteristic
45	M₃₂×C₅₃	Q₈	1	4	yes	complemented
46	C₂×C₁₆×C₅₃	Q₈	1	4	yes	complemented
47	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
48	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
49	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
50	#97×C₅₃	D₄	1	2	yes	—
51	C₄×C₈×C₅₃	C₂³	1	2	yes	—
52	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×Dic₅₃	—	4	2	no	—
54	C₂×Dic₁₀₆	—	4	4	no	—
55	M₁₆×C₅₃	Q₁₆	1	4	yes	—
56	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
58	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
59	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
60	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
61	C₄⋊C₄×C₅₃	D₈	1	2	yes	—
62	C₂²⋊C₄×C₅₃	D₈	1	2	yes	—
63	C₄²×C₅₃	C₂×D₄	1	2	yes	—
64	C₂²⋊Dic₅₃	—	4	2	no	—
65	C₄⋊Dic₅₃	—	2	2	no	—
66	C₄⋊Dic₅₃	—	2	4	no	—
67	C₄⋊Dic₅₃	—	2	2	no	—
68	C₄⋊Dic₅₃	—	4	2	no	—
69	C₄⋊Dic₅₃	—	4	4	no	—
70	#11719919	—	4	4	no	—
71	#11719919	—	4	4	no	—
72	C₄×Dic₅₃	—	4	2	no	—
73	C₁₆×C₅₃	C₂×Q₈	1	2	yes	—
74	C₁₆×C₅₃	C₂×Q₈	1	2	yes	—
75	C₁₆×C₅₃	C₂×Q₈	1	4	yes	—
76	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
78	C₂×C₄×C₅₃	C₂×D₈	1	4	yes	—
79	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
81	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
82	C₂×Dic₅₃	—	4	2	no	—
83	C₂×Dic₅₃	—	4	4	no	—
84	C₂×Dic₅₃	—	4	2	no	—
85	C₂×Dic₅₃	—	16	2	no	—
86	Dic₁₀₆	—	16	4	no	—
87	C₈×C₅₃	C₄⋊Q₈	1	2	yes	—
88	C₈×C₅₃	C₂×Q₁₆	1	4	yes	—
89	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
90	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
91	#33730	—	53	1	no	maximal, Sylow
92	C₂²×C₅₃	—	2	2	no	—
93	C₂²×C₅₃	#500	1	2	yes	—
94	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
95	C₄×C₅₃	—	2	4	no	—
96	C₄×C₅₃	#500	1	2	yes	—
97	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
98	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
99	Dic₅₃	—	8	2	no	—
100	Dic₅₃	—	8	4	no	—
101	Dic₅₃	—	8	2	no	—
102	#3164	—	53	2	no	—
103	#2255	—	53	4	no	—
104	#2254	—	53	2	no	—
105	#2253	—	53	2	no	—
106	#2159	—	53	2	no	—
107	#2158	—	53	1	no	—
108	C₂×#367	—	53	1	no	—
109	#2108	D₅₃	1	1	yes	characteristic
110	C₂×C₅₃	—	2	2	no	—
111	C₂×C₅₃	C₂×#500	1	1	yes	characteristic
112	C₂×C₅₃	#3416	1	2	yes	—
113	#544	—	106	2	no	—
114	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
115	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
116	#501	—	106	4	no	—
117	#500	—	53	4	no	—
118	#499	—	53	4	no	—
119	#428	—	106	2	no	—
120	#427	—	53	2	no	—
121	C₂×#107	—	53	2	no	—
122	#405	C₂×D₅₃	1	1	yes	characteristic
123	#367	—	53	2	no	—
124	#367	—	53	2	no	—
125	#367	—	53	1	no	—
126	#367	—	53	1	no	—
127	#367	—	53	4	no	—
128	#366	—	53	2	no	—
129	#366	—	53	4	no	—
130	#346	C₂×D₅₃	1	2	yes	—
131	C₄×C₁₆	C₂×D₅₃	1	2	yes	—
132	C₅₃	#33730	1	1	yes	characteristic, complemented, Sylow
133	C₂×M₁₆	D₂₁₂	1	1	yes	characteristic
134	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
135	C₄⋊Q₈	—	106	4	no	—
136	C₄⋊Q₈	—	212	2	no	—
137	#125	—	106	4	no	—
138	#125	—	212	2	no	—
139	C₂²⋊Q₈	—	212	4	no	—
140	#117	—	106	2	no	—
141	#117	D₂₁₂	1	1	yes	characteristic
142	C₂×C₄⋊C₄	—	106	2	no	—
143	M₃₂	Dic₁₀₆	1	4	yes	complemented
144	C₂×C₁₆	Dic₁₀₆	1	4	yes	complemented
145	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
146	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
147	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
148	#107	—	53	2	no	—
149	#107	—	53	4	no	—
150	#107	—	53	2	no	—
151	#107	—	106	2	no	—
152	#107	—	106	4	no	—
153	#106	—	106	2	no	—
154	#106	—	106	4	no	—
155	#97	D₂₁₂	1	2	yes	—
156	C₄×C₈	C₂²×D₅₃	1	2	yes	—
157	C₂×Q₈	—	212	4	no	—
158	C₂²×C₄	—	212	2	no	—
159	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
160	M₁₆	Dic₂₁₂	1	4	yes	—
161	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
162	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
163	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
164	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
165	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
166	C₄⋊C₄	—	106	2	no	—
167	C₄⋊C₄	—	106	4	no	—
168	C₄⋊C₄	—	106	2	no	—
169	C₄⋊C₄	—	212	2	no	—
170	C₄⋊C₄	—	212	4	no	—
171	C₄⋊C₄	—	212	4	no	—
172	C₄⋊C₄	—	212	4	no	—
173	C₄⋊C₄	D₄₂₄	1	2	yes	—
174	C₂²⋊C₄	—	212	2	no	—
175	C₂²⋊C₄	D₄₂₄	1	2	yes	—
176	C₄²	—	212	2	no	—
177	C₄²	C₂×D₂₁₂	1	2	yes	—
178	C₁₆	C₂×Dic₁₀₆	1	2	yes	—
179	C₁₆	C₂×Dic₁₀₆	1	2	yes	—
180	C₁₆	C₂×Dic₁₀₆	1	4	yes	—
181	C₂³	C₂×D₄₂₄	1	1	yes	characteristic
182	Q₈	—	848	4	no	—
183	C₂×C₄	—	848	2	no	—
184	C₂×C₄	—	212	2	no	—
185	C₂×C₄	—	212	2	no	—
186	C₂×C₄	—	212	4	no	—
187	C₂×C₄	#13133217	1	2	yes	—
188	C₂×C₄	C₂×D₄₂₄	1	4	yes	—
189	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
190	C₂×C₄	C₂×Dic₂₁₂	1	2	yes	—
191	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
192	C₈	#13133217	1	2	yes	—
193	C₈	C₂×Dic₂₁₂	1	4	yes	—
194	C₈	C₂²×Dic₁₀₆	1	1	yes	characteristic
195	C₈	C₂²×Dic₁₀₆	1	1	yes	characteristic
196	C₂²	—	2	2	no	—
197	C₂²	#16811028	1	2	yes	—
198	C₂²	C₂²×D₄₂₄	1	1	yes	characteristic
199	C₄	—	2	4	no	—
200	C₄	—	424	4	no	—
201	C₄	—	424	2	no	—
202	C₄	—	424	2	no	—
203	C₄	#16811028	1	2	yes	—
204	C₄	C₂×#13133217	1	1	yes	characteristic
205	C₄	C₂²×Dic₂₁₂	1	1	yes	characteristic
206	C₂	—	2	2	no	—
207	C₂	C₂×#16811028	1	1	yes	characteristic
208	C₂	#20287863	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₄₂₄
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2108	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂³×C₅₃	C₂×D₈
hypercenter	#2108	D₅₃
hypocenter	C₅₃	#33730
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
2. #20277001	C₂	semidirect
3. #20276997	C₂	semidirect
4. #20276994	C₂	semidirect
45. M₃₂×C₅₃	Q₈	semidirect
46. C₂×C₁₆×C₅₃	Q₈	semidirect
132. C₅₃	#33730	semidirect
143. M₃₂	Dic₁₀₆	semidirect
144. C₂×C₁₆	Dic₁₀₆	semidirect

Maximal subgroups

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

subgroup	index
#20285254	2
#20277001	2
#20276997	2
#20276994	2
#20276503	2
#20276499	2
C₂×#16810451	2
C₅₃×#2108	2
#33730	53