Subgroups — #27537131

Order 13568.

674 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810661	C₂	1	2	yes	maximal
2	#20276503	C₂	1	8	yes	maximal, complemented
3	#20276499	C₂	1	4	yes	maximal, complemented
4	C₂²×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133176	—	2	2	no	—
6	#16810661	—	2	2	no	—
7	#16810661	—	2	2	no	—
8	#16810661	C₂²	1	4	yes	—
9	#16810661	C₂²	1	4	yes	—
10	C₂×#13133151	C₂²	1	4	yes	—
11	C₂×#13133149	—	2	2	no	—
12	#16810451	C₂²	1	8	yes	complemented
13	#16810449	C₂²	1	8	yes	complemented
14	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×M₃₂×C₅₃	C₄	1	8	yes	complemented
16	C₂×M₃₂×C₅₃	C₂²	1	4	yes	—
17	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
18	#13133176	—	2	4	no	—
19	#13133176	—	2	4	no	—
20	#13133176	—	4	2	no	—
21	#13133176	—	4	2	no	—
22	C₂×C₄⋊Dic₅₃	—	2	4	no	—
23	C₂×C₂²⋊Dic₅₃	—	4	2	no	—
24	C₂×C₄×Dic₅₃	—	4	2	no	—
25	#13133151	—	2	4	no	—
26	#13133151	C₂³	1	8	yes	—
27	#13133149	—	2	4	no	—
28	#13133149	—	2	4	no	—
29	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₅₃	D₄	1	2	yes	—
32	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	Q₈	1	4	yes	—
34	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
36	M₃₂×C₅₃	C₂×C₄	1	16	yes	complemented
37	C₂×C₁₆×C₅₃	C₂×C₄	1	8	yes	complemented
38	C₂×C₁₆×C₅₃	C₂³	1	4	yes	—
39	C₂⁴×C₅₃	D₈	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
41	C₂²×C₄×C₅₃	D₈	1	2	yes	—
42	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	Q₁₆	1	4	yes	—
44	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
45	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×Dic₅₃	—	4	4	no	—
47	C₂×C₈×C₅₃	C₄⋊C₄	1	8	yes	—
48	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
49	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
50	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
51	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
53	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
54	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
55	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
56	C₂²⋊Dic₅₃	—	4	4	no	—
57	C₂²⋊Dic₅₃	—	4	4	no	—
58	C₄⋊Dic₅₃	—	2	8	no	—
59	C₄⋊Dic₅₃	—	4	4	no	—
60	C₄×Dic₅₃	—	4	4	no	—
61	C₄×Dic₅₃	—	4	4	no	—
62	C₁₆×C₅₃	C₂²×C₄	1	8	yes	—
63	C₂³×C₅₃	—	4	2	no	—
64	C₂³×C₅₃	#107	1	4	yes	—
65	C₂³×C₅₃	C₂×D₈	1	2	yes	—
66	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₅₃	#107	1	8	yes	—
68	C₂×C₄×C₅₃	#107	1	4	yes	—
69	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
70	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
71	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
72	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂×Q₁₆	1	4	yes	—
74	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
75	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
76	C₂×Dic₅₃	—	4	8	no	—
77	C₂×Dic₅₃	—	8	8	no	—
78	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
79	C₈×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
80	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
81	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
82	#33409	—	53	1	no	maximal, Sylow
83	C₂²×C₅₃	—	2	4	no	—
84	C₂²×C₅₃	—	2	2	no	—
85	C₂²×C₅₃	—	4	4	no	—
86	C₂²×C₅₃	C₂×#107	1	4	yes	—
87	C₂²×C₅₃	C₂×#107	1	2	yes	—
88	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
89	C₄×C₅₃	C₂×#107	1	4	yes	—
90	C₄×C₅₃	C₂×#107	1	2	yes	—
91	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
92	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
93	Dic₅₃	—	8	8	no	—
94	C₂²×M₃₂	D₅₃	1	1	yes	characteristic
95	C₂×#427	—	53	2	no	—
96	#2159	—	53	8	no	—
97	#2158	—	53	4	no	—
98	C₂×C₅₃	—	2	4	no	—
99	C₂×C₅₃	—	2	2	no	—
100	C₂×C₅₃	C₂²×#107	1	1	yes	characteristic
101	C₂×C₅₃	#3414	1	2	yes	—
102	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
103	C₂×#117	—	106	2	no	—
104	C₂×M₃₂	Dic₅₃	1	8	yes	complemented
105	C₂×M₃₂	C₂×D₅₃	1	4	yes	—
106	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
107	#427	—	53	4	no	—
108	#427	—	53	4	no	—
109	#427	—	106	2	no	—
110	#427	—	106	2	no	—
111	C₂×#107	—	53	4	no	—
112	C₂×#106	—	106	2	no	—
113	#367	—	53	8	no	—
114	#366	—	53	8	no	—
115	C₅₃	#33409	1	1	yes	characteristic, complemented, Sylow
116	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
117	C₂²×C₈	Dic₁₀₆	1	4	yes	—
118	C₂²×C₈	D₂₁₂	1	2	yes	—
119	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
120	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
121	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
122	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
123	#117	—	106	4	no	—
124	#117	—	106	4	no	—
125	#117	—	212	2	no	—
126	#117	—	212	2	no	—
127	C₂×C₄⋊C₄	—	106	4	no	—
128	C₂×C₂²⋊C₄	—	212	2	no	—
129	C₂×C₄²	—	212	2	no	—
130	M₃₂	C₂×Dic₅₃	1	16	yes	complemented
131	C₂×C₁₆	C₂×Dic₅₃	1	8	yes	complemented
132	C₂×C₁₆	C₂²×D₅₃	1	4	yes	—
133	#107	—	53	8	no	—
134	#107	—	106	4	no	—
135	#106	—	106	4	no	—
136	#106	—	106	4	no	—
137	C₂⁴	D₄₂₄	1	1	yes	characteristic
138	C₂²×C₄	—	212	4	no	—
139	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
140	C₂²×C₄	D₄₂₄	1	2	yes	—
141	C₂²×C₄	Dic₂₁₂	1	4	yes	—
142	C₂²×C₄	C₄⋊Dic₅₃	1	4	yes	—
143	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
144	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
145	C₂×C₈	C₄⋊Dic₅₃	1	8	yes	—
146	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
147	C₂×C₈	C₂×Dic₁₀₆	1	4	yes	—
148	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
149	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
150	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
151	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
152	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
153	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
154	C₄⋊C₄	—	106	8	no	—
155	C₄⋊C₄	—	212	4	no	—
156	C₂²⋊C₄	—	212	4	no	—
157	C₂²⋊C₄	—	212	4	no	—
158	C₄²	—	212	4	no	—
159	C₄²	—	212	4	no	—
160	C₁₆	C₂²×Dic₅₃	1	8	yes	—
161	C₂³	—	4	2	no	—
162	C₂³	#13133151	1	4	yes	—
163	C₂³	C₂×D₄₂₄	1	1	yes	characteristic
164	C₂³	C₂×D₄₂₄	1	2	yes	—
165	C₂×C₄	—	424	8	no	—
166	C₂×C₄	—	212	8	no	—
167	C₂×C₄	#13133151	1	4	yes	—
168	C₂×C₄	#13133151	1	8	yes	—
169	C₂×C₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
170	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
171	C₂×C₄	C₂×D₄₂₄	1	2	yes	—
172	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
173	C₂×C₄	C₂×Dic₂₁₂	1	4	yes	—
174	C₂×C₄	C₂×Dic₂₁₂	1	2	yes	—
175	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
176	C₈	C₂×C₄⋊Dic₅₃	1	2	yes	—
177	C₈	C₂×C₄⋊Dic₅₃	1	4	yes	—
178	C₈	C₂³×Dic₅₃	1	1	yes	characteristic
179	C₈	C₂²×Dic₁₀₆	1	1	yes	characteristic
180	C₂²	—	2	4	no	—
181	C₂²	—	2	2	no	—
182	C₂²	—	4	4	no	—
183	C₂²	C₂×#13133151	1	2	yes	—
184	C₂²	C₂×#13133151	1	4	yes	—
185	C₂²	C₂²×D₄₂₄	1	1	yes	characteristic
186	C₄	—	424	8	no	—
187	C₄	C₂×#13133151	1	4	yes	—
188	C₄	C₂×#13133151	1	2	yes	—
189	C₄	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
190	C₄	C₂²×Dic₂₁₂	1	1	yes	characteristic
191	C₂	—	2	4	no	—
192	C₂	—	2	2	no	—
193	C₂	C₂²×#13133151	1	1	yes	characteristic
194	C₂	#20287851	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄₂₄
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂²×M₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂⁴×C₅₃	D₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂²×M₃₂	D₅₃
hypocenter	C₅₃	#33409
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
2. #20276503	C₂	semidirect
3. #20276499	C₂	semidirect
12. #16810451	C₂²	semidirect
13. #16810449	C₂²	semidirect
15. C₂×M₃₂×C₅₃	C₄	semidirect
36. M₃₂×C₅₃	C₂×C₄	semidirect
37. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
104. C₂×M₃₂	Dic₅₃	semidirect
115. C₅₃	#33409	semidirect
130. M₃₂	C₂×Dic₅₃	semidirect
131. C₂×C₁₆	C₂×Dic₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810661	2
#20276503	2
#20276499	2
C₂²×M₃₂×C₅₃	2
#33409	53