Subgroups — #27590196

Order 13568.

4086 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	C₂×#16811173	C₂	1	6	yes	maximal, complemented
2	#20283113	C₂	1	1	yes	maximal, characteristic, complemented
3	#20283110	C₂	1	1	yes	maximal, characteristic, complemented
4	#20283072	C₂	1	6	yes	maximal, complemented
5	#20283060	C₂	1	8	yes	maximal, complemented
6	C₄×#13133214	C₂	1	8	yes	maximal, complemented
7	C₂×C₄³×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#11719945	C₂²	1	4	yes	—
9	#16811173	—	2	48	no	—
10	C₂×#13133217	C₂²	1	6	yes	—
11	C₂×#13133214	C₂²	1	8	yes	—
12	C₂×#13133214	C₂²	1	6	yes	—
13	C₂×#13133205	C₂²	1	12	yes	—
14	C₂×#13133191	C₂²	1	24	yes	—
15	C₂×C₄×D₂₁₂	C₂²	1	24	yes	—
16	#16810525	C₂²	1	48	yes	—
17	#16810493	C₂²	1	8	yes	—
18	C₂²×C₄²×C₅₃	C₂²	1	4	yes	—
19	C₂²×C₄²×C₅₃	C₂²	1	3	yes	—
20	C₄³×C₅₃	C₂²	1	4	yes	—
21	C₄³×C₅₃	C₂²	1	3	yes	—
22	C₄³×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×#11719945	—	2	48	no	—
24	C₂²×Dic₁₀₆	C₂³	1	4	yes	—
25	C₂²×D₂₁₂	C₂³	1	24	yes	—
26	C₂²×D₂₁₂	C₂³	1	4	yes	—
27	C₂²×C₂²⋊D₅₃	C₂³	1	8	yes	—
28	C₂²×C₄×D₅₃	C₂³	1	8	yes	—
29	#13133217	—	2	24	no	—
30	#13133214	—	2	24	no	—
31	#13133214	—	2	32	no	—
32	#13133205	—	2	48	no	—
33	#13133191	—	2	96	no	—
34	C₄×D₂₁₂	—	2	96	no	—
35	C₂×C₄⋊Dic₅₃	C₂³	1	24	yes	—
36	C₂×#11719921	C₂³	1	48	yes	—
37	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
38	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	—	2	16	no	—
40	C₂×C₄²×C₅₃	D₄	1	24	yes	complemented
41	C₂×C₄²×C₅₃	C₂³	1	12	yes	—
42	C₂×C₄²×C₅₃	C₂³	1	6	yes	—
43	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
44	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
45	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
46	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
47	C₂³×D₅₃	C₂⁴	1	8	yes	—
48	C₂²×C₄×C₅₃	—	2	24	no	—
49	C₂²×C₄×C₅₃	C₂×D₄	1	24	yes	—
50	C₂²×C₄×C₅₃	C₂×D₄	1	12	yes	—
51	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
52	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
55	C₂×C₂²⋊D₅₃	—	2	48	no	—
56	C₂²×Dic₅₃	C₂⁴	1	8	yes	—
57	#11719945	—	4	64	no	—
58	C₂×D₂₁₂	—	2	48	no	—
59	C₂×D₂₁₂	—	2	24	no	—
60	C₂×D₂₁₂	—	2	96	no	—
61	C₂×C₄×D₅₃	—	2	48	no	—
62	C₂×Dic₁₀₆	—	2	24	no	—
63	C₄²×C₅₃	—	2	24	no	—
64	C₄²×C₅₃	C₂×D₄	1	24	yes	—
65	C₄²×C₅₃	C₂×D₄	1	12	yes	—
66	C₄²×C₅₃	C₂×D₄	1	12	yes	—
67	C₄²×C₅₃	#41	1	16	yes	complemented
68	#11719921	—	2	96	no	—
69	C₄⋊Dic₅₃	—	2	48	no	—
70	C₂³×C₅₃	—	2	4	no	—
71	C₂³×C₅₃	C₂²×D₄	1	6	yes	—
72	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
73	C₂²×D₅₃	—	2	24	no	—
74	C₂²×D₅₃	—	2	32	no	—
75	C₂×C₄×C₅₃	—	2	12	no	—
76	C₂×C₄×C₅₃	—	2	48	no	—
77	C₂×C₄×C₅₃	#127	1	16	yes	complemented
78	C₂×C₄×C₅₃	C₂²×D₄	1	12	yes	—
79	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
80	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
81	C₂×C₄×C₅₃	C₂²×D₄	1	12	yes	—
82	C₂×C₄×C₅₃	C₂×#41	1	24	yes	—
83	C₂²⋊D₅₃	—	4	32	no	—
84	C₂×Dic₅₃	—	2	24	no	—
85	D₂₁₂	—	4	16	no	—
86	D₂₁₂	—	4	96	no	—
87	C₄×D₅₃	—	4	32	no	—
88	Dic₁₀₆	—	4	16	no	—
89	#36644	—	53	1	no	maximal, Sylow
90	C₂²×C₅₃	—	2	12	no	—
91	C₂²×C₅₃	C₂×#127	1	4	yes	—
92	C₂²×C₅₃	C₂³×D₄	1	3	yes	—
93	C₂²×C₅₃	C₂²×#41	1	4	yes	—
94	C₂×D₅₃	—	4	8	no	—
95	C₂×D₅₃	—	4	48	no	—
96	C₄×C₅₃	—	2	12	no	—
97	C₄×C₅₃	C₂×#127	1	4	yes	—
98	C₄×C₅₃	C₂×#127	1	4	yes	—
99	C₄×C₅₃	#532	1	24	yes	complemented
100	Dic₅₃	—	4	8	no	—
101	C₂×#532	—	53	6	no	—
102	#2868	—	53	1	no	—
103	#2867	—	53	1	no	—
104	#2858	—	53	6	no	—
105	#2856	—	53	8	no	—
106	C₄×#127	—	53	8	no	—
107	C₂×C₄³	D₅₃	1	1	yes	characteristic, complemented
108	C₂×C₅₃	—	2	4	no	—
109	C₂×C₅₃	C₂²×#127	1	1	yes	characteristic
110	C₂×C₅₃	C₂×#532	1	6	yes	—
111	D₅₃	—	8	8	no	—
112	C₂²×#41	—	53	4	no	—
113	#532	—	106	48	no	—
114	C₂×C₄⋊Q₈	—	53	6	no	—
115	C₂×#127	—	53	8	no	—
116	C₂×#127	—	53	6	no	—
117	C₂×#124	—	53	12	no	—
118	C₂×#121	—	53	24	no	—
119	C₂×C₄×D₄	—	53	24	no	—
120	C₂²×C₄²	C₂×D₅₃	1	4	yes	—
121	C₂²×C₄²	C₂×D₅₃	1	3	yes	—
122	#390	—	53	48	no	—
123	#384	—	53	8	no	—
124	C₄³	C₂×D₅₃	1	4	yes	—
125	C₄³	C₂×D₅₃	1	3	yes	—
126	C₄³	C₂×D₅₃	1	1	yes	characteristic
127	C₅₃	#36644	1	1	yes	characteristic, complemented, Sylow
128	C₂×#41	—	106	48	no	—
129	C₂²×Q₈	—	53	4	no	—
130	C₂²×D₄	—	53	8	no	—
131	C₂²×D₄	—	53	24	no	—
132	C₂²×D₄	—	53	4	no	—
133	C₂³×C₄	—	53	8	no	—
134	C₂³×C₄	C₂²×D₅₃	1	6	yes	—
135	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
136	C₄⋊Q₈	—	106	24	no	—
137	#127	—	106	24	no	—
138	#127	—	106	32	no	—
139	#124	—	106	48	no	—
140	#121	—	106	96	no	—
141	C₄×D₄	—	106	96	no	—
142	C₂×C₄⋊C₄	—	53	24	no	—
143	C₂×C₂²⋊C₄	—	53	48	no	—
144	C₂×C₄²	—	2	16	no	—
145	C₂×C₄²	D₂₁₂	1	24	yes	complemented
146	C₂×C₄²	C₂²×D₅₃	1	12	yes	—
147	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
148	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
149	C₂×C₄²	C₂²×D₅₃	1	6	yes	—
150	C₂×C₄²	C₂²×D₅₃	1	4	yes	—
151	C₂⁴	—	53	8	no	—
152	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
153	#41	—	212	64	no	—
154	C₂×Q₈	—	106	24	no	—
155	C₂×D₄	—	106	48	no	—
156	C₂×D₄	—	106	48	no	—
157	C₂×D₄	—	106	24	no	—
158	C₂×D₄	—	106	96	no	—
159	C₂²×C₄	—	2	24	no	—
160	C₂²×C₄	—	53	8	no	—
161	C₂²×C₄	—	106	48	no	—
162	C₂²×C₄	C₂×D₂₁₂	1	24	yes	—
163	C₂²×C₄	C₂×D₂₁₂	1	12	yes	—
164	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
165	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
166	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
167	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
168	C₄⋊C₄	—	106	48	no	—
169	C₂²⋊C₄	—	106	96	no	—
170	C₄²	—	2	24	no	—
171	C₄²	C₂×D₂₁₂	1	24	yes	—
172	C₄²	C₂×D₂₁₂	1	12	yes	—
173	C₄²	C₂×D₂₁₂	1	12	yes	—
174	C₄²	#11719945	1	16	yes	complemented
175	C₂³	—	2	4	no	—
176	C₂³	—	106	24	no	—
177	C₂³	—	106	32	no	—
178	C₂³	C₂²×D₂₁₂	1	6	yes	—
179	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
180	Q₈	—	212	16	no	—
181	D₄	—	212	32	no	—
182	D₄	—	212	16	no	—
183	D₄	—	212	96	no	—
184	C₂×C₄	—	2	12	no	—
185	C₂×C₄	—	2	48	no	—
186	C₂×C₄	—	106	24	no	—
187	C₂×C₄	—	212	32	no	—
188	C₂×C₄	#13133214	1	16	yes	complemented
189	C₂×C₄	C₂²×D₂₁₂	1	12	yes	—
190	C₂×C₄	C₂²×D₂₁₂	1	6	yes	—
191	C₂×C₄	C₂²×D₂₁₂	1	6	yes	—
192	C₂×C₄	C₂²×D₂₁₂	1	12	yes	—
193	C₂×C₄	C₂×#11719945	1	24	yes	—
194	C₂²	—	2	12	no	—
195	C₂²	—	212	8	no	—
196	C₂²	—	212	48	no	—
197	C₂²	C₂×#13133214	1	4	yes	—
198	C₂²	C₂³×D₂₁₂	1	3	yes	—
199	C₂²	C₂²×#11719945	1	4	yes	—
200	C₄	—	2	12	no	—
201	C₄	—	212	8	no	—
202	C₄	C₂×#13133214	1	4	yes	—
203	C₄	C₂×#13133214	1	4	yes	—
204	C₄	#16811173	1	24	yes	complemented
205	C₂	—	2	4	no	—
206	C₂	—	424	8	no	—
207	C₂	C₂²×#13133214	1	1	yes	characteristic
208	C₂	C₂×#16811173	1	6	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂³×D₅₃
commutator, socle, cosocle	C₂³×C₅₃	C₂⁵
Frattini	C₂³	C₂⁴×D₅₃
Fitting	C₂×C₄³×C₅₃	C₂
hypercenter	C₂×C₄³	D₅₃
hypocenter	C₅₃	#36644
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
1. C₂×#16811173	C₂	semidirect
2. #20283113	C₂	semidirect
3. #20283110	C₂	semidirect
4. #20283072	C₂	semidirect
5. #20283060	C₂	semidirect
6. C₄×#13133214	C₂	semidirect
7. C₂×C₄³×C₅₃	C₂	semidirect
40. C₂×C₄²×C₅₃	D₄	semidirect
67. C₄²×C₅₃	#41	semidirect
77. C₂×C₄×C₅₃	#127	semidirect
99. C₄×C₅₃	#532	semidirect
107. C₂×C₄³	D₅₃	semidirect
127. C₅₃	#36644	semidirect
145. C₂×C₄²	D₂₁₂	semidirect
174. C₄²	#11719945	semidirect
188. C₂×C₄	#13133214	semidirect
204. C₄	#16811173	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811173	2
#20283113	2
#20283110	2
#20283072	2
#20283060	2
C₄×#13133214	2
C₂×C₄³×C₅₃	2
#36644	53