Subgroups — #27371793

Order 13568.

922 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285294	C₂	1	4	yes	maximal, complemented
2	#20285143	C₂	1	2	yes	maximal, complemented
3	#20274333	C₂	1	1	yes	maximal, characteristic, complemented
4	#20274330	C₂	1	1	yes	maximal, characteristic, complemented
5	#20274321	C₂	1	2	yes	maximal, complemented
6	#20274307	C₂	1	4	yes	maximal, complemented
7	C₂×C₈²×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13133239	—	2	4	no	—
9	#16811173	C₂²	1	4	yes	—
10	#16811028	C₂²	1	4	yes	—
11	#16811007	—	2	4	no	—
12	#16811003	C₂²	1	2	yes	—
13	#16811000	C₂²	1	4	yes	—
14	#16811000	C₂²	1	2	yes	—
15	#16810996	C₂²	1	4	yes	—
16	#16810970	C₂²	1	8	yes	—
17	#16810859	—	2	8	no	—
18	C₄×D₄₂₄	—	2	8	no	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₈²×C₅₃	C₂²	1	2	yes	—
22	C₈²×C₅₃	C₂²	1	1	yes	characteristic
23	C₈²×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×#11719945	—	2	4	no	—
25	#13133239	—	4	16	no	—
26	C₂×Dic₂₁₂	—	2	4	no	—
27	C₂×#11719913	—	2	8	no	—
28	C₂×D₄₂₄	—	2	8	no	—
29	C₂×D₄₂₄	—	2	4	no	—
30	#13133217	C₂³	1	4	yes	—
31	#13133214	C₂³	1	4	yes	—
32	#13133205	—	2	4	no	—
33	#13133191	—	2	8	no	—
34	C₄×D₂₁₂	—	2	8	no	—
35	#13133151	—	2	8	no	—
36	#13133139	—	2	16	no	—
37	C₂²×C₈×C₅₃	D₄	1	4	yes	—
38	C₂²×C₈×C₅₃	D₄	1	2	yes	—
39	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
40	C₄×C₈×C₅₃	D₄	1	4	yes	—
41	C₄×C₈×C₅₃	D₄	1	2	yes	—
42	C₄×C₈×C₅₃	C₂³	1	2	yes	—
43	C₄×C₈×C₅₃	C₂³	1	2	yes	—
44	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
45	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
47	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₂²⋊D₅₃	—	4	4	no	—
49	#11719945	—	4	16	no	—
50	C₂×D₂₁₂	—	2	8	no	—
51	C₂×D₂₁₂	—	2	4	no	—
52	C₂×C₄×D₅₃	—	4	4	no	—
53	C₂×Dic₁₀₆	—	2	4	no	—
54	C₂×C₈×C₅₃	—	2	8	no	—
55	C₂×C₈×C₅₃	D₈	1	8	yes	complemented
56	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
57	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
58	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
59	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
60	C₄²×C₅₃	C₂×D₄	1	2	yes	—
61	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
63	#11719921	—	4	8	no	—
64	C₄⋊Dic₅₃	—	2	8	no	—
65	Dic₂₁₂	—	4	8	no	—
66	D₄₂₄	—	4	8	no	—
67	D₄₂₄	—	4	16	no	—
68	#11719913	—	4	16	no	—
69	C₂³×C₅₃	#127	1	1	yes	characteristic
70	C₂²×D₅₃	—	4	4	no	—
71	C₂×C₄×C₅₃	—	2	4	no	—
72	C₂×C₄×C₅₃	#127	1	2	yes	—
73	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂×D₈	1	4	yes	—
75	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
77	C₂²⋊D₅₃	—	8	8	no	—
78	C₂×Dic₅₃	—	4	4	no	—
79	D₂₁₂	—	4	8	no	—
80	D₂₁₂	—	4	16	no	—
81	C₄×D₅₃	—	8	8	no	—
82	Dic₁₀₆	—	4	8	no	—
83	C₈×C₅₃	—	2	4	no	—
84	C₈×C₅₃	C₂×D₈	1	4	yes	—
85	C₈×C₅₃	C₂×D₈	1	4	yes	—
86	C₈×C₅₃	#135	1	8	yes	complemented
87	#19603	—	53	1	no	maximal, Sylow
88	C₂²×C₅₃	—	2	2	no	—
89	C₂²×C₅₃	#493	1	2	yes	—
90	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
91	C₂×D₅₃	—	8	4	no	—
92	C₂×D₅₃	—	8	8	no	—
93	C₄×C₅₃	—	2	2	no	—
94	C₄×C₅₃	#493	1	2	yes	—
95	C₄×C₅₃	C₂²×D₈	1	2	yes	—
96	C₄×C₅₃	C₂×#135	1	4	yes	—
97	Dic₅₃	—	8	4	no	—
98	#3169	—	53	4	no	—
99	#3147	—	53	2	no	—
100	#1713	—	53	1	no	—
101	#1712	—	53	1	no	—
102	#1709	—	53	2	no	—
103	#1706	—	53	4	no	—
104	C₂×C₈²	D₅₃	1	1	yes	characteristic, complemented
105	C₂×C₅₃	—	2	2	no	—
106	C₂×C₅₃	C₂×#493	1	1	yes	characteristic
107	C₂×C₅₃	#3147	1	2	yes	—
108	D₅₃	—	16	4	no	—
109	C₂×#135	—	106	4	no	—
110	#532	—	53	4	no	—
111	#500	—	53	4	no	—
112	#495	—	106	4	no	—
113	#494	—	53	2	no	—
114	#493	—	53	2	no	—
115	#493	—	53	4	no	—
116	#492	—	53	4	no	—
117	#486	—	53	8	no	—
118	#466	—	106	8	no	—
119	C₄×D₈	—	106	8	no	—
120	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
121	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
122	C₈²	C₂×D₅₃	1	2	yes	—
123	C₈²	C₂×D₅₃	1	1	yes	characteristic
124	C₈²	C₂×D₅₃	1	1	yes	characteristic
125	C₅₃	#19603	1	1	yes	characteristic, complemented, Sylow
126	C₂×#41	—	106	4	no	—
127	#135	—	212	16	no	—
128	C₂×Q₁₆	—	106	4	no	—
129	C₂×SD₁₆	—	106	8	no	—
130	C₂×D₈	—	106	4	no	—
131	C₂×D₈	—	106	8	no	—
132	C₂²×C₈	D₂₁₂	1	4	yes	—
133	C₂²×C₈	D₂₁₂	1	2	yes	—
134	C₄⋊Q₈	—	53	4	no	—
135	#127	—	53	4	no	—
136	#124	—	106	4	no	—
137	#121	—	106	8	no	—
138	C₄×D₄	—	106	8	no	—
139	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
140	#107	—	106	8	no	—
141	#102	—	106	16	no	—
142	C₄×C₈	D₂₁₂	1	4	yes	—
143	C₄×C₈	D₂₁₂	1	2	yes	—
144	C₄×C₈	C₂²×D₅₃	1	2	yes	—
145	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
146	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
147	C₄×C₈	C₂²×D₅₃	1	2	yes	—
148	#41	—	212	16	no	—
149	C₂×Q₈	—	106	4	no	—
150	C₂×D₄	—	106	8	no	—
151	C₂×D₄	—	106	4	no	—
152	C₂×D₄	—	212	4	no	—
153	C₂²×C₄	—	212	4	no	—
154	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
155	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
156	Q₁₆	—	212	8	no	—
157	SD₁₆	—	212	16	no	—
158	D₈	—	212	8	no	—
159	D₈	—	212	16	no	—
160	C₂×C₈	—	2	8	no	—
161	C₂×C₈	D₄₂₄	1	8	yes	complemented
162	C₂×C₈	C₂×D₂₁₂	1	4	yes	—
163	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
164	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
165	C₂×C₈	C₂×D₂₁₂	1	4	yes	—
166	C₄⋊C₄	—	106	8	no	—
167	C₂²⋊C₄	—	212	8	no	—
168	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
169	C₄²	C₂×D₂₁₂	1	2	yes	—
170	C₄²	C₂³×D₅₃	1	1	yes	characteristic
171	C₂³	—	212	4	no	—
172	C₂³	#13133214	1	1	yes	characteristic
173	Q₈	—	212	8	no	—
174	D₄	—	424	8	no	—
175	D₄	—	212	8	no	—
176	D₄	—	212	16	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	424	8	no	—
179	C₂×C₄	—	212	4	no	—
180	C₂×C₄	#13133214	1	2	yes	—
181	C₂×C₄	#13133214	1	1	yes	characteristic
182	C₂×C₄	C₂×D₄₂₄	1	4	yes	—
183	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
184	C₂×C₄	C₂²×D₂₁₂	1	2	yes	—
185	C₈	—	2	4	no	—
186	C₈	C₂×D₄₂₄	1	4	yes	—
187	C₈	C₂×D₄₂₄	1	4	yes	—
188	C₈	#13133239	1	8	yes	complemented
189	C₂²	—	2	2	no	—
190	C₂²	—	424	4	no	—
191	C₂²	—	424	8	no	—
192	C₂²	#16811000	1	2	yes	—
193	C₂²	C₂×#13133214	1	1	yes	characteristic
194	C₄	—	2	2	no	—
195	C₄	—	424	4	no	—
196	C₄	#16811000	1	2	yes	—
197	C₄	C₂²×D₄₂₄	1	2	yes	—
198	C₄	C₂×#13133239	1	4	yes	—
199	C₂	—	2	2	no	—
200	C₂	—	848	4	no	—
201	C₂	C₂×#16811000	1	1	yes	characteristic
202	C₂	#20285143	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13133214
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₈²×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#127
hypercenter	C₂×C₈²	D₅₃
hypocenter	C₅₃	#19603
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
1. #20285294	C₂	semidirect
2. #20285143	C₂	semidirect
3. #20274333	C₂	semidirect
4. #20274330	C₂	semidirect
5. #20274321	C₂	semidirect
6. #20274307	C₂	semidirect
7. C₂×C₈²×C₅₃	C₂	semidirect
55. C₂×C₈×C₅₃	D₈	semidirect
86. C₈×C₅₃	#135	semidirect
104. C₂×C₈²	D₅₃	semidirect
125. C₅₃	#19603	semidirect
161. C₂×C₈	D₄₂₄	semidirect
188. C₈	#13133239	semidirect

Maximal subgroups

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

subgroup	index
#20285294	2
#20285143	2
#20274333	2
#20274330	2
#20274321	2
#20274307	2
C₂×C₈²×C₅₃	2
#19603	53