Subgroups — #27539498

Order 13568.

834 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811055	C₂	1	4	yes	maximal, complemented
2	#20285143	C₂	1	2	yes	maximal, complemented
3	#20276976	C₂	1	4	yes	maximal, complemented
4	#20276971	C₂	1	2	yes	maximal, complemented
5	#20276968	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276965	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13133239	—	2	2	no	—
9	C₂×#13133239	C₂²	1	4	yes	—
10	#16811173	—	2	2	no	—
11	#16811055	—	2	16	no	—
12	C₂×Dic₄₂₄	C₂²	1	4	yes	—
13	C₂×#13133162	C₂²	1	8	yes	—
14	C₂×D₈₄₈	C₂²	1	4	yes	—
15	#16811007	—	2	2	no	—
16	#16811007	C₂²	1	4	yes	—
17	#16811003	C₂²	1	2	yes	—
18	#16811000	C₂²	1	2	yes	—
19	#16810996	—	2	2	no	—
20	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
23	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
24	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
25	C₂×#11719945	—	2	4	no	—
26	#13133239	—	2	16	no	—
27	#13133239	—	4	8	no	—
28	C₂×Dic₂₁₂	—	2	2	no	—
29	C₂×Dic₂₁₂	C₂³	1	4	yes	—
30	C₂×#11719913	—	2	4	no	—
31	C₂×#11719913	—	2	4	no	—
32	C₂×D₄₂₄	—	2	2	no	—
33	C₂×D₄₂₄	C₂³	1	4	yes	—
34	#13133217	—	2	2	no	—
35	#13133214	—	2	2	no	—
36	#13133205	—	2	4	no	—
37	#13133191	—	4	4	no	—
38	C₄×D₂₁₂	—	4	4	no	—
39	Dic₄₂₄	—	2	8	no	—
40	#13133162	—	2	16	no	—
41	D₈₄₈	—	2	8	no	—
42	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
45	C₂×C₁₆×C₅₃	D₄	1	8	yes	complemented
46	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
47	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
48	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
49	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
50	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
51	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	D₈	1	2	yes	—
53	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₂²⋊D₅₃	—	4	4	no	—
55	#11719945	—	4	8	no	—
56	#11719945	—	4	8	no	—
57	C₂×D₂₁₂	—	2	4	no	—
58	C₂×D₂₁₂	—	4	4	no	—
59	C₂×C₄×D₅₃	—	4	4	no	—
60	C₂×Dic₁₀₆	—	2	4	no	—
61	C₂×C₈×C₅₃	D₈	1	4	yes	—
62	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
64	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
67	C₄²×C₅₃	D₈	1	2	yes	—
68	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
70	#11719921	—	4	8	no	—
71	C₄⋊Dic₅₃	—	4	4	no	—
72	Dic₂₁₂	—	2	8	no	—
73	Dic₂₁₂	—	4	4	no	—
74	D₄₂₄	—	2	8	no	—
75	D₄₂₄	—	4	4	no	—
76	#11719913	—	4	8	no	—
77	#11719913	—	4	8	no	—
78	C₁₆×C₅₃	C₂×D₄	1	4	yes	—
79	C₁₆×C₅₃	C₂×D₄	1	4	yes	—
80	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
81	C₂²×D₅₃	—	4	4	no	—
82	C₂×C₄×C₅₃	—	2	4	no	—
83	C₂×C₄×C₅₃	#127	1	2	yes	—
84	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
85	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
86	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
87	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
89	C₂²⋊D₅₃	—	8	8	no	—
90	C₂×Dic₅₃	—	4	4	no	—
91	D₂₁₂	—	4	4	no	—
92	D₂₁₂	—	4	4	no	—
93	D₂₁₂	—	8	8	no	—
94	C₄×D₅₃	—	8	8	no	—
95	Dic₁₀₆	—	4	4	no	—
96	Dic₁₀₆	—	4	4	no	—
97	C₈×C₅₃	#127	1	2	yes	—
98	C₈×C₅₃	C₂×D₈	1	2	yes	—
99	C₈×C₅₃	C₂×D₈	1	2	yes	—
100	C₈×C₅₃	C₂²×D₄	1	2	yes	—
101	#33713	—	53	1	no	maximal, Sylow
102	C₂²×C₅₃	—	2	2	no	—
103	C₂²×C₅₃	#493	1	2	yes	—
104	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
105	C₂×D₅₃	—	8	4	no	—
106	C₂×D₅₃	—	8	8	no	—
107	C₄×C₅₃	—	2	2	no	—
108	C₄×C₅₃	#493	1	2	yes	—
109	C₄×C₅₃	#508	1	4	yes	complemented
110	C₄×C₅₃	C₂×#127	1	1	yes	characteristic
111	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
112	Dic₅₃	—	8	4	no	—
113	C₂×#508	—	53	4	no	—
114	#3147	—	53	2	no	—
115	#2249	—	53	4	no	—
116	#2248	—	53	2	no	—
117	#2247	—	53	1	no	—
118	#2246	—	53	1	no	—
119	C₂×C₄×C₁₆	D₅₃	1	1	yes	characteristic, complemented
120	C₂×C₅₃	—	2	2	no	—
121	C₂×C₅₃	C₂×#493	1	1	yes	characteristic
122	C₂×C₅₃	C₂×#508	1	2	yes	—
123	D₅₃	—	16	4	no	—
124	C₂×#135	—	53	4	no	—
125	C₂×#135	—	106	2	no	—
126	#532	—	106	2	no	—
127	#508	—	106	16	no	—
128	C₂×Q₃₂	—	53	4	no	—
129	C₂×SD₃₂	—	53	8	no	—
130	C₂×D₁₆	—	53	4	no	—
131	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
132	#495	—	53	4	no	—
133	#495	—	106	2	no	—
134	#494	—	53	2	no	—
135	#493	—	53	2	no	—
136	#492	—	106	2	no	—
137	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
138	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
139	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
140	C₄×C₁₆	C₂×D₅₃	1	2	yes	—
141	C₅₃	#33713	1	1	yes	characteristic, complemented, Sylow
142	C₂×#41	—	106	4	no	—
143	#135	—	106	16	no	—
144	#135	—	212	8	no	—
145	C₂×Q₁₆	—	53	4	no	—
146	C₂×Q₁₆	—	106	2	no	—
147	C₂×SD₁₆	—	106	4	no	—
148	C₂×SD₁₆	—	106	4	no	—
149	C₂×D₈	—	53	4	no	—
150	C₂×D₈	—	106	2	no	—
151	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
152	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
153	C₄⋊Q₈	—	106	2	no	—
154	#127	—	106	2	no	—
155	#124	—	106	4	no	—
156	#121	—	212	4	no	—
157	C₄×D₄	—	212	4	no	—
158	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
159	Q₃₂	—	106	8	no	—
160	SD₃₂	—	106	16	no	—
161	D₁₆	—	106	8	no	—
162	C₂×C₁₆	D₂₁₂	1	8	yes	complemented
163	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
164	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
165	C₄×C₈	D₂₁₂	1	1	yes	characteristic
166	C₄×C₈	D₂₁₂	1	1	yes	characteristic
167	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
168	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
169	#41	—	212	8	no	—
170	#41	—	212	8	no	—
171	C₂×Q₈	—	106	4	no	—
172	C₂×D₄	—	106	4	no	—
173	C₂×D₄	—	212	4	no	—
174	C₂×D₄	—	212	4	no	—
175	C₂²×C₄	—	212	4	no	—
176	C₂²×C₄	D₄₂₄	1	2	yes	—
177	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
178	Q₁₆	—	106	8	no	—
179	Q₁₆	—	212	4	no	—
180	SD₁₆	—	212	8	no	—
181	SD₁₆	—	212	8	no	—
182	D₈	—	106	8	no	—
183	D₈	—	212	4	no	—
184	C₂×C₈	D₄₂₄	1	4	yes	—
185	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
186	C₂×C₈	C₂×D₂₁₂	1	4	yes	—
187	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
188	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
189	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
190	C₄⋊C₄	—	212	4	no	—
191	C₂²⋊C₄	—	212	8	no	—
192	C₄²	D₄₂₄	1	2	yes	—
193	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
194	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
195	C₁₆	C₂×D₂₁₂	1	4	yes	—
196	C₁₆	C₂×D₂₁₂	1	4	yes	—
197	C₂³	—	212	4	no	—
198	C₂³	C₂×D₄₂₄	1	1	yes	characteristic
199	Q₈	—	212	4	no	—
200	Q₈	—	212	4	no	—
201	D₄	—	424	8	no	—
202	D₄	—	424	8	no	—
203	D₄	—	212	4	no	—
204	D₄	—	212	4	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	424	8	no	—
207	C₂×C₄	—	212	4	no	—
208	C₂×C₄	#13133214	1	2	yes	—
209	C₂×C₄	C₂×D₄₂₄	1	2	yes	—
210	C₂×C₄	C₂×D₄₂₄	1	2	yes	—
211	C₂×C₄	C₂×D₄₂₄	1	2	yes	—
212	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
213	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
214	C₈	#13133214	1	2	yes	—
215	C₈	C₂×D₄₂₄	1	2	yes	—
216	C₈	C₂×D₄₂₄	1	2	yes	—
217	C₈	C₂²×D₂₁₂	1	2	yes	—
218	C₂²	—	2	2	no	—
219	C₂²	—	424	8	no	—
220	C₂²	—	424	4	no	—
221	C₂²	#16811000	1	2	yes	—
222	C₂²	C₂²×D₄₂₄	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	424	4	no	—
225	C₄	#16811000	1	2	yes	—
226	C₄	#16811055	1	4	yes	complemented
227	C₄	C₂×#13133214	1	1	yes	characteristic
228	C₄	C₂²×D₄₂₄	1	1	yes	characteristic
229	C₂	—	2	2	no	—
230	C₂	—	848	4	no	—
231	C₂	C₂×#16811000	1	1	yes	characteristic
232	C₂	C₂×#16811055	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄₂₄
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₄×C₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₈
hypercenter	C₂×C₄×C₁₆	D₅₃
hypocenter	C₅₃	#33713
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#16811055	C₂	semidirect
2. #20285143	C₂	semidirect
3. #20276976	C₂	semidirect
4. #20276971	C₂	semidirect
5. #20276968	C₂	semidirect
6. #20276965	C₂	semidirect
7. C₂×C₄×C₁₆×C₅₃	C₂	semidirect
45. C₂×C₁₆×C₅₃	D₄	semidirect
109. C₄×C₅₃	#508	semidirect
119. C₂×C₄×C₁₆	D₅₃	semidirect
141. C₅₃	#33713	semidirect
162. C₂×C₁₆	D₂₁₂	semidirect
226. C₄	#16811055	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811055	2
#20285143	2
#20276976	2
#20276971	2
#20276968	2
#20276965	2
C₂×C₄×C₁₆×C₅₃	2
#33713	53