Subgroups — #27538589

Order 13568.

714 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₈₄₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#20284363	C₂	1	2	yes	maximal, complemented
3	#20276818	C₂	1	4	yes	maximal, complemented
4	#20276801	C₂	1	4	yes	maximal, complemented
5	#20276499	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276432	C₂	1	2	yes	maximal, complemented
7	C₂×C₅₃×#504	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₄₂₄	C₂²	1	2	yes	—
9	#16811187	—	2	2	no	—
10	C₂×D₈₄₈	—	2	4	no	—
11	C₂×D₈₄₈	C₂²	1	2	yes	—
12	C₂×D₈₄₈	C₂²	1	2	yes	—
13	#16810876	—	2	4	no	—
14	#16810859	C₂²	1	8	yes	complemented
15	#16810661	C₂²	1	2	yes	—
16	#16810629	—	2	2	no	—
17	#16810451	C₂²	1	2	yes	complemented
18	#16810449	C₂²	1	2	yes	complemented
19	#16810425	C₂²	1	8	yes	complemented
20	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
21	C₅₃×#504	—	2	4	no	—
22	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×M₃₂×C₅₃	C₂²	1	2	yes	complemented
24	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
25	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	complemented
26	C₂²×D₂₁₂	—	2	2	no	—
27	C₂×D₄₂₄	—	2	2	no	—
28	C₂×D₄₂₄	—	2	4	no	—
29	C₂×D₄₂₄	—	2	4	no	—
30	C₂×D₄₂₄	C₂³	1	4	yes	—
31	#13133213	—	4	2	no	—
32	#13133206	—	4	2	no	—
33	#13133191	—	2	8	no	—
34	#13133185	—	4	4	no	—
35	#13133176	—	2	2	no	—
36	D₈₄₈	—	2	4	no	—
37	D₈₄₈	—	2	4	no	—
38	#13133151	C₂³	1	4	yes	—
39	#13133149	—	2	2	no	—
40	#13133139	—	2	8	no	—
41	C₂×#41×C₅₃	D₄	1	1	yes	characteristic
42	#131×C₅₃	—	2	2	no	—
43	#131×C₅₃	—	2	2	no	—
44	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
45	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
47	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
48	M₃₂×C₅₃	—	2	2	no	—
49	M₃₂×C₅₃	—	2	4	no	—
50	C₂×C₁₆×C₅₃	—	2	4	no	—
51	C₂×C₁₆×C₅₃	D₄	1	4	yes	complemented
52	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
53	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
54	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
55	C₂³×D₅₃	—	4	2	no	—
56	#41×C₅₃	—	2	2	no	—
57	#41×C₅₃	—	2	2	no	—
58	C₂×Q₈×C₅₃	D₈	1	1	yes	characteristic
59	C₂×D₄×C₅₃	D₈	1	2	yes	—
60	C₂×D₄×C₅₃	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₂²⋊D₅₃	—	4	8	no	—
64	C₂×D₂₁₂	—	2	4	no	—
65	C₂×D₂₁₂	—	4	4	no	—
66	C₂×D₂₁₂	—	4	4	no	—
67	C₂×D₂₁₂	—	4	2	no	—
68	M₁₆×C₅₃	—	2	2	no	—
69	M₁₆×C₅₃	—	2	4	no	—
70	C₂×C₈×C₅₃	—	2	2	no	—
71	C₂×C₈×C₅₃	—	2	2	no	—
72	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₅₃	#41	1	2	yes	—
76	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
77	C₂²⋊Dic₅₃	—	4	2	no	—
78	#11719921	—	4	8	no	—
79	C₄⋊Dic₅₃	—	2	4	no	—
80	C₄×Dic₅₃	—	4	2	no	—
81	D₄₂₄	—	2	4	no	—
82	D₄₂₄	—	2	4	no	—
83	D₄₂₄	—	4	4	no	—
84	C₁₆×C₅₃	—	2	2	no	—
85	C₁₆×C₅₃	C₂×D₄	1	2	yes	—
86	C₁₆×C₅₃	C₂×D₄	1	2	yes	—
87	C₂³×C₅₃	C₂×D₈	1	2	yes	—
88	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
89	C₂²×D₅₃	—	4	4	no	—
90	C₂²×D₅₃	—	8	4	no	—
91	C₂²×D₅₃	—	16	4	no	—
92	Q₈×C₅₃	—	2	2	no	—
93	D₄×C₅₃	—	2	4	no	—
94	D₄×C₅₃	—	2	2	no	—
95	C₂×C₄×C₅₃	—	2	2	no	—
96	C₂×C₄×C₅₃	—	2	2	no	—
97	C₂×C₄×C₅₃	#121	1	2	yes	—
98	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
99	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
100	C₂×C₄×C₅₃	#135	1	2	yes	—
101	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
102	C₂²⋊D₅₃	—	16	8	no	—
103	C₂×Dic₅₃	—	4	4	no	—
104	D₂₁₂	—	4	4	no	—
105	D₂₁₂	—	4	4	no	—
106	D₂₁₂	—	8	4	no	—
107	C₈×C₅₃	—	2	2	no	—
108	C₈×C₅₃	#121	1	2	yes	—
109	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
110	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
111	#33607	—	53	1	no	maximal, Sylow
112	C₂²×C₅₃	—	2	2	no	—
113	C₂²×C₅₃	—	2	4	no	—
114	C₂²×C₅₃	—	2	2	no	—
115	C₂²×C₅₃	#466	1	2	yes	—
116	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
117	C₂×D₅₃	—	8	8	no	—
118	C₂×D₅₃	—	8	4	no	—
119	C₂×D₅₃	—	16	8	no	—
120	C₄×C₅₃	—	2	2	no	—
121	C₄×C₅₃	#466	1	2	yes	—
122	C₄×C₅₃	C₂×#121	1	1	yes	characteristic
123	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
124	Dic₅₃	—	8	4	no	—
125	C₂²×D₁₆	—	53	1	no	—
126	C₂×#504	D₅₃	1	1	yes	characteristic, complemented
127	#3058	—	53	2	no	—
128	#2218	—	53	4	no	—
129	#2215	—	53	4	no	—
130	#2158	—	53	1	no	—
131	#2140	—	53	2	no	—
132	C₂×C₅₃	—	2	2	no	—
133	C₂×C₅₃	—	4	2	no	—
134	C₂×C₅₃	C₂×#466	1	1	yes	characteristic
135	C₂×C₅₃	#3415	1	2	yes	—
136	D₅₃	—	16	4	no	—
137	C₂²×D₈	—	53	2	no	—
138	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
139	#535	—	106	2	no	—
140	C₂×D₁₆	—	53	2	no	—
141	C₂×D₁₆	—	53	2	no	—
142	C₂×D₁₆	—	106	4	no	—
143	#504	—	2	4	no	—
144	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
145	C₂×M₃₂	C₂×D₅₃	1	2	yes	complemented
146	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
147	C₂²×C₁₆	C₂×D₅₃	1	2	yes	complemented
148	#469	—	106	4	no	—
149	#466	—	53	8	no	—
150	#427	—	53	2	no	—
151	#418	—	106	2	no	—
152	#367	—	53	2	no	—
153	#366	—	53	2	no	—
154	#357	—	53	8	no	—
155	C₅₃	#33607	1	1	yes	characteristic, complemented, Sylow
156	C₂×#41	D₂₁₂	1	1	yes	characteristic
157	C₂²×D₄	—	106	2	no	—
158	C₂×D₈	—	53	4	no	—
159	C₂×D₈	—	106	2	no	—
160	C₂×D₈	—	106	4	no	—
161	C₂×D₈	—	106	4	no	—
162	#131	—	2	2	no	—
163	#131	—	2	2	no	—
164	C₂×M₁₆	D₂₁₂	1	1	yes	characteristic
165	C₂×M₁₆	D₂₁₂	1	2	yes	—
166	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
167	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
168	#127	—	212	2	no	—
169	#124	—	212	2	no	—
170	#121	—	106	8	no	—
171	#120	—	212	4	no	—
172	#117	—	106	2	no	—
173	D₁₆	—	106	4	no	—
174	D₁₆	—	106	4	no	—
175	M₃₂	—	2	2	no	—
176	M₃₂	—	2	4	no	—
177	C₂×C₁₆	—	2	4	no	—
178	C₂×C₁₆	C₂²⋊D₅₃	1	4	yes	complemented
179	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
180	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
181	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
182	#107	—	53	4	no	—
183	#106	—	106	2	no	—
184	#102	—	106	8	no	—
185	C₂⁴	—	212	2	no	—
186	#41	—	2	2	no	—
187	#41	—	2	2	no	—
188	C₂×Q₈	D₄₂₄	1	1	yes	characteristic
189	C₂×D₄	—	106	4	no	—
190	C₂×D₄	—	212	8	no	—
191	C₂×D₄	—	212	4	no	—
192	C₂×D₄	—	212	2	no	—
193	C₂×D₄	—	212	4	no	—
194	C₂×D₄	D₄₂₄	1	2	yes	—
195	C₂×D₄	D₄₂₄	1	1	yes	characteristic
196	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
197	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
198	D₈	—	106	4	no	—
199	D₈	—	106	4	no	—
200	D₈	—	212	4	no	—
201	M₁₆	—	2	2	no	—
202	M₁₆	—	2	4	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
206	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
207	C₂×C₈	#11719945	1	2	yes	—
208	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
209	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
210	C₄⋊C₄	—	106	4	no	—
211	C₂²⋊C₄	—	212	2	no	—
212	C₂²⋊C₄	—	212	8	no	—
213	C₄²	—	212	2	no	—
214	C₁₆	—	2	2	no	—
215	C₁₆	C₂×C₂²⋊D₅₃	1	2	yes	—
216	C₁₆	C₂×C₂²⋊D₅₃	1	2	yes	—
217	C₂³	—	848	4	no	—
218	C₂³	—	424	4	no	—
219	C₂³	—	212	4	no	—
220	C₂³	C₂×D₄₂₄	1	2	yes	—
221	C₂³	C₂×D₄₂₄	1	1	yes	characteristic
222	Q₈	—	2	2	no	—
223	D₄	—	2	4	no	—
224	D₄	—	2	2	no	—
225	D₄	—	848	8	no	—
226	D₄	—	424	4	no	—
227	D₄	—	212	4	no	—
228	D₄	—	212	4	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	212	4	no	—
232	C₂×C₄	#13133191	1	2	yes	—
233	C₂×C₄	C₂×D₄₂₄	1	2	yes	—
234	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
235	C₂×C₄	#13133239	1	2	yes	—
236	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
237	C₈	—	2	2	no	—
238	C₈	#13133191	1	2	yes	—
239	C₈	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
240	C₈	C₂×#11719945	1	1	yes	characteristic
241	C₂²	—	2	2	no	—
242	C₂²	—	2	4	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	848	8	no	—
245	C₂²	—	424	8	no	—
246	C₂²	—	424	4	no	—
247	C₂²	#16810859	1	2	yes	—
248	C₂²	C₂²×D₄₂₄	1	1	yes	characteristic
249	C₄	—	2	2	no	—
250	C₄	—	424	4	no	—
251	C₄	#16810859	1	2	yes	—
252	C₄	C₂×#13133191	1	1	yes	characteristic
253	C₄	C₂×#13133239	1	1	yes	characteristic
254	C₂	—	2	2	no	—
255	C₂	—	4	2	no	—
256	C₂	—	848	4	no	—
257	C₂	C₂×#16810859	1	1	yes	characteristic
258	C₂	#20287856	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₄₂₄
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#504	C₂
socle	C₂²×C₅₃	C₂²×D₈
hypercenter	C₂×#504	D₅₃
hypocenter	C₅₃	#33607
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
1. C₂²×D₈₄₈	C₂	semidirect
2. #20284363	C₂	semidirect
3. #20276818	C₂	semidirect
4. #20276801	C₂	semidirect
5. #20276499	C₂	semidirect
6. #20276432	C₂	semidirect
7. C₂×C₅₃×#504	C₂	semidirect
14. #16810859	C₂²	semidirect
17. #16810451	C₂²	semidirect
18. #16810449	C₂²	semidirect
19. #16810425	C₂²	semidirect
23. C₂×M₃₂×C₅₃	C₂²	semidirect
25. C₂²×C₁₆×C₅₃	C₂²	semidirect
51. C₂×C₁₆×C₅₃	D₄	semidirect
126. C₂×#504	D₅₃	semidirect
145. C₂×M₃₂	C₂×D₅₃	semidirect
147. C₂²×C₁₆	C₂×D₅₃	semidirect
155. C₅₃	#33607	semidirect
178. C₂×C₁₆	C₂²⋊D₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₈₄₈	2
#20284363	2
#20276818	2
#20276801	2
#20276499	2
#20276432	2
C₂×C₅₃×#504	2
#33607	53