Subgroups — C₂×#20276504 #27537127

Order 13568.

718 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×#13133157	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13133151	C₂	1	1	yes	maximal, characteristic
3	#20276504	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16810450	C₂	1	2	yes	maximal, complemented
5	C₂×#16810448	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₃×#427	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₄⋊Dic₅₃	—	2	1	no	—
8	C₂×#13133157	C₄	1	8	yes	complemented
9	C₂×#13133157	C₂²	1	4	yes	—
10	C₂²×#11719908	C₂²	1	2	yes	—
11	C₂×#13133151	—	2	1	no	—
12	C₂×#13133151	—	2	2	no	—
13	C₂×#13133151	C₂²	1	2	yes	—
14	C₂×#13133151	C₂²	1	4	yes	—
15	#16810450	C₂²	1	8	yes	complemented
16	#16810448	C₂²	1	8	yes	complemented
17	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×#117×C₅₃	—	2	1	no	—
19	C₅₃×#427	—	2	2	no	—
20	C₅₃×#427	C₂²	1	4	yes	—
21	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×#106×C₅₃	—	2	1	no	—
24	C₂³×Dic₅₃	—	4	1	no	—
25	C₂×C₄⋊Dic₅₃	—	2	2	no	—
26	C₂×C₄⋊Dic₅₃	—	2	4	no	—
27	C₂×C₄⋊Dic₅₃	—	4	1	no	—
28	C₂×C₄⋊Dic₅₃	—	4	2	no	—
29	#13133157	C₂×C₄	1	16	yes	complemented
30	C₂×#11719908	C₂×C₄	1	8	yes	complemented
31	C₂×#11719908	C₂³	1	4	yes	—
32	#13133151	—	2	4	no	—
33	#13133151	—	2	2	no	—
34	#13133151	C₂³	1	4	yes	—
35	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
37	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	D₄	1	2	yes	—
39	C₂²×C₈×C₅₃	Q₈	1	4	yes	—
40	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
42	#117×C₅₃	—	2	4	no	—
43	#117×C₅₃	—	4	2	no	—
44	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
45	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
46	C₂×C₂²⋊C₄×C₅₃	—	4	1	no	—
47	C₂×C₄²×C₅₃	—	4	1	no	—
48	#107×C₅₃	—	2	1	no	—
49	#107×C₅₃	—	2	1	no	—
50	#107×C₅₃	C₂³	1	2	yes	—
51	#107×C₅₃	C₂³	1	2	yes	—
52	#106×C₅₃	—	2	4	no	—
53	C₂⁴×C₅₃	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	—	4	1	no	—
55	C₂²×C₄×C₅₃	—	4	1	no	—
56	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
57	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	D₈	1	2	yes	—
59	C₂²×C₄×C₅₃	Q₁₆	1	4	yes	—
60	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
62	C₂²×Dic₅₃	—	4	4	no	—
63	C₂²×Dic₅₃	—	4	2	no	—
64	C₂²×Dic₅₃	—	8	4	no	—
65	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
66	C₂×C₈×C₅₃	C₄⋊C₄	1	8	yes	—
67	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
69	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
72	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
73	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
74	C₄⋊C₄×C₅₃	—	2	2	no	—
75	C₄⋊C₄×C₅₃	—	2	2	no	—
76	C₄⋊C₄×C₅₃	—	4	1	no	—
77	C₄⋊C₄×C₅₃	—	4	1	no	—
78	C₂²⋊C₄×C₅₃	—	4	4	no	—
79	C₄²×C₅₃	—	4	4	no	—
80	C₄⋊Dic₅₃	—	2	4	no	—
81	C₄⋊Dic₅₃	—	4	4	no	—
82	C₄⋊Dic₅₃	—	4	2	no	—
83	#11719908	C₂²×C₄	1	8	yes	complemented
84	C₂³×C₅₃	—	2	2	no	—
85	C₂³×C₅₃	—	2	2	no	—
86	C₂³×C₅₃	#107	1	4	yes	—
87	C₂³×C₅₃	C₂×D₈	1	2	yes	—
88	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₅₃	—	4	2	no	—
90	C₂×C₄×C₅₃	—	4	2	no	—
91	C₂×C₄×C₅₃	—	8	2	no	—
92	C₂×C₄×C₅₃	—	8	2	no	—
93	C₂×C₄×C₅₃	#107	1	8	yes	—
94	C₂×C₄×C₅₃	#107	1	4	yes	—
95	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
96	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
97	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
98	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×Q₁₆	1	4	yes	—
100	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
101	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
102	C₂×Dic₅₃	—	4	4	no	—
103	C₂×Dic₅₃	—	8	8	no	—
104	C₂×Dic₅₃	—	8	4	no	—
105	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
106	C₈×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
107	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
108	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
109	C₂×#2159	—	53	1	no	maximal, Sylow
110	C₂²×C₅₃	—	2	2	no	—
111	C₂²×C₅₃	—	2	2	no	—
112	C₂²×C₅₃	—	2	8	no	—
113	C₂²×C₅₃	C₂×#107	1	4	yes	—
114	C₂²×C₅₃	C₂×#107	1	2	yes	—
115	C₂²×C₅₃	#509	1	2	yes	—
116	C₂²×C₅₃	#510	1	2	yes	—
117	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
118	C₄×C₅₃	—	8	2	no	—
119	C₄×C₅₃	—	8	2	no	—
120	C₄×C₅₃	C₂×#107	1	4	yes	—
121	C₄×C₅₃	C₂×#107	1	2	yes	—
122	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
123	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
124	Dic₅₃	—	8	4	no	—
125	C₂²×M₃₂	—	53	1	no	—
126	C₂×#427	D₅₃	1	1	yes	characteristic
127	C₂²×#107	—	53	1	no	—
128	#2159	—	53	8	no	—
129	C₂×#367	—	53	2	no	—
130	C₂×#366	—	53	2	no	—
131	C₂×C₅₃	—	2	4	no	—
132	C₂×C₅₃	#2159	1	4	yes	complemented
133	C₂×C₅₃	C₂²×#107	1	1	yes	characteristic
134	C₂×C₅₃	C₂×#509	1	1	yes	characteristic
135	C₂×C₅₃	C₂×#510	1	1	yes	characteristic
136	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
137	C₂×#117	—	2	1	no	—
138	C₂²×C₄⋊C₄	—	106	1	no	—
139	C₂×M₃₂	—	53	8	no	—
140	C₂×M₃₂	—	53	4	no	—
141	C₂²×C₁₆	—	53	2	no	—
142	#427	—	2	2	no	—
143	#427	C₂×D₅₃	1	4	yes	—
144	C₂×#107	—	53	2	no	—
145	C₂×#107	—	53	4	no	—
146	C₂×#107	—	106	1	no	—
147	C₂×#107	—	106	2	no	—
148	C₂×#107	C₂×D₅₃	1	1	yes	characteristic
149	C₂×#107	C₂×D₅₃	1	1	yes	characteristic
150	C₂×#106	—	2	1	no	—
151	#367	—	53	8	no	—
152	#366	—	53	8	no	—
153	C₅₃	C₂×#2159	1	1	yes	characteristic, complemented, Sylow
154	C₂³×C₄	—	212	1	no	—
155	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
156	C₂²×C₈	Dic₁₀₆	1	4	yes	—
157	C₂²×C₈	C₄×D₅₃	1	4	yes	—
158	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
159	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
160	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
161	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
162	#117	—	2	4	no	—
163	#117	—	4	2	no	—
164	C₂×C₄⋊C₄	—	2	1	no	—
165	C₂×C₄⋊C₄	—	2	1	no	—
166	C₂×C₄⋊C₄	—	106	4	no	—
167	C₂×C₄⋊C₄	—	106	2	no	—
168	C₂×C₄⋊C₄	—	212	2	no	—
169	C₂×C₄⋊C₄	—	212	1	no	—
170	C₂×C₂²⋊C₄	—	4	1	no	—
171	C₂×C₄²	—	4	1	no	—
172	M₃₂	—	53	16	no	—
173	C₂×C₁₆	—	53	8	no	—
174	C₂×C₁₆	—	53	4	no	—
175	#107	—	2	1	no	—
176	#107	—	2	1	no	—
177	#107	—	53	4	no	—
178	#107	—	106	4	no	—
179	#107	—	106	2	no	—
180	#107	C₂²×D₅₃	1	2	yes	—
181	#107	C₂²×D₅₃	1	2	yes	—
182	#106	—	2	4	no	—
183	C₂⁴	#11719922	1	1	yes	characteristic
184	C₂²×C₄	—	4	1	no	—
185	C₂²×C₄	—	4	1	no	—
186	C₂²×C₄	—	424	4	no	—
187	C₂²×C₄	—	212	4	no	—
188	C₂²×C₄	—	212	2	no	—
189	C₂²×C₄	#11719919	1	4	yes	—
190	C₂²×C₄	#11719922	1	2	yes	—
191	C₂²×C₄	#11719922	1	1	yes	characteristic
192	C₂²×C₄	#11719925	1	4	yes	—
193	C₂²×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
194	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
195	C₂×C₈	#11719919	1	4	yes	—
196	C₂×C₈	#11719919	1	8	yes	—
197	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
198	C₂×C₈	C₂×Dic₁₀₆	1	4	yes	—
199	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
200	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
201	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
202	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
203	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	4	1	no	—
207	C₄⋊C₄	—	4	1	no	—
208	C₄⋊C₄	—	106	4	no	—
209	C₄⋊C₄	—	212	4	no	—
210	C₄⋊C₄	—	212	2	no	—
211	C₂²⋊C₄	—	4	4	no	—
212	C₄²	—	4	4	no	—
213	C₁₆	—	53	8	no	—
214	C₂³	—	2	2	no	—
215	C₂³	—	2	2	no	—
216	C₂³	#13133150	1	4	yes	—
217	C₂³	C₂×#11719922	1	2	yes	—
218	C₂³	C₂×#11719922	1	1	yes	characteristic
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	8	2	no	—
222	C₂×C₄	—	8	2	no	—
223	C₂×C₄	—	424	8	no	—
224	C₂×C₄	—	424	4	no	—
225	C₂×C₄	—	212	4	no	—
226	C₂×C₄	#13133150	1	8	yes	—
227	C₂×C₄	#13133150	1	4	yes	—
228	C₂×C₄	C₂×#11719919	1	4	yes	—
229	C₂×C₄	C₂×#11719919	1	2	yes	—
230	C₂×C₄	C₂×#11719922	1	2	yes	—
231	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
232	C₂×C₄	C₂×#11719925	1	4	yes	—
233	C₂×C₄	C₂×#11719925	1	2	yes	—
234	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
235	C₈	C₂×#11719919	1	2	yes	—
236	C₈	C₂×#11719919	1	4	yes	—
237	C₈	C₂²×C₄×D₅₃	1	1	yes	characteristic
238	C₈	C₂²×Dic₁₀₆	1	1	yes	characteristic
239	C₂²	—	2	8	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	2	no	—
242	C₂²	C₂×#13133150	1	4	yes	—
243	C₂²	C₂×#13133150	1	2	yes	—
244	C₂²	#16811058	1	2	yes	—
245	C₂²	#16811063	1	2	yes	—
246	C₂²	C₂²×#11719922	1	1	yes	characteristic
247	C₄	—	8	2	no	—
248	C₄	—	8	2	no	—
249	C₄	—	424	4	no	—
250	C₄	C₂×#13133150	1	4	yes	—
251	C₄	C₂×#13133150	1	2	yes	—
252	C₄	C₂²×#11719919	1	1	yes	characteristic
253	C₄	C₂²×#11719925	1	1	yes	characteristic
254	C₂	—	2	4	no	—
255	C₂	#20276504	1	4	yes	direct product factor
256	C₂	C₂²×#13133150	1	1	yes	characteristic
257	C₂	C₂×#16811058	1	1	yes	characteristic
258	C₂	C₂×#16811063	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11719922
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#427	C₂
socle	C₂³×C₅₃	C₂×D₈
prime radical	C₂⁴×C₅₃	D₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×#427	D₅₃
hypocenter	C₅₃	C₂×#2159
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
3. #20276504	C₂	direct (N × H)
4. C₂×#16810450	C₂	semidirect
5. C₂×#16810448	C₂	semidirect
8. C₂×#13133157	C₄	semidirect
15. #16810450	C₂²	semidirect
16. #16810448	C₂²	semidirect
29. #13133157	C₂×C₄	semidirect
30. C₂×#11719908	C₂×C₄	semidirect
83. #11719908	C₂²×C₄	semidirect
132. C₂×C₅₃	#2159	semidirect
153. C₅₃	C₂×#2159	semidirect
255. C₂	#20276504	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#13133157	2
C₂²×#13133151	2
#20276504	2
C₂×#16810450	2
C₂×#16810448	2
C₂×C₅₃×#427	2
C₂×#2159	53

Subgroups — C2×#20276504 #27537127

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276504 #27537127