Subgroups — C₂×#20291074 #28328216

Order 13568.

726 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	#20291074	C₂	1	8	yes	maximal, direct product factor
2	C₂×#16811544	C₂	1	2	yes	maximal, complemented
3	C₂×#16811518	C₂	1	2	yes	maximal
4	C₂×C₄×#11719910	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719910	C₂²	1	4	yes	—
7	#16811544	C₂²	1	8	yes	complemented
8	#16811518	C₂²	1	8	yes	—
9	C₄×#11719910	C₂²	1	8	yes	complemented
10	C₂²×C₈×D₅₃	C₄	1	1	yes	characteristic
11	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₄²×D₅₃	C₄	1	1	yes	characteristic
13	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
16	C₄×C₈×D₅₃	C₄	1	4	yes	—
17	C₄×C₈×D₅₃	C₂²	1	4	yes	—
18	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
19	C₂×#11719910	—	2	8	no	—
20	C₂×#11719910	C₂³	1	8	yes	—
21	C₂²×C₄×D₅₃	C₈	1	1	yes	characteristic
22	C₂²×C₄×D₅₃	C₈	1	1	yes	characteristic
23	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
25	C₂²×#88322	C₂³	1	1	yes	characteristic
26	C₂×C₈×D₅₃	—	2	4	no	—
27	C₂×C₈×D₅₃	—	2	4	no	—
28	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
29	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
30	C₂×C₈×D₅₃	C₂³	1	2	yes	—
31	C₂×C₄×Dic₅₃	C₈	1	1	yes	characteristic
32	C₂×C₄×Dic₅₃	C₈	1	1	yes	characteristic
33	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
34	C₄²×D₅₃	C₈	1	4	yes	—
35	C₄²×D₅₃	C₂×C₄	1	4	yes	—
36	C₄×#88322	C₂×C₄	1	2	yes	—
37	C₄×#88322	C₂³	1	2	yes	—
38	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
39	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
40	C₈×Dic₅₃	C₂³	1	2	yes	—
41	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
45	C₂³×D₅₃	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
50	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
51	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
52	C₂×C₄×D₅₃	—	2	4	no	—
53	C₂×C₄×D₅₃	—	2	2	no	—
54	C₂×C₄×D₅₃	—	2	2	no	—
55	C₂×C₄×D₅₃	C₁₆	1	8	yes	complemented
56	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
57	C₂×C₄×D₅₃	C₂×C₈	1	4	yes	—
58	C₂×C₄×D₅₃	C₂×C₈	1	4	yes	—
59	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
60	C₂×C₈×C₅₃	—	2	2	no	—
61	C₂×C₈×C₅₃	—	2	2	no	—
62	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
63	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
64	C₄²×C₅₃	C₂×C₈	1	2	yes	—
65	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
66	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
67	C₄×Dic₅₃	C₂×C₈	1	4	yes	—
68	C₄×Dic₅₃	C₂×C₈	1	4	yes	—
69	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
70	C₂×#88322	M₁₆	1	4	yes	—
71	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
72	C₂×#88322	C₂²×C₄	1	2	yes	—
73	C₂×#88322	#41	1	4	yes	—
74	C₂×#88322	C₂⁴	1	1	yes	characteristic
75	C₈×D₅₃	—	2	4	no	—
76	C₈×D₅₃	—	2	8	no	—
77	C₈×D₅₃	—	2	4	no	—
78	#11719910	—	2	8	no	—
79	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
80	C₂²×D₅₃	—	2	2	no	—
81	C₂²×D₅₃	—	2	2	no	—
82	C₂²×D₅₃	C₂×C₁₆	1	4	yes	—
83	C₂²×D₅₃	C₂²×C₈	1	2	yes	—
84	C₂×C₄×C₅₃	C₂×C₁₆	1	4	yes	—
85	C₂×C₄×C₅₃	M₃₂	1	4	yes	—
86	C₂×C₄×C₅₃	#117	1	2	yes	—
87	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
89	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
90	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
91	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
92	C₂×Dic₅₃	—	2	2	no	—
93	C₂×Dic₅₃	—	2	2	no	—
94	C₂×Dic₅₃	C₂×C₁₆	1	4	yes	—
95	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
96	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
97	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
98	C₄×D₅₃	—	2	2	no	—
99	C₄×D₅₃	—	2	2	no	—
100	C₄×D₅₃	—	2	4	no	—
101	C₄×D₅₃	—	2	8	no	—
102	C₄×D₅₃	C₂×C₁₆	1	16	yes	complemented
103	C₈×C₅₃	—	2	2	no	—
104	C₈×C₅₃	—	2	2	no	—
105	#88322	#117	1	2	yes	—
106	#88322	C₂×M₁₆	1	4	yes	—
107	#88322	C₂×#41	1	2	yes	—
108	C₂×#2162	—	53	1	no	maximal, Sylow
109	C₂²×C₅₃	#431	1	2	yes	—
110	C₂²×C₅₃	C₂²×C₁₆	1	2	yes	—
111	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
112	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
113	C₂×D₅₃	—	2	2	no	—
114	C₂×D₅₃	—	2	2	no	—
115	C₂×D₅₃	—	2	8	no	—
116	C₂×D₅₃	C₂²×C₁₆	1	4	yes	—
117	C₄×C₅₃	#431	1	2	yes	—
118	C₄×C₅₃	C₂²×C₁₆	1	4	yes	—
119	C₄×C₅₃	C₂×M₃₂	1	4	yes	—
120	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
121	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
122	Dic₅₃	—	2	2	no	—
123	Dic₅₃	—	2	2	no	—
124	Dic₅₃	C₂²×C₁₆	1	4	yes	—
125	C₂²×C₄×C₈	—	53	1	no	—
126	#2162	—	53	8	no	—
127	C₂×C₄⋊C₁₆	—	53	2	no	—
128	C₂×C₂²⋊C₁₆	—	53	2	no	—
129	C₂×C₄×C₁₆	—	53	2	no	—
130	C₂×C₅₃	#2162	1	4	yes	complemented
131	C₂×C₅₃	C₂×#431	1	1	yes	characteristic
132	C₂×C₅₃	C₂³×C₁₆	1	1	yes	characteristic
133	C₂×C₅₃	C₂²×M₃₂	1	1	yes	characteristic
134	D₅₃	—	2	4	no	—
135	C₂³×C₈	—	53	1	no	—
136	C₂³×C₈	—	53	1	no	—
137	C₂²×C₄²	—	53	1	no	—
138	C₂²×C₁₆	—	53	4	no	—
139	C₂×C₄×C₈	—	53	1	no	—
140	C₂×C₄×C₈	—	53	1	no	—
141	C₂×C₄×C₈	—	53	1	no	—
142	C₂×C₄×C₈	—	53	4	no	—
143	C₂×C₄×C₈	—	53	4	no	—
144	C₂×C₄×C₈	#6157	1	1	yes	characteristic
145	C₄⋊C₁₆	—	53	8	no	—
146	C₂²⋊C₁₆	—	53	8	no	—
147	C₄×C₁₆	—	53	8	no	—
148	C₅₃	C₂×#2162	1	1	yes	characteristic, complemented, Sylow
149	C₂³×C₄	—	53	1	no	—
150	C₂³×C₄	—	53	1	no	—
151	C₂³×C₄	—	53	1	no	—
152	C₂²×C₈	—	53	1	no	—
153	C₂²×C₈	—	53	1	no	—
154	C₂²×C₈	—	53	2	no	—
155	C₂²×C₈	—	53	2	no	—
156	C₂²×C₈	—	53	4	no	—
157	C₂²×C₈	—	106	4	no	—
158	C₂²×C₈	—	106	4	no	—
159	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
160	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
161	C₂×C₄²	—	53	1	no	—
162	C₂×C₄²	—	53	1	no	—
163	C₂×C₄²	—	53	1	no	—
164	C₂×C₄²	—	53	4	no	—
165	C₂×C₄²	—	53	4	no	—
166	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
167	C₂×C₁₆	—	53	8	no	—
168	C₂×C₁₆	—	106	8	no	—
169	C₄×C₈	—	53	2	no	—
170	C₄×C₈	—	53	4	no	—
171	C₄×C₈	—	53	2	no	—
172	C₄×C₈	—	53	2	no	—
173	C₄×C₈	—	53	2	no	—
174	C₄×C₈	C₂×#6157	1	4	yes	—
175	C₂⁴	—	53	1	no	—
176	C₂²×C₄	—	53	1	no	—
177	C₂²×C₄	—	53	8	no	—
178	C₂²×C₄	—	53	4	no	—
179	C₂²×C₄	—	53	1	no	—
180	C₂²×C₄	—	53	1	no	—
181	C₂²×C₄	—	53	2	no	—
182	C₂²×C₄	—	53	4	no	—
183	C₂²×C₄	—	53	2	no	—
184	C₂²×C₄	—	106	4	no	—
185	C₂²×C₄	—	106	2	no	—
186	C₂²×C₄	—	106	2	no	—
187	C₂²×C₄	D₅₃⋊C₈	1	1	yes	characteristic
188	C₂²×C₄	D₅₃⋊C₈	1	1	yes	characteristic
189	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	53	4	no	—
193	C₂×C₈	—	53	4	no	—
194	C₂×C₈	—	53	1	no	—
195	C₂×C₈	—	53	1	no	—
196	C₂×C₈	—	53	2	no	—
197	C₂×C₈	—	106	4	no	—
198	C₂×C₈	—	106	4	no	—
199	C₂×C₈	—	106	8	no	—
200	C₂×C₈	C₂²×#6157	1	2	yes	—
201	C₂×C₈	C₂²×#6157	1	2	yes	—
202	C₄²	—	53	2	no	—
203	C₄²	—	53	4	no	—
204	C₄²	—	53	4	no	—
205	C₄²	—	53	2	no	—
206	C₄²	D₅₃⋊C₈	1	2	yes	—
207	C₄²	C₂²×#6157	1	2	yes	—
208	C₁₆	—	106	8	no	—
209	C₂³	—	53	4	no	—
210	C₂³	—	53	2	no	—
211	C₂³	—	106	2	no	—
212	C₂³	—	106	2	no	—
213	C₂³	C₂×D₅₃⋊C₈	1	1	yes	characteristic
214	C₂×C₄	—	53	4	no	—
215	C₂×C₄	—	53	2	no	—
216	C₂×C₄	—	53	16	no	—
217	C₂×C₄	—	53	2	no	—
218	C₂×C₄	—	53	2	no	—
219	C₂×C₄	—	106	2	no	—
220	C₂×C₄	—	106	2	no	—
221	C₂×C₄	—	106	4	no	—
222	C₂×C₄	—	106	8	no	—
223	C₂×C₄	—	106	2	no	—
224	C₂×C₄	—	106	2	no	—
225	C₂×C₄	D₅₃⋊C₁₆	1	4	yes	—
226	C₂×C₄	#13133292	1	4	yes	—
227	C₂×C₄	#13133296	1	2	yes	—
228	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
229	C₂×C₄	C₂×D₅₃⋊C₈	1	2	yes	—
230	C₂×C₄	C₂×D₅₃⋊C₈	1	2	yes	—
231	C₂×C₄	D₅₃⋊M₁₆	1	2	yes	—
232	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
233	C₈	—	2	2	no	—
234	C₈	—	2	2	no	—
235	C₈	—	53	2	no	—
236	C₈	—	53	2	no	—
237	C₈	—	53	4	no	—
238	C₂²	—	53	4	no	—
239	C₂²	—	106	8	no	—
240	C₂²	—	106	2	no	—
241	C₂²	—	106	2	no	—
242	C₂²	#16811641	1	2	yes	—
243	C₂²	C₂×D₅₃⋊C₁₆	1	2	yes	—
244	C₂²	C₂×#13133292	1	2	yes	—
245	C₂²	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
246	C₄	—	53	4	no	—
247	C₄	—	106	2	no	—
248	C₄	—	106	2	no	—
249	C₄	#16811641	1	2	yes	—
250	C₄	C₂×D₅₃⋊C₁₆	1	4	yes	—
251	C₄	C₂×#13133292	1	4	yes	—
252	C₄	C₂×#13133296	1	1	yes	characteristic
253	C₄	C₂×D₅₃⋊M₁₆	1	1	yes	characteristic
254	C₂	—	106	4	no	—
255	C₂	#20291074	1	4	yes	direct product factor
256	C₂	C₂×#16811641	1	1	yes	characteristic
257	C₂	C₂²×D₅₃⋊C₁₆	1	1	yes	characteristic
258	C₂	C₂²×#13133292	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂×#6157
commutator	C₂×C₅₃	C₂³×C₁₆
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₂×C₄×C₈×C₅₃	C₄
socle	C₂³×C₅₃	C₂²×C₈
prime radical	C₂³×D₅₃	C₂×C₈
cosocle	C₂×#88322	C₂⁴
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	C₂×#2162
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
1. #20291074	C₂	direct (N × H)
2. C₂×#16811544	C₂	semidirect
4. C₂×C₄×#11719910	C₂	semidirect
7. #16811544	C₂²	semidirect
9. C₄×#11719910	C₂²	semidirect
55. C₂×C₄×D₅₃	C₁₆	semidirect
102. C₄×D₅₃	C₂×C₁₆	semidirect
130. C₂×C₅₃	#2162	semidirect
148. C₅₃	C₂×#2162	semidirect
255. C₂	#20291074	direct (N × H)

Maximal subgroups

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

subgroup	index
#20291074	2
C₂×#16811544	2
C₂×#16811518	2
C₂×C₄×#11719910	2
C₂×C₄×C₈×D₅₃	2
C₂×#2162	53

Subgroups — C2×#20291074 #28328216

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20291074 #28328216