Subgroups — C₂×#20274879 #27384499

Order 13568.

1182 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133151	C₂	1	2	yes	maximal
2	C₂²×#13133149	C₂	1	2	yes	maximal
3	C₂×#16810489	C₂	1	2	yes	maximal
4	#20274879	C₂	1	8	yes	maximal, direct product factor
5	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄⋊Dic₅₃	C₂²	1	4	yes	—
7	C₂×#13133151	—	2	8	no	—
8	C₂×#13133151	C₂²	1	8	yes	—
9	C₂×#13133149	—	2	8	no	—
10	C₂×#13133149	C₂²	1	8	yes	—
11	#16810489	—	2	4	no	—
12	#16810489	C₂²	1	8	yes	—
13	C₂×#13133124	—	2	4	no	—
14	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₅₃	C₄	1	8	yes	complemented
18	C₂×C₄×C₈×C₅₃	C₂²	1	4	yes	—
19	C₂³×Dic₅₃	—	2	4	no	—
20	C₂×C₄⋊Dic₅₃	—	2	4	no	—
21	C₂×C₄⋊Dic₅₃	—	2	8	no	—
22	C₂×C₄⋊Dic₅₃	—	2	8	no	—
23	C₂×C₄⋊Dic₅₃	C₂³	1	8	yes	—
24	#13133151	—	2	16	no	—
25	#13133149	—	2	16	no	—
26	#13133124	—	2	16	no	—
27	C₂³×C₄×C₅₃	D₄	1	2	yes	—
28	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₅₃	D₄	1	4	yes	—
32	C₂²×C₈×C₅₃	D₄	1	4	yes	—
33	C₂²×C₈×C₅₃	Q₈	1	4	yes	—
34	C₂²×C₈×C₅₃	Q₈	1	4	yes	—
35	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
36	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
37	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₅₃	D₄	1	2	yes	—
39	C₂×C₄²×C₅₃	Q₈	1	4	yes	—
40	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
41	C₄×C₈×C₅₃	C₂×C₄	1	16	yes	complemented
42	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	—	2	8	no	—
44	C₂²×C₄×C₅₃	C₄⋊C₄	1	8	yes	—
45	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
46	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
47	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
49	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
51	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
52	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
53	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
54	C₂²×Dic₅₃	—	2	8	no	—
55	C₂²×Dic₅₃	—	4	8	no	—
56	C₂²×Dic₅₃	—	4	16	no	—
57	C₂×C₈×C₅₃	C₄⋊C₄	1	16	yes	—
58	C₂×C₈×C₅₃	C₄⋊C₄	1	16	yes	—
59	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
60	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
61	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
62	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
63	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
64	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
65	C₄²×C₅₃	C₄⋊C₄	1	8	yes	—
66	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
67	C₄²×C₅₃	C₂×Q₈	1	4	yes	—
68	C₄⋊Dic₅₃	—	2	8	no	—
69	C₄⋊Dic₅₃	—	2	8	no	—
70	C₄⋊Dic₅₃	—	4	8	no	—
71	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
72	C₂³×C₅₃	#127	1	2	yes	—
73	C₂³×C₅₃	C₄⋊Q₈	1	2	yes	—
74	C₂³×C₅₃	#135	1	8	yes	—
75	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
76	C₂×C₄×C₅₃	—	2	8	no	—
77	C₂×C₄×C₅₃	—	2	16	no	—
78	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	8	yes	—
79	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
80	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
81	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
82	C₂×C₄×C₅₃	#127	1	2	yes	—
83	C₂×C₄×C₅₃	C₄⋊Q₈	1	4	yes	—
84	C₂×C₄×C₅₃	C₄⋊Q₈	1	4	yes	—
85	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
86	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
92	C₂×Dic₅₃	—	4	8	no	—
93	C₂×Dic₅₃	—	4	16	no	—
94	C₂×Dic₅₃	—	8	16	no	—
95	C₈×C₅₃	C₂×C₄⋊C₄	1	8	yes	—
96	C₈×C₅₃	C₂×C₄⋊C₄	1	8	yes	—
97	C₂×#1846	—	53	1	no	maximal, Sylow
98	C₂²×C₅₃	#384	1	4	yes	—
99	C₂²×C₅₃	#427	1	8	yes	—
100	C₂²×C₅₃	#495	1	8	yes	—
101	C₂²×C₅₃	#499	1	8	yes	—
102	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
103	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
104	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
105	C₂²×C₅₃	C₂×#135	1	4	yes	—
106	C₄×C₅₃	—	2	8	no	—
107	C₄×C₅₃	#384	1	4	yes	—
108	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
109	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
110	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
111	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
112	Dic₅₃	—	8	8	no	—
113	C₂²×#107	—	53	2	no	—
114	C₂²×#106	—	53	2	no	—
115	C₂²×C₄×C₈	D₅₃	1	1	yes	characteristic
116	C₂×#382	—	53	2	no	—
117	#1846	—	53	8	no	—
118	C₂×C₅₃	#1846	1	8	yes	complemented
119	C₂×C₅₃	C₂×#384	1	1	yes	characteristic
120	C₂×C₅₃	C₂×#427	1	2	yes	—
121	C₂×C₅₃	C₂×#495	1	2	yes	—
122	C₂×C₅₃	C₂×#499	1	2	yes	—
123	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
124	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
125	C₂²×C₄⋊C₄	—	53	4	no	—
126	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
127	C₂×#107	—	53	8	no	—
128	C₂×#107	—	106	8	no	—
129	C₂×#106	—	53	8	no	—
130	C₂×#106	—	106	8	no	—
131	C₂×C₄×C₈	Dic₅₃	1	8	yes	complemented
132	C₂×C₄×C₈	C₂×D₅₃	1	4	yes	—
133	#382	—	53	8	no	—
134	#382	—	106	4	no	—
135	C₂×#95	—	106	4	no	—
136	C₅₃	C₂×#1846	1	1	yes	characteristic, complemented, Sylow
137	C₂³×C₄	—	106	4	no	—
138	C₂³×C₄	D₂₁₂	1	2	yes	—
139	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
140	C₂²×C₈	Dic₁₀₆	1	4	yes	—
141	C₂²×C₈	Dic₁₀₆	1	4	yes	—
142	C₂²×C₈	D₂₁₂	1	4	yes	—
143	C₂²×C₈	D₂₁₂	1	4	yes	—
144	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
145	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
146	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
147	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
148	C₂×C₄⋊C₄	—	53	8	no	—
149	C₂×C₄⋊C₄	—	106	8	no	—
150	C₂×C₄⋊C₄	—	106	8	no	—
151	C₂×C₄⋊C₄	—	106	4	no	—
152	C₂×C₄²	Dic₁₀₆	1	4	yes	—
153	C₂×C₄²	D₂₁₂	1	2	yes	—
154	C₂×C₄²	C₂×Dic₅₃	1	4	yes	—
155	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
156	#107	—	106	16	no	—
157	#106	—	106	16	no	—
158	C₄×C₈	C₂×Dic₅₃	1	16	yes	complemented
159	#95	—	106	16	no	—
160	C₂⁴	C₂×D₂₁₂	1	1	yes	characteristic
161	C₂²×C₄	—	2	8	no	—
162	C₂²×C₄	—	106	8	no	—
163	C₂²×C₄	—	212	8	no	—
164	C₂²×C₄	—	212	16	no	—
165	C₂²×C₄	C₄⋊Dic₅₃	1	8	yes	—
166	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
167	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
168	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
169	C₂²×C₄	C₂×D₂₁₂	1	4	yes	—
170	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
171	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
172	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
173	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
174	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
175	C₂×C₈	C₄⋊Dic₅₃	1	16	yes	—
176	C₂×C₈	C₄⋊Dic₅₃	1	16	yes	—
177	C₂×C₈	C₂×Dic₁₀₆	1	4	yes	—
178	C₂×C₈	C₂×Dic₁₀₆	1	4	yes	—
179	C₂×C₈	C₂×D₂₁₂	1	4	yes	—
180	C₂×C₈	C₂×D₂₁₂	1	4	yes	—
181	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
182	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
183	C₄⋊C₄	—	106	8	no	—
184	C₄⋊C₄	—	106	8	no	—
185	C₄⋊C₄	—	212	8	no	—
186	C₄²	C₄⋊Dic₅₃	1	8	yes	—
187	C₄²	C₂×Dic₁₀₆	1	4	yes	—
188	C₄²	C₂²×Dic₅₃	1	4	yes	—
189	C₂³	C₂×C₄⋊Dic₅₃	1	2	yes	—
190	C₂³	#13133214	1	2	yes	—
191	C₂³	#13133217	1	2	yes	—
192	C₂³	#13133239	1	8	yes	—
193	C₂³	C₂²×D₂₁₂	1	1	yes	characteristic
194	C₂×C₄	—	2	8	no	—
195	C₂×C₄	—	2	16	no	—
196	C₂×C₄	—	424	16	no	—
197	C₂×C₄	—	212	16	no	—
198	C₂×C₄	—	212	8	no	—
199	C₂×C₄	C₂×C₄⋊Dic₅₃	1	8	yes	—
200	C₂×C₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
201	C₂×C₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
202	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
203	C₂×C₄	#13133214	1	2	yes	—
204	C₂×C₄	#13133217	1	4	yes	—
205	C₂×C₄	#13133217	1	4	yes	—
206	C₂×C₄	#13133217	1	2	yes	—
207	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
208	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
209	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
210	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
211	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
212	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
213	C₈	C₂×C₄⋊Dic₅₃	1	8	yes	—
214	C₈	C₂×C₄⋊Dic₅₃	1	8	yes	—
215	C₂²	#16810493	1	4	yes	—
216	C₂²	#16810661	1	8	yes	—
217	C₂²	#16811007	1	8	yes	—
218	C₂²	#16811025	1	8	yes	—
219	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
220	C₂²	C₂×#13133214	1	1	yes	characteristic
221	C₂²	C₂×#13133217	1	1	yes	characteristic
222	C₂²	C₂×#13133239	1	4	yes	—
223	C₄	—	2	8	no	—
224	C₄	—	424	8	no	—
225	C₄	#16810493	1	4	yes	—
226	C₄	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
227	C₄	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
228	C₄	C₂×#13133217	1	1	yes	characteristic
229	C₄	C₂×#13133217	1	1	yes	characteristic
230	C₂	#20274879	1	8	yes	direct product factor
231	C₂	C₂×#16810493	1	1	yes	characteristic
232	C₂	C₂×#16810661	1	2	yes	—
233	C₂	C₂×#16811007	1	2	yes	—
234	C₂	C₂×#16811025	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂²×C₄×C₈×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	C₂×D₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂²×C₄×C₈	D₅₃
hypocenter	C₅₃	C₂×#1846
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
4. #20274879	C₂	direct (N × H)
17. C₂×C₄×C₈×C₅₃	C₄	semidirect
41. C₄×C₈×C₅₃	C₂×C₄	semidirect
118. C₂×C₅₃	#1846	semidirect
131. C₂×C₄×C₈	Dic₅₃	semidirect
136. C₅₃	C₂×#1846	semidirect
158. C₄×C₈	C₂×Dic₅₃	semidirect
230. C₂	#20274879	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×#13133151	2
C₂²×#13133149	2
C₂×#16810489	2
#20274879	2
C₂²×C₄×C₈×C₅₃	2
C₂×#1846	53

Subgroups — C2×#20274879 #27384499

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20274879 #27384499