Subgroups — C₂×#20274887 #27384548

Order 13568.

1046 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133153	C₂	1	4	yes	maximal
2	C₂×#16810647	C₂	1	2	yes	maximal
3	#20274887	C₂	1	8	yes	maximal, direct product factor
4	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719917	C₂²	1	4	yes	—
6	C₂×#13133153	—	2	16	no	—
7	C₂×#13133153	C₂²	1	16	yes	—
8	#16810647	—	2	4	no	—
9	#16810647	C₂²	1	8	yes	—
10	C₂×C₄⋊#88322	—	2	4	no	—
11	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
12	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₅₃	C₄	1	4	yes	—
15	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	4	yes	—
17	C₂×#11719917	—	2	4	no	—
18	C₂×#11719917	—	2	16	no	—
19	C₂×#11719917	C₂³	1	8	yes	—
20	C₂²×#88322	—	2	4	no	—
21	#13133153	—	2	32	no	—
22	C₄⋊#88322	—	2	16	no	—
23	C₂³×C₄×C₅₃	Q₈	1	2	yes	—
24	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
25	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₅₃	D₄	1	8	yes	—
28	C₂²×C₈×C₅₃	Q₈	1	8	yes	—
29	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
30	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	D₄	1	2	yes	—
34	C₂×C₄²×C₅₃	D₄	1	2	yes	—
35	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
36	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
37	C₄×C₈×C₅₃	C₂×C₄	1	8	yes	—
38	C₄×C₈×C₅₃	C₂×C₄	1	8	yes	—
39	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	—	2	8	no	—
41	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
42	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
43	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
44	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
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	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
50	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
52	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
53	C₂×C₈×C₅₃	C₄⋊C₄	1	16	yes	—
54	C₂×C₈×C₅₃	C₄⋊C₄	1	16	yes	—
55	C₂×C₈×C₅₃	C₂²×C₄	1	8	yes	—
56	C₂×C₈×C₅₃	C₂×D₄	1	8	yes	—
57	C₂×C₈×C₅₃	C₂×Q₈	1	8	yes	—
58	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
59	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
60	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
61	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
62	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
63	C₄²×C₅₃	C₂×D₄	1	4	yes	—
64	#11719917	—	2	16	no	—
65	#11719917	—	4	8	no	—
66	C₂×#88322	—	2	8	no	—
67	C₂×#88322	—	4	8	no	—
68	C₂³×C₅₃	—	2	4	no	—
69	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂³×C₅₃	C₄⋊Q₈	1	4	yes	—
72	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
73	C₂×C₄×C₅₃	—	2	4	no	—
74	C₂×C₄×C₅₃	—	2	8	no	—
75	C₂×C₄×C₅₃	—	2	8	no	—
76	C₂×C₄×C₅₃	#108	1	8	yes	complemented
77	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
78	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
79	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
80	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
83	C₂×C₄×C₅₃	#127	1	2	yes	—
84	C₂×C₄×C₅₃	#127	1	2	yes	—
85	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
86	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
87	C₂×C₄×C₅₃	C₄⋊Q₈	1	4	yes	—
88	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
90	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
92	C₈×C₅₃	C₂×C₄⋊C₄	1	16	yes	—
93	#88322	—	4	8	no	—
94	C₂×#1848	—	53	1	no	maximal, Sylow
95	C₂²×C₅₃	—	2	8	no	—
96	C₂²×C₅₃	—	2	2	no	—
97	C₂²×C₅₃	—	2	2	no	—
98	C₂²×C₅₃	#384	1	2	yes	—
99	C₂²×C₅₃	#384	1	2	yes	—
100	C₂²×C₅₃	C₂×#108	1	4	yes	—
101	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
102	C₂²×C₅₃	C₂×C₄⋊Q₈	1	2	yes	—
103	C₄×C₅₃	—	2	4	no	—
104	C₄×C₅₃	#384	1	2	yes	—
105	C₄×C₅₃	#384	1	2	yes	—
106	C₄×C₅₃	C₂×#108	1	8	yes	complemented
107	C₄×C₅₃	C₂²×C₄⋊C₄	1	2	yes	—
108	C₄×C₅₃	C₂×#127	1	1	yes	characteristic
109	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
110	C₂²×#108	—	53	4	no	—
111	C₂×#423	—	53	2	no	—
112	C₂²×C₄×C₈	D₅₃	1	1	yes	characteristic
113	#1848	—	53	8	no	—
114	C₂×C₅₃	—	2	2	no	—
115	C₂×C₅₃	—	2	2	no	—
116	C₂×C₅₃	#1848	1	4	yes	complemented
117	C₂×C₅₃	C₂×#384	1	1	yes	characteristic
118	C₂×C₅₃	C₂²×#108	1	2	yes	—
119	C₂²×M₁₆	—	53	4	no	—
120	C₂³×C₈	C₂×D₅₃	1	2	yes	—
121	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
122	C₂×#108	—	53	16	no	—
123	C₂×#108	—	106	16	no	—
124	#423	—	53	8	no	—
125	#423	—	106	4	no	—
126	C₂×C₄⋊C₈	—	106	4	no	—
127	C₂×C₄×C₈	Dic₅₃	1	2	yes	—
128	C₂×C₄×C₈	Dic₅₃	1	4	yes	—
129	C₂×C₄×C₈	Dic₅₃	1	2	yes	—
130	C₂×C₄×C₈	C₂×D₅₃	1	4	yes	—
131	C₅₃	C₂×#1848	1	1	yes	characteristic, complemented, Sylow
132	C₂³×C₄	Dic₁₀₆	1	2	yes	—
133	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
134	C₂×M₁₆	—	53	8	no	—
135	C₂×M₁₆	—	106	4	no	—
136	C₂×M₁₆	—	106	16	no	—
137	C₂²×C₈	—	106	4	no	—
138	C₂²×C₈	Dic₁₀₆	1	8	yes	—
139	C₂²×C₈	D₂₁₂	1	8	yes	—
140	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
141	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
142	C₂²×C₈	C₂²×D₅₃	1	4	yes	—
143	C₂×C₄²	Dic₁₀₆	1	2	yes	—
144	C₂×C₄²	D₂₁₂	1	2	yes	—
145	C₂×C₄²	D₂₁₂	1	2	yes	—
146	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
147	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
148	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
149	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
150	#108	—	106	32	no	—
151	C₄⋊C₈	—	106	16	no	—
152	C₄×C₈	C₂×Dic₅₃	1	8	yes	—
153	C₄×C₈	C₂×Dic₅₃	1	8	yes	—
154	C₂⁴	C₂×Dic₁₀₆	1	1	yes	characteristic
155	C₂²×C₄	—	2	8	no	—
156	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
157	C₂²×C₄	C₄⋊Dic₅₃	1	4	yes	—
158	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
159	C₂²×C₄	C₂×Dic₁₀₆	1	4	yes	—
160	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
161	C₂²×C₄	C₂×Dic₁₀₆	1	4	yes	—
162	C₂²×C₄	C₂×D₂₁₂	1	4	yes	—
163	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
164	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
165	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
166	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
167	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
168	M₁₆	—	106	16	no	—
169	M₁₆	—	212	8	no	—
170	C₂×C₈	—	106	8	no	—
171	C₂×C₈	—	212	8	no	—
172	C₂×C₈	C₄⋊Dic₅₃	1	16	yes	—
173	C₂×C₈	C₄⋊Dic₅₃	1	16	yes	—
174	C₂×C₈	C₂×Dic₁₀₆	1	8	yes	—
175	C₂×C₈	C₂×D₂₁₂	1	8	yes	—
176	C₂×C₈	C₂²×Dic₅₃	1	8	yes	—
177	C₄²	C₄⋊Dic₅₃	1	2	yes	—
178	C₄²	C₄⋊Dic₅₃	1	4	yes	—
179	C₄²	C₄⋊Dic₅₃	1	2	yes	—
180	C₄²	C₂×D₂₁₂	1	4	yes	—
181	C₄²	C₂²×Dic₅₃	1	2	yes	—
182	C₄²	C₂²×Dic₅₃	1	2	yes	—
183	C₂³	—	2	4	no	—
184	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
185	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
186	C₂³	#13133217	1	4	yes	—
187	C₂³	C₂²×Dic₁₀₆	1	1	yes	characteristic
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	2	8	no	—
190	C₂×C₄	—	2	8	no	—
191	C₂×C₄	#13133153	1	8	yes	complemented
192	C₂×C₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
193	C₂×C₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
194	C₂×C₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
195	C₂×C₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
196	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
197	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
198	C₂×C₄	#13133214	1	2	yes	—
199	C₂×C₄	#13133214	1	2	yes	—
200	C₂×C₄	#13133217	1	2	yes	—
201	C₂×C₄	#13133217	1	2	yes	—
202	C₂×C₄	#13133217	1	4	yes	—
203	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
204	C₂×C₄	C₂²×D₂₁₂	1	2	yes	—
205	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
206	C₂×C₄	C₂²×Dic₁₀₆	1	2	yes	—
207	C₈	—	212	8	no	—
208	C₈	C₂×C₄⋊Dic₅₃	1	16	yes	—
209	C₂²	—	2	2	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	8	no	—
212	C₂²	#16810493	1	2	yes	—
213	C₂²	#16810493	1	2	yes	—
214	C₂²	C₂×#13133153	1	4	yes	—
215	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
216	C₂²	C₂×#13133217	1	2	yes	—
217	C₄	—	2	4	no	—
218	C₄	#16810493	1	2	yes	—
219	C₄	#16810493	1	2	yes	—
220	C₄	C₂×#13133153	1	8	yes	complemented
221	C₄	C₂²×C₄⋊Dic₅₃	1	2	yes	—
222	C₄	C₂×#13133214	1	1	yes	characteristic
223	C₄	C₂×#13133217	1	1	yes	characteristic
224	C₂	—	2	2	no	—
225	C₂	—	2	2	no	—
226	C₂	#20274887	1	4	yes	direct product factor
227	C₂	C₂×#16810493	1	1	yes	characteristic
228	C₂	C₂²×#13133153	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂²×C₄×C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₂×Q₈
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂²×C₄×C₈	D₅₃
hypocenter	C₅₃	C₂×#1848
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #20274887	C₂	direct (N × H)
76. C₂×C₄×C₅₃	#108	semidirect
106. C₄×C₅₃	C₂×#108	semidirect
116. C₂×C₅₃	#1848	semidirect
131. C₅₃	C₂×#1848	semidirect
191. C₂×C₄	#13133153	semidirect
220. C₄	C₂×#13133153	semidirect
226. C₂	#20274887	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×#13133153	2
C₂×#16810647	2
#20274887	2
C₂²×C₄×C₈×C₅₃	2
C₂×#1848	53

Subgroups — C2×#20274887 #27384548

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20274887 #27384548