Subgroups — C₂×#20274886 #27384547

Order 13568.

1046 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×#13133152	C₂	1	4	yes	maximal
2	C₂×#16810647	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×#88322	C₂	1	1	yes	maximal, characteristic
4	#20274886	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅₃×#423	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719917	C₂²	1	2	yes	—
7	C₂³×#88322	C₂²	1	2	yes	—
8	C₂×#13133152	—	2	16	no	—
9	C₂×#13133152	C₂²	1	16	yes	—
10	#16810647	—	2	2	no	—
11	#16810647	C₂²	1	4	yes	—
12	C₂×C₄⋊#88322	—	2	2	no	—
13	C₂×C₄×#88322	C₄	1	2	yes	—
14	C₂×C₄×#88322	C₄	1	4	yes	—
15	C₂×C₄×#88322	C₄	1	2	yes	—
16	C₂×C₄×#88322	C₂²	1	4	yes	—
17	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
18	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#423	—	2	2	no	—
20	C₅₃×#423	C₂²	1	4	yes	—
21	C₂×C₄⋊C₈×C₅₃	—	2	2	no	—
22	C₂×#11719917	—	2	2	no	—
23	C₂×#11719917	—	2	8	no	—
24	C₂×#11719917	C₂³	1	4	yes	—
25	C₂²×#88322	—	2	2	no	—
26	C₂²×#88322	C₂×C₄	1	4	yes	—
27	C₂²×#88322	C₂×C₄	1	4	yes	—
28	C₂²×#88322	D₄	1	8	yes	—
29	C₂²×#88322	Q₈	1	8	yes	—
30	C₂²×#88322	C₂³	1	4	yes	—
31	#13133152	—	2	32	no	—
32	C₄⋊#88322	—	2	8	no	—
33	C₄×#88322	C₂×C₄	1	8	yes	—
34	C₄×#88322	C₂×C₄	1	8	yes	—
35	C₂³×C₄×C₅₃	Q₈	1	2	yes	—
36	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
37	C₂×M₁₆×C₅₃	—	2	2	no	—
38	C₂×M₁₆×C₅₃	—	2	8	no	—
39	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
40	C₂²×C₈×C₅₃	—	2	2	no	—
41	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₅₃	D₄	1	2	yes	—
45	C₂×C₄²×C₅₃	D₄	1	2	yes	—
46	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
47	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
48	C₄⋊C₈×C₅₃	—	2	8	no	—
49	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	—	2	8	no	—
51	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
53	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
57	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
60	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
62	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
63	M₁₆×C₅₃	—	2	8	no	—
64	M₁₆×C₅₃	—	4	4	no	—
65	C₂×C₈×C₅₃	—	2	4	no	—
66	C₂×C₈×C₅₃	—	4	4	no	—
67	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
68	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
69	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
70	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
71	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
72	C₄²×C₅₃	C₂×D₄	1	4	yes	—
73	#11719917	—	2	8	no	—
74	#11719917	—	4	4	no	—
75	C₂×#88322	—	2	4	no	—
76	C₂×#88322	—	4	4	no	—
77	C₂×#88322	C₄⋊C₄	1	16	yes	—
78	C₂×#88322	C₄⋊C₄	1	16	yes	—
79	C₂×#88322	C₂²×C₄	1	8	yes	—
80	C₂×#88322	C₂×D₄	1	8	yes	—
81	C₂×#88322	C₂×Q₈	1	8	yes	—
82	C₂³×C₅₃	—	2	4	no	—
83	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂³×C₅₃	C₄⋊Q₈	1	4	yes	—
86	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
87	C₂×C₄×C₅₃	—	2	4	no	—
88	C₂×C₄×C₅₃	—	2	8	no	—
89	C₂×C₄×C₅₃	—	2	8	no	—
90	C₂×C₄×C₅₃	#108	1	8	yes	complemented
91	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
92	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
93	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
94	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
97	C₂×C₄×C₅₃	#127	1	2	yes	—
98	C₂×C₄×C₅₃	#127	1	2	yes	—
99	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
100	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
101	C₂×C₄×C₅₃	C₄⋊Q₈	1	4	yes	—
102	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
104	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
106	C₈×C₅₃	—	4	4	no	—
107	#88322	—	4	4	no	—
108	#88322	C₂×C₄⋊C₄	1	16	yes	—
109	C₂×#1848	—	53	1	no	maximal, Sylow
110	C₂²×C₅₃	—	2	8	no	—
111	C₂²×C₅₃	—	2	2	no	—
112	C₂²×C₅₃	—	2	2	no	—
113	C₂²×C₅₃	#384	1	2	yes	—
114	C₂²×C₅₃	#384	1	2	yes	—
115	C₂²×C₅₃	C₂×#108	1	4	yes	—
116	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
117	C₂²×C₅₃	C₂×C₄⋊Q₈	1	2	yes	—
118	C₄×C₅₃	—	2	4	no	—
119	C₄×C₅₃	#384	1	2	yes	—
120	C₄×C₅₃	#384	1	2	yes	—
121	C₄×C₅₃	C₂×#108	1	8	yes	complemented
122	C₄×C₅₃	C₂²×C₄⋊C₄	1	2	yes	—
123	C₄×C₅₃	C₂×#127	1	1	yes	characteristic
124	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
125	C₂²×#108	—	53	4	no	—
126	C₂×#423	—	53	1	no	—
127	C₂×#423	D₅₃	1	1	yes	characteristic
128	C₂²×C₄×C₈	—	53	1	no	—
129	#1848	—	53	8	no	—
130	C₂×C₅₃	—	2	2	no	—
131	C₂×C₅₃	—	2	2	no	—
132	C₂×C₅₃	#1848	1	4	yes	complemented
133	C₂×C₅₃	C₂×#384	1	1	yes	characteristic
134	C₂×C₅₃	C₂²×#108	1	2	yes	—
135	C₂²×M₁₆	—	53	2	no	—
136	C₂²×M₁₆	C₂×D₅₃	1	2	yes	—
137	C₂³×C₈	—	53	2	no	—
138	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
139	C₂×#108	—	53	16	no	—
140	C₂×#108	—	106	16	no	—
141	#423	—	2	2	no	—
142	#423	—	53	4	no	—
143	#423	—	106	2	no	—
144	#423	C₂×D₅₃	1	4	yes	—
145	C₂×C₄⋊C₈	—	2	2	no	—
146	C₂×C₄⋊C₈	—	106	2	no	—
147	C₂×C₄×C₈	—	53	2	no	—
148	C₂×C₄×C₈	—	53	4	no	—
149	C₂×C₄×C₈	—	53	2	no	—
150	C₂×C₄×C₈	—	53	4	no	—
151	C₅₃	C₂×#1848	1	1	yes	characteristic, complemented, Sylow
152	C₂³×C₄	Dic₁₀₆	1	2	yes	—
153	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
154	C₂×M₁₆	—	2	2	no	—
155	C₂×M₁₆	—	2	8	no	—
156	C₂×M₁₆	—	53	4	no	—
157	C₂×M₁₆	—	106	2	no	—
158	C₂×M₁₆	—	106	8	no	—
159	C₂×M₁₆	C₂²×D₅₃	1	4	yes	—
160	C₂²×C₈	—	2	2	no	—
161	C₂²×C₈	—	53	4	no	—
162	C₂²×C₈	—	53	4	no	—
163	C₂²×C₈	—	53	8	no	—
164	C₂²×C₈	—	53	8	no	—
165	C₂²×C₈	—	53	4	no	—
166	C₂²×C₈	—	106	2	no	—
167	C₂×C₄²	Dic₁₀₆	1	2	yes	—
168	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
169	C₂×C₄²	C₄×D₅₃	1	2	yes	—
170	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
171	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
172	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
173	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
174	#108	—	106	32	no	—
175	C₄⋊C₈	—	2	8	no	—
176	C₄⋊C₈	—	106	8	no	—
177	C₄×C₈	—	53	8	no	—
178	C₄×C₈	—	53	8	no	—
179	C₂⁴	C₂×Dic₁₀₆	1	1	yes	characteristic
180	C₂²×C₄	—	2	8	no	—
181	C₂²×C₄	#11719919	1	2	yes	—
182	C₂²×C₄	#11719919	1	4	yes	—
183	C₂²×C₄	#11719919	1	2	yes	—
184	C₂²×C₄	C₂×Dic₁₀₆	1	4	yes	—
185	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
186	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
187	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
188	C₂²×C₄	D₄×D₅₃	1	4	yes	—
189	C₂²×C₄	Q₈×D₅₃	1	4	yes	—
190	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
192	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
193	M₁₆	—	2	8	no	—
194	M₁₆	—	4	4	no	—
195	M₁₆	—	106	8	no	—
196	M₁₆	—	212	4	no	—
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	—	4	4	no	—
199	C₂×C₈	—	53	16	no	—
200	C₂×C₈	—	53	16	no	—
201	C₂×C₈	—	53	8	no	—
202	C₂×C₈	—	53	8	no	—
203	C₂×C₈	—	53	8	no	—
204	C₂×C₈	—	106	4	no	—
205	C₂×C₈	—	212	4	no	—
206	C₄²	#11719919	1	2	yes	—
207	C₄²	#11719919	1	4	yes	—
208	C₄²	#11719919	1	2	yes	—
209	C₄²	C₂×C₄×D₅₃	1	2	yes	—
210	C₄²	C₂×C₄×D₅₃	1	2	yes	—
211	C₄²	C₂×C₂²⋊D₅₃	1	4	yes	—
212	C₂³	—	2	4	no	—
213	C₂³	C₂×#11719919	1	1	yes	characteristic
214	C₂³	C₂×#11719919	1	1	yes	characteristic
215	C₂³	#13133216	1	4	yes	—
216	C₂³	C₂²×Dic₁₀₆	1	1	yes	characteristic
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	2	8	no	—
219	C₂×C₄	—	2	8	no	—
220	C₂×C₄	#13133152	1	8	yes	complemented
221	C₂×C₄	C₄⋊C₄×D₅₃	1	4	yes	—
222	C₂×C₄	C₄⋊C₄×D₅₃	1	4	yes	—
223	C₂×C₄	C₂×#11719919	1	4	yes	—
224	C₂×C₄	C₂×#11719919	1	4	yes	—
225	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
226	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
227	C₂×C₄	#13133213	1	2	yes	—
228	C₂×C₄	#13133213	1	2	yes	—
229	C₂×C₄	#13133215	1	2	yes	—
230	C₂×C₄	#13133215	1	2	yes	—
231	C₂×C₄	#13133216	1	4	yes	—
232	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
233	C₂×C₄	C₂×D₄×D₅₃	1	2	yes	—
234	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
235	C₂×C₄	C₂×Q₈×D₅₃	1	2	yes	—
236	C₈	—	4	4	no	—
237	C₈	—	53	16	no	—
238	C₈	—	212	4	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	8	no	—
242	C₂²	#16810492	1	2	yes	—
243	C₂²	#16810492	1	2	yes	—
244	C₂²	C₂×#13133152	1	4	yes	—
245	C₂²	C₂²×#11719919	1	1	yes	characteristic
246	C₂²	C₂×#13133216	1	2	yes	—
247	C₄	—	2	4	no	—
248	C₄	#16810492	1	2	yes	—
249	C₄	#16810492	1	2	yes	—
250	C₄	C₂×#13133152	1	8	yes	complemented
251	C₄	C₂×C₄⋊C₄×D₅₃	1	2	yes	—
252	C₄	C₂×#13133213	1	1	yes	characteristic
253	C₄	C₂×#13133215	1	1	yes	characteristic
254	C₂	—	2	2	no	—
255	C₂	—	2	2	no	—
256	C₂	#20274886	1	4	yes	direct product factor
257	C₂	C₂×#16810492	1	1	yes	characteristic
258	C₂	C₂²×#13133152	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#423	C₂
socle	C₂³×C₅₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₂×Q₈
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#423	D₅₃
hypocenter	C₅₃	C₂×#1848
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
4. #20274886	C₂	direct (N × H)
90. C₂×C₄×C₅₃	#108	semidirect
121. C₄×C₅₃	C₂×#108	semidirect
132. C₂×C₅₃	#1848	semidirect
151. C₅₃	C₂×#1848	semidirect
220. C₂×C₄	#13133152	semidirect
250. C₄	C₂×#13133152	semidirect
256. C₂	#20274886	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×#13133152	2
C₂×#16810647	2
C₂²×C₄×#88322	2
#20274886	2
C₂×C₅₃×#423	2
C₂×#1848	53

Subgroups — C2×#20274886 #27384547

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20274886 #27384547