Subgroups — C₂×#20272819 #27299004

Order 13568.

554 conjugacy classes of subgroups.

The subgroup lattice has 270 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	2	yes	maximal
2	#20272819	C₂	1	4	yes	maximal, direct product factor
3	C₂²×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11719917	—	2	2	no	—
5	C₂×#13133153	—	2	2	no	—
6	C₂×#13133153	—	2	2	no	—
7	C₂×#13133153	—	2	2	no	—
8	C₂×#13133153	C₄	1	4	yes	—
9	C₂×#13133153	C₄	1	4	yes	—
10	C₂×#13133153	C₂²	1	4	yes	—
11	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×M₃₂×C₅₃	—	2	2	no	—
13	C₂×M₃₂×C₅₃	—	2	2	no	—
14	C₂×M₃₂×C₅₃	C₄	1	2	yes	—
15	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
16	C₂²×C₁₆×C₅₃	C₄	1	2	yes	—
17	C₂×#11719917	—	2	4	no	—
18	C₂×#11719917	—	2	4	no	—
19	C₂×#11719917	—	2	4	no	—
20	C₂×#11719917	—	4	2	no	—
21	C₂×#11719917	—	4	2	no	—
22	C₂×#11719917	—	4	2	no	—
23	C₂²×#88322	—	4	2	no	—
24	#13133153	—	2	4	no	—
25	#13133153	—	2	4	no	—
26	#13133153	—	2	4	no	—
27	#13133153	C₂×C₄	1	8	yes	—
28	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	D₄	1	2	yes	—
34	C₂²×C₈×C₅₃	D₄	1	2	yes	—
35	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
39	M₃₂×C₅₃	—	2	4	no	—
40	M₃₂×C₅₃	—	2	4	no	—
41	C₂×C₁₆×C₅₃	—	2	2	no	—
42	C₂×C₁₆×C₅₃	—	2	2	no	—
43	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
44	C₂⁴×C₅₃	Q₁₆	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
52	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
53	C₂²×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
56	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
57	C₂×C₈×C₅₃	C₄²	1	2	yes	—
58	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
59	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
60	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
66	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
67	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
68	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
69	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
73	#11719917	—	2	8	no	—
74	#11719917	—	4	4	no	—
75	#11719917	—	4	4	no	—
76	#11719917	—	4	4	no	—
77	C₂×#88322	—	4	4	no	—
78	C₂×#88322	—	4	4	no	—
79	C₂×#88322	—	4	4	no	—
80	C₁₆×C₅₃	—	2	4	no	—
81	C₂³×C₅₃	—	4	2	no	—
82	C₂³×C₅₃	#103	1	2	yes	—
83	C₂³×C₅₃	#103	1	2	yes	—
84	C₂³×C₅₃	#107	1	1	yes	characteristic
85	C₂³×C₅₃	#107	1	1	yes	characteristic
86	C₂³×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
87	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
88	C₂×C₄×C₅₃	#95	1	2	yes	—
89	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
90	C₂×C₄×C₅₃	#102	1	2	yes	—
91	C₂×C₄×C₅₃	#102	1	2	yes	—
92	C₂×C₄×C₅₃	#102	1	2	yes	—
93	C₂×C₄×C₅₃	#102	1	2	yes	—
94	C₂×C₄×C₅₃	#103	1	2	yes	—
95	C₂×C₄×C₅₃	#103	1	2	yes	—
96	C₂×C₄×C₅₃	#106	1	2	yes	—
97	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
98	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
99	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
100	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
102	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
105	C₈×C₅₃	#95	1	1	yes	characteristic
106	C₈×C₅₃	#95	1	1	yes	characteristic
107	C₈×C₅₃	#95	1	2	yes	—
108	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
109	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
111	#88322	—	4	8	no	—
112	C₂×#1385	—	53	1	no	maximal, Sylow
113	C₂²×C₅₃	—	2	4	no	—
114	C₂²×C₅₃	—	2	2	no	—
115	C₂²×C₅₃	—	4	2	no	—
116	C₂²×C₅₃	—	4	1	no	—
117	C₂²×C₅₃	—	4	1	no	—
118	C₂²×C₅₃	#340	1	2	yes	—
119	C₂²×C₅₃	#340	1	1	yes	characteristic
120	C₂²×C₅₃	#340	1	1	yes	characteristic
121	C₂²×C₅₃	C₂×#103	1	2	yes	—
122	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
123	C₄×C₅₃	#340	1	2	yes	—
124	C₄×C₅₃	#340	1	1	yes	characteristic
125	C₄×C₅₃	#340	1	1	yes	characteristic
126	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
127	C₄×C₅₃	C₂×#102	1	2	yes	—
128	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
129	C₂²×M₃₂	D₅₃	1	1	yes	characteristic
130	C₂²×#108	—	53	2	no	—
131	#1385	—	53	4	no	—
132	C₂×C₅₃	—	2	2	no	—
133	C₂×C₅₃	—	2	1	no	—
134	C₂×C₅₃	—	2	2	no	—
135	C₂×C₅₃	—	2	1	no	—
136	C₂×C₅₃	#1385	1	2	yes	complemented
137	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
138	C₂²×M₁₆	—	106	2	no	—
139	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
140	C₂×M₃₂	—	2	2	no	—
141	C₂×M₃₂	—	2	2	no	—
142	C₂×M₃₂	Dic₅₃	1	2	yes	—
143	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
144	C₂²×C₁₆	Dic₅₃	1	2	yes	—
145	C₂×#108	—	53	4	no	—
146	C₂×#108	—	53	4	no	—
147	C₂×#108	—	53	4	no	—
148	C₂×#108	—	106	2	no	—
149	C₂×#108	—	106	2	no	—
150	C₂×#108	—	106	2	no	—
151	C₅₃	C₂×#1385	1	1	yes	characteristic, complemented, Sylow
152	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
153	C₂×M₁₆	—	106	4	no	—
154	C₂×M₁₆	—	106	4	no	—
155	C₂×M₁₆	—	106	4	no	—
156	C₂×M₁₆	—	212	2	no	—
157	C₂×M₁₆	—	212	2	no	—
158	C₂×M₁₆	—	212	2	no	—
159	C₂²×C₈	—	212	2	no	—
160	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
161	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
162	C₂²×C₈	C₄×D₅₃	1	2	yes	—
163	C₂²×C₈	C₄×D₅₃	1	2	yes	—
164	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
165	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
166	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
167	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
168	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
169	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
170	M₃₂	—	2	4	no	—
171	M₃₂	—	2	4	no	—
172	C₂×C₁₆	—	2	2	no	—
173	C₂×C₁₆	—	2	2	no	—
174	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	—
175	#108	—	53	8	no	—
176	#108	—	106	4	no	—
177	#108	—	106	4	no	—
178	#108	—	106	4	no	—
179	C₂⁴	Dic₂₁₂	1	1	yes	characteristic
180	C₂²×C₄	#11719913	1	2	yes	—
181	C₂²×C₄	#11719913	1	2	yes	—
182	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
183	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
184	C₂²×C₄	Dic₂₁₂	1	1	yes	characteristic
185	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
186	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
187	C₂²×C₄	#11719921	1	2	yes	—
188	C₂²×C₄	#11719921	1	2	yes	—
189	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
190	M₁₆	—	106	8	no	—
191	M₁₆	—	212	4	no	—
192	M₁₆	—	212	4	no	—
193	M₁₆	—	212	4	no	—
194	C₂×C₈	—	212	4	no	—
195	C₂×C₈	—	212	4	no	—
196	C₂×C₈	—	212	4	no	—
197	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
198	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
199	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
200	C₂×C₈	#11719919	1	2	yes	—
201	C₂×C₈	#11719919	1	2	yes	—
202	C₂×C₈	#11719919	1	2	yes	—
203	C₂×C₈	#11719919	1	2	yes	—
204	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
205	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
206	C₂×C₈	#11719921	1	2	yes	—
207	C₂×C₈	#11719921	1	2	yes	—
208	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
209	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
210	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
211	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
212	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
213	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
214	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
215	C₁₆	—	2	4	no	—
216	C₂³	—	4	2	no	—
217	C₂³	#13133142	1	2	yes	—
218	C₂³	#13133142	1	2	yes	—
219	C₂³	#13133151	1	1	yes	characteristic
220	C₂³	#13133151	1	1	yes	characteristic
221	C₂³	C₂×Dic₂₁₂	1	1	yes	characteristic
222	C₂×C₄	#13133124	1	2	yes	—
223	C₂×C₄	#13133124	1	1	yes	characteristic
224	C₂×C₄	#13133124	1	1	yes	characteristic
225	C₂×C₄	#13133139	1	2	yes	—
226	C₂×C₄	#13133139	1	2	yes	—
227	C₂×C₄	#13133139	1	2	yes	—
228	C₂×C₄	#13133139	1	2	yes	—
229	C₂×C₄	#13133142	1	2	yes	—
230	C₂×C₄	#13133142	1	2	yes	—
231	C₂×C₄	#13133149	1	1	yes	characteristic
232	C₂×C₄	#13133149	1	2	yes	—
233	C₂×C₄	#13133149	1	1	yes	characteristic
234	C₂×C₄	#13133151	1	1	yes	characteristic
235	C₂×C₄	#13133151	1	1	yes	characteristic
236	C₂×C₄	C₂×#11719921	1	2	yes	—
237	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
238	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
239	C₂×C₄	C₂×#11719913	1	2	yes	—
240	C₈	—	212	8	no	—
241	C₈	#13133124	1	1	yes	characteristic
242	C₈	#13133124	1	1	yes	characteristic
243	C₈	#13133124	1	2	yes	—
244	C₈	C₂×C₄×Dic₅₃	1	1	yes	characteristic
245	C₈	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
246	C₈	C₂×#11719919	1	2	yes	—
247	C₂²	—	2	4	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	4	2	no	—
250	C₂²	—	4	1	no	—
251	C₂²	—	4	1	no	—
252	C₂²	#16810386	1	2	yes	—
253	C₂²	#16810386	1	1	yes	characteristic
254	C₂²	#16810386	1	1	yes	characteristic
255	C₂²	C₂×#13133142	1	2	yes	—
256	C₂²	C₂×#13133151	1	1	yes	characteristic
257	C₄	#16810386	1	2	yes	—
258	C₄	#16810386	1	1	yes	characteristic
259	C₄	#16810386	1	1	yes	characteristic
260	C₄	C₂×#13133124	1	1	yes	characteristic
261	C₄	C₂×#13133139	1	2	yes	—
262	C₄	C₂×#13133149	1	1	yes	characteristic
263	C₂	—	2	2	no	—
264	C₂	—	2	2	no	—
265	C₂	—	2	1	no	—
266	C₂	—	2	1	no	—
267	C₂	#20272819	1	2	yes	direct product factor
268	C₂	C₂×#16810386	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133149
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂²×M₃₂×C₅₃	C₂
socle	C₂²×C₅₃	#340
prime radical	C₂⁴×C₅₃	Q₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂²×M₃₂	D₅₃
hypocenter	C₅₃	C₂×#1385
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
2. #20272819	C₂	direct (N × H)
136. C₂×C₅₃	#1385	semidirect
151. C₅₃	C₂×#1385	semidirect
267. C₂	#20272819	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
#20272819	2
C₂²×M₃₂×C₅₃	2
C₂×#1385	53

Subgroups — C2×#20272819 #27299004

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20272819 #27299004