Subgroups — C₂×#20274852 #27384317

Order 13568.

870 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810647	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×#88322	C₂	1	1	yes	maximal, characteristic
3	#20274852	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16810387	C₂	1	4	yes	maximal
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	#16810647	—	2	2	no	—
9	#16810647	C₂²	1	4	yes	—
10	C₂×C₄⋊#88322	—	2	2	no	—
11	C₂×C₄×#88322	C₄	1	2	yes	—
12	C₂×C₄×#88322	C₄	1	4	yes	—
13	C₂×C₄×#88322	C₄	1	2	yes	—
14	C₂×C₄×#88322	C₂²	1	4	yes	—
15	#16810387	C₂²	1	16	yes	—
16	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
17	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#423	—	2	2	no	—
19	C₅₃×#423	C₂²	1	4	yes	—
20	C₂×C₄⋊C₈×C₅₃	—	2	2	no	—
21	C₂×#11719917	—	2	2	no	—
22	C₂×#11719917	—	2	8	no	—
23	C₂×#11719917	C₂³	1	4	yes	—
24	C₂²×#88322	—	2	2	no	—
25	C₂²×#88322	—	2	4	no	—
26	C₂²×#88322	—	2	4	no	—
27	C₂²×#88322	C₂×C₄	1	4	yes	—
28	C₂²×#88322	C₂×C₄	1	4	yes	—
29	C₂²×#88322	C₂³	1	4	yes	—
30	C₄⋊#88322	—	2	8	no	—
31	C₄×#88322	C₂×C₄	1	8	yes	—
32	C₄×#88322	C₂×C₄	1	8	yes	—
33	C₂³×C₄×C₅₃	Q₈	1	2	yes	—
34	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₅₃	—	2	2	no	—
36	C₂×M₁₆×C₅₃	—	2	8	no	—
37	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
38	C₂²×C₈×C₅₃	—	2	2	no	—
39	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	D₄	1	2	yes	—
43	C₂×C₄²×C₅₃	D₄	1	2	yes	—
44	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
45	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
46	C₄⋊C₈×C₅₃	—	2	8	no	—
47	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	—	2	8	no	—
49	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
50	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
51	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
57	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	#41	1	4	yes	—
59	C₂²×C₄×C₅₃	#41	1	4	yes	—
60	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
61	M₁₆×C₅₃	—	2	8	no	—
62	M₁₆×C₅₃	—	4	4	no	—
63	C₂×C₈×C₅₃	—	2	4	no	—
64	C₂×C₈×C₅₃	—	4	4	no	—
65	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
66	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
67	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
68	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
69	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
70	C₄²×C₅₃	C₂×D₄	1	4	yes	—
71	#11719917	—	2	8	no	—
72	#11719917	—	4	4	no	—
73	C₂×#88322	—	2	8	no	—
74	C₂×#88322	—	2	8	no	—
75	C₂×#88322	—	2	4	no	—
76	C₂×#88322	—	2	4	no	—
77	C₂×#88322	—	2	4	no	—
78	C₂×#88322	—	4	4	no	—
79	C₂×#88322	C₂²×C₄	1	8	yes	—
80	C₂³×C₅₃	—	2	4	no	—
81	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂³×C₅₃	#125	1	4	yes	—
84	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
85	C₂×C₄×C₅₃	—	2	4	no	—
86	C₂×C₄×C₅₃	—	2	8	no	—
87	C₂×C₄×C₅₃	—	2	8	no	—
88	C₂×C₄×C₅₃	#108	1	8	yes	—
89	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
90	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
93	C₂×C₄×C₅₃	#117	1	4	yes	—
94	C₂×C₄×C₅₃	#117	1	4	yes	—
95	C₂×C₄×C₅₃	#124	1	2	yes	—
96	C₂×C₄×C₅₃	#124	1	2	yes	—
97	C₂×C₄×C₅₃	#124	1	2	yes	—
98	C₂×C₄×C₅₃	#124	1	2	yes	—
99	C₂×C₄×C₅₃	#125	1	4	yes	—
100	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
103	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
104	C₈×C₅₃	—	4	4	no	—
105	#88322	—	2	8	no	—
106	#88322	—	4	4	no	—
107	C₂×#1837	—	53	1	no	maximal, Sylow
108	C₂²×C₅₃	—	2	8	no	—
109	C₂²×C₅₃	—	2	2	no	—
110	C₂²×C₅₃	—	2	2	no	—
111	C₂²×C₅₃	#382	1	2	yes	—
112	C₂²×C₅₃	#382	1	2	yes	—
113	C₂²×C₅₃	C₂×#108	1	4	yes	—
114	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
115	C₂²×C₅₃	C₂×#125	1	2	yes	—
116	C₄×C₅₃	—	2	4	no	—
117	C₄×C₅₃	#382	1	2	yes	—
118	C₄×C₅₃	#382	1	2	yes	—
119	C₄×C₅₃	C₂×#108	1	8	yes	—
120	C₄×C₅₃	C₂×#117	1	2	yes	—
121	C₄×C₅₃	C₂×#124	1	1	yes	characteristic
122	C₄×C₅₃	C₂×#124	1	1	yes	characteristic
123	C₂×#423	—	53	1	no	—
124	C₂×#423	D₅₃	1	1	yes	characteristic
125	C₂²×C₄×C₈	—	53	1	no	—
126	#1837	—	53	8	no	—
127	C₂×#341	—	53	4	no	—
128	C₂×C₅₃	—	2	2	no	—
129	C₂×C₅₃	—	2	2	no	—
130	C₂×C₅₃	#1837	1	4	yes	complemented
131	C₂×C₅₃	C₂×#382	1	1	yes	characteristic
132	C₂×C₅₃	C₂²×#108	1	2	yes	—
133	C₂²×M₁₆	—	53	2	no	—
134	C₂²×M₁₆	C₂×D₅₃	1	2	yes	—
135	C₂³×C₈	—	53	2	no	—
136	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
137	#423	—	2	2	no	—
138	#423	—	53	4	no	—
139	#423	—	106	2	no	—
140	#423	C₂×D₅₃	1	4	yes	—
141	C₂×C₄⋊C₈	—	2	2	no	—
142	C₂×C₄⋊C₈	—	106	2	no	—
143	C₂×C₄×C₈	—	53	2	no	—
144	C₂×C₄×C₈	—	53	4	no	—
145	C₂×C₄×C₈	—	53	2	no	—
146	C₂×C₄×C₈	—	53	4	no	—
147	#341	—	53	16	no	—
148	C₅₃	C₂×#1837	1	1	yes	characteristic, complemented, Sylow
149	C₂³×C₄	Dic₁₀₆	1	2	yes	—
150	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
151	C₂×M₁₆	—	2	2	no	—
152	C₂×M₁₆	—	2	8	no	—
153	C₂×M₁₆	—	53	4	no	—
154	C₂×M₁₆	—	106	2	no	—
155	C₂×M₁₆	—	106	8	no	—
156	C₂×M₁₆	C₂²×D₅₃	1	4	yes	—
157	C₂²×C₈	—	2	2	no	—
158	C₂²×C₈	—	53	4	no	—
159	C₂²×C₈	—	53	4	no	—
160	C₂²×C₈	—	53	4	no	—
161	C₂²×C₈	—	106	4	no	—
162	C₂²×C₈	—	106	4	no	—
163	C₂²×C₈	—	106	2	no	—
164	C₂×C₄²	Dic₁₀₆	1	2	yes	—
165	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
166	C₂×C₄²	C₄×D₅₃	1	2	yes	—
167	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
168	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
169	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
170	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
171	C₄⋊C₈	—	2	8	no	—
172	C₄⋊C₈	—	106	8	no	—
173	C₄×C₈	—	53	8	no	—
174	C₄×C₈	—	53	8	no	—
175	C₂⁴	C₂×Dic₁₀₆	1	1	yes	characteristic
176	C₂²×C₄	—	2	8	no	—
177	C₂²×C₄	#11719919	1	2	yes	—
178	C₂²×C₄	#11719919	1	4	yes	—
179	C₂²×C₄	#11719919	1	2	yes	—
180	C₂²×C₄	C₂×Dic₁₀₆	1	4	yes	—
181	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
182	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
183	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
184	C₂²×C₄	#11719947	1	4	yes	—
185	C₂²×C₄	#11719949	1	4	yes	—
186	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
187	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
188	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
189	M₁₆	—	2	8	no	—
190	M₁₆	—	4	4	no	—
191	M₁₆	—	106	8	no	—
192	M₁₆	—	212	4	no	—
193	C₂×C₈	—	2	4	no	—
194	C₂×C₈	—	4	4	no	—
195	C₂×C₈	—	53	8	no	—
196	C₂×C₈	—	106	8	no	—
197	C₂×C₈	—	106	8	no	—
198	C₂×C₈	—	106	4	no	—
199	C₂×C₈	—	106	4	no	—
200	C₂×C₈	—	106	4	no	—
201	C₂×C₈	—	212	4	no	—
202	C₄²	#11719919	1	2	yes	—
203	C₄²	#11719919	1	4	yes	—
204	C₄²	#11719919	1	2	yes	—
205	C₄²	C₂×C₄×D₅₃	1	2	yes	—
206	C₄²	C₂×C₄×D₅₃	1	2	yes	—
207	C₄²	C₂×C₂²⋊D₅₃	1	4	yes	—
208	C₂³	—	2	4	no	—
209	C₂³	C₂×#11719919	1	1	yes	characteristic
210	C₂³	C₂×#11719919	1	1	yes	characteristic
211	C₂³	#13133208	1	4	yes	—
212	C₂³	C₂²×Dic₁₀₆	1	1	yes	characteristic
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	8	no	—
215	C₂×C₄	—	2	8	no	—
216	C₂×C₄	#13133152	1	8	yes	—
217	C₂×C₄	C₂×#11719919	1	4	yes	—
218	C₂×C₄	C₂×#11719919	1	4	yes	—
219	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
220	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
221	C₂×C₄	#13133175	1	4	yes	—
222	C₂×C₄	#13133175	1	4	yes	—
223	C₂×C₄	#13133204	1	2	yes	—
224	C₂×C₄	#13133204	1	2	yes	—
225	C₂×C₄	#13133206	1	2	yes	—
226	C₂×C₄	#13133206	1	2	yes	—
227	C₂×C₄	#13133208	1	4	yes	—
228	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
229	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
230	C₂×C₄	C₂×#11719947	1	2	yes	—
231	C₂×C₄	C₂×#11719949	1	2	yes	—
232	C₈	—	4	4	no	—
233	C₈	—	106	8	no	—
234	C₈	—	212	4	no	—
235	C₂²	—	2	8	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	2	2	no	—
238	C₂²	#16810488	1	2	yes	—
239	C₂²	#16810488	1	2	yes	—
240	C₂²	C₂×#13133152	1	4	yes	—
241	C₂²	C₂²×#11719919	1	1	yes	characteristic
242	C₂²	C₂×#13133208	1	2	yes	—
243	C₄	—	2	4	no	—
244	C₄	#16810488	1	2	yes	—
245	C₄	#16810488	1	2	yes	—
246	C₄	C₂×#13133152	1	8	yes	—
247	C₄	C₂×#13133175	1	2	yes	—
248	C₄	C₂×#13133204	1	1	yes	characteristic
249	C₄	C₂×#13133206	1	1	yes	characteristic
250	C₂	—	2	2	no	—
251	C₂	—	2	2	no	—
252	C₂	#20274852	1	4	yes	direct product factor
253	C₂	C₂×#16810488	1	1	yes	characteristic
254	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×#1837
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
3. #20274852	C₂	direct (N × H)
130. C₂×C₅₃	#1837	semidirect
148. C₅₃	C₂×#1837	semidirect
252. C₂	#20274852	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16810647	2
C₂²×C₄×#88322	2
#20274852	2
C₂×#16810387	2
C₂×C₅₃×#423	2
C₂×#1837	53

Subgroups — C2×#20274852 #27384317

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20274852 #27384317