Subgroups — C₂×#20276311 #27536726

Order 13568.

854 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133157	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810578	C₂	1	1	yes	maximal, characteristic
3	#20276311	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16810398	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₆×Dic₅₃	C₂	1	2	yes	maximal, complemented
6	C₂³×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11719917	C₄	1	1	yes	characteristic
8	C₂×#13133176	C₄	1	1	yes	characteristic
9	C₂×#13133157	C₄	1	8	yes	complemented
10	C₂×#13133157	C₂²	1	4	yes	—
11	C₂²×#11719908	C₂²	1	2	yes	—
12	#16810578	C₄	1	4	yes	—
13	#16810578	C₂²	1	4	yes	—
14	C₂×#13133127	C₄	1	2	yes	—
15	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
16	#16810398	C₂²	1	8	yes	complemented
17	C₁₆×Dic₅₃	C₂²	1	8	yes	complemented
18	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂²×C₁₆×C₅₃	C₄	1	8	yes	complemented
20	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
22	C₂²×C₁₆×C₅₃	C₂²	1	4	yes	—
23	C₂×#11719917	C₈	1	8	yes	—
24	C₂×#11719917	C₂×C₄	1	4	yes	—
25	C₂²×#88322	C₂×C₄	1	2	yes	—
26	#13133176	C₈	1	4	yes	—
27	#13133176	C₂×C₄	1	4	yes	—
28	C₂×C₄⋊Dic₅₃	C₈	1	2	yes	—
29	C₂×C₂²⋊Dic₅₃	C₈	1	2	yes	—
30	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
31	#13133157	C₂×C₄	1	16	yes	complemented
32	C₂×#11719908	C₂×C₄	1	8	yes	—
33	C₂×#11719908	C₂³	1	4	yes	—
34	#13133127	C₂×C₄	1	8	yes	—
35	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
36	C₈×Dic₅₃	C₂³	1	4	yes	—
37	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
39	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
40	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
44	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
45	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
46	C₂×C₁₆×C₅₃	C₂×C₄	1	16	yes	complemented
47	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
48	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
49	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
51	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
54	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
56	C₂²×Dic₅₃	C₂×C₈	1	4	yes	—
57	C₂×C₈×C₅₃	C₄²	1	4	yes	—
58	C₂×C₈×C₅₃	C₄²	1	8	yes	—
59	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
61	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
62	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
63	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
64	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
65	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
66	C₂²⋊Dic₅₃	C₂×C₈	1	8	yes	—
67	C₄⋊Dic₅₃	C₂×C₈	1	8	yes	—
68	C₄×Dic₅₃	C₂×C₈	1	4	yes	—
69	C₄×Dic₅₃	C₂²×C₄	1	4	yes	—
70	#11719917	C₂×C₈	1	16	yes	—
71	C₂×#88322	C₂×C₈	1	8	yes	—
72	C₂×#88322	C₂²×C₄	1	4	yes	—
73	C₁₆×C₅₃	C₂²×C₄	1	4	yes	—
74	C₁₆×C₅₃	C₂²×C₄	1	4	yes	—
75	#11719908	C₂²×C₄	1	8	yes	—
76	C₂³×C₅₃	—	2	4	no	—
77	C₂³×C₅₃	C₄×C₈	1	4	yes	—
78	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
79	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₄×C₈	1	8	yes	—
81	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
82	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
83	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
84	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
85	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
86	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
88	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
89	C₂×Dic₅₃	—	2	8	no	—
90	C₂×Dic₅₃	C₂²×C₈	1	8	yes	—
91	C₈×C₅₃	C₂×C₄²	1	2	yes	—
92	C₈×C₅₃	C₂×C₄²	1	4	yes	—
93	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
94	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
95	#88322	C₂²×C₈	1	8	yes	—
96	C₂×#2108	—	53	1	no	maximal, Sylow
97	C₂²×C₅₃	—	2	8	no	—
98	C₂²×C₅₃	—	2	4	no	—
99	C₂²×C₅₃	C₂×C₄×C₈	1	4	yes	—
100	C₂²×C₅₃	C₂×C₄×C₈	1	2	yes	—
101	C₂²×C₅₃	#504	1	4	yes	—
102	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
103	C₄×C₅₃	C₂×C₄×C₈	1	2	yes	—
104	C₄×C₅₃	C₂×C₄×C₈	1	4	yes	—
105	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
106	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
107	Dic₅₃	—	2	8	no	—
108	C₂²×M₃₂	—	53	1	no	—
109	C₂³×C₁₆	D₅₃	1	1	yes	characteristic
110	C₂×#405	—	53	1	no	—
111	#2108	—	53	8	no	—
112	C₂×#346	—	53	2	no	—
113	C₂×C₄×C₁₆	—	53	2	no	—
114	C₂×C₅₃	—	2	4	no	—
115	C₂×C₅₃	#2108	1	4	yes	complemented
116	C₂×C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic
117	C₂×C₅₃	C₂×#504	1	2	yes	—
118	C₂²×M₁₆	—	53	1	no	—
119	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
120	C₂×#117	—	53	1	no	—
121	C₂×M₃₂	—	53	8	no	—
122	C₂×M₃₂	—	53	4	no	—
123	C₂²×C₁₆	—	53	2	no	—
124	C₂²×C₁₆	Dic₅₃	1	8	yes	complemented
125	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
126	C₂²×C₁₆	C₂×D₅₃	1	4	yes	—
127	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
128	#405	—	53	4	no	—
129	#405	—	53	4	no	—
130	C₂×#97	—	53	2	no	—
131	C₂×C₄×C₈	—	53	2	no	—
132	#346	—	53	8	no	—
133	C₄×C₁₆	—	53	8	no	—
134	C₅₃	C₂×#2108	1	1	yes	characteristic, complemented, Sylow
135	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
136	C₂×M₁₆	—	53	4	no	—
137	C₂×M₁₆	—	53	8	no	—
138	C₂²×C₈	—	53	2	no	—
139	C₂²×C₈	C₄×D₅₃	1	4	yes	—
140	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
141	C₂²×C₈	C₄×D₅₃	1	2	yes	—
142	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
143	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
144	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
145	#117	—	53	4	no	—
146	#117	—	53	4	no	—
147	C₂×C₄⋊C₄	—	53	2	no	—
148	C₂×C₂²⋊C₄	—	53	2	no	—
149	C₂×C₄²	—	53	2	no	—
150	M₃₂	—	53	16	no	—
151	C₂×C₁₆	—	53	8	no	—
152	C₂×C₁₆	—	53	4	no	—
153	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	—
154	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	—
155	C₂×C₁₆	C₂×Dic₅₃	1	16	yes	complemented
156	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
157	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
158	#97	—	53	8	no	—
159	C₄×C₈	—	53	4	no	—
160	C₄×C₈	—	53	4	no	—
161	C₂⁴	C₈×D₅₃	1	1	yes	characteristic
162	C₂²×C₄	—	53	4	no	—
163	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
164	C₂²×C₄	C₈×D₅₃	1	2	yes	—
165	C₂²×C₄	C₈×D₅₃	1	4	yes	—
166	C₂²×C₄	C₄×Dic₅₃	1	4	yes	—
167	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
168	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
169	M₁₆	—	53	16	no	—
170	C₂×C₈	—	53	8	no	—
171	C₂×C₈	—	53	4	no	—
172	C₂×C₈	C₄×Dic₅₃	1	4	yes	—
173	C₂×C₈	C₄×Dic₅₃	1	8	yes	—
174	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
175	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
176	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
177	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
178	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
179	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
180	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
181	C₄⋊C₄	—	53	8	no	—
182	C₂²⋊C₄	—	53	8	no	—
183	C₄²	—	53	4	no	—
184	C₄²	—	53	4	no	—
185	C₁₆	—	53	8	no	—
186	C₁₆	C₂²×Dic₅₃	1	4	yes	—
187	C₁₆	C₂²×Dic₅₃	1	4	yes	—
188	C₂³	—	2	4	no	—
189	C₂³	C₈×Dic₅₃	1	4	yes	—
190	C₂³	C₂×C₈×D₅₃	1	2	yes	—
191	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
192	C₂×C₄	—	53	8	no	—
193	C₂×C₄	—	106	8	no	—
194	C₂×C₄	C₈×Dic₅₃	1	8	yes	—
195	C₂×C₄	C₈×Dic₅₃	1	4	yes	—
196	C₂×C₄	C₂×C₄×Dic₅₃	1	4	yes	—
197	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
198	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
199	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
200	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
201	C₂×C₄	C₂×C₈×D₅₃	1	4	yes	—
202	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
203	C₈	—	53	8	no	—
204	C₈	C₂×C₄×Dic₅₃	1	2	yes	—
205	C₈	C₂×C₄×Dic₅₃	1	4	yes	—
206	C₈	C₂²×C₄×D₅₃	1	1	yes	characteristic
207	C₈	C₂³×Dic₅₃	1	1	yes	characteristic
208	C₂²	—	2	4	no	—
209	C₂²	—	2	8	no	—
210	C₂²	C₂×C₈×Dic₅₃	1	4	yes	—
211	C₂²	C₂×C₈×Dic₅₃	1	2	yes	—
212	C₂²	#16811039	1	4	yes	—
213	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
214	C₄	—	106	8	no	—
215	C₄	C₂×C₈×Dic₅₃	1	2	yes	—
216	C₄	C₂×C₈×Dic₅₃	1	4	yes	—
217	C₄	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
218	C₄	C₂²×C₈×D₅₃	1	1	yes	characteristic
219	C₂	—	2	4	no	—
220	C₂	#20276311	1	4	yes	direct product factor
221	C₂	C₂²×C₈×Dic₅₃	1	1	yes	characteristic
222	C₂	C₂×#16811039	1	2	yes	—

	subgroup	G/H
center	C₂²×C₁₆	C₂×D₅₃
commutator	C₂×C₅₃	C₂²×C₄×C₈
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂³×C₁₆×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×C₈
prime radical	C₂⁴×C₅₃	C₂×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂³×C₁₆	D₅₃
hypocenter	C₅₃	C₂×#2108
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₁₆ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
3. #20276311	C₂	direct (N × H)
4. C₂×#16810398	C₂	semidirect
5. C₂×C₁₆×Dic₅₃	C₂	semidirect
9. C₂×#13133157	C₄	semidirect
16. #16810398	C₂²	semidirect
17. C₁₆×Dic₅₃	C₂²	semidirect
19. C₂²×C₁₆×C₅₃	C₄	semidirect
31. #13133157	C₂×C₄	semidirect
46. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
115. C₂×C₅₃	#2108	semidirect
124. C₂²×C₁₆	Dic₅₃	semidirect
134. C₅₃	C₂×#2108	semidirect
155. C₂×C₁₆	C₂×Dic₅₃	semidirect
220. C₂	#20276311	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×#13133157	2
C₂×#16810578	2
#20276311	2
C₂×#16810398	2
C₂×C₁₆×Dic₅₃	2
C₂³×C₁₆×C₅₃	2
C₂×#2108	53

Subgroups — C2×#20276311 #27536726

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276311 #27536726