Subgroups — C₂×#20273012 #27349712

Order 13568.

866 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810676	C₂	1	2	yes	maximal
2	#20273012	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16810345	C₂	1	2	yes	maximal, complemented
4	C₂×#16810344	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
7	#16810676	C₄	1	8	yes	—
8	#16810676	C₂²	1	8	yes	—
9	C₂×C₄⋊#88322	C₄	1	4	yes	—
10	C₂×C₂²⋊#88322	C₄	1	4	yes	—
11	C₂×C₄×#88322	C₂²	1	2	yes	—
12	C₂×C₄×#88322	C₂²	1	2	yes	—
13	#16810345	C₂²	1	8	yes	complemented
14	#16810344	C₂²	1	4	yes	complemented
15	C₈×#88322	C₂²	1	4	yes	complemented
16	C₂²×M₁₆×C₅₃	C₄	1	2	yes	—
17	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
18	C₄×M₁₆×C₅₃	C₄	1	4	yes	—
19	C₄×M₁₆×C₅₃	C₂²	1	4	yes	—
20	C₂×#97×C₅₃	C₄	1	2	yes	—
21	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂²×#88322	C₂×C₄	1	4	yes	—
24	C₂²×#88322	C₂×C₄	1	4	yes	—
25	C₄⋊#88322	C₂×C₄	1	16	yes	—
26	C₂²⋊#88322	C₂×C₄	1	16	yes	—
27	C₄×#88322	C₂×C₄	1	4	yes	—
28	C₄×#88322	C₂×C₄	1	4	yes	—
29	C₄×#88322	C₂³	1	4	yes	—
30	C₄×#88322	C₂³	1	4	yes	—
31	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
32	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×M₁₆×C₅₃	—	2	8	no	—
34	C₂×M₁₆×C₅₃	C₂×C₄	1	8	yes	—
35	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
39	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
42	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
43	#97×C₅₃	C₂×C₄	1	8	yes	—
44	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₅₃	C₂³	1	2	yes	—
47	C₄×C₈×C₅₃	C₂³	1	2	yes	—
48	C₂⁴×C₅₃	C₄²	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	—	2	4	no	—
50	C₂²×C₄×C₅₃	—	2	8	no	—
51	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
53	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
54	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
55	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
56	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
59	M₁₆×C₅₃	—	2	16	no	—
60	C₂×C₈×C₅₃	—	2	4	no	—
61	C₂×C₈×C₅₃	M₁₆	1	8	yes	—
62	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
63	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
64	C₄²×C₅₃	C₄²	1	2	yes	—
65	C₄²×C₅₃	C₄²	1	4	yes	—
66	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
67	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
68	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
70	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
71	C₂×#88322	—	2	8	no	—
72	C₂×#88322	—	2	8	no	—
73	C₂×#88322	C₂²×C₄	1	8	yes	—
74	C₂×#88322	C₂²×C₄	1	8	yes	—
75	C₂³×C₅₃	—	2	2	no	—
76	C₂³×C₅₃	—	2	4	no	—
77	C₂³×C₅₃	C₂×C₄²	1	2	yes	—
78	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
79	C₂×C₄×C₅₃	—	2	2	no	—
80	C₂×C₄×C₅₃	—	2	4	no	—
81	C₂×C₄×C₅₃	—	2	4	no	—
82	C₂×C₄×C₅₃	—	2	8	no	—
83	C₂×C₄×C₅₃	—	2	4	no	—
84	C₂×C₄×C₅₃	—	2	8	no	—
85	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
86	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
87	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
89	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
90	C₂×C₄×C₅₃	#131	1	8	yes	—
91	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
92	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
93	C₈×C₅₃	—	2	4	no	—
94	C₈×C₅₃	C₂×M₁₆	1	8	yes	—
95	#88322	—	2	8	no	—
96	#88322	—	2	8	no	—
97	C₂×#1453	—	53	1	no	maximal, Sylow
98	C₂²×C₅₃	—	2	2	no	—
99	C₂²×C₅₃	—	2	4	no	—
100	C₂²×C₅₃	—	4	4	no	—
101	C₂²×C₅₃	C₄×M₁₆	1	2	yes	—
102	C₂²×C₅₃	#405	1	4	yes	—
103	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
104	C₄×C₅₃	—	2	2	no	—
105	C₄×C₅₃	—	2	4	no	—
106	C₄×C₅₃	C₄×M₁₆	1	2	yes	—
107	C₄×C₅₃	#405	1	4	yes	—
108	C₄×C₅₃	C₂²×M₁₆	1	2	yes	—
109	C₄×C₅₃	C₂×#131	1	4	yes	—
110	C₂×#431	—	53	2	no	—
111	C₂×C₄×M₁₆	D₅₃	1	1	yes	characteristic
112	#1453	—	53	8	no	—
113	C₂×#322	—	53	1	no	—
114	C₂×#322	—	53	2	no	—
115	C₂×C₈²	—	53	1	no	—
116	C₂×C₅₃	—	4	2	no	—
117	C₂×C₅₃	#1453	1	4	yes	complemented
118	C₂×C₅₃	C₂×C₄×M₁₆	1	1	yes	characteristic
119	C₂×C₅₃	C₂×#405	1	2	yes	—
120	C₂²×M₁₆	Dic₅₃	1	2	yes	—
121	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
122	#431	—	53	8	no	—
123	#431	—	53	8	no	—
124	C₂×C₄⋊C₈	—	53	4	no	—
125	C₂×C₂²⋊C₈	—	53	4	no	—
126	C₄×M₁₆	Dic₅₃	1	4	yes	—
127	C₄×M₁₆	C₂×D₅₃	1	4	yes	—
128	C₂×#97	Dic₅₃	1	2	yes	—
129	C₂×C₄×C₈	—	53	2	no	—
130	C₂×C₄×C₈	—	53	2	no	—
131	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
132	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
133	#322	—	53	4	no	—
134	#322	—	53	8	no	—
135	C₈²	—	53	4	no	—
136	C₅₃	C₂×#1453	1	1	yes	characteristic, complemented, Sylow
137	C₂³×C₄	C₄×D₅₃	1	2	yes	—
138	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
139	C₂×M₁₆	—	2	8	no	—
140	C₂×M₁₆	C₂×Dic₅₃	1	8	yes	—
141	C₂²×C₈	—	53	4	no	—
142	C₂²×C₈	—	53	4	no	—
143	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
144	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
145	C₂×C₄²	C₄×D₅₃	1	2	yes	—
146	C₂×C₄²	C₄×D₅₃	1	4	yes	—
147	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
148	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
149	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
150	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
151	C₄⋊C₈	—	53	16	no	—
152	C₂²⋊C₈	—	53	16	no	—
153	#97	C₂×Dic₅₃	1	8	yes	—
154	C₄×C₈	—	53	4	no	—
155	C₄×C₈	—	53	4	no	—
156	C₄×C₈	—	53	4	no	—
157	C₄×C₈	—	53	4	no	—
158	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
159	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
160	C₄×C₈	C₂²×D₅₃	1	2	yes	—
161	C₄×C₈	C₂²×D₅₃	1	2	yes	—
162	C₂⁴	C₄×Dic₅₃	1	1	yes	characteristic
163	C₂²×C₄	—	2	4	no	—
164	C₂²×C₄	—	2	8	no	—
165	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
166	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
167	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
168	C₂²×C₄	C₄×Dic₅₃	1	4	yes	—
169	C₂²×C₄	C₂×C₄×D₅₃	1	4	yes	—
170	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
171	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
172	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
173	M₁₆	—	2	16	no	—
174	C₂×C₈	—	2	4	no	—
175	C₂×C₈	—	53	8	no	—
176	C₂×C₈	—	53	8	no	—
177	C₂×C₈	—	106	8	no	—
178	C₂×C₈	—	106	8	no	—
179	C₂×C₈	#11719917	1	8	yes	—
180	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
181	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
182	C₄²	C₄×Dic₅₃	1	4	yes	—
183	C₄²	C₄×Dic₅₃	1	2	yes	—
184	C₄²	C₂×C₄×D₅₃	1	2	yes	—
185	C₄²	C₂×C₄×D₅₃	1	4	yes	—
186	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
187	C₄²	C₂²×Dic₅₃	1	2	yes	—
188	C₄²	C₂³×D₅₃	1	1	yes	characteristic
189	C₂³	—	2	2	no	—
190	C₂³	—	2	4	no	—
191	C₂³	C₂×C₄×Dic₅₃	1	2	yes	—
192	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	2	8	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	8	no	—
199	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
200	C₂×C₄	C₂×C₄×Dic₅₃	1	4	yes	—
201	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
202	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
203	C₂×C₄	C₂×#11719917	1	4	yes	—
204	C₂×C₄	#13133225	1	8	yes	—
205	C₂×C₄	C₂²×C₄×D₅₃	1	2	yes	—
206	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
207	C₈	—	2	4	no	—
208	C₈	—	106	8	no	—
209	C₈	—	106	8	no	—
210	C₈	C₂×#11719917	1	8	yes	—
211	C₂²	—	2	2	no	—
212	C₂²	—	2	4	no	—
213	C₂²	—	4	4	no	—
214	C₂²	C₄×#11719917	1	2	yes	—
215	C₂²	#16810581	1	4	yes	—
216	C₂²	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
217	C₄	—	2	2	no	—
218	C₄	—	2	4	no	—
219	C₄	C₄×#11719917	1	2	yes	—
220	C₄	#16810581	1	4	yes	—
221	C₄	C₂²×#11719917	1	2	yes	—
222	C₄	C₂×#13133225	1	4	yes	—
223	C₂	—	4	2	no	—
224	C₂	#20273012	1	4	yes	direct product factor
225	C₂	C₂×C₄×#11719917	1	1	yes	characteristic
226	C₂	C₂×#16810581	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂²×C₄²
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₄×M₁₆×C₅₃	C₂
socle	C₂³×C₅₃	C₂×C₄²
prime radical	C₂⁴×C₅₃	C₄²
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₅₃
hypocenter	C₅₃	C₂×#1453
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #20273012	C₂	direct (N × H)
3. C₂×#16810345	C₂	semidirect
4. C₂×#16810344	C₂	semidirect
5. C₂×C₈×#88322	C₂	semidirect
13. #16810345	C₂²	semidirect
14. #16810344	C₂²	semidirect
15. C₈×#88322	C₂²	semidirect
117. C₂×C₅₃	#1453	semidirect
136. C₅₃	C₂×#1453	semidirect
224. C₂	#20273012	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×#16810676	2
#20273012	2
C₂×#16810345	2
C₂×#16810344	2
C₂×C₈×#88322	2
C₂×C₄×M₁₆×C₅₃	2
C₂×#1453	53

Subgroups — C2×#20273012 #27349712

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20273012 #27349712