Subgroups — C₂×#20273018 #27349798

Order 13568.

994 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11719917	C₂	1	2	yes	maximal
2	#20273018	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	C₂²×#11719917	C₄	1	4	yes	—
8	C₄×#11719917	C₄	1	8	yes	—
9	C₄×#11719917	C₂²	1	8	yes	—
10	C₂×#13133128	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	C₂×#11719917	—	2	16	no	—
24	C₂×#11719917	C₂×C₄	1	16	yes	—
25	C₂²×#88322	C₂×C₄	1	4	yes	—
26	C₂²×#88322	C₂×C₄	1	4	yes	—
27	#13133128	C₂×C₄	1	16	yes	—
28	C₄×#88322	C₂×C₄	1	4	yes	—
29	C₄×#88322	C₂×C₄	1	4	yes	—
30	C₄×#88322	C₂³	1	4	yes	—
31	C₄×#88322	C₂³	1	4	yes	—
32	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
33	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×M₁₆×C₅₃	—	2	8	no	—
35	C₂×M₁₆×C₅₃	C₂×C₄	1	8	yes	—
36	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
43	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
44	#97×C₅₃	C₂×C₄	1	8	yes	—
45	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₅₃	C₂³	1	2	yes	—
48	C₄×C₈×C₅₃	C₂³	1	2	yes	—
49	C₂⁴×C₅₃	C₄²	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	—	2	4	no	—
51	C₂²×C₄×C₅₃	—	2	8	no	—
52	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
54	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
55	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
56	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
57	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
60	M₁₆×C₅₃	—	2	16	no	—
61	C₂×C₈×C₅₃	—	2	4	no	—
62	C₂×C₈×C₅₃	M₁₆	1	8	yes	complemented
63	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
64	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
65	C₄²×C₅₃	C₄²	1	4	yes	—
66	C₄²×C₅₃	C₄²	1	2	yes	—
67	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
68	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
69	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
71	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
72	#11719917	—	2	32	no	—
73	C₂×#88322	—	2	8	no	—
74	C₂×#88322	M₁₆	1	16	yes	complemented
75	C₂×#88322	C₂²×C₄	1	8	yes	—
76	C₂×#88322	C₂²×C₄	1	8	yes	—
77	C₂³×C₅₃	—	2	4	no	—
78	C₂³×C₅₃	—	2	2	no	—
79	C₂³×C₅₃	C₂×C₄²	1	2	yes	—
80	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
81	C₂×C₄×C₅₃	—	2	4	no	—
82	C₂×C₄×C₅₃	—	2	2	no	—
83	C₂×C₄×C₅₃	—	2	8	no	—
84	C₂×C₄×C₅₃	—	2	4	no	—
85	C₂×C₄×C₅₃	—	2	8	no	—
86	C₂×C₄×C₅₃	—	2	4	no	—
87	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
88	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
89	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
90	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂×M₁₆	1	8	yes	—
92	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
93	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
94	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
95	C₈×C₅₃	—	2	4	no	—
96	C₈×C₅₃	C₂×M₁₆	1	8	yes	complemented
97	#88322	—	2	8	no	—
98	#88322	C₂×M₁₆	1	16	yes	complemented
99	C₂×#1455	—	53	1	no	maximal, Sylow
100	C₂²×C₅₃	—	2	4	no	—
101	C₂²×C₅₃	—	2	2	no	—
102	C₂²×C₅₃	—	4	4	no	—
103	C₂²×C₅₃	C₄×M₁₆	1	4	yes	—
104	C₂²×C₅₃	C₄×M₁₆	1	2	yes	—
105	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
106	C₄×C₅₃	—	2	4	no	—
107	C₄×C₅₃	—	2	2	no	—
108	C₄×C₅₃	C₄×M₁₆	1	4	yes	—
109	C₄×C₅₃	C₄×M₁₆	1	2	yes	—
110	C₄×C₅₃	C₂²×M₁₆	1	4	yes	—
111	C₄×C₅₃	C₂²×M₁₆	1	2	yes	—
112	C₂×C₄×M₁₆	—	53	2	no	—
113	C₂×C₄×M₁₆	D₅₃	1	1	yes	characteristic
114	#1455	—	53	8	no	—
115	C₂×#322	—	53	1	no	—
116	C₂×#322	—	53	2	no	—
117	C₂×C₈²	—	53	1	no	—
118	C₂×C₅₃	—	4	2	no	—
119	C₂×C₅₃	#1455	1	4	yes	complemented
120	C₂×C₅₃	C₂×C₄×M₁₆	1	2	yes	—
121	C₂×C₅₃	C₂×C₄×M₁₆	1	1	yes	characteristic
122	C₂²×M₁₆	—	53	4	no	—
123	C₂²×M₁₆	Dic₅₃	1	2	yes	—
124	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
125	C₄×M₁₆	—	53	8	no	—
126	C₄×M₁₆	—	53	8	no	—
127	C₄×M₁₆	Dic₅₃	1	4	yes	—
128	C₄×M₁₆	C₂×D₅₃	1	4	yes	—
129	C₂×#97	—	53	4	no	—
130	C₂×#97	Dic₅₃	1	2	yes	—
131	C₂×C₄×C₈	—	53	2	no	—
132	C₂×C₄×C₈	—	53	2	no	—
133	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
134	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
135	#322	—	53	4	no	—
136	#322	—	53	8	no	—
137	C₈²	—	53	4	no	—
138	C₅₃	C₂×#1455	1	1	yes	characteristic, complemented, Sylow
139	C₂³×C₄	C₄×D₅₃	1	2	yes	—
140	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
141	C₂×M₁₆	—	2	8	no	—
142	C₂×M₁₆	—	53	16	no	—
143	C₂×M₁₆	—	106	16	no	—
144	C₂×M₁₆	C₂×Dic₅₃	1	8	yes	—
145	C₂²×C₈	—	53	4	no	—
146	C₂²×C₈	—	53	4	no	—
147	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
148	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
149	C₂×C₄²	C₄×D₅₃	1	2	yes	—
150	C₂×C₄²	C₄×D₅₃	1	4	yes	—
151	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
152	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
153	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
154	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
155	#97	—	53	16	no	—
156	#97	C₂×Dic₅₃	1	8	yes	—
157	C₄×C₈	—	53	4	no	—
158	C₄×C₈	—	53	4	no	—
159	C₄×C₈	—	53	4	no	—
160	C₄×C₈	—	53	4	no	—
161	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
162	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
163	C₄×C₈	C₂²×D₅₃	1	2	yes	—
164	C₄×C₈	C₂²×D₅₃	1	2	yes	—
165	C₂⁴	C₄×Dic₅₃	1	1	yes	characteristic
166	C₂²×C₄	—	2	8	no	—
167	C₂²×C₄	—	2	4	no	—
168	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
169	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
170	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
171	C₂²×C₄	C₄×Dic₅₃	1	4	yes	—
172	C₂²×C₄	C₂×C₄×D₅₃	1	4	yes	—
173	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
174	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
175	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
176	M₁₆	—	2	16	no	—
177	M₁₆	—	106	32	no	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	53	16	no	—
180	C₂×C₈	—	53	8	no	—
181	C₂×C₈	—	53	8	no	—
182	C₂×C₈	—	106	8	no	—
183	C₂×C₈	#11719917	1	8	yes	complemented
184	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
185	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
186	C₄²	C₄×Dic₅₃	1	4	yes	—
187	C₄²	C₄×Dic₅₃	1	2	yes	—
188	C₄²	C₂×C₄×D₅₃	1	2	yes	—
189	C₄²	C₂×C₄×D₅₃	1	4	yes	—
190	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
191	C₄²	C₂²×Dic₅₃	1	2	yes	—
192	C₄²	C₂³×D₅₃	1	1	yes	characteristic
193	C₂³	—	2	4	no	—
194	C₂³	—	2	2	no	—
195	C₂³	C₂×C₄×Dic₅₃	1	2	yes	—
196	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	8	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	8	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
204	C₂×C₄	C₂×C₄×Dic₅₃	1	4	yes	—
205	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
206	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
207	C₂×C₄	M₁₆×D₅₃	1	8	yes	—
208	C₂×C₄	C₂×#11719917	1	4	yes	—
209	C₂×C₄	C₂²×C₄×D₅₃	1	2	yes	—
210	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
211	C₈	—	2	4	no	—
212	C₈	—	53	16	no	—
213	C₈	—	106	8	no	—
214	C₈	C₂×#11719917	1	8	yes	complemented
215	C₂²	—	2	4	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	4	4	no	—
218	C₂²	M₁₆×Dic₅₃	1	4	yes	—
219	C₂²	C₄×#11719917	1	2	yes	—
220	C₂²	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
221	C₄	—	2	4	no	—
222	C₄	—	2	2	no	—
223	C₄	M₁₆×Dic₅₃	1	4	yes	—
224	C₄	C₄×#11719917	1	2	yes	—
225	C₄	C₂×M₁₆×D₅₃	1	4	yes	—
226	C₄	C₂²×#11719917	1	2	yes	—
227	C₂	—	4	2	no	—
228	C₂	#20273018	1	4	yes	direct product factor
229	C₂	C₂×M₁₆×Dic₅₃	1	2	yes	—
230	C₂	C₂×C₄×#11719917	1	1	yes	characteristic

	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×#1455
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. #20273018	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
62. C₂×C₈×C₅₃	M₁₆	semidirect
74. C₂×#88322	M₁₆	semidirect
96. C₈×C₅₃	C₂×M₁₆	semidirect
98. #88322	C₂×M₁₆	semidirect
119. C₂×C₅₃	#1455	semidirect
138. C₅₃	C₂×#1455	semidirect
183. C₂×C₈	#11719917	semidirect
214. C₈	C₂×#11719917	semidirect
228. C₂	#20273018	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×C₄×#11719917	2
#20273018	2
C₂×#16810345	2
C₂×#16810344	2
C₂×C₈×#88322	2
C₂×C₄×M₁₆×C₅₃	2
C₂×#1455	53

Subgroups — C2×#20273018 #27349798

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20273018 #27349798