Subgroups — C₂×#16810391 #20274450

Order 6784.

432 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#11719917	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	#16810391	C₂	1	4	yes	maximal, direct product factor
4	C₂²×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₅₃	C₄	1	2	yes	—
6	C₂×#11719917	—	2	2	no	—
7	C₂×#11719917	—	2	1	no	—
8	C₂×#11719917	C₄	1	4	yes	complemented
9	C₂×#11719917	C₂²	1	2	yes	—
10	C₂²×#88322	—	2	1	no	—
11	C₂×C₄⋊Dic₅₃	—	2	1	no	—
12	C₂×C₄⋊Dic₅₃	—	2	1	no	—
13	C₂×C₄⋊Dic₅₃	—	2	1	no	—
14	C₂×C₄⋊Dic₅₃	—	2	1	no	—
15	C₂×C₄⋊Dic₅₃	C₄	1	2	yes	—
16	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
17	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×M₁₆×C₅₃	—	2	2	no	—
19	C₂×M₁₆×C₅₃	—	2	1	no	—
20	C₂×M₁₆×C₅₃	C₄	1	4	yes	complemented
21	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
22	C₂²×C₈×C₅₃	—	2	1	no	—
23	C₂⁴×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₅₃	D₄	1	2	yes	—
28	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	Q₈	1	2	yes	—
33	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
34	C₂²×Dic₅₃	—	2	2	no	—
35	C₂²×Dic₅₃	—	2	2	no	—
36	C₂²×Dic₅₃	—	4	4	no	—
37	C₂²×Dic₅₃	C₂×C₄	1	4	yes	—
38	M₁₆×C₅₃	—	2	4	no	—
39	M₁₆×C₅₃	—	2	2	no	—
40	M₁₆×C₅₃	C₂×C₄	1	4	yes	complemented
41	C₂×C₈×C₅₃	—	2	4	no	—
42	C₂×C₈×C₅₃	—	2	2	no	—
43	C₄⋊Dic₅₃	—	2	2	no	—
44	C₄⋊Dic₅₃	—	2	2	no	—
45	C₄⋊Dic₅₃	—	2	2	no	—
46	C₄⋊Dic₅₃	—	2	2	no	—
47	#11719917	—	2	4	no	—
48	#11719917	—	2	2	no	—
49	#11719917	C₂×C₄	1	4	yes	complemented
50	C₂×#88322	—	2	4	no	—
51	C₂×#88322	—	2	2	no	—
52	C₂³×C₅₃	—	2	2	no	—
53	C₂³×C₅₃	—	2	2	no	—
54	C₂³×C₅₃	C₄²	1	2	yes	—
55	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂³×C₅₃	C₄⋊C₄	1	2	yes	—
58	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₅₃	C₄²	1	2	yes	—
60	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
63	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
64	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
65	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
66	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
67	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
68	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
69	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
70	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
71	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
77	C₂×Dic₅₃	—	2	4	no	—
78	C₂×Dic₅₃	—	4	4	no	—
79	C₂×Dic₅₃	—	4	4	no	—
80	C₂×Dic₅₃	—	4	4	no	—
81	C₈×C₅₃	—	2	4	no	—
82	#88322	—	2	4	no	—
83	C₂²×C₅₃	—	2	4	no	—
84	C₂²×C₅₃	—	2	4	no	—
85	C₂²×C₅₃	—	2	1	no	—
86	C₂²×C₅₃	—	2	1	no	—
87	C₂²×C₅₃	—	2	1	no	—
88	C₂²×C₅₃	—	2	1	no	—
89	C₂²×C₅₃	#95	1	2	yes	—
90	C₂²×C₅₃	#95	1	2	yes	—
91	C₂²×C₅₃	#100	1	2	yes	—
92	C₂²×C₅₃	#101	1	2	yes	—
93	C₂²×C₅₃	C₂×C₄²	1	1	yes	characteristic
94	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₄×C₅₃	#95	1	2	yes	—
97	C₄×C₅₃	#95	1	2	yes	—
98	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
102	Dic₅₃	—	4	4	no	—
103	C₂×#343	—	53	1	no	maximal, Sylow
104	C₂×C₅₃	—	2	2	no	—
105	C₂×C₅₃	—	2	1	no	—
106	C₂×C₅₃	—	2	1	no	—
107	C₂×C₅₃	#343	1	4	yes	complemented
108	C₂×C₅₃	C₂×#95	1	1	yes	characteristic
109	C₂×C₅₃	C₂×#100	1	1	yes	characteristic
110	C₂×C₅₃	C₂×#101	1	1	yes	characteristic
111	C₂²×M₁₆	—	53	1	no	—
112	C₂²×M₁₆	D₅₃	1	1	yes	characteristic
113	C₂²×C₄⋊C₄	—	53	1	no	—
114	#343	—	53	4	no	—
115	C₅₃	C₂×#343	1	1	yes	characteristic, complemented, Sylow
116	C₂³×C₄	—	53	2	no	—
117	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
118	C₂×M₁₆	—	2	1	no	—
119	C₂×M₁₆	—	2	2	no	—
120	C₂×M₁₆	—	53	4	no	—
121	C₂×M₁₆	—	53	2	no	—
122	C₂×M₁₆	—	106	1	no	—
123	C₂×M₁₆	—	106	2	no	—
124	C₂×M₁₆	Dic₅₃	1	4	yes	complemented
125	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
126	C₂²×C₈	—	2	1	no	—
127	C₂²×C₈	—	106	1	no	—
128	C₂×C₄⋊C₄	—	53	2	no	—
129	C₂×C₄⋊C₄	—	53	2	no	—
130	C₂×C₄⋊C₄	—	106	1	no	—
131	C₂×C₄⋊C₄	—	106	1	no	—
132	C₂×C₄⋊C₄	—	106	1	no	—
133	C₂×C₄⋊C₄	—	106	1	no	—
134	C₂⁴	C₄×D₅₃	1	1	yes	characteristic
135	C₂²×C₄	—	53	4	no	—
136	C₂²×C₄	—	106	2	no	—
137	C₂²×C₄	—	106	2	no	—
138	C₂²×C₄	—	212	4	no	—
139	C₂²×C₄	Dic₁₀₆	1	2	yes	—
140	C₂²×C₄	C₄×D₅₃	1	2	yes	—
141	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
142	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
143	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
144	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
145	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
146	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
147	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
148	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
149	M₁₆	—	2	2	no	—
150	M₁₆	—	2	4	no	—
151	M₁₆	—	53	4	no	—
152	M₁₆	—	106	2	no	—
153	M₁₆	—	106	4	no	—
154	M₁₆	C₂×Dic₅₃	1	4	yes	complemented
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	4	no	—
157	C₂×C₈	—	106	2	no	—
158	C₂×C₈	—	106	4	no	—
159	C₄⋊C₄	—	106	2	no	—
160	C₄⋊C₄	—	106	2	no	—
161	C₄⋊C₄	—	106	2	no	—
162	C₄⋊C₄	—	106	2	no	—
163	C₂³	—	2	2	no	—
164	C₂³	—	2	2	no	—
165	C₂³	C₄×Dic₅₃	1	2	yes	—
166	C₂³	#11719919	1	2	yes	—
167	C₂³	#11719921	1	1	yes	characteristic
168	C₂³	#11719921	1	1	yes	characteristic
169	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
170	C₂×C₄	—	106	4	no	—
171	C₂×C₄	—	212	4	no	—
172	C₂×C₄	—	212	4	no	—
173	C₂×C₄	—	212	4	no	—
174	C₂×C₄	C₄×Dic₅₃	1	2	yes	—
175	C₂×C₄	#11719919	1	2	yes	—
176	C₂×C₄	#11719919	1	2	yes	—
177	C₂×C₄	#11719919	1	2	yes	—
178	C₂×C₄	C₄⋊Dic₅₃	1	2	yes	—
179	C₂×C₄	C₄⋊Dic₅₃	1	2	yes	—
180	C₂×C₄	#11719921	1	1	yes	characteristic
181	C₂×C₄	#11719921	1	1	yes	characteristic
182	C₂×C₄	#11719921	1	2	yes	—
183	C₂×C₄	#11719921	1	2	yes	—
184	C₂×C₄	C₂²⋊Dic₅₃	1	2	yes	—
185	C₂×C₄	C₂²⋊Dic₅₃	1	2	yes	—
186	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
187	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
188	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
189	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
190	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
191	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
192	C₈	—	2	4	no	—
193	C₈	—	106	4	no	—
194	C₂²	—	2	1	no	—
195	C₂²	—	2	1	no	—
196	C₂²	—	2	1	no	—
197	C₂²	—	2	1	no	—
198	C₂²	—	2	4	no	—
199	C₂²	—	2	4	no	—
200	C₂²	#13133124	1	2	yes	—
201	C₂²	#13133124	1	2	yes	—
202	C₂²	#13133133	1	2	yes	—
203	C₂²	#13133135	1	2	yes	—
204	C₂²	C₂×C₄×Dic₅₃	1	1	yes	characteristic
205	C₂²	C₂×#11719921	1	1	yes	characteristic
206	C₂²	C₂×#11719919	1	1	yes	characteristic
207	C₄	—	212	4	no	—
208	C₄	#13133124	1	2	yes	—
209	C₄	#13133124	1	2	yes	—
210	C₄	C₂×#11719921	1	1	yes	characteristic
211	C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
212	C₄	C₂×#11719919	1	1	yes	characteristic
213	C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
214	C₂	—	2	2	no	—
215	C₂	—	2	1	no	—
216	C₂	—	2	1	no	—
217	C₂	#16810391	1	4	yes	direct product factor
218	C₂	C₂×#13133124	1	1	yes	characteristic
219	C₂	C₂×#13133133	1	1	yes	characteristic
220	C₂	C₂×#13133135	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#11719921
commutator	C₂²×C₅₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂²×M₁₆×C₅₃	C₂
socle	C₂³×C₅₃	C₂²⋊C₄
hypercenter	C₂²×M₁₆	D₅₃
hypocenter	C₅₃	C₂×#343
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. #16810391	C₂	direct (N × H)
8. C₂×#11719917	C₄	semidirect
20. C₂×M₁₆×C₅₃	C₄	semidirect
40. M₁₆×C₅₃	C₂×C₄	semidirect
49. #11719917	C₂×C₄	semidirect
107. C₂×C₅₃	#343	semidirect
115. C₅₃	C₂×#343	semidirect
124. C₂×M₁₆	Dic₅₃	semidirect
154. M₁₆	C₂×Dic₅₃	semidirect
217. C₂	#16810391	direct (N × H)

Subgroups — C2×#16810391 #20274450

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16810391 #20274450