Subgroups — C₂×#16810382 #20274420

Order 6784.

488 conjugacy classes of subgroups.

The subgroup lattice has 214 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	#16810382	C₂	1	4	yes	maximal, direct product factor
4	C₂×#117×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₅₃	—	2	1	no	—
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	2	no	—
12	C₂×C₄×Dic₅₃	—	2	1	no	—
13	C₂×C₄×Dic₅₃	C₄	1	4	yes	complemented
14	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
15	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
16	#117×C₅₃	—	2	1	no	—
17	#117×C₅₃	—	2	1	no	—
18	#117×C₅₃	C₄	1	2	yes	complemented
19	#117×C₅₃	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₅₃	C₄	1	2	yes	complemented
21	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
22	C₂×C₄²×C₅₃	—	2	1	no	—
23	C₂⁴×C₅₃	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₅₃	—	2	2	no	—
25	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	D₄	1	2	yes	—
31	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	Q₈	1	2	yes	—
34	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
35	C₂²×Dic₅₃	—	2	8	no	—
36	C₂²×Dic₅₃	—	2	4	no	—
37	C₂²×Dic₅₃	—	2	2	no	—
38	C₄⋊C₄×C₅₃	—	2	2	no	—
39	C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	complemented
40	C₂²⋊C₄×C₅₃	—	2	2	no	—
41	C₂²⋊C₄×C₅₃	—	2	2	no	—
42	C₄²×C₅₃	—	2	2	no	—
43	C₄²×C₅₃	—	2	2	no	—
44	C₄×Dic₅₃	—	2	4	no	—
45	C₄×Dic₅₃	—	2	2	no	—
46	C₄×Dic₅₃	C₂×C₄	1	4	yes	complemented
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	4	no	—
53	C₂³×C₅₃	C₂²⋊C₄	1	2	yes	—
54	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂³×C₅₃	C₄⋊C₄	1	2	yes	—
57	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₅₃	—	2	4	no	—
59	C₂×C₄×C₅₃	—	4	4	no	—
60	C₂×C₄×C₅₃	C₄²	1	2	yes	—
61	C₂×C₄×C₅₃	C₄²	1	2	yes	—
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	1	yes	characteristic
66	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
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	C₄⋊C₄	1	2	yes	—
72	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
78	C₂×Dic₅₃	—	2	16	no	—
79	C₂×Dic₅₃	—	2	8	no	—
80	C₂×Dic₅₃	—	2	4	no	—
81	#88322	—	2	4	no	—
82	C₂²×C₅₃	—	2	4	no	—
83	C₂²×C₅₃	—	2	4	no	—
84	C₂²×C₅₃	—	2	2	no	—
85	C₂²×C₅₃	—	2	2	no	—
86	C₂²×C₅₃	#95	1	2	yes	—
87	C₂²×C₅₃	#95	1	2	yes	—
88	C₂²×C₅₃	#104	1	4	yes	—
89	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₄×C₅₃	—	4	4	no	—
93	C₄×C₅₃	#95	1	2	yes	—
94	C₄×C₅₃	#95	1	2	yes	—
95	C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
96	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
99	Dic₅₃	—	2	8	no	—
100	C₂×#339	—	53	1	no	maximal, Sylow
101	C₂×C₅₃	—	2	1	no	—
102	C₂×C₅₃	—	2	2	no	—
103	C₂×C₅₃	—	2	1	no	—
104	C₂×C₅₃	#339	1	4	yes	complemented
105	C₂×C₅₃	C₂×#95	1	1	yes	characteristic
106	C₂×C₅₃	C₂×#104	1	2	yes	—
107	C₂²×M₁₆	—	53	1	no	—
108	C₂×#117	D₅₃	1	1	yes	characteristic
109	C₂²×C₄²	—	53	1	no	—
110	#339	—	53	4	no	—
111	C₅₃	C₂×#339	1	1	yes	characteristic, complemented, Sylow
112	C₂³×C₄	—	106	1	no	—
113	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
114	C₂×M₁₆	—	53	4	no	—
115	C₂×M₁₆	—	53	2	no	—
116	C₂×M₁₆	—	106	2	no	—
117	C₂×M₁₆	—	106	1	no	—
118	C₂²×C₈	—	106	1	no	—
119	#117	—	2	1	no	—
120	#117	—	2	1	no	—
121	#117	Dic₅₃	1	2	yes	complemented
122	#117	C₂×D₅₃	1	2	yes	—
123	C₂×C₄⋊C₄	Dic₅₃	1	2	yes	complemented
124	C₂×C₂²⋊C₄	—	2	1	no	—
125	C₂×C₄²	—	2	1	no	—
126	C₂×C₄²	—	53	4	no	—
127	C₂×C₄²	—	53	2	no	—
128	C₂×C₄²	—	106	2	no	—
129	C₂×C₄²	—	106	1	no	—
130	C₂⁴	C₂²⋊D₅₃	1	1	yes	characteristic
131	C₂²×C₄	—	2	2	no	—
132	C₂²×C₄	—	106	8	no	—
133	C₂²×C₄	—	106	2	no	—
134	C₂²×C₄	—	106	4	no	—
135	C₂²×C₄	Dic₁₀₆	1	2	yes	—
136	C₂²×C₄	C₄×D₅₃	1	2	yes	—
137	C₂²×C₄	C₄×D₅₃	1	2	yes	—
138	C₂²×C₄	D₂₁₂	1	2	yes	—
139	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
140	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
141	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
142	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
143	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
144	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
145	M₁₆	—	53	4	no	—
146	M₁₆	—	106	2	no	—
147	M₁₆	—	106	4	no	—
148	C₂×C₈	—	106	2	no	—
149	C₂×C₈	—	106	4	no	—
150	C₄⋊C₄	—	2	2	no	—
151	C₄⋊C₄	C₂×Dic₅₃	1	4	yes	complemented
152	C₂²⋊C₄	—	2	2	no	—
153	C₂²⋊C₄	—	2	2	no	—
154	C₄²	—	2	2	no	—
155	C₄²	—	2	2	no	—
156	C₄²	—	53	4	no	—
157	C₄²	—	106	2	no	—
158	C₄²	—	106	4	no	—
159	C₂³	—	2	4	no	—
160	C₂³	#11719919	1	2	yes	—
161	C₂³	#11719921	1	2	yes	—
162	C₂³	C₂²⋊Dic₅₃	1	1	yes	characteristic
163	C₂³	C₂²⋊Dic₅₃	1	1	yes	characteristic
164	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	—	106	8	no	—
168	C₂×C₄	—	106	16	no	—
169	C₂×C₄	—	106	4	no	—
170	C₂×C₄	C₄×Dic₅₃	1	2	yes	—
171	C₂×C₄	C₄×Dic₅₃	1	2	yes	—
172	C₂×C₄	#11719919	1	2	yes	—
173	C₂×C₄	#11719919	1	2	yes	—
174	C₂×C₄	#11719919	1	2	yes	—
175	C₂×C₄	C₄⋊Dic₅₃	1	2	yes	—
176	C₂×C₄	C₄⋊Dic₅₃	1	2	yes	—
177	C₂×C₄	#11719921	1	2	yes	—
178	C₂×C₄	#11719921	1	2	yes	—
179	C₂×C₄	#11719921	1	2	yes	—
180	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
181	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
182	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
183	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
184	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
185	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
186	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
187	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
188	C₈	—	106	4	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	2	2	no	—
191	C₂²	—	2	4	no	—
192	C₂²	—	2	4	no	—
193	C₂²	#13133124	1	2	yes	—
194	C₂²	#13133124	1	2	yes	—
195	C₂²	#13133144	1	4	yes	—
196	C₂²	C₂×#11719921	1	1	yes	characteristic
197	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
198	C₂²	C₂×#11719919	1	1	yes	characteristic
199	C₄	—	4	4	no	—
200	C₄	—	106	8	no	—
201	C₄	#13133124	1	2	yes	—
202	C₄	#13133124	1	2	yes	—
203	C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
204	C₄	C₂×#11719921	1	1	yes	characteristic
205	C₄	C₂×#11719919	1	1	yes	characteristic
206	C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
207	C₂	—	2	1	no	—
208	C₂	—	2	2	no	—
209	C₂	—	2	1	no	—
210	C₂	#16810382	1	4	yes	direct product factor
211	C₂	C₂×#13133124	1	1	yes	characteristic
212	C₂	C₂×#13133144	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×#117×C₅₃	C₂
socle	C₂³×C₅₃	C₂²⋊C₄
hypercenter	C₂×#117	D₅₃
hypocenter	C₅₃	C₂×#339
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. #16810382	C₂	direct (N × H)
8. C₂×#11719917	C₄	semidirect
13. C₂×C₄×Dic₅₃	C₄	semidirect
18. #117×C₅₃	C₄	semidirect
20. C₂×C₄⋊C₄×C₅₃	C₄	semidirect
39. C₄⋊C₄×C₅₃	C₂×C₄	semidirect
46. C₄×Dic₅₃	C₂×C₄	semidirect
49. #11719917	C₂×C₄	semidirect
104. C₂×C₅₃	#339	semidirect
111. C₅₃	C₂×#339	semidirect
121. #117	Dic₅₃	semidirect
123. C₂×C₄⋊C₄	Dic₅₃	semidirect
151. C₄⋊C₄	C₂×Dic₅₃	semidirect
210. C₂	#16810382	direct (N × H)

Subgroups — C2×#16810382 #20274420

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16810382 #20274420