Subgroups — C₂×#16811091 #20287894

Order 6784.

1728 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	C₂²×#11719945	C₂	1	1	yes	maximal, characteristic
2	#16811091	C₂	1	16	yes	maximal, direct product factor
3	C₂×#13133182	C₂	1	2	yes	maximal, complemented
4	C₂×#13133179	C₂	1	2	yes	maximal, complemented
5	C₂×#13133177	C₂	1	4	yes	maximal, complemented
6	C₂×#13133174	C₂	1	2	yes	maximal, complemented
7	C₂²×C₄×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄²×D₅₃	C₂	1	2	yes	maximal, complemented
9	C₂×#117×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#11719945	C₄	1	16	yes	complemented
11	C₂×#11719945	C₂²	1	8	yes	—
12	C₂²×Dic₁₀₆	C₂²	1	1	yes	characteristic
13	C₂²×D₂₁₂	C₂²	1	1	yes	characteristic
14	C₂²×C₂²⋊D₅₃	C₂²	1	2	yes	—
15	C₂³×Dic₅₃	C₂²	1	2	yes	—
16	C₂²×C₄×D₅₃	C₂²	1	4	yes	—
17	C₂²×C₄×D₅₃	C₂²	1	2	yes	—
18	#13133182	C₂²	1	16	yes	complemented
19	#13133179	C₂²	1	16	yes	complemented
20	#13133177	C₂²	1	32	yes	complemented
21	#13133174	C₂²	1	16	yes	complemented
22	C₂×#11719919	C₂²	1	4	yes	—
23	C₂×#11719921	C₂²	1	4	yes	—
24	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×Dic₅₃	C₂²	1	4	yes	—
26	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
27	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
28	C₂×C₄×Dic₅₃	C₂²	1	8	yes	complemented
29	C₄²×D₅₃	C₂²	1	16	yes	complemented
30	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
31	#117×C₅₃	C₂²	1	8	yes	complemented
32	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
33	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
34	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
35	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
36	C₂³×D₅₃	C₂³	1	2	yes	—
37	C₂²×C₄×C₅₃	C₂×C₄	1	8	yes	complemented
38	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
39	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
42	C₂×C₂²⋊D₅₃	C₂×C₄	1	16	yes	complemented
43	C₂×C₂²⋊D₅₃	C₂³	1	8	yes	—
44	C₂²×Dic₅₃	—	2	8	no	—
45	C₂²×Dic₅₃	C₂³	1	2	yes	—
46	C₂²×Dic₅₃	C₂³	1	8	yes	—
47	C₂²×Dic₅₃	C₂³	1	2	yes	—
48	C₂²×Dic₅₃	C₂³	1	2	yes	—
49	#11719945	C₂×C₄	1	64	yes	complemented
50	C₂×D₂₁₂	C₂×C₄	1	8	yes	complemented
51	C₂×D₂₁₂	C₂³	1	4	yes	—
52	C₂×C₄×D₅₃	—	2	16	no	—
53	C₂×C₄×D₅₃	C₂×C₄	1	16	yes	complemented
54	C₂×C₄×D₅₃	C₂³	1	16	yes	—
55	C₂×C₄×D₅₃	C₂³	1	8	yes	—
56	C₂×Dic₁₀₆	C₂×C₄	1	8	yes	complemented
57	C₂×Dic₁₀₆	C₂³	1	4	yes	—
58	C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
59	C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
60	C₄²×C₅₃	C₂³	1	8	yes	—
61	#11719921	C₂³	1	16	yes	—
62	#11719919	C₂³	1	16	yes	—
63	C₄×Dic₅₃	C₂³	1	4	yes	—
64	C₄×Dic₅₃	C₂³	1	16	yes	—
65	C₄×Dic₅₃	C₂³	1	4	yes	—
66	C₄×Dic₅₃	C₂³	1	8	yes	—
67	C₂³×C₅₃	—	2	4	no	—
68	C₂³×C₅₃	C₂²×C₄	1	4	yes	—
69	C₂³×C₅₃	C₂⁴	1	2	yes	—
70	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
71	C₂²×D₅₃	—	2	8	no	—
72	C₂²×D₅₃	C₂²×C₄	1	8	yes	—
73	C₂²×D₅₃	C₂⁴	1	4	yes	—
74	C₂×C₄×C₅₃	—	2	8	no	—
75	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
76	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
77	C₂×C₄×C₅₃	C₂²×C₄	1	16	yes	complemented
78	C₂×C₄×C₅₃	C₂⁴	1	8	yes	—
79	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
80	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
81	C₂²⋊D₅₃	C₂²×C₄	1	32	yes	complemented
82	C₂×Dic₅₃	—	2	4	no	—
83	C₂×Dic₅₃	—	2	16	no	—
84	C₂×Dic₅₃	C₂²×C₄	1	8	yes	—
85	C₂×Dic₅₃	#41	1	8	yes	complemented
86	C₂×Dic₅₃	C₂⁴	1	4	yes	—
87	C₂×Dic₅₃	C₂⁴	1	4	yes	—
88	C₂×Dic₅₃	C₂⁴	1	4	yes	—
89	D₂₁₂	C₂²×C₄	1	16	yes	complemented
90	C₄×D₅₃	—	2	32	no	—
91	C₄×D₅₃	C₂²×C₄	1	32	yes	complemented
92	Dic₁₀₆	C₂²×C₄	1	16	yes	complemented
93	C₂²×C₅₃	—	2	8	no	—
94	C₂²×C₅₃	—	2	4	no	—
95	C₂²×C₅₃	C₂³×C₄	1	4	yes	—
96	C₂²×C₅₃	C₂³×C₄	1	2	yes	—
97	C₂²×C₅₃	C₂×#41	1	4	yes	—
98	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
99	C₂×D₅₃	—	2	16	no	—
100	C₂×D₅₃	—	2	8	no	—
101	C₂×D₅₃	C₂³×C₄	1	8	yes	—
102	C₄×C₅₃	—	2	8	no	—
103	C₄×C₅₃	C₂³×C₄	1	4	yes	—
104	C₄×C₅₃	C₂³×C₄	1	4	yes	—
105	Dic₅₃	—	2	4	no	—
106	Dic₅₃	C₂³×C₄	1	8	yes	—
107	Dic₅₃	C₂×#41	1	8	yes	complemented
108	C₂×C₄×#41	—	53	1	no	maximal, Sylow
109	C₂×C₅₃	—	2	4	no	—
110	C₂×C₅₃	C₄×#41	1	4	yes	complemented
111	C₂×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
112	C₂×C₅₃	C₂²×#41	1	2	yes	—
113	D₅₃	—	2	8	no	—
114	C₂²×#41	—	53	1	no	—
115	C₄×#41	—	53	16	no	—
116	C₂×C₄×Q₈	—	53	2	no	—
117	C₂×C₄×D₄	—	53	2	no	—
118	C₂×C₄×D₄	—	53	4	no	—
119	C₂×#117	—	53	2	no	—
120	C₂×#117	D₅₃	1	1	yes	characteristic, complemented
121	C₂²×C₄²	—	53	1	no	—
122	C₂²×C₄²	—	53	2	no	—
123	C₅₃	C₂×C₄×#41	1	1	yes	characteristic, complemented, Sylow
124	C₂×#41	—	53	16	no	—
125	C₂×#41	—	53	8	no	—
126	C₂²×Q₈	—	53	1	no	—
127	C₂²×D₄	—	53	2	no	—
128	C₂²×D₄	—	53	1	no	—
129	C₂³×C₄	—	53	2	no	—
130	C₂³×C₄	—	53	2	no	—
131	C₂³×C₄	—	53	4	no	—
132	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
133	C₄×Q₈	—	53	16	no	—
134	C₄×D₄	—	53	16	no	—
135	C₄×D₄	—	53	32	no	—
136	#117	—	53	16	no	—
137	#117	C₂×D₅₃	1	8	yes	complemented
138	C₂×C₄⋊C₄	—	53	4	no	—
139	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
140	C₂×C₂²⋊C₄	—	53	4	no	—
141	C₂×C₂²⋊C₄	C₂×D₅₃	1	2	yes	—
142	C₂×C₄²	—	53	4	no	—
143	C₂×C₄²	—	53	2	no	—
144	C₂×C₄²	—	53	8	no	—
145	C₂×C₄²	—	53	1	no	—
146	C₂×C₄²	—	53	1	no	—
147	C₂×C₄²	—	53	16	no	—
148	C₂×C₄²	C₂×D₅₃	1	2	yes	—
149	C₂⁴	—	53	2	no	—
150	C₂⁴	C₂²×D₅₃	1	1	yes	characteristic
151	#41	—	53	64	no	—
152	C₂×Q₈	—	53	8	no	—
153	C₂×Q₈	—	53	4	no	—
154	C₂×D₄	—	53	16	no	—
155	C₂×D₄	—	53	8	no	—
156	C₂×D₄	—	53	8	no	—
157	C₂×D₄	—	53	4	no	—
158	C₂²×C₄	—	53	2	no	—
159	C₂²×C₄	—	53	2	no	—
160	C₂²×C₄	—	53	16	no	—
161	C₂²×C₄	—	53	8	no	—
162	C₂²×C₄	—	53	2	no	—
163	C₂²×C₄	—	53	8	no	—
164	C₂²×C₄	—	53	16	no	—
165	C₂²×C₄	—	106	16	no	—
166	C₂²×C₄	—	106	8	no	—
167	C₂²×C₄	C₄×D₅₃	1	8	yes	complemented
168	C₂²×C₄	C₂²×D₅₃	1	4	yes	—
169	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
170	C₂²×C₄	C₂²×D₅₃	1	4	yes	—
171	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
172	C₄⋊C₄	—	53	16	no	—
173	C₄⋊C₄	C₂²×D₅₃	1	8	yes	—
174	C₂²⋊C₄	—	53	16	no	—
175	C₂²⋊C₄	C₂²×D₅₃	1	8	yes	—
176	C₄²	—	53	4	no	—
177	C₄²	—	53	4	no	—
178	C₄²	—	53	16	no	—
179	C₄²	—	53	8	no	—
180	C₄²	C₂²×D₅₃	1	8	yes	—
181	C₂³	—	2	4	no	—
182	C₂³	—	53	8	no	—
183	C₂³	—	53	4	no	—
184	C₂³	—	106	8	no	—
185	C₂³	C₂×C₄×D₅₃	1	4	yes	—
186	C₂³	C₂³×D₅₃	1	2	yes	—
187	C₂³	C₂³×D₅₃	1	1	yes	characteristic
188	Q₈	—	53	16	no	—
189	D₄	—	53	32	no	—
190	D₄	—	53	16	no	—
191	C₂×C₄	—	2	8	no	—
192	C₂×C₄	—	53	8	no	—
193	C₂×C₄	—	53	8	no	—
194	C₂×C₄	—	53	4	no	—
195	C₂×C₄	—	53	4	no	—
196	C₂×C₄	—	53	32	no	—
197	C₂×C₄	—	53	4	no	—
198	C₂×C₄	—	106	32	no	—
199	C₂×C₄	—	106	16	no	—
200	C₂×C₄	—	106	4	no	—
201	C₂×C₄	C₂×C₄×D₅₃	1	4	yes	—
202	C₂×C₄	C₂×C₄×D₅₃	1	4	yes	—
203	C₂×C₄	C₂×C₄×D₅₃	1	16	yes	complemented
204	C₂×C₄	C₂³×D₅₃	1	8	yes	—
205	C₂×C₄	C₂³×D₅₃	1	2	yes	—
206	C₂×C₄	C₂³×D₅₃	1	2	yes	—
207	C₂²	—	2	8	no	—
208	C₂²	—	2	4	no	—
209	C₂²	—	53	8	no	—
210	C₂²	—	106	16	no	—
211	C₂²	—	106	8	no	—
212	C₂²	C₂²×C₄×D₅₃	1	4	yes	—
213	C₂²	C₂²×C₄×D₅₃	1	2	yes	—
214	C₂²	#41×D₅₃	1	4	yes	—
215	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
216	C₄	—	2	8	no	—
217	C₄	—	53	8	no	—
218	C₄	—	53	8	no	—
219	C₄	—	106	4	no	—
220	C₄	C₂²×C₄×D₅₃	1	4	yes	—
221	C₄	C₂²×C₄×D₅₃	1	4	yes	—
222	C₂	—	2	4	no	—
223	C₂	—	106	8	no	—
224	C₂	#16811091	1	4	yes	direct product factor
225	C₂	C₂³×C₄×D₅₃	1	1	yes	characteristic
226	C₂	C₂×#41×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₅₃
commutator	C₂×C₅₃	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₅₃
Fitting	C₂×#117×C₅₃	C₂
socle	C₂³×C₅₃	C₂⁴
hypercenter	C₂×#117	D₅₃
hypocenter	C₅₃	C₂×C₄×#41
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
2. #16811091	C₂	direct (N × H)
3. C₂×#13133182	C₂	semidirect
4. C₂×#13133179	C₂	semidirect
5. C₂×#13133177	C₂	semidirect
6. C₂×#13133174	C₂	semidirect
7. C₂²×C₄×Dic₅₃	C₂	semidirect
8. C₂×C₄²×D₅₃	C₂	semidirect
9. C₂×#117×C₅₃	C₂	semidirect
10. C₂×#11719945	C₄	semidirect
18. #13133182	C₂²	semidirect
19. #13133179	C₂²	semidirect
20. #13133177	C₂²	semidirect
21. #13133174	C₂²	semidirect
28. C₂×C₄×Dic₅₃	C₂²	semidirect
29. C₄²×D₅₃	C₂²	semidirect
31. #117×C₅₃	C₂²	semidirect
37. C₂²×C₄×C₅₃	C₂×C₄	semidirect
42. C₂×C₂²⋊D₅₃	C₂×C₄	semidirect
49. #11719945	C₂×C₄	semidirect
50. C₂×D₂₁₂	C₂×C₄	semidirect
53. C₂×C₄×D₅₃	C₂×C₄	semidirect
56. C₂×Dic₁₀₆	C₂×C₄	semidirect
77. C₂×C₄×C₅₃	C₂²×C₄	semidirect
81. C₂²⋊D₅₃	C₂²×C₄	semidirect
85. C₂×Dic₅₃	#41	semidirect
89. D₂₁₂	C₂²×C₄	semidirect
91. C₄×D₅₃	C₂²×C₄	semidirect
92. Dic₁₀₆	C₂²×C₄	semidirect
107. Dic₅₃	C₂×#41	semidirect
110. C₂×C₅₃	C₄×#41	semidirect
120. C₂×#117	D₅₃	semidirect
123. C₅₃	C₂×C₄×#41	semidirect
137. #117	C₂×D₅₃	semidirect
167. C₂²×C₄	C₄×D₅₃	semidirect
203. C₂×C₄	C₂×C₄×D₅₃	semidirect
224. C₂	#16811091	direct (N × H)

Subgroups — C2×#16811091 #20287894

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16811091 #20287894