Subgroups — C₂×#16811192 #20288026

Order 6784.

1896 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	C₂²×#11719945	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×D₂₁₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#16811192	C₂	1	16	yes	maximal, direct product factor
4	C₂×#13133214	C₂	1	2	yes	maximal, complemented
5	C₂×#13133205	C₂	1	2	yes	maximal, complemented
6	C₂×#13133192	C₂	1	4	yes	maximal, complemented
7	C₂×#13133187	C₂	1	4	yes	maximal, complemented
8	C₂×#117×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁴×D₅₃	C₂²	1	2	yes	—
10	C₂×#11719945	—	2	8	no	—
11	C₂×#11719945	C₂²	1	8	yes	complemented
12	C₂²×Dic₁₀₆	C₂²	1	1	yes	characteristic, complemented
13	C₂²×D₂₁₂	—	2	8	no	—
14	C₂²×D₂₁₂	C₂²	1	2	yes	—
15	C₂²×D₂₁₂	C₂²	1	2	yes	—
16	C₂²×D₂₁₂	C₂²	1	8	yes	complemented
17	C₂²×D₂₁₂	C₂²	1	1	yes	characteristic, complemented
18	C₂²×D₂₁₂	C₂²	1	8	yes	complemented
19	C₂²×C₂²⋊D₅₃	C₂²	1	2	yes	—
20	C₂²×C₄×D₅₃	C₂²	1	2	yes	complemented
21	#13133214	C₂²	1	16	yes	complemented
22	#13133205	C₂²	1	16	yes	complemented
23	#13133192	C₂²	1	32	yes	complemented
24	#13133187	C₂²	1	32	yes	complemented
25	C₂×#11719921	C₂²	1	8	yes	complemented
26	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
27	#117×C₅₃	C₂²	1	8	yes	complemented
28	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	complemented
29	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
30	C₂×C₄²×C₅₃	C₂²	1	2	yes	complemented
31	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
32	C₂³×D₅₃	—	2	8	no	—
33	C₂³×D₅₃	—	4	8	no	—
34	C₂³×D₅₃	C₂³	1	4	yes	—
35	C₂³×D₅₃	C₂³	1	2	yes	—
36	C₂³×D₅₃	C₂³	1	8	yes	—
37	C₂²×C₄×C₅₃	D₄	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	—	2	8	no	—
43	C₂×C₂²⋊D₅₃	C₂³	1	8	yes	—
44	C₂²×Dic₅₃	C₂³	1	2	yes	—
45	#11719945	—	2	32	no	—
46	C₂×D₂₁₂	—	2	8	no	—
47	C₂×D₂₁₂	—	2	8	no	—
48	C₂×D₂₁₂	—	2	4	no	—
49	C₂×D₂₁₂	—	2	32	no	—
50	C₂×D₂₁₂	—	4	16	no	—
51	C₂×D₂₁₂	C₂³	1	8	yes	—
52	C₂×D₂₁₂	C₂³	1	8	yes	—
53	C₂×D₂₁₂	C₂³	1	4	yes	complemented
54	C₂×D₂₁₂	C₂³	1	32	yes	complemented
55	C₂×C₄×D₅₃	—	2	8	no	—
56	C₂×C₄×D₅₃	C₂³	1	8	yes	complemented
57	C₂×Dic₁₀₆	—	2	4	no	—
58	C₂×Dic₁₀₆	C₂³	1	4	yes	complemented
59	C₄⋊C₄×C₅₃	C₂³	1	8	yes	complemented
60	C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
61	C₄²×C₅₃	C₂³	1	8	yes	complemented
62	#11719921	C₂³	1	32	yes	complemented
63	C₂³×C₅₃	—	2	4	no	—
64	C₂³×C₅₃	C₂×D₄	1	4	yes	—
65	C₂³×C₅₃	C₂⁴	1	2	yes	—
66	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
67	C₂²×D₅₃	—	2	4	no	—
68	C₂²×D₅₃	—	2	16	no	—
69	C₂²×D₅₃	—	2	8	no	—
70	C₂²×D₅₃	—	2	8	no	—
71	C₂²×D₅₃	—	2	16	no	—
72	C₂²×D₅₃	—	4	16	no	—
73	C₂²×D₅₃	—	4	32	no	—
74	C₂²×D₅₃	C₂⁴	1	4	yes	—
75	C₂²×D₅₃	C₂⁴	1	8	yes	—
76	C₂×C₄×C₅₃	—	2	8	no	—
77	C₂×C₄×C₅₃	C₂×D₄	1	4	yes	—
78	C₂×C₄×C₅₃	C₂×D₄	1	4	yes	—
79	C₂×C₄×C₅₃	C₂×D₄	1	16	yes	complemented
80	C₂×C₄×C₅₃	C₂⁴	1	8	yes	—
81	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
82	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
83	C₂²⋊D₅₃	—	2	16	no	—
84	C₂×Dic₅₃	—	2	4	no	—
85	C₂×Dic₅₃	C₂⁴	1	4	yes	—
86	D₂₁₂	—	2	16	no	—
87	D₂₁₂	—	2	16	no	—
88	D₂₁₂	—	2	8	no	—
89	D₂₁₂	—	4	32	no	—
90	C₄×D₅₃	—	2	16	no	—
91	Dic₁₀₆	—	2	8	no	—
92	C₂²×C₅₃	—	2	8	no	—
93	C₂²×C₅₃	—	2	4	no	—
94	C₂²×C₅₃	C₂²×D₄	1	4	yes	—
95	C₂²×C₅₃	C₂²×D₄	1	2	yes	—
96	C₂²×C₅₃	H(5,𝔽₂)	1	4	yes	—
97	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
98	C₂×D₅₃	—	2	4	no	—
99	C₂×D₅₃	—	2	8	no	—
100	C₂×D₅₃	—	2	8	no	—
101	C₂×D₅₃	—	2	16	no	—
102	C₂×D₅₃	—	4	32	no	—
103	C₂×D₅₃	—	4	8	no	—
104	C₂×D₅₃	—	4	16	no	—
105	C₄×C₅₃	—	2	8	no	—
106	C₄×C₅₃	C₂²×D₄	1	4	yes	—
107	C₄×C₅₃	C₂²×D₄	1	4	yes	—
108	Dic₅₃	—	2	4	no	—
109	C₂×#535	—	53	1	no	maximal, Sylow
110	C₂×C₅₃	—	2	4	no	—
111	C₂×C₅₃	#535	1	4	yes	complemented
112	C₂×C₅₃	C₂³×D₄	1	1	yes	characteristic
113	C₂×C₅₃	C₂×H(5,𝔽₂)	1	2	yes	—
114	D₅₃	—	4	4	no	—
115	D₅₃	—	4	8	no	—
116	C₂²×#41	—	53	1	no	—
117	C₂³×D₄	—	53	1	no	—
118	#535	—	53	16	no	—
119	C₂×#127	—	53	2	no	—
120	C₂×#124	—	53	2	no	—
121	C₂×#121	—	53	4	no	—
122	C₂×#120	—	53	4	no	—
123	C₂×#117	D₅₃	1	1	yes	characteristic, complemented
124	C₅₃	C₂×#535	1	1	yes	characteristic, complemented, Sylow
125	C₂⁵	—	53	2	no	—
126	C₂×#41	—	53	8	no	—
127	C₂×#41	—	106	8	no	—
128	C₂²×Q₈	—	53	1	no	—
129	C₂²×D₄	—	53	2	no	—
130	C₂²×D₄	—	53	2	no	—
131	C₂²×D₄	—	53	2	no	—
132	C₂²×D₄	—	53	1	no	—
133	C₂²×D₄	—	53	8	no	—
134	C₂²×D₄	—	53	8	no	—
135	C₂²×D₄	—	106	8	no	—
136	C₂³×C₄	—	53	2	no	—
137	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
138	#127	—	53	16	no	—
139	#124	—	53	16	no	—
140	#121	—	53	32	no	—
141	#120	—	53	32	no	—
142	#117	C₂×D₅₃	1	8	yes	complemented
143	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	complemented
144	C₂×C₂²⋊C₄	—	53	8	no	—
145	C₂×C₂²⋊C₄	C₂×D₅₃	1	2	yes	—
146	C₂×C₄²	C₂×D₅₃	1	2	yes	complemented
147	C₂⁴	—	53	4	no	—
148	C₂⁴	—	53	2	no	—
149	C₂⁴	—	53	8	no	—
150	C₂⁴	—	106	8	no	—
151	C₂⁴	—	212	8	no	—
152	C₂⁴	C₂²×D₅₃	1	1	yes	characteristic
153	#41	—	106	32	no	—
154	C₂×Q₈	—	53	4	no	—
155	C₂×Q₈	—	106	4	no	—
156	C₂×D₄	—	53	8	no	—
157	C₂×D₄	—	53	8	no	—
158	C₂×D₄	—	53	8	no	—
159	C₂×D₄	—	53	4	no	—
160	C₂×D₄	—	53	32	no	—
161	C₂×D₄	—	106	8	no	—
162	C₂×D₄	—	106	8	no	—
163	C₂×D₄	—	106	8	no	—
164	C₂×D₄	—	106	4	no	—
165	C₂×D₄	—	106	32	no	—
166	C₂×D₄	—	212	16	no	—
167	C₂²×C₄	—	53	2	no	—
168	C₂²×C₄	—	53	8	no	—
169	C₂²×C₄	—	106	8	no	—
170	C₂²×C₄	D₂₁₂	1	8	yes	complemented
171	C₂²×C₄	C₂²×D₅₃	1	4	yes	—
172	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
173	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
174	C₂²×C₄	C₂²×D₅₃	1	4	yes	—
175	C₄⋊C₄	C₂²×D₅₃	1	8	yes	complemented
176	C₂²⋊C₄	—	53	32	no	—
177	C₂²⋊C₄	C₂²×D₅₃	1	8	yes	—
178	C₄²	C₂²×D₅₃	1	8	yes	complemented
179	C₂³	—	2	4	no	—
180	C₂³	—	53	8	no	—
181	C₂³	—	53	4	no	—
182	C₂³	—	106	4	no	—
183	C₂³	—	106	16	no	—
184	C₂³	—	106	8	no	—
185	C₂³	—	106	8	no	—
186	C₂³	—	106	16	no	—
187	C₂³	—	212	16	no	—
188	C₂³	—	212	32	no	—
189	C₂³	C₂×D₂₁₂	1	4	yes	—
190	C₂³	C₂³×D₅₃	1	2	yes	—
191	C₂³	C₂³×D₅₃	1	1	yes	characteristic
192	Q₈	—	106	8	no	—
193	D₄	—	106	16	no	—
194	D₄	—	106	16	no	—
195	D₄	—	106	16	no	—
196	D₄	—	106	8	no	—
197	D₄	—	212	32	no	—
198	C₂×C₄	—	2	8	no	—
199	C₂×C₄	—	53	4	no	—
200	C₂×C₄	—	106	4	no	—
201	C₂×C₄	—	106	16	no	—
202	C₂×C₄	C₂×D₂₁₂	1	4	yes	—
203	C₂×C₄	C₂×D₂₁₂	1	4	yes	—
204	C₂×C₄	C₂×D₂₁₂	1	16	yes	complemented
205	C₂×C₄	C₂³×D₅₃	1	8	yes	—
206	C₂×C₄	C₂³×D₅₃	1	2	yes	—
207	C₂×C₄	C₂³×D₅₃	1	2	yes	—
208	C₂²	—	2	8	no	—
209	C₂²	—	2	4	no	—
210	C₂²	—	106	4	no	—
211	C₂²	—	106	8	no	—
212	C₂²	—	106	8	no	—
213	C₂²	—	106	16	no	—
214	C₂²	—	212	32	no	—
215	C₂²	—	212	16	no	—
216	C₂²	—	212	8	no	—
217	C₂²	C₂²×D₂₁₂	1	4	yes	—
218	C₂²	C₂²×D₂₁₂	1	2	yes	—
219	C₂²	#13133263	1	4	yes	—
220	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
221	C₄	—	2	8	no	—
222	C₄	—	106	4	no	—
223	C₄	C₂²×D₂₁₂	1	4	yes	—
224	C₄	C₂²×D₂₁₂	1	4	yes	—
225	C₂	—	2	4	no	—
226	C₂	—	212	8	no	—
227	C₂	—	212	4	no	—
228	C₂	#16811192	1	4	yes	direct product factor
229	C₂	C₂³×D₂₁₂	1	1	yes	characteristic
230	C₂	C₂×#13133263	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂⁵
Frattini	C₂²	C₂⁴×D₅₃
Fitting	C₂×#117×C₅₃	C₂
socle	C₂³×C₅₃	C₂⁴
hypercenter	C₂×#117	D₅₃
hypocenter	C₅₃	C₂×#535
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
1. C₂²×#11719945	C₂	semidirect
2. C₂³×D₂₁₂	C₂	semidirect
3. #16811192	C₂	direct (N × H)
4. C₂×#13133214	C₂	semidirect
5. C₂×#13133205	C₂	semidirect
6. C₂×#13133192	C₂	semidirect
7. C₂×#13133187	C₂	semidirect
8. C₂×#117×C₅₃	C₂	semidirect
11. C₂×#11719945	C₂²	semidirect
12. C₂²×Dic₁₀₆	C₂²	semidirect
16. C₂²×D₂₁₂	C₂²	semidirect
17. C₂²×D₂₁₂	C₂²	semidirect
18. C₂²×D₂₁₂	C₂²	semidirect
20. C₂²×C₄×D₅₃	C₂²	semidirect
21. #13133214	C₂²	semidirect
22. #13133205	C₂²	semidirect
23. #13133192	C₂²	semidirect
24. #13133187	C₂²	semidirect
25. C₂×#11719921	C₂²	semidirect
27. #117×C₅₃	C₂²	semidirect
28. C₂×C₄⋊C₄×C₅₃	C₂²	semidirect
30. C₂×C₄²×C₅₃	C₂²	semidirect
37. C₂²×C₄×C₅₃	D₄	semidirect
53. C₂×D₂₁₂	C₂³	semidirect
54. C₂×D₂₁₂	C₂³	semidirect
56. C₂×C₄×D₅₃	C₂³	semidirect
58. C₂×Dic₁₀₆	C₂³	semidirect
59. C₄⋊C₄×C₅₃	C₂³	semidirect
61. C₄²×C₅₃	C₂³	semidirect
62. #11719921	C₂³	semidirect
79. C₂×C₄×C₅₃	C₂×D₄	semidirect
111. C₂×C₅₃	#535	semidirect
123. C₂×#117	D₅₃	semidirect
124. C₅₃	C₂×#535	semidirect
142. #117	C₂×D₅₃	semidirect
143. C₂×C₄⋊C₄	C₂×D₅₃	semidirect
146. C₂×C₄²	C₂×D₅₃	semidirect
170. C₂²×C₄	D₂₁₂	semidirect
175. C₄⋊C₄	C₂²×D₅₃	semidirect
178. C₄²	C₂²×D₅₃	semidirect
204. C₂×C₄	C₂×D₂₁₂	semidirect
228. C₂	#16811192	direct (N × H)

Subgroups — C2×#16811192 #20288026

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16811192 #20288026