Subgroups — C₂×#16811586 #20291273

Order 6784.

592 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×D₅₃⋊M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133296	C₂	1	1	yes	maximal, characteristic
3	#16811586	C₂	1	8	yes	maximal, direct product factor
4	C₂×#13133271	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×#6157	C₂	1	2	yes	maximal, complemented
6	C₂²×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂×#11719936	C₄	1	2	yes	complemented
8	D₅₃⋊M₁₆	C₄	1	4	yes	complemented
9	D₅₃⋊M₁₆	C₂²	1	4	yes	—
10	C₂×#11719940	C₄	1	2	yes	complemented
11	C₂×D₅₃⋊C₈	C₂²	1	2	yes	—
12	#13133296	C₄	1	4	yes	—
13	#13133296	C₂²	1	4	yes	—
14	C₂×C₄⋊#6157	C₄	1	2	yes	—
15	C₂×C₂²⋊#6157	C₄	1	2	yes	—
16	C₂×C₄×#6157	C₂²	1	2	yes	—
17	#13133271	C₂²	1	8	yes	complemented
18	C₈×#6157	C₂²	1	8	yes	complemented
19	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
20	C₂²×#88322	C₄	1	1	yes	characteristic, complemented
21	C₂×C₈×D₅₃	C₄	1	4	yes	complemented
22	C₂×C₈×D₅₃	C₄	1	1	yes	characteristic, complemented
23	C₂×C₈×D₅₃	C₄	1	1	yes	characteristic, complemented
24	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
25	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
26	C₂×C₈×D₅₃	C₂²	1	4	yes	—
27	C₂²×C₈×C₅₃	C₄	1	1	yes	characteristic, complemented
28	C₂³×D₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×#6157	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
34	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
35	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
37	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
39	C₂×C₄×D₅₃	C₂³	1	2	yes	—
40	C₂²⋊#6157	C₂×C₄	1	8	yes	—
41	#11719940	C₂×C₄	1	8	yes	complemented
42	C₂×#88324	C₂×C₄	1	4	yes	—
43	C₄⋊#6157	C₂×C₄	1	8	yes	—
44	C₄×#6157	C₂×C₄	1	4	yes	—
45	C₄×#6157	C₂³	1	4	yes	—
46	#11719936	C₂×C₄	1	8	yes	complemented
47	D₅₃⋊C₈	C₂×C₄	1	4	yes	—
48	D₅₃⋊C₈	C₂³	1	4	yes	—
49	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×C₅₃	C₂×C₄	1	4	yes	complemented
52	C₂×#88322	C₂×C₄	1	1	yes	characteristic
53	C₂×#88322	C₂×C₄	1	1	yes	characteristic
54	C₂×#88322	C₂×C₄	1	4	yes	complemented
55	C₈×D₅₃	C₂×C₄	1	2	yes	—
56	C₈×D₅₃	C₂×C₄	1	2	yes	—
57	C₈×D₅₃	C₂×C₄	1	4	yes	complemented
58	C₈×D₅₃	C₂×C₄	1	4	yes	complemented
59	C₈×D₅₃	C₂³	1	2	yes	—
60	C₈×D₅₃	C₂³	1	2	yes	—
61	C₂³×C₅₃	C₄²	1	1	yes	characteristic
62	C₂²×D₅₃	—	2	4	no	—
63	C₂²×D₅₃	C₄²	1	1	yes	characteristic
64	C₂²×D₅₃	C₄²	1	2	yes	—
65	C₂²×D₅₃	C₂²×C₄	1	2	yes	—
66	C₂²×D₅₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×#6157	—	2	8	no	—
68	C₂×#6157	C₂²×C₄	1	8	yes	—
69	C₂×C₄×C₅₃	C₄²	1	2	yes	—
70	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
73	C₂×Dic₅₃	C₄²	1	2	yes	—
74	C₂×Dic₅₃	C₄²	1	1	yes	characteristic
75	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
77	C₄×D₅₃	C₄²	1	2	yes	—
78	C₄×D₅₃	C₄²	1	2	yes	—
79	C₄×D₅₃	C₄²	1	2	yes	—
80	C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₄×D₅₃	C₂²×C₄	1	2	yes	—
82	C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₄×D₅₃	C₂²×C₄	1	2	yes	—
84	C₄×D₅₃	C₂²×C₄	1	2	yes	—
85	C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₄×D₅₃	C₂⁴	1	1	yes	characteristic
87	#88324	C₂²×C₄	1	8	yes	—
88	C₈×C₅₃	C₂²×C₄	1	2	yes	—
89	C₈×C₅₃	C₂²×C₄	1	2	yes	—
90	#88322	C₂²×C₄	1	2	yes	—
91	#88322	C₂²×C₄	1	2	yes	—
92	C₂²×C₅₃	—	2	2	no	—
93	C₂²×C₅₃	C₂×C₄²	1	2	yes	—
94	C₂²×C₅₃	C₂×C₄²	1	1	yes	characteristic
95	C₂×D₅₃	—	2	2	no	—
96	C₂×D₅₃	—	2	4	no	—
97	C₂×D₅₃	—	2	4	no	—
98	C₂×D₅₃	C₂×C₄²	1	2	yes	—
99	C₂×D₅₃	C₂×C₄²	1	1	yes	characteristic
100	C₂×D₅₃	#131	1	4	yes	—
101	C₂×D₅₃	C₂³×C₄	1	1	yes	characteristic
102	#6157	—	2	8	no	—
103	C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
104	C₄×C₅₃	C₂×C₄²	1	2	yes	—
105	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
106	Dic₅₃	C₂×C₄²	1	1	yes	characteristic
107	Dic₅₃	C₂×C₄²	1	2	yes	—
108	Dic₅₃	C₂³×C₄	1	1	yes	characteristic
109	C₂×#405	—	53	1	no	maximal, Sylow
110	C₂×C₅₃	—	2	2	no	—
111	C₂×C₅₃	#405	1	2	yes	complemented
112	C₂×C₅₃	C₂²×C₄²	1	1	yes	characteristic
113	D₅₃	—	2	2	no	—
114	D₅₃	#405	1	2	yes	complemented
115	D₅₃	C₂×#131	1	2	yes	—
116	C₂²×M₁₆	—	53	1	no	—
117	C₂³×C₈	—	53	1	no	—
118	C₂×#117	—	53	1	no	—
119	#405	—	53	8	no	—
120	C₂×#97	—	53	2	no	—
121	C₂×C₄×C₈	—	53	2	no	—
122	C₅₃	C₂×#405	1	1	yes	characteristic, complemented, Sylow
123	C₂³×C₄	—	53	1	no	—
124	C₂×M₁₆	—	53	4	no	—
125	C₂×M₁₆	—	53	2	no	—
126	C₂×M₁₆	—	53	4	no	—
127	C₂×M₁₆	—	53	2	no	—
128	C₂²×C₈	—	53	2	no	—
129	C₂²×C₈	—	53	1	no	—
130	C₂²×C₈	—	53	1	no	—
131	C₂²×C₈	—	53	1	no	—
132	C₂²×C₈	—	53	1	no	—
133	C₂²×C₈	—	53	1	no	—
134	C₂²×C₈	—	53	4	no	—
135	C₂²×C₈	—	53	4	no	—
136	C₂²×C₈	#6157	1	1	yes	characteristic, complemented
137	#117	—	53	4	no	—
138	#117	—	53	4	no	—
139	C₂×C₄⋊C₄	—	53	2	no	—
140	C₂×C₂²⋊C₄	—	53	2	no	—
141	C₂×C₄²	—	53	2	no	—
142	#97	—	53	8	no	—
143	C₄×C₈	—	53	8	no	—
144	C₂⁴	—	53	1	no	—
145	C₂²×C₄	—	53	1	no	—
146	C₂²×C₄	—	53	1	no	—
147	C₂²×C₄	—	53	2	no	—
148	C₂²×C₄	—	53	1	no	—
149	C₂²×C₄	—	53	2	no	—
150	C₂²×C₄	—	53	1	no	—
151	C₂²×C₄	—	53	2	no	—
152	C₂²×C₄	—	53	1	no	—
153	C₂²×C₄	—	53	2	no	—
154	C₂²×C₄	—	53	4	no	—
155	C₂²×C₄	C₂×#6157	1	1	yes	characteristic
156	M₁₆	—	53	8	no	—
157	M₁₆	—	53	8	no	—
158	C₂×C₈	—	53	4	no	—
159	C₂×C₈	—	53	4	no	—
160	C₂×C₈	—	53	4	no	—
161	C₂×C₈	—	53	1	no	—
162	C₂×C₈	—	53	1	no	—
163	C₂×C₈	—	53	4	no	—
164	C₂×C₈	—	53	2	no	—
165	C₂×C₈	—	53	2	no	—
166	C₂×C₈	—	53	2	no	—
167	C₂×C₈	—	53	2	no	—
168	C₂×C₈	—	53	4	no	—
169	C₂×C₈	—	53	4	no	—
170	C₂×C₈	C₂×#6157	1	1	yes	characteristic
171	C₂×C₈	C₂×#6157	1	1	yes	characteristic
172	C₂×C₈	C₂×#6157	1	4	yes	complemented
173	C₄⋊C₄	—	53	8	no	—
174	C₂²⋊C₄	—	53	8	no	—
175	C₄²	—	53	4	no	—
176	C₄²	—	53	4	no	—
177	C₂³	—	53	1	no	—
178	C₂³	—	53	2	no	—
179	C₂³	—	53	1	no	—
180	C₂³	—	53	2	no	—
181	C₂³	—	106	4	no	—
182	C₂³	C₄×#6157	1	1	yes	characteristic
183	C₂×C₄	—	53	2	no	—
184	C₂×C₄	—	53	1	no	—
185	C₂×C₄	—	53	1	no	—
186	C₂×C₄	—	53	2	no	—
187	C₂×C₄	—	53	2	no	—
188	C₂×C₄	—	53	2	no	—
189	C₂×C₄	—	53	2	no	—
190	C₂×C₄	—	53	1	no	—
191	C₂×C₄	—	53	2	no	—
192	C₂×C₄	—	53	1	no	—
193	C₂×C₄	—	53	2	no	—
194	C₂×C₄	—	53	1	no	—
195	C₂×C₄	—	53	2	no	—
196	C₂×C₄	—	53	1	no	—
197	C₂×C₄	—	53	8	no	—
198	C₂×C₄	—	106	8	no	—
199	C₂×C₄	C₄×#6157	1	2	yes	—
200	C₂×C₄	C₄×#6157	1	1	yes	characteristic
201	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
202	C₂×C₄	C₂²×#6157	1	2	yes	—
203	C₈	—	53	8	no	—
204	C₈	—	53	2	no	—
205	C₈	—	53	2	no	—
206	C₈	C₂²×#6157	1	2	yes	—
207	C₈	C₂²×#6157	1	2	yes	—
208	C₂²	—	2	2	no	—
209	C₂²	—	53	4	no	—
210	C₂²	—	53	1	no	—
211	C₂²	—	53	1	no	—
212	C₂²	—	53	2	no	—
213	C₂²	—	106	4	no	—
214	C₂²	—	106	4	no	—
215	C₂²	—	106	2	no	—
216	C₂²	C₂×C₄×#6157	1	2	yes	—
217	C₂²	C₂×C₄×#6157	1	1	yes	characteristic
218	C₄	—	53	1	no	—
219	C₄	—	53	1	no	—
220	C₄	—	53	2	no	—
221	C₄	—	106	8	no	—
222	C₄	C₂×C₄×#6157	1	1	yes	characteristic
223	C₄	C₂×C₄×#6157	1	2	yes	—
224	C₄	C₂³×#6157	1	1	yes	characteristic
225	C₂	—	2	2	no	—
226	C₂	—	53	2	no	—
227	C₂	—	53	2	no	—
228	C₂	—	106	2	no	—
229	C₂	#16811586	1	2	yes	direct product factor
230	C₂	C₂²×C₄×#6157	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×#6157
commutator	C₂×C₅₃	C₂²×C₄²
Frattini	C₄	C₂³×#6157
Fitting	C₂²×C₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₄²
hypercenter	C₂²×C₈	#6157
hypocenter	C₅₃	C₂×#405
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
3. #16811586	C₂	direct (N × H)
4. C₂×#13133271	C₂	semidirect
5. C₂×C₈×#6157	C₂	semidirect
7. C₂×#11719936	C₄	semidirect
8. D₅₃⋊M₁₆	C₄	semidirect
10. C₂×#11719940	C₄	semidirect
17. #13133271	C₂²	semidirect
18. C₈×#6157	C₂²	semidirect
20. C₂²×#88322	C₄	semidirect
21. C₂×C₈×D₅₃	C₄	semidirect
22. C₂×C₈×D₅₃	C₄	semidirect
23. C₂×C₈×D₅₃	C₄	semidirect
27. C₂²×C₈×C₅₃	C₄	semidirect
41. #11719940	C₂×C₄	semidirect
46. #11719936	C₂×C₄	semidirect
51. C₂×C₈×C₅₃	C₂×C₄	semidirect
54. C₂×#88322	C₂×C₄	semidirect
57. C₈×D₅₃	C₂×C₄	semidirect
58. C₈×D₅₃	C₂×C₄	semidirect
111. C₂×C₅₃	#405	semidirect
114. D₅₃	#405	semidirect
122. C₅₃	C₂×#405	semidirect
136. C₂²×C₈	#6157	semidirect
172. C₂×C₈	C₂×#6157	semidirect
229. C₂	#16811586	direct (N × H)

Subgroups — C2×#16811586 #20291273

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16811586 #20291273