Subgroups — C₂×#16811589 #20291277

Order 6784.

592 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₅₃⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133296	C₂	1	1	yes	maximal, characteristic
3	#16811589	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	C₂×M₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂×D₅₃⋊C₈	C₄	1	2	yes	complemented
8	C₂×D₅₃⋊C₈	C₄	1	4	yes	complemented
9	C₂×D₅₃⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×D₅₃⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×D₅₃⋊C₈	C₂²	1	4	yes	—
12	C₂²×#88324	C₄	1	2	yes	complemented
13	#13133296	C₄	1	4	yes	—
14	#13133296	C₂²	1	4	yes	—
15	C₂×C₄⋊#6157	C₄	1	2	yes	—
16	C₂×C₂²⋊#6157	C₄	1	2	yes	—
17	C₂×C₄×#6157	C₂²	1	2	yes	—
18	#13133271	C₂²	1	8	yes	complemented
19	C₈×#6157	C₂²	1	8	yes	complemented
20	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
21	C₂×#11719917	C₄	1	1	yes	characteristic, complemented
22	C₂×#11719912	C₄	1	2	yes	complemented
23	M₁₆×D₅₃	C₄	1	4	yes	complemented
24	M₁₆×D₅₃	C₂²	1	4	yes	—
25	C₂×C₈×D₅₃	C₂²	1	2	yes	—
26	C₂×M₁₆×C₅₃	C₄	1	1	yes	characteristic, complemented
27	C₂³×D₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×#6157	C₂×C₄	1	4	yes	—
29	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
33	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
34	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
36	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
38	C₂×C₄×D₅₃	C₂³	1	2	yes	—
39	C₂²⋊#6157	C₂×C₄	1	8	yes	—
40	C₂×#88324	C₂×C₄	1	2	yes	—
41	C₂×#88324	C₂×C₄	1	2	yes	—
42	C₂×#88324	C₂×C₄	1	8	yes	complemented
43	C₄⋊#6157	C₂×C₄	1	8	yes	—
44	C₄×#6157	C₂×C₄	1	4	yes	—
45	C₄×#6157	C₂³	1	4	yes	—
46	D₅₃⋊C₈	C₂×C₄	1	2	yes	—
47	D₅₃⋊C₈	C₂×C₄	1	2	yes	—
48	D₅₃⋊C₈	C₂×C₄	1	8	yes	complemented
49	D₅₃⋊C₈	C₂³	1	2	yes	—
50	D₅₃⋊C₈	C₂³	1	2	yes	—
51	M₁₆×C₅₃	C₂×C₄	1	4	yes	complemented
52	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
53	#11719917	C₂×C₄	1	4	yes	complemented
54	C₂×#88322	C₂×C₄	1	2	yes	—
55	#11719912	C₂×C₄	1	8	yes	complemented
56	C₈×D₅₃	C₂×C₄	1	4	yes	—
57	C₈×D₅₃	C₂³	1	4	yes	—
58	C₂³×C₅₃	C₄²	1	1	yes	characteristic
59	C₂²×D₅₃	—	2	4	no	—
60	C₂²×D₅₃	C₄²	1	1	yes	characteristic
61	C₂²×D₅₃	C₄²	1	2	yes	—
62	C₂²×D₅₃	C₂²×C₄	1	2	yes	—
63	C₂²×D₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂×#6157	—	2	8	no	—
65	C₂×#6157	C₂²×C₄	1	8	yes	—
66	C₂×C₄×C₅₃	C₄²	1	2	yes	—
67	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
69	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×Dic₅₃	C₄²	1	2	yes	—
71	C₂×Dic₅₃	C₄²	1	1	yes	characteristic
72	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
74	C₄×D₅₃	C₄²	1	2	yes	—
75	C₄×D₅₃	C₄²	1	2	yes	—
76	C₄×D₅₃	C₄²	1	2	yes	—
77	C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
78	C₄×D₅₃	C₂²×C₄	1	2	yes	—
79	C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₄×D₅₃	C₂²×C₄	1	2	yes	—
81	C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₄×D₅₃	C₂²×C₄	1	2	yes	—
83	C₄×D₅₃	C₂⁴	1	1	yes	characteristic
84	#88324	C₂²×C₄	1	4	yes	—
85	#88324	C₂²×C₄	1	4	yes	—
86	C₈×C₅₃	C₂²×C₄	1	4	yes	—
87	#88322	C₂²×C₄	1	4	yes	—
88	C₂²×C₅₃	—	2	2	no	—
89	C₂²×C₅₃	C₂×C₄²	1	2	yes	—
90	C₂²×C₅₃	C₂×C₄²	1	1	yes	characteristic
91	C₂×D₅₃	—	2	2	no	—
92	C₂×D₅₃	—	2	4	no	—
93	C₂×D₅₃	—	2	4	no	—
94	C₂×D₅₃	C₂×C₄²	1	2	yes	—
95	C₂×D₅₃	C₂×C₄²	1	1	yes	characteristic
96	C₂×D₅₃	#131	1	4	yes	—
97	C₂×D₅₃	C₂³×C₄	1	1	yes	characteristic
98	#6157	—	2	8	no	—
99	C₄×C₅₃	C₂×C₄²	1	2	yes	—
100	C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
101	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
102	Dic₅₃	C₂×C₄²	1	1	yes	characteristic
103	Dic₅₃	C₂×C₄²	1	2	yes	—
104	Dic₅₃	C₂³×C₄	1	1	yes	characteristic
105	C₂×#405	—	53	1	no	maximal, Sylow
106	C₂×C₅₃	—	2	2	no	—
107	C₂×C₅₃	#405	1	2	yes	complemented
108	C₂×C₅₃	C₂²×C₄²	1	1	yes	characteristic
109	D₅₃	—	2	2	no	—
110	D₅₃	#405	1	2	yes	complemented
111	D₅₃	C₂×#131	1	2	yes	—
112	C₂²×M₁₆	—	53	1	no	—
113	C₂³×C₈	—	53	1	no	—
114	C₂×#117	—	53	1	no	—
115	#405	—	53	8	no	—
116	C₂×#97	—	53	2	no	—
117	C₂×C₄×C₈	—	53	2	no	—
118	C₅₃	C₂×#405	1	1	yes	characteristic, complemented, Sylow
119	C₂³×C₄	—	53	1	no	—
120	C₂×M₁₆	—	53	1	no	—
121	C₂×M₁₆	—	53	4	no	—
122	C₂×M₁₆	—	53	4	no	—
123	C₂×M₁₆	—	53	2	no	—
124	C₂×M₁₆	#6157	1	1	yes	characteristic, complemented
125	C₂²×C₈	—	53	1	no	—
126	C₂²×C₈	—	53	1	no	—
127	C₂²×C₈	—	53	4	no	—
128	C₂²×C₈	—	53	2	no	—
129	C₂²×C₈	—	53	2	no	—
130	C₂²×C₈	—	53	4	no	—
131	C₂²×C₈	—	53	2	no	—
132	#117	—	53	4	no	—
133	#117	—	53	4	no	—
134	C₂×C₄⋊C₄	—	53	2	no	—
135	C₂×C₂²⋊C₄	—	53	2	no	—
136	C₂×C₄²	—	53	2	no	—
137	#97	—	53	8	no	—
138	C₄×C₈	—	53	8	no	—
139	C₂⁴	—	53	1	no	—
140	C₂²×C₄	—	53	1	no	—
141	C₂²×C₄	—	53	1	no	—
142	C₂²×C₄	—	53	2	no	—
143	C₂²×C₄	—	53	2	no	—
144	C₂²×C₄	—	53	1	no	—
145	C₂²×C₄	—	53	2	no	—
146	C₂²×C₄	—	53	2	no	—
147	C₂²×C₄	—	53	1	no	—
148	C₂²×C₄	—	53	1	no	—
149	C₂²×C₄	—	53	4	no	—
150	C₂²×C₄	C₂×#6157	1	1	yes	characteristic
151	M₁₆	—	53	4	no	—
152	M₁₆	—	53	8	no	—
153	M₁₆	C₂×#6157	1	4	yes	complemented
154	C₂×C₈	—	53	2	no	—
155	C₂×C₈	—	53	2	no	—
156	C₂×C₈	—	53	8	no	—
157	C₂×C₈	—	53	2	no	—
158	C₂×C₈	—	53	2	no	—
159	C₂×C₈	—	53	2	no	—
160	C₂×C₈	—	53	2	no	—
161	C₂×C₈	—	53	8	no	—
162	C₂×C₈	—	53	2	no	—
163	C₂×C₈	—	53	4	no	—
164	C₂×C₈	—	53	4	no	—
165	C₂×C₈	C₂×#6157	1	2	yes	—
166	C₄⋊C₄	—	53	8	no	—
167	C₂²⋊C₄	—	53	8	no	—
168	C₄²	—	53	4	no	—
169	C₄²	—	53	4	no	—
170	C₂³	—	53	1	no	—
171	C₂³	—	53	2	no	—
172	C₂³	—	53	1	no	—
173	C₂³	—	53	2	no	—
174	C₂³	—	106	4	no	—
175	C₂³	C₄×#6157	1	1	yes	characteristic
176	C₂×C₄	—	53	2	no	—
177	C₂×C₄	—	53	1	no	—
178	C₂×C₄	—	53	2	no	—
179	C₂×C₄	—	53	1	no	—
180	C₂×C₄	—	53	2	no	—
181	C₂×C₄	—	53	2	no	—
182	C₂×C₄	—	53	2	no	—
183	C₂×C₄	—	53	1	no	—
184	C₂×C₄	—	53	2	no	—
185	C₂×C₄	—	53	2	no	—
186	C₂×C₄	—	53	1	no	—
187	C₂×C₄	—	53	1	no	—
188	C₂×C₄	—	53	2	no	—
189	C₂×C₄	—	53	1	no	—
190	C₂×C₄	—	53	8	no	—
191	C₂×C₄	—	106	8	no	—
192	C₂×C₄	C₄×#6157	1	1	yes	characteristic
193	C₂×C₄	C₄×#6157	1	2	yes	—
194	C₂×C₄	C₂²×#6157	1	2	yes	—
195	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
196	C₈	—	53	4	no	—
197	C₈	—	53	4	no	—
198	C₈	—	53	4	no	—
199	C₈	C₂²×#6157	1	4	yes	—
200	C₂²	—	2	2	no	—
201	C₂²	—	53	4	no	—
202	C₂²	—	53	1	no	—
203	C₂²	—	53	1	no	—
204	C₂²	—	53	2	no	—
205	C₂²	—	106	4	no	—
206	C₂²	—	106	4	no	—
207	C₂²	—	106	2	no	—
208	C₂²	C₂×C₄×#6157	1	2	yes	—
209	C₂²	C₂×C₄×#6157	1	1	yes	characteristic
210	C₄	—	53	1	no	—
211	C₄	—	53	1	no	—
212	C₄	—	53	2	no	—
213	C₄	—	106	8	no	—
214	C₄	C₂×C₄×#6157	1	2	yes	—
215	C₄	C₂×C₄×#6157	1	1	yes	characteristic
216	C₄	C₂³×#6157	1	1	yes	characteristic
217	C₂	—	2	2	no	—
218	C₂	—	53	2	no	—
219	C₂	—	53	2	no	—
220	C₂	—	106	2	no	—
221	C₂	#16811589	1	2	yes	direct product factor
222	C₂	C₂²×C₄×#6157	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×#6157
commutator	C₂×C₅₃	C₂²×C₄²
Frattini	C₄	C₂³×#6157
Fitting	C₂×M₁₆×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₄²
hypercenter	C₂×M₁₆	#6157
hypocenter	C₅₃	C₂×#405
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. #16811589	C₂	direct (N × H)
4. C₂×#13133271	C₂	semidirect
5. C₂×C₈×#6157	C₂	semidirect
7. C₂×D₅₃⋊C₈	C₄	semidirect
8. C₂×D₅₃⋊C₈	C₄	semidirect
12. C₂²×#88324	C₄	semidirect
18. #13133271	C₂²	semidirect
19. C₈×#6157	C₂²	semidirect
21. C₂×#11719917	C₄	semidirect
22. C₂×#11719912	C₄	semidirect
23. M₁₆×D₅₃	C₄	semidirect
26. C₂×M₁₆×C₅₃	C₄	semidirect
42. C₂×#88324	C₂×C₄	semidirect
48. D₅₃⋊C₈	C₂×C₄	semidirect
51. M₁₆×C₅₃	C₂×C₄	semidirect
53. #11719917	C₂×C₄	semidirect
55. #11719912	C₂×C₄	semidirect
107. C₂×C₅₃	#405	semidirect
110. D₅₃	#405	semidirect
118. C₅₃	C₂×#405	semidirect
124. C₂×M₁₆	#6157	semidirect
153. M₁₆	C₂×#6157	semidirect
221. C₂	#16811589	direct (N × H)

Subgroups — C2×#16811589 #20291277

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16811589 #20291277