Subgroups — C₂×#16811334 #20288250

Order 6784.

1200 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811334	C₂	1	16	yes	maximal, direct product factor
2	C₂×#13133208	C₂	1	2	yes	maximal, complemented
3	C₂×#13133197	C₂	1	2	yes	maximal, complemented
4	C₂×#13133193	C₂	1	4	yes	maximal, complemented
5	C₂×C₄×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₂₁₂	C₂	1	1	yes	maximal, characteristic
7	C₂×#13133179	C₂	1	2	yes	maximal
8	C₂×C₄⋊C₄×D₅₃	C₂	1	2	yes	maximal, complemented
9	C₂×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Dic₁₀₆	C₂²	1	1	yes	characteristic
11	C₂²×D₂₁₂	C₂²	1	1	yes	characteristic
12	C₂²×C₄×D₅₃	C₂²	1	2	yes	—
13	C₂²×C₄×D₅₃	C₂²	1	4	yes	—
14	#13133208	C₂²	1	16	yes	complemented
15	#13133197	C₂²	1	16	yes	complemented
16	#13133193	C₂²	1	32	yes	complemented
17	C₄×Dic₁₀₆	C₂²	1	8	yes	complemented
18	C₄×D₂₁₂	C₂²	1	8	yes	—
19	#13133179	C₂²	1	16	yes	—
20	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Dic₅₃	C₂²	1	4	yes	—
23	C₂×#11719919	C₂²	1	4	yes	—
24	C₂×#11719919	C₂²	1	2	yes	—
25	C₄⋊C₄×D₅₃	C₂²	1	16	yes	complemented
26	C₂×#11719921	C₂²	1	4	yes	—
27	C₂×#11719921	C₂²	1	2	yes	—
28	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
29	C₂²×Q₈×C₅₃	C₂²	1	2	yes	—
30	C₄⋊Q₈×C₅₃	C₂²	1	8	yes	complemented
31	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
32	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
33	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
34	C₂³×D₅₃	C₂³	1	2	yes	—
35	C₂×Q₈×C₅₃	—	4	4	no	—
36	C₂×Q₈×C₅₃	C₂³	1	8	yes	—
37	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
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×Dic₅₃	C₂³	1	2	yes	—
42	C₂²×Dic₅₃	C₂³	1	2	yes	—
43	C₂²×Dic₅₃	C₂³	1	2	yes	—
44	C₂×D₂₁₂	Q₈	1	8	yes	complemented
45	C₂×D₂₁₂	C₂³	1	4	yes	—
46	C₂×C₄×D₅₃	—	2	8	no	—
47	C₂×C₄×D₅₃	—	4	8	no	—
48	C₂×C₄×D₅₃	C₂³	1	8	yes	—
49	C₂×C₄×D₅₃	C₂³	1	16	yes	—
50	C₂×Dic₁₀₆	—	2	4	no	—
51	C₂×Dic₁₀₆	C₂³	1	4	yes	—
52	C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
53	C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
54	C₄²×C₅₃	C₂³	1	4	yes	—
55	#11719921	C₂³	1	16	yes	—
56	#11719921	C₂³	1	8	yes	—
57	C₄⋊Dic₅₃	C₂³	1	4	yes	—
58	C₄⋊Dic₅₃	C₂³	1	4	yes	—
59	C₄⋊Dic₅₃	C₂³	1	16	yes	—
60	#11719919	C₂³	1	16	yes	—
61	#11719919	C₂³	1	8	yes	—
62	C₄×Dic₅₃	C₂³	1	8	yes	—
63	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
64	C₂²×D₅₃	—	2	4	no	—
65	C₂²×D₅₃	—	2	4	no	—
66	C₂²×D₅₃	C₂×Q₈	1	8	yes	—
67	C₂²×D₅₃	C₂⁴	1	4	yes	—
68	Q₈×C₅₃	—	4	8	no	—
69	C₂×C₄×C₅₃	—	2	8	no	—
70	C₂×C₄×C₅₃	—	2	2	no	—
71	C₂×C₄×C₅₃	C₂×Q₈	1	4	yes	—
72	C₂×C₄×C₅₃	#41	1	4	yes	—
73	C₂×C₄×C₅₃	C₂⁴	1	8	yes	—
74	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
75	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
76	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
77	C₂×Dic₅₃	—	2	4	no	—
78	C₂×Dic₅₃	—	2	4	no	—
79	C₂×Dic₅₃	—	2	4	no	—
80	C₂×Dic₅₃	C₂⁴	1	4	yes	—
81	C₂×Dic₅₃	C₂⁴	1	4	yes	—
82	C₂×Dic₅₃	C₂⁴	1	4	yes	—
83	D₂₁₂	C₂×Q₈	1	16	yes	complemented
84	C₄×D₅₃	—	2	16	no	—
85	C₄×D₅₃	—	4	16	no	—
86	Dic₁₀₆	—	2	8	no	—
87	C₂²×C₅₃	C₂²×Q₈	1	2	yes	—
88	C₂²×C₅₃	C₂×#41	1	2	yes	—
89	C₂²×C₅₃	H(5,𝔽₂)	1	2	yes	—
90	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
91	C₂×D₅₃	—	2	16	no	—
92	C₂×D₅₃	—	2	4	no	—
93	C₂×D₅₃	—	2	4	no	—
94	C₂×D₅₃	C₂²×Q₈	1	8	yes	—
95	C₄×C₅₃	—	2	8	no	—
96	C₄×C₅₃	—	2	2	no	—
97	C₄×C₅₃	C₂²×Q₈	1	4	yes	—
98	C₄×C₅₃	C₂×#41	1	4	yes	—
99	Dic₅₃	—	2	4	no	—
100	Dic₅₃	—	2	4	no	—
101	Dic₅₃	—	2	4	no	—
102	C₂×#554	—	53	1	no	maximal, Sylow
103	C₂×C₅₃	#554	1	4	yes	complemented
104	C₂×C₅₃	C₂³×Q₈	1	1	yes	characteristic
105	C₂×C₅₃	C₂²×#41	1	1	yes	characteristic
106	C₂×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
107	D₅₃	—	2	8	no	—
108	#554	—	53	16	no	—
109	C₂×C₄⋊Q₈	D₅₃	1	1	yes	characteristic, complemented
110	C₂×#125	—	53	2	no	—
111	C₂×C₂²⋊Q₈	—	53	4	no	—
112	C₂×C₂²⋊Q₈	—	53	2	no	—
113	C₂×C₄×Q₈	—	53	1	no	—
114	C₂×C₄×D₄	—	53	2	no	—
115	C₂×C₄×D₄	—	53	1	no	—
116	C₂²×C₄⋊C₄	—	53	2	no	—
117	C₅₃	C₂×#554	1	1	yes	characteristic, complemented, Sylow
118	C₂²×Q₈	—	53	1	no	—
119	C₂²×Q₈	C₂×D₅₃	1	2	yes	—
120	C₂²×D₄	—	53	1	no	—
121	C₂³×C₄	—	53	2	no	—
122	C₂³×C₄	—	53	4	no	—
123	C₄⋊Q₈	C₂×D₅₃	1	8	yes	complemented
124	#125	—	53	16	no	—
125	C₂²⋊Q₈	—	53	32	no	—
126	C₂²⋊Q₈	—	53	16	no	—
127	C₄×Q₈	—	53	8	no	—
128	C₄×D₄	—	53	16	no	—
129	C₄×D₄	—	53	8	no	—
130	C₂×C₄⋊C₄	—	53	1	no	—
131	C₂×C₄⋊C₄	—	53	4	no	—
132	C₂×C₄⋊C₄	—	53	1	no	—
133	C₂×C₄⋊C₄	—	53	2	no	—
134	C₂×C₄⋊C₄	—	53	4	no	—
135	C₂×C₄⋊C₄	—	53	16	no	—
136	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
137	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
138	C₂×C₂²⋊C₄	—	53	4	no	—
139	C₂×C₂²⋊C₄	—	53	2	no	—
140	C₂×C₄²	—	53	2	no	—
141	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
142	C₂⁴	—	53	2	no	—
143	C₂×Q₈	—	4	4	no	—
144	C₂×Q₈	—	53	4	no	—
145	C₂×Q₈	—	106	4	no	—
146	C₂×Q₈	C₂²×D₅₃	1	8	yes	—
147	C₂×D₄	—	53	8	no	—
148	C₂×D₄	—	53	4	no	—
149	C₂²×C₄	—	53	2	no	—
150	C₂²×C₄	—	53	2	no	—
151	C₂²×C₄	—	53	2	no	—
152	C₂²×C₄	—	53	8	no	—
153	C₂²×C₄	—	53	16	no	—
154	C₂²×C₄	—	106	8	no	—
155	C₂²×C₄	—	212	8	no	—
156	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
157	C₂²×C₄	C₂²×D₅₃	1	4	yes	—
158	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
159	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
160	C₄⋊C₄	—	53	4	no	—
161	C₄⋊C₄	—	53	16	no	—
162	C₄⋊C₄	—	53	4	no	—
163	C₄⋊C₄	—	53	8	no	—
164	C₄⋊C₄	—	53	16	no	—
165	C₄⋊C₄	C₂²×D₅₃	1	8	yes	—
166	C₄⋊C₄	C₂²×D₅₃	1	8	yes	—
167	C₂²⋊C₄	—	53	16	no	—
168	C₂²⋊C₄	—	53	8	no	—
169	C₄²	—	53	8	no	—
170	C₄²	C₂²×D₅₃	1	4	yes	—
171	C₂³	—	53	8	no	—
172	C₂³	—	53	4	no	—
173	C₂³	—	106	4	no	—
174	C₂³	—	106	4	no	—
175	C₂³	C₂³×D₅₃	1	1	yes	characteristic
176	Q₈	—	4	8	no	—
177	Q₈	—	106	8	no	—
178	D₄	—	53	16	no	—
179	C₂×C₄	—	2	8	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	53	4	no	—
182	C₂×C₄	—	53	4	no	—
183	C₂×C₄	—	53	4	no	—
184	C₂×C₄	—	106	4	no	—
185	C₂×C₄	—	106	4	no	—
186	C₂×C₄	—	106	4	no	—
187	C₂×C₄	—	106	16	no	—
188	C₂×C₄	—	212	16	no	—
189	C₂×C₄	Q₈×D₅₃	1	4	yes	—
190	C₂×C₄	#11719949	1	4	yes	—
191	C₂×C₄	C₂³×D₅₃	1	2	yes	—
192	C₂×C₄	C₂³×D₅₃	1	8	yes	—
193	C₂×C₄	C₂³×D₅₃	1	2	yes	—
194	C₂×C₄	C₂³×D₅₃	1	2	yes	—
195	C₂²	—	53	8	no	—
196	C₂²	—	106	16	no	—
197	C₂²	—	106	4	no	—
198	C₂²	—	106	4	no	—
199	C₂²	C₂×Q₈×D₅₃	1	2	yes	—
200	C₂²	C₂×#11719949	1	2	yes	—
201	C₂²	#13133262	1	2	yes	—
202	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
203	C₄	—	2	8	no	—
204	C₄	—	2	2	no	—
205	C₄	—	106	4	no	—
206	C₄	—	106	4	no	—
207	C₄	—	106	4	no	—
208	C₄	C₂×Q₈×D₅₃	1	4	yes	—
209	C₄	C₂×#11719949	1	4	yes	—
210	C₂	—	106	8	no	—
211	C₂	#16811334	1	4	yes	direct product factor
212	C₂	C₂²×Q₈×D₅₃	1	1	yes	characteristic
213	C₂	C₂²×#11719949	1	1	yes	characteristic
214	C₂	C₂×#13133262	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂⁵
Frattini	C₂²	C₂⁴×D₅₃
Fitting	C₂×C₄⋊Q₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂⁴
hypercenter	C₂×C₄⋊Q₈	D₅₃
hypocenter	C₅₃	C₂×#554
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #16811334	C₂	direct (N × H)
2. C₂×#13133208	C₂	semidirect
3. C₂×#13133197	C₂	semidirect
4. C₂×#13133193	C₂	semidirect
5. C₂×C₄×Dic₁₀₆	C₂	semidirect
8. C₂×C₄⋊C₄×D₅₃	C₂	semidirect
9. C₂×C₄⋊Q₈×C₅₃	C₂	semidirect
14. #13133208	C₂²	semidirect
15. #13133197	C₂²	semidirect
16. #13133193	C₂²	semidirect
17. C₄×Dic₁₀₆	C₂²	semidirect
25. C₄⋊C₄×D₅₃	C₂²	semidirect
30. C₄⋊Q₈×C₅₃	C₂²	semidirect
44. C₂×D₂₁₂	Q₈	semidirect
83. D₂₁₂	C₂×Q₈	semidirect
103. C₂×C₅₃	#554	semidirect
109. C₂×C₄⋊Q₈	D₅₃	semidirect
117. C₅₃	C₂×#554	semidirect
123. C₄⋊Q₈	C₂×D₅₃	semidirect
211. C₂	#16811334	direct (N × H)

Subgroups — C2×#16811334 #20288250

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16811334 #20288250