Subgroups — #20283666

Order 6784.

388 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
2	#16810752	C₂	1	4	yes	maximal, complemented
3	#16810750	C₂	1	2	yes	maximal, complemented
4	#16810747	C₂	1	2	yes	maximal, complemented
5	#16810646	C₂	1	2	yes	maximal
6	C₄×#11719917	C₂	1	1	yes	maximal, characteristic
7	M₁₆×Dic₅₃	C₂	1	2	yes	maximal
8	C₅₃×#423	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₀₆	C₄	1	1	yes	characteristic
10	C₂×#11719917	C₂²	1	1	yes	characteristic
11	C₂×#11719917	C₂²	1	1	yes	characteristic
12	C₄×Dic₁₀₆	C₄	1	1	yes	characteristic
13	C₄×Dic₁₀₆	C₄	1	2	yes	—
14	C₄×Dic₁₀₆	C₄	1	1	yes	characteristic
15	C₄×Dic₁₀₆	C₂²	1	2	yes	—
16	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
17	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₅₃	C₄	1	1	yes	characteristic
19	C₂×#11719919	C₄	1	2	yes	—
20	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
21	#13133146	C₂²	1	4	yes	—
22	#13133146	C₂²	1	4	yes	—
23	#13133128	C₂²	1	2	yes	—
24	#13133127	C₂²	1	4	yes	—
25	C₄×#88322	C₂²	1	2	yes	—
26	C₈×Dic₅₃	C₂²	1	4	yes	—
27	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
28	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
29	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
30	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
31	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
34	C₂²×Dic₅₃	C₂×C₄	1	2	yes	—
35	C₂²×Dic₅₃	C₂×C₄	1	2	yes	—
36	C₂×Dic₁₀₆	—	2	4	no	—
37	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
38	C₂×Dic₁₀₆	C₂×C₄	1	2	yes	—
39	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
40	M₁₆×C₅₃	—	2	4	no	—
41	C₂×C₈×C₅₃	C₂³	1	4	yes	—
42	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₄²×C₅₃	C₂³	1	1	yes	characteristic
45	C₄²×C₅₃	C₂³	1	1	yes	characteristic
46	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
47	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
48	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
49	#11719919	C₂×C₄	1	2	yes	—
50	#11719919	C₂×C₄	1	2	yes	—
51	#11719919	C₂×C₄	1	2	yes	—
52	#11719919	C₂×C₄	1	2	yes	—
53	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
54	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
55	C₄×Dic₅₃	C₂³	1	2	yes	—
56	C₄×Dic₅₃	C₂³	1	2	yes	—
57	#11719917	—	2	2	no	—
58	#11719917	Q₈	1	4	yes	complemented
59	C₂×#88322	C₂³	1	2	yes	—
60	C₂×#88322	C₂³	1	2	yes	—
61	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₅₃	—	2	2	no	—
63	C₂×C₄×C₅₃	—	4	1	no	—
64	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
70	C₂×C₄×C₅₃	#41	1	2	yes	—
71	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
72	C₂×Dic₅₃	—	2	4	no	—
73	C₂×Dic₅₃	—	4	2	no	—
74	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
75	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
76	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
77	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
78	Dic₁₀₆	—	2	2	no	—
79	Dic₁₀₆	—	2	4	no	—
80	Dic₁₀₆	M₁₆	1	4	yes	complemented
81	C₈×C₅₃	—	2	4	no	—
82	#88322	—	2	2	no	—
83	#88322	C₂×Q₈	1	4	yes	complemented
84	C₂²×C₅₃	—	2	1	no	—
85	C₂²×C₅₃	—	2	1	no	—
86	C₂²×C₅₃	C₄×Q₈	1	2	yes	—
87	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
88	C₄×C₅₃	—	2	1	no	—
89	C₄×C₅₃	—	2	1	no	—
90	C₄×C₅₃	—	2	1	no	—
91	C₄×C₅₃	C₄×Q₈	1	2	yes	—
92	C₄×C₅₃	C₂×M₁₆	1	2	yes	—
93	C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
94	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
95	Dic₅₃	—	2	2	no	—
96	Dic₅₃	—	2	2	no	—
97	Dic₅₃	—	2	2	no	—
98	Dic₅₃	C₂×M₁₆	1	4	yes	—
99	Q₈×M₁₆	—	53	1	no	maximal, Sylow
100	C₂×C₅₃	—	2	2	no	—
101	C₂×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
102	C₂×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
103	C₂×C₅₃	#568	1	1	yes	characteristic
104	C₂×C₄×Q₈	—	53	1	no	—
105	#446	—	53	4	no	—
106	#446	—	53	2	no	—
107	C₈×Q₈	—	53	2	no	—
108	#423	—	53	2	no	—
109	#423	D₅₃	1	1	yes	characteristic
110	C₄×M₁₆	—	53	2	no	—
111	C₄×M₁₆	—	53	1	no	—
112	C₅₃	Q₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
113	C₂²×Q₈	—	53	1	no	—
114	C₂×M₁₆	—	53	1	no	—
115	C₂×M₁₆	—	53	1	no	—
116	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
117	C₄×Q₈	—	53	1	no	—
118	C₄×Q₈	—	53	2	no	—
119	C₄×Q₈	—	53	1	no	—
120	C₄×Q₈	—	53	1	no	—
121	C₄×Q₈	—	53	2	no	—
122	C₄×Q₈	—	53	1	no	—
123	C₂×C₄⋊C₄	—	53	2	no	—
124	C₂×C₄⋊C₄	—	53	1	no	—
125	C₂×C₄²	—	53	2	no	—
126	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
127	C₄⋊C₈	—	53	4	no	—
128	C₄⋊C₈	—	53	4	no	—
129	C₄⋊C₈	C₂×D₅₃	1	2	yes	—
130	C₄⋊C₈	C₂×D₅₃	1	2	yes	—
131	#97	—	53	4	no	—
132	#97	—	53	2	no	—
133	C₄×C₈	—	53	4	no	—
134	C₄×C₈	—	53	2	no	—
135	C₂×Q₈	—	53	1	no	—
136	C₂×Q₈	—	53	2	no	—
137	C₂×Q₈	—	53	1	no	—
138	C₂×Q₈	—	106	4	no	—
139	C₂²×C₄	—	53	2	no	—
140	C₂²×C₄	—	53	2	no	—
141	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
142	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
143	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
144	M₁₆	—	2	4	no	—
145	M₁₆	—	53	4	no	—
146	M₁₆	—	106	2	no	—
147	C₂×C₈	—	53	2	no	—
148	C₂×C₈	—	53	2	no	—
149	C₂×C₈	C₂²×D₅₃	1	4	yes	—
150	C₄⋊C₄	—	53	2	no	—
151	C₄⋊C₄	—	53	2	no	—
152	C₄⋊C₄	—	53	2	no	—
153	C₄⋊C₄	—	53	2	no	—
154	C₄⋊C₄	—	53	2	no	—
155	C₄⋊C₄	—	53	1	no	—
156	C₄⋊C₄	—	53	1	no	—
157	C₄²	—	53	2	no	—
158	C₄²	—	53	2	no	—
159	C₄²	—	53	2	no	—
160	C₄²	—	53	2	no	—
161	C₄²	C₄×D₅₃	1	1	yes	characteristic
162	C₄²	C₄×D₅₃	1	1	yes	characteristic
163	C₄²	C₂²×D₅₃	1	1	yes	characteristic
164	C₄²	C₂²×D₅₃	1	1	yes	characteristic
165	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
166	Q₈	—	53	4	no	—
167	Q₈	—	106	4	no	—
168	Q₈	—	106	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	4	1	no	—
171	C₂×C₄	—	53	2	no	—
172	C₂×C₄	—	53	2	no	—
173	C₂×C₄	—	53	2	no	—
174	C₂×C₄	—	53	2	no	—
175	C₂×C₄	—	106	4	no	—
176	C₂×C₄	—	212	2	no	—
177	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
178	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
179	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
180	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
181	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
182	C₂×C₄	#11719947	1	2	yes	—
183	C₂×C₄	Q₈×D₅₃	1	2	yes	—
184	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
185	C₈	—	2	4	no	—
186	C₈	—	53	4	no	—
187	C₈	—	106	2	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	2	1	no	—
190	C₂²	#13133182	1	2	yes	—
191	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
192	C₄	—	2	1	no	—
193	C₄	—	2	1	no	—
194	C₄	—	2	1	no	—
195	C₄	—	53	4	no	—
196	C₄	—	106	2	no	—
197	C₄	—	106	2	no	—
198	C₄	—	106	2	no	—
199	C₄	#13133182	1	2	yes	—
200	C₄	M₁₆×D₅₃	1	2	yes	—
201	C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
202	C₄	C₂×#11719947	1	1	yes	characteristic
203	C₂	—	2	2	no	—
204	C₂	C₂×#13133182	1	1	yes	characteristic
205	C₂	C₂×M₁₆×D₅₃	1	1	yes	characteristic
206	C₂	#16811389	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₅₃×#423	C₂
hypercenter	#423	D₅₃
hypocenter	C₅₃	Q₈×M₁₆
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #423

Complements

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

N	H ≅ G/N	type
2. #16810752	C₂	semidirect
3. #16810750	C₂	semidirect
4. #16810747	C₂	semidirect
58. #11719917	Q₈	semidirect
80. Dic₁₀₆	M₁₆	semidirect
83. #88322	C₂×Q₈	semidirect
112. C₅₃	Q₈×M₁₆	semidirect