Subgroups — #20283380

Order 6784.

392 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#11719945	C₂	1	1	yes	maximal, characteristic
2	#16810683	C₂	1	4	yes	maximal, complemented
3	#16810679	C₂	1	2	yes	maximal, complemented
4	#16810675	C₂	1	2	yes	maximal, complemented
5	#16810647	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810589	C₂	1	2	yes	maximal
7	#16810578	C₂	1	2	yes	maximal, complemented
8	C₂×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#11719945	C₂²	1	1	yes	characteristic
10	C₂×#11719917	C₂²	1	2	yes	—
11	C₄×Dic₁₀₆	C₄	1	1	yes	characteristic
12	C₄×Dic₁₀₆	C₄	1	1	yes	characteristic
13	C₄×D₂₁₂	C₄	1	1	yes	characteristic
14	C₄×D₂₁₂	C₄	1	1	yes	characteristic
15	C₄×C₂²⋊D₅₃	C₄	1	2	yes	—
16	C₄×C₂²⋊D₅₃	C₄	1	2	yes	—
17	#13133176	C₂²	1	1	yes	characteristic
18	#13133174	C₂²	1	2	yes	—
19	C₄²×D₅₃	C₂²	1	2	yes	—
20	C₄⋊#88322	C₂²	1	2	yes	—
21	C₄⋊#88322	C₂²	1	2	yes	—
22	#13133129	C₂²	1	4	yes	—
23	#13133129	C₂²	1	4	yes	—
24	#13133127	C₂²	1	4	yes	—
25	C₈×Dic₅₃	C₂²	1	4	yes	—
26	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
27	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
28	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
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	C₂×C₂²⋊D₅₃	C₂×C₄	1	2	yes	—
35	#11719945	—	2	2	no	—
36	#11719945	—	2	2	no	—
37	C₂×D₂₁₂	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
39	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
40	C₂×C₄×D₅₃	C₂³	1	2	yes	—
41	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
42	C₂×C₈×C₅₃	—	2	4	no	—
43	C₂×C₈×C₅₃	C₂³	1	2	yes	—
44	C₂×C₈×C₅₃	C₂³	1	2	yes	—
45	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₄²×C₅₃	C₂³	1	1	yes	characteristic
48	C₄²×C₅₃	C₂³	1	1	yes	characteristic
49	C₂²⋊Dic₅₃	C₂×C₄	1	2	yes	—
50	#11719921	C₂×C₄	1	2	yes	—
51	#11719921	C₂×C₄	1	2	yes	—
52	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
53	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
54	#11719919	C₂×C₄	1	2	yes	—
55	#11719919	C₂×C₄	1	2	yes	—
56	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
57	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
58	C₄×Dic₅₃	C₂³	1	2	yes	—
59	#11719917	—	2	4	no	—
60	C₂×#88322	C₂³	1	4	yes	—
61	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×D₅₃	C₂²×C₄	1	2	yes	—
63	C₂×C₄×C₅₃	—	2	2	no	—
64	C₂×C₄×C₅₃	—	4	1	no	—
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	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×C₅₃	#41	1	2	yes	—
71	C₂×C₄×C₅₃	#41	1	2	yes	—
72	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
73	C₂²⋊D₅₃	—	2	4	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	D₂₁₂	—	2	2	no	—
78	C₄×D₅₃	—	2	4	no	—
79	C₄×D₅₃	—	2	2	no	—
80	C₄×D₅₃	—	2	2	no	—
81	C₄×D₅₃	—	4	2	no	—
82	Dic₁₀₆	—	2	2	no	—
83	C₈×C₅₃	—	2	2	no	—
84	C₈×C₅₃	—	2	2	no	—
85	#88322	—	2	4	no	—
86	C₂²×C₅₃	—	2	1	no	—
87	C₂²×C₅₃	—	2	1	no	—
88	C₂²×C₅₃	#117	1	2	yes	—
89	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
90	C₂×D₅₃	—	2	2	no	—
91	C₂×D₅₃	—	2	2	no	—
92	C₂×D₅₃	—	2	2	no	—
93	C₄×C₅₃	—	2	1	no	—
94	C₄×C₅₃	—	2	1	no	—
95	C₄×C₅₃	—	2	1	no	—
96	C₄×C₅₃	#117	1	2	yes	—
97	C₄×C₅₃	#131	1	2	yes	—
98	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
99	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
100	Dic₅₃	—	2	2	no	—
101	Dic₅₃	—	2	2	no	—
102	Dic₅₃	—	2	2	no	—
103	#2925	—	53	1	no	maximal, Sylow
104	C₂×C₅₃	—	2	2	no	—
105	C₂×C₅₃	C₂×#117	1	1	yes	characteristic
106	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
107	C₂×C₅₃	#568	1	1	yes	characteristic
108	D₅₃	—	4	2	no	—
109	C₄×#41	—	53	1	no	—
110	#433	—	53	4	no	—
111	#432	—	53	2	no	—
112	#431	—	53	2	no	—
113	#423	—	53	1	no	—
114	C₂×C₄⋊C₈	D₅₃	1	1	yes	characteristic, complemented
115	#408	—	53	2	no	—
116	#405	—	53	2	no	—
117	C₅₃	#2925	1	1	yes	characteristic, complemented, Sylow
118	C₂×#41	—	53	1	no	—
119	C₂×M₁₆	—	53	2	no	—
120	C₂²×C₈	C₂×D₅₃	1	2	yes	—
121	C₄×Q₈	—	53	1	no	—
122	C₄×Q₈	—	53	1	no	—
123	C₄×D₄	—	53	2	no	—
124	C₄×D₄	—	53	2	no	—
125	C₄×D₄	—	53	1	no	—
126	C₄×D₄	—	53	1	no	—
127	#117	—	53	1	no	—
128	#117	—	53	2	no	—
129	C₂×C₄²	—	53	2	no	—
130	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
131	C₄⋊C₈	—	53	2	no	—
132	C₄⋊C₈	—	53	2	no	—
133	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
134	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
135	C₄⋊C₈	C₂×D₅₃	1	2	yes	—
136	C₂²⋊C₈	—	53	4	no	—
137	C₂²⋊C₈	—	53	4	no	—
138	#97	—	53	4	no	—
139	C₄×C₈	—	53	4	no	—
140	#41	—	106	2	no	—
141	#41	—	106	2	no	—
142	C₂×Q₈	—	53	1	no	—
143	C₂×D₄	—	53	2	no	—
144	C₂×D₄	—	53	1	no	—
145	C₂²×C₄	—	53	2	no	—
146	C₂²×C₄	—	53	2	no	—
147	C₂²×C₄	—	53	2	no	—
148	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
149	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
150	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
151	M₁₆	—	106	4	no	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	—	53	4	no	—
154	C₂×C₈	C₂²×D₅₃	1	2	yes	—
155	C₂×C₈	C₂²×D₅₃	1	2	yes	—
156	C₄⋊C₄	—	53	2	no	—
157	C₄⋊C₄	—	53	2	no	—
158	C₄⋊C₄	—	53	1	no	—
159	C₄⋊C₄	—	53	1	no	—
160	C₂²⋊C₄	—	53	2	no	—
161	C₂²⋊C₄	—	53	2	no	—
162	C₂²⋊C₄	—	53	2	no	—
163	C₄²	—	53	2	no	—
164	C₄²	—	53	2	no	—
165	C₄²	—	53	2	no	—
166	C₄²	C₄×D₅₃	1	1	yes	characteristic
167	C₄²	C₄×D₅₃	1	1	yes	characteristic
168	C₄²	C₂²×D₅₃	1	1	yes	characteristic
169	C₄²	C₂²×D₅₃	1	1	yes	characteristic
170	C₂³	—	53	2	no	—
171	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
172	Q₈	—	106	2	no	—
173	D₄	—	106	4	no	—
174	D₄	—	106	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	4	1	no	—
177	C₂×C₄	—	53	2	no	—
178	C₂×C₄	—	53	2	no	—
179	C₂×C₄	—	53	2	no	—
180	C₂×C₄	—	106	4	no	—
181	C₂×C₄	—	106	2	no	—
182	C₂×C₄	—	106	2	no	—
183	C₂×C₄	—	212	2	no	—
184	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
185	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
186	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
187	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
188	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
189	C₂×C₄	#11719947	1	2	yes	—
190	C₂×C₄	#11719949	1	2	yes	—
191	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
192	C₈	—	2	2	no	—
193	C₈	—	2	2	no	—
194	C₈	—	106	4	no	—
195	C₂²	—	2	1	no	—
196	C₂²	—	2	1	no	—
197	C₂²	—	106	2	no	—
198	C₂²	—	106	2	no	—
199	C₂²	—	106	2	no	—
200	C₂²	#13133175	1	2	yes	—
201	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
202	C₄	—	2	1	no	—
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	106	2	no	—
206	C₄	—	106	2	no	—
207	C₄	—	106	2	no	—
208	C₄	#13133175	1	2	yes	—
209	C₄	#13133223	1	2	yes	—
210	C₄	C₂×#11719947	1	1	yes	characteristic
211	C₄	C₂×#11719949	1	1	yes	characteristic
212	C₂	—	2	2	no	—
213	C₂	—	212	2	no	—
214	C₂	C₂×#13133175	1	1	yes	characteristic
215	C₂	C₂×#13133223	1	1	yes	characteristic
216	C₂	#16811389	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₂×C₄⋊C₈×C₅₃	C₂
hypercenter	C₂×C₄⋊C₈	D₅₃
hypocenter	C₅₃	#2925
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. #16810683	C₂	semidirect
3. #16810679	C₂	semidirect
4. #16810675	C₂	semidirect
5. #16810647	C₂	semidirect
7. #16810578	C₂	semidirect
8. C₂×C₄⋊C₈×C₅₃	C₂	semidirect
114. C₂×C₄⋊C₈	D₅₃	semidirect
117. C₅₃	#2925	semidirect