Subgroups — #20283356

Order 6784.

400 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811091	C₂	1	1	yes	maximal, characteristic
2	#16810685	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810684	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810682	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810675	C₂	1	2	yes	maximal, complemented
6	#16810673	C₂	1	2	yes	maximal, complemented
7	#16810648	C₂	1	2	yes	maximal, complemented
8	#16810588	C₂	1	2	yes	maximal
9	#16810575	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#433	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11719945	C₂²	1	1	yes	characteristic
13	C₂×#11719912	C₂²	1	2	yes	—
14	C₂×C₈×D₅₃	C₂²	1	2	yes	—
15	#13133182	C₄	1	2	yes	—
16	#13133179	C₄	1	2	yes	—
17	#13133177	C₄	1	2	yes	—
18	#13133177	C₄	1	2	yes	—
19	#13133174	C₂²	1	1	yes	characteristic
20	#13133174	C₂²	1	1	yes	characteristic
21	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
22	C₄²×D₅₃	C₂²	1	1	yes	characteristic
23	C₄²×D₅₃	C₂²	1	1	yes	characteristic
24	C₄⋊#88322	C₂²	1	2	yes	—
25	#13133146	C₂²	1	2	yes	—
26	#13133146	C₂²	1	2	yes	—
27	C₂²⋊#88322	C₂²	1	2	yes	—
28	#13133129	C₂²	1	2	yes	—
29	#13133129	C₂²	1	2	yes	—
30	#13133128	C₂²	1	1	yes	characteristic
31	C₄×#88322	C₂²	1	1	yes	characteristic
32	C₈×Dic₅₃	C₂²	1	2	yes	—
33	C₈×Dic₅₃	C₂²	1	2	yes	—
34	#117×C₅₃	C₂²	1	1	yes	characteristic
35	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
36	C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
37	#97×C₅₃	C₂²	1	1	yes	characteristic
38	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
40	C₂×C₂²⋊D₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₂²⋊D₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×Dic₅₃	C₂×C₄	1	2	yes	—
43	#11719945	—	2	4	no	—
44	C₂×D₂₁₂	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
46	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
47	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
48	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
49	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×C₅₃	C₂³	1	2	yes	—
51	C₂×C₈×C₅₃	C₂³	1	2	yes	—
52	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
53	C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
54	C₄²×C₅₃	C₂³	1	1	yes	characteristic
55	C₄²×C₅₃	C₂³	1	1	yes	characteristic
56	#11719921	C₂×C₄	1	2	yes	—
57	#11719921	C₂×C₄	1	2	yes	—
58	#11719919	C₂×C₄	1	2	yes	—
59	#11719919	C₂×C₄	1	2	yes	—
60	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
61	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
62	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
63	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
64	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
65	C₂×#88322	C₂³	1	2	yes	—
66	C₂×#88322	C₂³	1	2	yes	—
67	#11719912	—	2	4	no	—
68	C₈×D₅₃	—	2	4	no	—
69	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×D₅₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×D₅₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₅₃	—	2	2	no	—
73	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
75	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
77	C₂²⋊D₅₃	—	2	2	no	—
78	C₂²⋊D₅₃	—	2	2	no	—
79	C₂×Dic₅₃	—	2	4	no	—
80	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
82	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
84	D₂₁₂	—	2	2	no	—
85	C₄×D₅₃	—	2	2	no	—
86	C₄×D₅₃	—	2	4	no	—
87	C₄×D₅₃	—	2	4	no	—
88	C₄×D₅₃	#41	1	4	yes	—
89	Dic₁₀₆	—	2	2	no	—
90	C₈×C₅₃	—	2	2	no	—
91	C₈×C₅₃	—	2	2	no	—
92	#88322	—	2	2	no	—
93	#88322	—	2	2	no	—
94	C₂²×C₅₃	—	2	1	no	—
95	C₂²×C₅₃	—	2	2	no	—
96	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
97	C₂×D₅₃	—	2	2	no	—
98	C₂×D₅₃	—	2	1	no	—
99	C₂×D₅₃	—	2	2	no	—
100	C₂×D₅₃	#117	1	2	yes	—
101	C₄×C₅₃	—	2	1	no	—
102	C₄×C₅₃	—	2	2	no	—
103	C₄×C₅₃	—	2	2	no	—
104	C₄×C₅₃	C₂×#41	1	2	yes	—
105	Dic₅₃	—	2	1	no	—
106	Dic₅₃	—	2	2	no	—
107	Dic₅₃	#117	1	2	yes	—
108	Dic₅₃	#131	1	4	yes	—
109	#2922	—	53	1	no	maximal, Sylow
110	C₂×C₅₃	—	4	1	no	—
111	C₂×C₅₃	C₂×#117	1	1	yes	characteristic
112	C₂×C₅₃	C₂×#131	1	2	yes	—
113	D₅₃	—	2	2	no	—
114	D₅₃	—	4	1	no	—
115	C₄×#41	—	53	1	no	—
116	#433	—	53	1	no	—
117	#433	—	53	1	no	—
118	#433	—	53	1	no	—
119	#433	D₅₃	1	1	yes	characteristic, complemented
120	#431	—	53	2	no	—
121	#431	—	53	2	no	—
122	#424	—	53	2	no	—
123	#408	—	53	2	no	—
124	C₄×M₁₆	—	53	1	no	—
125	C₂×C₄×C₈	—	53	1	no	—
126	C₅₃	#2922	1	1	yes	characteristic, complemented, Sylow
127	C₂×#41	—	53	1	no	—
128	C₂×M₁₆	—	53	2	no	—
129	C₂²×C₈	—	53	2	no	—
130	C₄×Q₈	—	53	2	no	—
131	C₄×D₄	—	53	2	no	—
132	C₄×D₄	—	53	2	no	—
133	C₄×D₄	—	53	2	no	—
134	#117	—	53	1	no	—
135	#117	—	53	1	no	—
136	#117	C₂×D₅₃	1	1	yes	characteristic
137	C₂×C₄²	—	53	1	no	—
138	C₂×C₄²	—	53	1	no	—
139	C₂×C₄²	—	53	1	no	—
140	C₄⋊C₈	—	53	2	no	—
141	C₄⋊C₈	—	53	2	no	—
142	C₄⋊C₈	—	53	2	no	—
143	C₄⋊C₈	C₂×D₅₃	1	2	yes	—
144	C₂²⋊C₈	—	53	2	no	—
145	C₂²⋊C₈	—	53	2	no	—
146	C₂²⋊C₈	—	53	2	no	—
147	C₂²⋊C₈	C₂×D₅₃	1	2	yes	—
148	#97	—	53	1	no	—
149	#97	C₂×D₅₃	1	1	yes	characteristic
150	C₄×C₈	—	53	2	no	—
151	C₄×C₈	—	53	2	no	—
152	C₄×C₈	—	53	1	no	—
153	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
154	#41	—	106	4	no	—
155	C₂×Q₈	—	53	1	no	—
156	C₂×D₄	—	53	1	no	—
157	C₂×D₄	—	53	1	no	—
158	C₂×D₄	—	53	1	no	—
159	C₂²×C₄	—	53	1	no	—
160	C₂²×C₄	—	53	2	no	—
161	C₂²×C₄	—	53	2	no	—
162	C₂²×C₄	—	53	2	no	—
163	C₂²×C₄	—	53	1	no	—
164	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
165	M₁₆	—	106	4	no	—
166	C₂×C₈	—	53	2	no	—
167	C₂×C₈	—	53	2	no	—
168	C₂×C₈	—	106	4	no	—
169	C₂×C₈	C₂²×D₅₃	1	2	yes	—
170	C₂×C₈	C₂²×D₅₃	1	2	yes	—
171	C₄⋊C₄	—	53	2	no	—
172	C₄⋊C₄	—	53	2	no	—
173	C₄⋊C₄	C₄×D₅₃	1	2	yes	—
174	C₂²⋊C₄	—	53	2	no	—
175	C₂²⋊C₄	—	53	2	no	—
176	C₂²⋊C₄	C₄×D₅₃	1	2	yes	—
177	C₄²	—	53	2	no	—
178	C₄²	—	53	2	no	—
179	C₄²	—	53	2	no	—
180	C₄²	—	53	1	no	—
181	C₄²	—	53	1	no	—
182	C₄²	C₂²×D₅₃	1	1	yes	characteristic
183	C₄²	C₂²×D₅₃	1	1	yes	characteristic
184	C₂³	—	53	1	no	—
185	C₂³	—	53	1	no	—
186	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
187	Q₈	—	106	2	no	—
188	D₄	—	106	2	no	—
189	D₄	—	106	2	no	—
190	D₄	—	106	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	53	2	no	—
193	C₂×C₄	—	53	1	no	—
194	C₂×C₄	—	53	4	no	—
195	C₂×C₄	—	53	2	no	—
196	C₂×C₄	—	53	1	no	—
197	C₂×C₄	—	106	4	no	—
198	C₂×C₄	—	106	4	no	—
199	C₂×C₄	—	106	4	no	—
200	C₂×C₄	—	106	2	no	—
201	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
202	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
203	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
204	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
205	C₈	—	2	2	no	—
206	C₈	—	2	2	no	—
207	C₈	—	106	2	no	—
208	C₈	—	106	2	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	53	2	no	—
212	C₂²	—	106	2	no	—
213	C₂²	—	106	1	no	—
214	C₂²	—	106	2	no	—
215	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
216	C₄	—	2	1	no	—
217	C₄	—	2	2	no	—
218	C₄	—	2	2	no	—
219	C₄	—	53	2	no	—
220	C₄	—	53	4	no	—
221	C₄	—	106	1	no	—
222	C₄	—	106	2	no	—
223	C₄	#41×D₅₃	1	2	yes	—
224	C₂	—	4	1	no	—
225	C₂	—	106	2	no	—
226	C₂	—	212	1	no	—
227	C₂	#117×D₅₃	1	1	yes	characteristic
228	C₂	#131×D₅₃	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₅₃×#433	C₂
hypercenter	#433	D₅₃
hypocenter	C₅₃	#2922
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #433

Complements

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

N	H ≅ G/N	type
2. #16810685	C₂	semidirect
3. #16810684	C₂	semidirect
4. #16810682	C₂	semidirect
5. #16810675	C₂	semidirect
6. #16810673	C₂	semidirect
7. #16810648	C₂	semidirect
9. #16810575	C₂	semidirect
10. C₄×C₈×D₅₃	C₂	semidirect
11. C₅₃×#433	C₂	semidirect
119. #433	D₅₃	semidirect
126. C₅₃	#2922	semidirect