Subgroups — #20283304

Order 6784.

492 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#11719917	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133176	C₂	1	1	yes	maximal, characteristic
3	#16810666	C₂	1	4	yes	maximal, complemented
4	#16810663	C₂	1	4	yes	maximal, complemented
5	#16810659	C₂	1	4	yes	maximal, complemented
6	C₂×#117×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂×#11719917	C₄	1	8	yes	complemented
8	C₂×#11719917	C₂²	1	2	yes	—
9	C₂×#11719917	C₂²	1	2	yes	—
10	C₂²×#88322	C₂²	1	2	yes	—
11	#13133176	—	2	1	no	—
12	#13133176	—	2	1	no	—
13	#13133176	C₂²	1	2	yes	—
14	#13133176	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
16	C₂×C₂²⋊Dic₅₃	—	2	1	no	—
17	C₂×C₄×Dic₅₃	—	2	1	no	—
18	#13133150	C₂²	1	8	yes	complemented
19	#13133148	C₂²	1	8	yes	complemented
20	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
21	#117×C₅₃	—	2	1	no	—
22	#117×C₅₃	—	2	1	no	—
23	#117×C₅₃	C₂²	1	2	yes	—
24	#117×C₅₃	C₂²	1	2	yes	—
25	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
27	C₂×C₄²×C₅₃	—	2	1	no	—
28	C₂⁴×C₅₃	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	—	2	2	no	—
30	C₂²×C₄×C₅₃	C₂×C₄	1	4	yes	—
31	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	Q₈	1	4	yes	—
35	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
38	C₂²×Dic₅₃	—	2	2	no	—
39	C₄⋊C₄×C₅₃	—	2	2	no	—
40	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
41	C₂²⋊C₄×C₅₃	—	2	2	no	—
42	C₂²⋊C₄×C₅₃	—	2	2	no	—
43	C₄²×C₅₃	—	2	2	no	—
44	C₄²×C₅₃	—	2	2	no	—
45	C₂²⋊Dic₅₃	—	2	2	no	—
46	C₂²⋊Dic₅₃	—	2	2	no	—
47	C₄⋊Dic₅₃	—	2	2	no	—
48	C₄⋊Dic₅₃	C₂³	1	4	yes	—
49	C₄×Dic₅₃	—	2	2	no	—
50	C₄×Dic₅₃	—	2	2	no	—
51	#11719917	C₂×C₄	1	8	yes	complemented
52	#11719917	C₂×C₄	1	8	yes	complemented
53	C₂×#88322	C₂×C₄	1	8	yes	complemented
54	C₂×#88322	C₂³	1	4	yes	—
55	C₂³×C₅₃	—	4	2	no	—
56	C₂³×C₅₃	C₄⋊C₄	1	4	yes	—
57	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₅₃	—	2	4	no	—
61	C₂×C₄×C₅₃	—	4	4	no	—
62	C₂×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
63	C₂×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
64	C₂×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
65	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
67	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
68	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
72	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
73	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
74	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
75	C₂×Dic₅₃	—	2	4	no	—
76	C₂×Dic₅₃	—	4	4	no	—
77	#88322	C₂²×C₄	1	8	yes	—
78	C₂²×C₅₃	—	2	2	no	—
79	C₂²×C₅₃	—	2	2	no	—
80	C₂²×C₅₃	—	2	2	no	—
81	C₂²×C₅₃	—	4	4	no	—
82	C₂²×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
83	C₂²×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
84	C₂²×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
85	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
86	C₄×C₅₃	—	4	4	no	—
87	C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
88	C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
89	C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
90	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
91	C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
92	Dic₅₃	—	4	4	no	—
93	#2911	—	53	1	no	maximal, Sylow
94	C₂×C₅₃	—	2	2	no	—
95	C₂×C₅₃	—	2	2	no	—
96	C₂×C₅₃	—	2	2	no	—
97	C₂×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₅₃	#575	1	2	yes	—
99	C₂²×M₁₆	—	53	1	no	—
100	C₂×#117	—	53	1	no	—
101	C₂×#117	D₅₃	1	1	yes	characteristic
102	#428	—	53	4	no	—
103	#428	—	53	4	no	—
104	#427	—	53	4	no	—
105	C₅₃	#2911	1	1	yes	characteristic, complemented, Sylow
106	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
107	C₂×M₁₆	—	53	8	no	—
108	C₂×M₁₆	—	53	2	no	—
109	C₂×M₁₆	—	53	2	no	—
110	C₂²×C₈	—	53	2	no	—
111	#117	—	2	1	no	—
112	#117	—	2	1	no	—
113	#117	—	53	2	no	—
114	#117	—	53	2	no	—
115	#117	—	106	1	no	—
116	#117	—	106	1	no	—
117	#117	C₂×D₅₃	1	2	yes	—
118	#117	C₂×D₅₃	1	2	yes	—
119	C₂×C₄⋊C₄	—	53	2	no	—
120	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
121	C₂×C₂²⋊C₄	—	2	1	no	—
122	C₂×C₂²⋊C₄	—	106	1	no	—
123	C₂×C₄²	—	2	1	no	—
124	C₂×C₄²	—	106	1	no	—
125	#107	—	53	8	no	—
126	#106	—	53	8	no	—
127	C₂⁴	C₂²⋊D₅₃	1	1	yes	characteristic
128	C₂²×C₄	—	2	2	no	—
129	C₂²×C₄	—	106	2	no	—
130	C₂²×C₄	Dic₁₀₆	1	4	yes	—
131	C₂²×C₄	C₄×D₅₃	1	4	yes	—
132	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
133	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
134	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
135	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
136	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
137	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
138	M₁₆	—	53	8	no	—
139	M₁₆	—	53	8	no	—
140	C₂×C₈	—	53	8	no	—
141	C₂×C₈	—	53	4	no	—
142	C₄⋊C₄	—	2	2	no	—
143	C₄⋊C₄	—	53	4	no	—
144	C₄⋊C₄	—	106	2	no	—
145	C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
146	C₂²⋊C₄	—	2	2	no	—
147	C₂²⋊C₄	—	2	2	no	—
148	C₂²⋊C₄	—	106	2	no	—
149	C₂²⋊C₄	—	106	2	no	—
150	C₄²	—	2	2	no	—
151	C₄²	—	2	2	no	—
152	C₄²	—	106	2	no	—
153	C₄²	—	106	2	no	—
154	C₂³	—	4	2	no	—
155	C₂³	#11719919	1	4	yes	—
156	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
157	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
158	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	4	4	no	—
161	C₂×C₄	—	106	4	no	—
162	C₂×C₄	—	212	4	no	—
163	C₂×C₄	#11719919	1	4	yes	—
164	C₂×C₄	#11719919	1	4	yes	—
165	C₂×C₄	#11719919	1	4	yes	—
166	C₂×C₄	C₂×Dic₁₀₆	1	2	yes	—
167	C₂×C₄	C₂×Dic₁₀₆	1	2	yes	—
168	C₂×C₄	C₂×Dic₁₀₆	1	2	yes	—
169	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
170	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
171	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
172	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
173	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
174	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
175	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
176	C₈	—	53	8	no	—
177	C₂²	—	2	2	no	—
178	C₂²	—	2	2	no	—
179	C₂²	—	2	2	no	—
180	C₂²	—	4	4	no	—
181	C₂²	C₂×#11719919	1	2	yes	—
182	C₂²	C₂×#11719919	1	2	yes	—
183	C₂²	C₂×#11719919	1	2	yes	—
184	C₂²	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
185	C₄	—	4	4	no	—
186	C₄	—	212	4	no	—
187	C₄	C₂×#11719919	1	2	yes	—
188	C₄	C₂×#11719919	1	2	yes	—
189	C₄	C₂×#11719919	1	2	yes	—
190	C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
191	C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
192	C₂	—	2	2	no	—
193	C₂	—	2	2	no	—
194	C₂	—	2	2	no	—
195	C₂	C₂²×#11719919	1	1	yes	characteristic
196	C₂	#16811421	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₂×#117×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	C₂×#117	D₅₃
hypocenter	C₅₃	#2911
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. #16810666	C₂	semidirect
4. #16810663	C₂	semidirect
5. #16810659	C₂	semidirect
7. C₂×#11719917	C₄	semidirect
18. #13133150	C₂²	semidirect
19. #13133148	C₂²	semidirect
51. #11719917	C₂×C₄	semidirect
52. #11719917	C₂×C₄	semidirect
53. C₂×#88322	C₂×C₄	semidirect
105. C₅₃	#2911	semidirect