Subgroups — #20283682

Order 6784.

412 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810754	C₂	1	2	yes	maximal, complemented
2	#16810701	C₂	1	2	yes	maximal, complemented
3	#16810695	C₂	1	4	yes	maximal, complemented
4	#16810676	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810676	C₂	1	2	yes	maximal, complemented
6	C₄×#11719917	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×#88322	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#41×C₅₃	C₂	1	1	yes	maximal, characteristic
9	C₂×#11719917	C₂²	1	4	yes	—
10	C₂²×#88322	C₂²	1	2	yes	—
11	C₄⋊#88322	C₂²	1	4	yes	—
12	C₄⋊#88322	C₂²	1	2	yes	—
13	C₂²⋊#88322	C₂²	1	2	yes	—
14	C₂²⋊#88322	C₂²	1	4	yes	—
15	#13133128	C₂²	1	4	yes	—
16	C₄×#88322	C₂²	1	2	yes	—
17	C₄×#88322	C₂²	1	1	yes	characteristic
18	C₄×#88322	C₂²	1	1	yes	characteristic
19	C₄×#88322	C₂²	1	2	yes	—
20	C₂×#41×C₅₃	C₄	1	1	yes	characteristic
21	C₄×Q₈×C₅₃	C₄	1	1	yes	characteristic
22	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₅₃	C₄	1	1	yes	characteristic
24	C₄×D₄×C₅₃	C₄	1	2	yes	—
25	C₄×D₄×C₅₃	C₂²	1	2	yes	—
26	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
27	#117×C₅₃	C₄	1	2	yes	—
28	#117×C₅₃	C₄	1	1	yes	characteristic
29	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
31	#41×C₅₃	—	2	4	no	—
32	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×D₄×C₅₃	C₂×C₄	1	2	yes	—
34	C₂×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
36	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
37	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
41	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
42	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
44	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
46	C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
47	C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
48	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₄²×C₅₃	C₂×C₄	1	2	yes	—
50	C₄²×C₅₃	C₂×C₄	1	2	yes	—
51	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₄²×C₅₃	C₂³	1	1	yes	characteristic
53	C₄²×C₅₃	C₂³	1	2	yes	—
54	C₄²×C₅₃	C₂³	1	1	yes	characteristic
55	#11719917	—	2	8	no	—
56	C₂×#88322	—	2	4	no	—
57	C₂×#88322	C₂³	1	2	yes	—
58	C₂×#88322	C₂³	1	4	yes	—
59	C₂×#88322	C₂³	1	2	yes	—
60	C₂³×C₅₃	C₂²×C₄	1	2	yes	—
61	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	Q₈×C₅₃	—	2	2	no	—
63	D₄×C₅₃	—	2	4	no	—
64	D₄×C₅₃	—	2	2	no	—
65	C₂×C₄×C₅₃	—	2	4	no	—
66	C₂×C₄×C₅₃	—	2	2	no	—
67	C₂×C₄×C₅₃	—	2	4	no	—
68	C₂×C₄×C₅₃	—	2	2	no	—
69	C₂×C₄×C₅₃	—	2	4	no	—
70	C₂×C₄×C₅₃	M₁₆	1	4	yes	complemented
71	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
72	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
74	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
80	#88322	—	2	2	no	—
81	#88322	—	2	4	no	—
82	#88322	#41	1	4	yes	complemented
83	C₂²×C₅₃	—	2	1	no	—
84	C₂²×C₅₃	—	2	1	no	—
85	C₂²×C₅₃	—	2	2	no	—
86	C₂²×C₅₃	—	2	2	no	—
87	C₂²×C₅₃	—	2	2	no	—
88	C₂²×C₅₃	C₂×M₁₆	1	2	yes	—
89	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
90	C₄×C₅₃	—	2	1	no	—
91	C₄×C₅₃	—	2	2	no	—
92	C₄×C₅₃	—	2	2	no	—
93	C₄×C₅₃	—	2	1	no	—
94	C₄×C₅₃	—	2	2	no	—
95	C₄×C₅₃	C₂×M₁₆	1	2	yes	—
96	C₄×C₅₃	C₂×M₁₆	1	2	yes	—
97	C₄×C₅₃	#131	1	2	yes	—
98	C₄×C₅₃	C₂×#41	1	2	yes	—
99	#2965	—	53	1	no	maximal, Sylow
100	C₂×C₅₃	—	2	2	no	—
101	C₂×C₅₃	—	4	2	no	—
102	C₂×C₅₃	C₄×#41	1	1	yes	characteristic
103	C₂×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
104	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
105	C₄×#41	D₅₃	1	1	yes	characteristic
106	#446	—	53	2	no	—
107	#436	—	53	2	no	—
108	#435	—	53	4	no	—
109	#431	—	53	2	no	—
110	#431	—	53	1	no	—
111	C₄×M₁₆	—	53	2	no	—
112	C₂×C₄×C₈	—	53	1	no	—
113	C₅₃	#2965	1	1	yes	characteristic, complemented, Sylow
114	C₂×#41	Dic₅₃	1	1	yes	characteristic
115	C₂×M₁₆	—	53	4	no	—
116	C₂²×C₈	—	53	2	no	—
117	C₄×Q₈	Dic₅₃	1	1	yes	characteristic
118	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
119	C₄×D₄	Dic₅₃	1	2	yes	—
120	C₄×D₄	Dic₅₃	1	1	yes	characteristic
121	C₄×D₄	C₂×D₅₃	1	2	yes	—
122	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
123	#117	Dic₅₃	1	2	yes	—
124	#117	Dic₅₃	1	1	yes	characteristic
125	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
126	C₂×C₄²	C₂×D₅₃	1	2	yes	—
127	C₄⋊C₈	—	53	4	no	—
128	C₄⋊C₈	—	53	2	no	—
129	C₂²⋊C₈	—	53	2	no	—
130	C₂²⋊C₈	—	53	4	no	—
131	#97	—	53	4	no	—
132	C₄×C₈	—	53	2	no	—
133	C₄×C₈	—	53	1	no	—
134	C₄×C₈	—	53	1	no	—
135	C₄×C₈	—	53	2	no	—
136	#41	—	2	4	no	—
137	C₂×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
138	C₂×D₄	C₂×Dic₅₃	1	2	yes	—
139	C₂×D₄	C₂×Dic₅₃	1	1	yes	characteristic
140	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
141	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
142	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
143	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
144	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
145	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
146	M₁₆	—	106	8	no	—
147	C₂×C₈	—	53	2	no	—
148	C₂×C₈	—	53	2	no	—
149	C₂×C₈	—	53	4	no	—
150	C₂×C₈	—	106	4	no	—
151	C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
152	C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
153	C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
154	C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
155	C₂²⋊C₄	C₂×Dic₅₃	1	2	yes	—
156	C₂²⋊C₄	C₂×Dic₅₃	1	2	yes	—
157	C₂²⋊C₄	C₂×Dic₅₃	1	2	yes	—
158	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
159	C₄²	C₂×Dic₅₃	1	2	yes	—
160	C₄²	C₂×Dic₅₃	1	2	yes	—
161	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
162	C₄²	C₂²×D₅₃	1	1	yes	characteristic
163	C₄²	C₂²×D₅₃	1	2	yes	—
164	C₄²	C₂²×D₅₃	1	1	yes	characteristic
165	C₂³	C₂²×Dic₅₃	1	2	yes	—
166	C₂³	C₂²×Dic₅₃	1	1	yes	characteristic
167	Q₈	—	2	2	no	—
168	D₄	—	2	4	no	—
169	D₄	—	2	2	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	#11719917	1	4	yes	complemented
176	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
177	C₂×C₄	C₂²×Dic₅₃	1	2	yes	—
178	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
179	C₂×C₄	C₂²×Dic₅₃	1	2	yes	—
180	C₂×C₄	C₂²×Dic₅₃	1	2	yes	—
181	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
182	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
183	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
184	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
185	C₈	—	53	4	no	—
186	C₈	—	106	2	no	—
187	C₈	—	106	4	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	2	1	no	—
190	C₂²	—	2	2	no	—
191	C₂²	—	2	2	no	—
192	C₂²	—	2	2	no	—
193	C₂²	C₂×#11719917	1	2	yes	—
194	C₂²	C₂³×Dic₅₃	1	1	yes	characteristic
195	C₄	—	2	1	no	—
196	C₄	—	2	2	no	—
197	C₄	—	2	2	no	—
198	C₄	—	2	1	no	—
199	C₄	—	2	2	no	—
200	C₄	C₂×#11719917	1	2	yes	—
201	C₄	C₂×#11719917	1	2	yes	—
202	C₄	#13133224	1	2	yes	—
203	C₄	#41×D₅₃	1	2	yes	—
204	C₂	—	2	2	no	—
205	C₂	—	4	2	no	—
206	C₂	#41×Dic₅₃	1	1	yes	characteristic
207	C₂	C₂²×#11719917	1	1	yes	characteristic
208	C₂	C₂×#13133224	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₄×#41×C₅₃	C₂
hypercenter	C₄×#41	D₅₃
hypocenter	C₅₃	#2965
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. #16810754	C₂	semidirect
2. #16810701	C₂	semidirect
3. #16810695	C₂	semidirect
4. #16810676	C₂	semidirect
5. #16810676	C₂	semidirect
6. C₄×#11719917	C₂	semidirect
7. C₂×C₄×#88322	C₂	semidirect
70. C₂×C₄×C₅₃	M₁₆	semidirect
82. #88322	#41	semidirect
113. C₅₃	#2965	semidirect
175. C₂×C₄	#11719917	semidirect