Subgroups — #20283658

Order 6784.

356 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811180	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	#16810649	C₂	1	2	yes	maximal
6	#16810578	C₂	1	2	yes	maximal
7	C₄×#11719917	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
9	C₂×#11719917	C₂²	1	1	yes	characteristic
10	C₂×#11719917	C₂²	1	1	yes	characteristic
11	#13133217	C₄	1	1	yes	characteristic
12	#13133217	C₄	1	1	yes	characteristic
13	#13133209	C₄	1	2	yes	—
14	#13133196	C₄	1	2	yes	—
15	#13133196	C₄	1	2	yes	—
16	C₄×Dic₁₀₆	C₂²	1	2	yes	—
17	C₄×Dic₁₀₆	C₂²	1	2	yes	—
18	#13133176	C₂²	1	2	yes	—
19	#13133146	C₂²	1	4	yes	—
20	#13133146	C₂²	1	4	yes	—
21	#13133128	C₂²	1	2	yes	—
22	#13133127	C₂²	1	4	yes	—
23	C₄×#88322	C₂²	1	2	yes	—
24	C₈×Dic₅₃	C₂²	1	4	yes	—
25	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
26	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
27	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
28	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
33	C₂×Dic₁₀₆	C₂×C₄	1	2	yes	—
34	C₂×Dic₁₀₆	C₂×C₄	1	2	yes	—
35	C₂×C₈×C₅₃	—	2	4	no	—
36	C₂×C₈×C₅₃	C₂³	1	2	yes	—
37	C₂×C₈×C₅₃	C₂³	1	2	yes	—
38	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₄²×C₅₃	C₂³	1	1	yes	characteristic
41	C₄²×C₅₃	C₂³	1	1	yes	characteristic
42	C₂²⋊Dic₅₃	C₂×C₄	1	4	yes	—
43	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
44	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
45	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
46	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
47	#11719919	C₂×C₄	1	4	yes	—
48	#11719919	C₂×C₄	1	4	yes	—
49	C₄×Dic₅₃	C₂³	1	4	yes	—
50	#11719917	—	2	2	no	—
51	#11719917	Q₈	1	4	yes	complemented
52	C₂×#88322	C₂³	1	2	yes	—
53	C₂×#88322	C₂³	1	2	yes	—
54	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₅₃	—	2	2	no	—
56	C₂×C₄×C₅₃	—	4	1	no	—
57	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
63	C₂×C₄×C₅₃	#41	1	2	yes	—
64	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
65	C₂×Dic₅₃	C₂²×C₄	1	4	yes	—
66	C₂×Dic₅₃	C₂²×C₄	1	4	yes	—
67	Dic₁₀₆	—	2	4	no	—
68	Dic₁₀₆	—	4	2	no	—
69	C₈×C₅₃	—	2	2	no	—
70	C₈×C₅₃	—	2	2	no	—
71	#88322	—	2	2	no	—
72	#88322	C₂×Q₈	1	4	yes	—
73	C₂²×C₅₃	—	2	1	no	—
74	C₂²×C₅₃	—	2	1	no	—
75	C₂²×C₅₃	C₄×Q₈	1	2	yes	—
76	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
77	C₄×C₅₃	—	2	1	no	—
78	C₄×C₅₃	—	2	1	no	—
79	C₄×C₅₃	—	2	1	no	—
80	C₄×C₅₃	C₄×Q₈	1	2	yes	—
81	C₄×C₅₃	#131	1	2	yes	—
82	C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
83	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
84	Dic₅₃	—	2	4	no	—
85	Dic₅₃	—	2	4	no	—
86	#2962	—	53	1	no	maximal, Sylow
87	C₂×C₅₃	—	2	2	no	—
88	C₂×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
89	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
90	C₂×C₅₃	#568	1	1	yes	characteristic
91	#533	—	53	1	no	—
92	#446	—	53	4	no	—
93	#446	—	53	2	no	—
94	C₈×Q₈	—	53	2	no	—
95	#424	—	53	2	no	—
96	C₂×C₄⋊C₈	D₅₃	1	1	yes	characteristic
97	#405	—	53	2	no	—
98	C₄×M₁₆	—	53	1	no	—
99	C₅₃	#2962	1	1	yes	characteristic, complemented, Sylow
100	C₂×M₁₆	—	53	1	no	—
101	C₂×M₁₆	—	53	1	no	—
102	C₂²×C₈	C₂×D₅₃	1	2	yes	—
103	C₄⋊Q₈	—	53	1	no	—
104	C₄⋊Q₈	—	53	1	no	—
105	#125	—	53	2	no	—
106	C₂²⋊Q₈	—	53	2	no	—
107	C₂²⋊Q₈	—	53	2	no	—
108	C₄×Q₈	—	53	2	no	—
109	C₄×Q₈	—	53	2	no	—
110	#117	—	53	2	no	—
111	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
112	C₄⋊C₈	—	53	4	no	—
113	C₄⋊C₈	—	53	4	no	—
114	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
115	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
116	C₄⋊C₈	C₂×D₅₃	1	2	yes	—
117	#97	—	53	4	no	—
118	#97	—	53	2	no	—
119	C₄×C₈	—	53	4	no	—
120	C₄×C₈	—	53	2	no	—
121	C₂×Q₈	—	53	2	no	—
122	C₂×Q₈	—	53	2	no	—
123	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
124	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
125	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
126	M₁₆	—	53	4	no	—
127	M₁₆	—	106	2	no	—
128	C₂×C₈	—	2	4	no	—
129	C₂×C₈	—	53	2	no	—
130	C₂×C₈	—	53	2	no	—
131	C₂×C₈	C₂²×D₅₃	1	2	yes	—
132	C₂×C₈	C₂²×D₅₃	1	2	yes	—
133	C₄⋊C₄	—	53	2	no	—
134	C₄⋊C₄	—	53	2	no	—
135	C₄⋊C₄	—	53	4	no	—
136	C₄⋊C₄	—	53	4	no	—
137	C₄⋊C₄	—	53	2	no	—
138	C₄⋊C₄	—	53	2	no	—
139	C₂²⋊C₄	—	53	4	no	—
140	C₄²	—	53	4	no	—
141	C₄²	C₄×D₅₃	1	1	yes	characteristic
142	C₄²	C₄×D₅₃	1	1	yes	characteristic
143	C₄²	C₂²×D₅₃	1	1	yes	characteristic
144	C₄²	C₂²×D₅₃	1	1	yes	characteristic
145	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
146	Q₈	—	106	4	no	—
147	Q₈	—	212	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	53	4	no	—
151	C₂×C₄	—	53	4	no	—
152	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
154	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
155	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
156	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
157	C₂×C₄	#11719947	1	2	yes	—
158	C₂×C₄	Q₈×D₅₃	1	2	yes	—
159	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
160	C₈	—	2	2	no	—
161	C₈	—	2	2	no	—
162	C₈	—	53	4	no	—
163	C₈	—	106	2	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	#13133182	1	2	yes	—
167	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	106	4	no	—
172	C₄	—	106	4	no	—
173	C₄	#13133182	1	2	yes	—
174	C₄	#13133223	1	2	yes	—
175	C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
176	C₄	C₂×#11719947	1	1	yes	characteristic
177	C₂	—	2	2	no	—
178	C₂	C₂×#13133182	1	1	yes	characteristic
179	C₂	C₂×#13133223	1	1	yes	characteristic
180	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	#2962
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. #16810752	C₂	semidirect
3. #16810750	C₂	semidirect
4. #16810747	C₂	semidirect
51. #11719917	Q₈	semidirect
99. C₅₃	#2962	semidirect