Finite Group Databasebeta

Subgroups — #20290598

Order 6784.

260 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811633	C₂	1	2	yes	maximal, complemented
2	#16811602	C₂	1	2	yes	maximal
3	#16811491	C₂	1	2	yes	maximal, complemented
4	C₂×#13133225	C₂	1	1	yes	maximal, characteristic
5	#13133296	C₂²	1	4	yes	—
6	#13133286	—	2	2	no	—
7	#13133285	—	2	2	no	—
8	#13133276	—	2	2	no	—
9	#13133275	—	2	2	no	—
10	C₂×#11719945	C₂²	1	1	yes	characteristic
11	#13133225	—	2	2	no	—
12	#13133225	—	2	2	no	—
13	C₂×#11719912	C₄	1	1	yes	characteristic, complemented
14	C₂×#11719912	C₂²	1	1	yes	characteristic
15	C₂²×#88322	C₄	1	1	yes	characteristic, complemented
16	C₂×C₈×D₅₃	C₄	1	1	yes	characteristic, complemented
17	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
18	C₂×M₁₆×C₅₃	C₄	1	1	yes	characteristic, complemented
19	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₂²⋊D₅₃	D₄	1	1	yes	characteristic
21	C₂×C₂²⋊D₅₃	Q₈	1	1	yes	characteristic
22	#11719945	—	2	2	no	—
23	#11719945	—	2	2	no	—
24	C₂×D₂₁₂	D₄	1	1	yes	characteristic
25	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
27	C₂×Dic₁₀₆	Q₈	1	1	yes	characteristic
28	C₂²⋊#6157	—	2	4	no	—
29	C₄⋊#6157	—	2	4	no	—
30	C₄×#6157	—	2	4	no	—
31	M₁₆×C₅₃	—	2	2	no	—
32	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×#88322	—	2	1	no	—
35	C₂×#88322	—	2	1	no	—
36	C₂×#88322	C₂×C₄	1	1	yes	characteristic
37	C₂×#88322	C₂×C₄	1	1	yes	characteristic
38	#11719912	—	2	2	no	—
39	#11719912	—	2	2	no	—
40	C₈×D₅₃	—	2	1	no	—
41	C₈×D₅₃	—	2	1	no	—
42	C₈×D₅₃	D₄	1	2	yes	—
43	C₈×D₅₃	D₄	1	2	yes	—
44	C₂³×C₅₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×D₅₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×D₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂×#6157	—	4	4	no	—
48	C₂×C₄×C₅₃	—	2	1	no	—
49	C₂×C₄×C₅₃	—	2	1	no	—
50	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²⋊D₅₃	—	2	2	no	—
53	C₂²⋊D₅₃	—	2	2	no	—
54	C₂×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
56	D₂₁₂	—	2	2	no	—
57	C₄×D₅₃	—	2	1	no	—
58	C₄×D₅₃	—	2	1	no	—
59	C₄×D₅₃	C₂×D₄	1	2	yes	—
60	C₄×D₅₃	#41	1	2	yes	—
61	Dic₁₀₆	—	2	2	no	—
62	C₈×C₅₃	—	2	1	no	—
63	C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
64	#88322	—	2	1	no	—
65	#88322	C₂²⋊C₄	1	1	yes	characteristic
66	#88322	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₅₃	—	2	1	no	—
68	C₂²×C₅₃	—	4	1	no	—
69	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×D₅₃	—	2	1	no	—
71	C₂×D₅₃	—	4	1	no	—
72	C₂×D₅₃	—	4	1	no	—
73	C₂×D₅₃	#121	1	2	yes	—
74	#6157	—	8	4	no	—
75	C₄×C₅₃	—	2	1	no	—
76	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₄×C₅₃	#117	1	1	yes	characteristic
78	Dic₅₃	—	2	1	no	—
79	Dic₅₃	C₂²⋊Q₈	1	2	yes	—
80	#1812	—	53	1	no	maximal, Sylow
81	C₂×C₅₃	—	2	1	no	—
82	C₂×C₅₃	—	4	1	no	—
83	C₂×C₅₃	#380	1	1	yes	characteristic
84	D₅₃	—	2	2	no	—
85	D₅₃	—	4	1	no	—
86	C₂×#131	—	53	1	no	—
87	#427	—	53	2	no	—
88	#410	—	53	2	no	—
89	H(3,ℤ/4)	—	53	2	no	—
90	C₅₃	#1812	1	1	yes	characteristic, complemented, Sylow
91	C₂×#41	—	53	1	no	—
92	#131	—	106	2	no	—
93	#131	—	106	2	no	—
94	C₂×M₁₆	—	53	1	no	—
95	C₂×M₁₆	—	53	1	no	—
96	C₂×M₁₆	#6157	1	1	yes	characteristic, complemented
97	C₂²×C₈	—	53	1	no	—
98	C₂²×C₈	—	53	1	no	—
99	C₂²×C₈	—	53	1	no	—
100	#117	—	53	4	no	—
101	#107	—	106	2	no	—
102	#106	—	106	2	no	—
103	#99	—	106	2	no	—
104	#99	—	106	2	no	—
105	#41	—	106	2	no	—
106	#41	—	106	2	no	—
107	C₂×Q₈	—	53	1	no	—
108	C₂×D₄	—	53	1	no	—
109	C₂×D₄	—	53	1	no	—
110	C₂×D₄	—	53	1	no	—
111	C₂²×C₄	—	53	1	no	—
112	C₂²×C₄	—	53	1	no	—
113	C₂²×C₄	C₂×#6157	1	1	yes	characteristic
114	M₁₆	—	2	2	no	—
115	M₁₆	—	106	2	no	—
116	M₁₆	—	106	2	no	—
117	C₂×C₈	—	53	1	no	—
118	C₂×C₈	—	53	2	no	—
119	C₂×C₈	—	53	2	no	—
120	C₂×C₈	—	53	1	no	—
121	C₂×C₈	—	106	1	no	—
122	C₂×C₈	—	106	1	no	—
123	C₂×C₈	—	106	1	no	—
124	C₂×C₈	—	106	1	no	—
125	C₂×C₈	C₂×#6157	1	1	yes	characteristic
126	C₂×C₈	C₂×#6157	1	1	yes	characteristic
127	C₄⋊C₄	—	106	4	no	—
128	C₂²⋊C₄	—	106	4	no	—
129	C₄²	—	106	4	no	—
130	C₂³	—	53	1	no	—
131	C₂³	—	53	1	no	—
132	C₂³	C₄⋊#6157	1	1	yes	characteristic
133	Q₈	—	106	2	no	—
134	D₄	—	106	2	no	—
135	D₄	—	106	2	no	—
136	D₄	—	106	2	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	53	1	no	—
140	C₂×C₄	—	53	2	no	—
141	C₂×C₄	—	53	2	no	—
142	C₂×C₄	—	53	1	no	—
143	C₂×C₄	—	106	1	no	—
144	C₂×C₄	—	106	1	no	—
145	C₂×C₄	—	212	4	no	—
146	C₂×C₄	C₄⋊#6157	1	1	yes	characteristic
147	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
148	C₈	—	2	1	no	—
149	C₈	—	53	1	no	—
150	C₈	—	53	1	no	—
151	C₈	—	106	1	no	—
152	C₈	C₂²⋊#6157	1	2	yes	—
153	C₂²	—	2	1	no	—
154	C₂²	—	4	1	no	—
155	C₂²	—	53	2	no	—
156	C₂²	—	106	1	no	—
157	C₂²	—	212	1	no	—
158	C₂²	—	212	1	no	—
159	C₂²	C₂×C₄⋊#6157	1	1	yes	characteristic
160	C₄	—	2	1	no	—
161	C₄	—	53	2	no	—
162	C₄	—	106	1	no	—
163	C₄	—	424	4	no	—
164	C₄	C₂×C₂²⋊#6157	1	1	yes	characteristic
165	C₄	#13133296	1	1	yes	characteristic
166	C₂	—	2	1	no	—
167	C₂	—	4	1	no	—
168	C₂	—	106	2	no	—
169	C₂	—	212	1	no	—
170	C₂	#16811565	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#13133296
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6157
Fitting	C₂×M₁₆×C₅₃	C₄
socle	C₂×C₅₃	#380
hypercenter	C₂×M₁₆	#6157
hypocenter	C₅₃	#1812
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
1. #16811633	C₂	semidirect
3. #16811491	C₂	semidirect
13. C₂×#11719912	C₄	semidirect
15. C₂²×#88322	C₄	semidirect
16. C₂×C₈×D₅₃	C₄	semidirect
18. C₂×M₁₆×C₅₃	C₄	semidirect
90. C₅₃	#1812	semidirect
96. C₂×M₁₆	#6157	semidirect