Finite Group Databasebeta

Subgroups — #20276715

Order 6784.

270 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 41 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	#16811365	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810810	C₂	1	2	yes	maximal, complemented
3	#16810418	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810411	C₂	1	2	yes	maximal, complemented
5	C₅₃×#455	C₂	1	1	yes	maximal, characteristic, complemented
6	#13133206	—	2	1	no	—
7	#13133205	—	2	1	no	—
8	#13133203	—	2	2	no	—
9	#13133187	—	2	2	no	—
10	#13133185	—	2	2	no	—
11	#13133185	C₂²	1	2	yes	—
12	#13133132	—	2	2	no	—
13	#13133132	C₂²	1	2	yes	—
14	H(5,𝔽₂)×C₅₃	C₂²	1	1	yes	characteristic
15	#137×C₅₃	—	2	1	no	—
16	#124×C₅₃	C₂²	1	1	yes	characteristic
17	#104×C₅₃	—	2	1	no	—
18	#100×C₅₃	C₂²	1	1	yes	characteristic
19	C₂³×D₅₃	D₄	1	2	yes	complemented
20	#41×C₅₃	—	2	1	no	—
21	#41×C₅₃	—	4	1	no	—
22	C₂×Q₈×C₅₃	D₄	1	1	yes	characteristic, complemented
23	C₂×D₄×C₅₃	—	2	2	no	—
24	C₂×D₄×C₅₃	—	4	1	no	—
25	C₂×D₄×C₅₃	C₂³	1	1	yes	characteristic
26	C₂×C₂²⋊D₅₃	—	2	2	no	—
27	C₂×C₂²⋊D₅₃	—	4	2	no	—
28	C₂×D₂₁₂	—	2	1	no	—
29	C₂×D₂₁₂	—	2	1	no	—
30	C₂×Dic₁₀₆	—	2	1	no	—
31	Q₁₆×C₅₃	—	4	1	no	—
32	SD₁₆×C₅₃	—	4	1	no	—
33	M₁₆×C₅₃	—	2	1	no	—
34	C₂²⋊C₄×C₅₃	—	2	2	no	—
35	C₄²×C₅₃	D₄	1	1	yes	characteristic, complemented
36	C₂²⋊Dic₅₃	—	2	2	no	—
37	C₂²⋊Dic₅₃	D₄	1	2	yes	complemented
38	#11719921	—	2	2	no	—
39	#11719921	—	2	2	no	—
40	#11719921	—	2	2	no	—
41	#11719921	—	4	2	no	—
42	C₄×Dic₅₃	—	2	1	no	—
43	C₂³×C₅₃	—	2	2	no	—
44	C₂³×C₅₃	C₂×D₄	1	2	yes	—
45	C₂²×D₅₃	—	2	2	no	—
46	C₂²×D₅₃	—	4	2	no	—
47	C₂²×D₅₃	—	8	2	no	—
48	Q₈×C₅₃	—	2	1	no	—
49	Q₈×C₅₃	—	4	1	no	—
50	D₄×C₅₃	—	2	1	no	—
51	D₄×C₅₃	—	4	1	no	—
52	D₄×C₅₃	—	4	2	no	—
53	D₄×C₅₃	—	4	1	no	—
54	C₂×C₄×C₅₃	—	2	2	no	—
55	C₂×C₄×C₅₃	—	2	1	no	—
56	C₂×C₄×C₅₃	—	2	1	no	—
57	C₂×C₄×C₅₃	—	4	1	no	—
58	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂²⋊D₅₃	—	8	2	no	—
60	C₂²⋊D₅₃	—	8	4	no	—
61	C₂×Dic₅₃	—	2	2	no	—
62	C₂×Dic₅₃	—	4	2	no	—
63	D₂₁₂	—	8	1	no	—
64	D₂₁₂	—	8	1	no	—
65	Dic₁₀₆	—	8	1	no	—
66	C₈×C₅₃	—	4	1	no	—
67	C₂²×C₅₃	—	2	2	no	—
68	C₂²×C₅₃	—	4	2	no	—
69	C₂²×C₅₃	—	4	1	no	—
70	C₂²×C₅₃	—	8	2	no	—
71	C₂²×C₅₃	#120	1	1	yes	characteristic
72	C₂×D₅₃	—	4	2	no	—
73	C₂×D₅₃	—	4	2	no	—
74	C₂×D₅₃	—	4	4	no	—
75	C₂×D₅₃	—	4	2	no	—
76	C₂×D₅₃	—	8	2	no	—
77	C₄×C₅₃	—	2	1	no	—
78	C₄×C₅₃	—	4	1	no	—
79	C₄×C₅₃	—	4	1	no	—
80	C₄×C₅₃	—	4	1	no	—
81	Dic₅₃	—	4	2	no	—
82	Dic₅₃	—	8	2	no	—
83	#2199	—	53	1	no	maximal, Sylow
84	C₂×C₅₃	—	2	1	no	—
85	C₂×C₅₃	—	4	2	no	—
86	C₂×C₅₃	—	8	1	no	—
87	C₂×C₅₃	B(4,𝔽₂)	1	1	yes	characteristic
88	D₅₃	—	8	2	no	—
89	H(3,𝔽₄)	—	53	1	no	—
90	B(4,𝔽₂)	—	53	2	no	—
91	#455	D₅₃	1	1	yes	characteristic, complemented
92	#354	—	53	1	no	—
93	AΣL(1,𝔽₁₆)	—	53	2	no	—
94	C₅₃	#2199	1	1	yes	characteristic, complemented, Sylow
95	H(5,𝔽₂)	C₂×D₅₃	1	1	yes	characteristic
96	#137	—	2	1	no	—
97	#124	—	106	2	no	—
98	#124	—	106	1	no	—
99	#124	—	106	1	no	—
100	#124	C₂×D₅₃	1	1	yes	characteristic
101	#120	—	53	2	no	—
102	#120	—	106	2	no	—
103	#120	—	106	2	no	—
104	#104	—	2	1	no	—
105	#100	C₂×D₅₃	1	1	yes	characteristic
106	#99	—	53	2	no	—
107	#99	—	106	2	no	—
108	C₂⁴	—	53	2	no	—
109	#41	—	2	1	no	—
110	#41	—	4	1	no	—
111	C₂×Q₈	—	106	1	no	—
112	C₂×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic, complemented
113	C₂×D₄	—	2	2	no	—
114	C₂×D₄	—	4	1	no	—
115	C₂×D₄	—	106	1	no	—
116	C₂×D₄	—	106	1	no	—
117	C₂×D₄	—	106	2	no	—
118	C₂×D₄	—	212	2	no	—
119	C₂×D₄	C₂²×D₅₃	1	1	yes	characteristic
120	Q₁₆	—	4	1	no	—
121	SD₁₆	—	4	1	no	—
122	M₁₆	—	2	1	no	—
123	C₂²⋊C₄	—	2	2	no	—
124	C₂²⋊C₄	—	53	2	no	—
125	C₂²⋊C₄	—	106	2	no	—
126	C₂²⋊C₄	—	106	2	no	—
127	C₂²⋊C₄	—	106	2	no	—
128	C₂²⋊C₄	—	106	2	no	—
129	C₂²⋊C₄	—	212	2	no	—
130	C₄²	—	106	1	no	—
131	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic, complemented
132	C₂³	—	2	2	no	—
133	C₂³	—	106	2	no	—
134	C₂³	—	424	2	no	—
135	C₂³	—	212	2	no	—
136	C₂³	D₄×D₅₃	1	2	yes	—
137	Q₈	—	2	1	no	—
138	Q₈	—	4	1	no	—
139	Q₈	—	424	1	no	—
140	D₄	—	2	1	no	—
141	D₄	—	4	1	no	—
142	D₄	—	4	2	no	—
143	D₄	—	4	1	no	—
144	D₄	—	424	1	no	—
145	D₄	—	424	1	no	—
146	D₄	—	424	4	no	—
147	D₄	—	424	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	—	106	2	no	—
153	C₂×C₄	—	212	2	no	—
154	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
155	C₈	—	4	1	no	—
156	C₂²	—	2	2	no	—
157	C₂²	—	4	2	no	—
158	C₂²	—	4	1	no	—
159	C₂²	—	8	2	no	—
160	C₂²	—	424	2	no	—
161	C₂²	—	212	2	no	—
162	C₂²	—	212	2	no	—
163	C₂²	—	212	4	no	—
164	C₂²	—	212	2	no	—
165	C₂²	#13133185	1	1	yes	characteristic
166	C₄	—	2	1	no	—
167	C₄	—	4	1	no	—
168	C₄	—	4	1	no	—
169	C₄	—	4	1	no	—
170	C₄	—	424	2	no	—
171	C₄	—	212	2	no	—
172	C₂	—	2	1	no	—
173	C₂	—	4	2	no	—
174	C₂	—	8	1	no	—
175	C₂	—	424	2	no	—
176	C₂	#16810809	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16810809
commutator	C₂×D₄×C₅₃	C₂³
Frattini	C₂×D₄	C₂²×D₅₃
Fitting	C₅₃×#455	C₂
socle	C₂×C₅₃	B(4,𝔽₂)
hypercenter	#455	D₅₃
hypocenter	C₅₃	#2199
LCS	G ▷ C₂×D₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ #455

Complements

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

N	H ≅ G/N	type
1. #16811365	C₂	semidirect
2. #16810810	C₂	semidirect
3. #16810418	C₂	semidirect
4. #16810411	C₂	semidirect
5. C₅₃×#455	C₂	semidirect
19. C₂³×D₅₃	D₄	semidirect
22. C₂×Q₈×C₅₃	D₄	semidirect
35. C₄²×C₅₃	D₄	semidirect
37. C₂²⋊Dic₅₃	D₄	semidirect
91. #455	D₅₃	semidirect
94. C₅₃	#2199	semidirect
112. C₂×Q₈	C₂²⋊D₅₃	semidirect
131. C₄²	C₂²⋊D₅₃	semidirect