Finite Group Databasebeta

Subgroups — #20290959

Order 6784.

220 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 78 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	C₂×#13133291	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13133291	C₂	1	1	yes	maximal, characteristic, complemented
3	#16811522	C₂	1	4	yes	maximal, complemented
4	C₂×#13133225	C₂	1	1	yes	maximal, characteristic
5	#13133291	—	2	1	no	—
6	#13133291	—	2	1	no	—
7	#13133291	—	2	1	no	—
8	#13133291	C₂²	1	2	yes	complemented
9	C₂×#11719910	—	2	1	no	—
10	C₂×#11719910	C₂²	1	2	yes	complemented
11	C₂×#11719945	C₄	1	1	yes	characteristic
12	#13133225	—	2	1	no	—
13	#13133225	—	2	1	no	—
14	#13133225	C₄	1	2	yes	—
15	#13133225	C₂²	1	2	yes	—
16	C₂×#11719912	—	2	1	no	—
17	C₂²×#88322	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₅₃	—	2	1	no	—
19	C₂×M₁₆×C₅₃	C₄	1	1	yes	characteristic
20	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₂²⋊D₅₃	—	2	1	no	—
22	#11719945	—	2	1	no	—
23	#11719945	—	2	1	no	—
24	#11719945	C₈	1	2	yes	—
25	#11719945	C₂×C₄	1	2	yes	—
26	C₂×D₂₁₂	C₈	1	1	yes	characteristic
27	C₂×C₄×D₅₃	—	2	1	no	—
28	C₂×Dic₁₀₆	C₈	1	1	yes	characteristic
29	M₁₆×C₅₃	—	2	1	no	—
30	M₁₆×C₅₃	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₅₃	—	2	1	no	—
32	C₂×#88322	C₂×C₄	1	1	yes	characteristic
33	C₂×#88322	D₄	1	1	yes	characteristic
34	C₂×#88322	D₄	1	1	yes	characteristic
35	C₂×#88322	D₄	1	1	yes	characteristic
36	C₂×#88322	D₄	1	1	yes	characteristic
37	C₂×#88322	C₂³	1	1	yes	characteristic
38	#11719912	—	2	2	no	—
39	#11719912	—	2	2	no	—
40	C₈×D₅₃	—	2	2	no	—
41	C₈×D₅₃	—	2	2	no	—
42	#11719910	—	2	2	no	—
43	#11719910	—	2	2	no	—
44	C₂³×C₅₃	M₁₆	1	1	yes	characteristic
45	C₂²×D₅₃	—	2	1	no	—
46	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
50	C₂×C₄×C₅₃	M₁₆	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	—	2	1	no	—
55	D₂₁₂	—	2	1	no	—
56	D₂₁₂	C₂×C₈	1	2	yes	—
57	C₄×D₅₃	—	2	2	no	—
58	C₄×D₅₃	—	2	2	no	—
59	Dic₁₀₆	—	2	1	no	—
60	Dic₁₀₆	C₂×C₈	1	2	yes	—
61	C₈×C₅₃	—	2	2	no	—
62	#88322	C₂²⋊C₄	1	1	yes	characteristic
63	#88322	C₂²⋊C₄	1	1	yes	characteristic
64	#88322	C₂×D₄	1	1	yes	characteristic
65	#88322	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₅₃	—	4	1	no	—
67	C₂²×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
68	C₂²×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
69	C₂²×C₅₃	C₂×M₁₆	1	1	yes	characteristic
70	C₂×D₅₃	—	2	2	no	—
71	C₂×D₅₃	—	4	2	no	—
72	C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
73	C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
74	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
76	Dic₅₃	—	2	2	no	—
77	#2117	—	53	1	no	maximal, Sylow
78	C₂×C₅₃	—	2	1	no	—
79	C₂×C₅₃	—	2	1	no	—
80	C₂×C₅₃	—	2	1	no	—
81	C₂×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
82	D₅₃	—	4	2	no	—
83	C₂×#131	—	53	1	no	—
84	C₂×M₃₂	—	53	1	no	—
85	C₂×M₃₂	—	53	1	no	—
86	#350	—	53	4	no	—
87	C₅₃	#2117	1	1	yes	characteristic, complemented, Sylow
88	C₂×#41	—	53	1	no	—
89	#131	—	53	2	no	—
90	#131	—	53	2	no	—
91	#131	—	106	1	no	—
92	#131	—	106	1	no	—
93	C₂×M₁₆	—	106	1	no	—
94	C₂×M₁₆	#6157	1	1	yes	characteristic
95	C₂²×C₈	—	53	1	no	—
96	C₂²×C₈	—	106	1	no	—
97	M₃₂	—	53	2	no	—
98	M₃₂	—	106	1	no	—
99	M₃₂	—	106	1	no	—
100	M₃₂	—	106	1	no	—
101	C₂×C₁₆	—	53	2	no	—
102	C₂×C₁₆	—	106	1	no	—
103	#41	—	53	2	no	—
104	#41	—	53	2	no	—
105	#41	—	106	1	no	—
106	#41	—	106	1	no	—
107	C₂×Q₈	—	53	1	no	—
108	C₂×D₄	—	53	1	no	—
109	C₂×D₄	—	106	1	no	—
110	C₂²×C₄	—	106	1	no	—
111	C₂²×C₄	C₂×#6157	1	1	yes	characteristic
112	M₁₆	—	2	1	no	—
113	M₁₆	—	106	2	no	—
114	M₁₆	—	106	2	no	—
115	M₁₆	C₂×#6157	1	2	yes	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	53	1	no	—
118	C₂×C₈	—	53	1	no	—
119	C₂×C₈	—	53	1	no	—
120	C₂×C₈	—	53	1	no	—
121	C₂×C₈	—	53	1	no	—
122	C₂×C₈	—	53	1	no	—
123	C₂×C₈	—	106	2	no	—
124	C₂×C₈	—	106	2	no	—
125	C₁₆	—	106	2	no	—
126	C₁₆	—	106	2	no	—
127	C₂³	—	106	1	no	—
128	C₂³	#11719936	1	1	yes	characteristic
129	Q₈	—	53	2	no	—
130	Q₈	—	106	1	no	—
131	D₄	—	53	2	no	—
132	D₄	—	106	2	no	—
133	D₄	—	106	2	no	—
134	D₄	—	106	1	no	—
135	C₂×C₄	—	106	1	no	—
136	C₂×C₄	—	106	2	no	—
137	C₂×C₄	—	106	2	no	—
138	C₂×C₄	D₅₃⋊C₈	1	1	yes	characteristic
139	C₂×C₄	D₅₃⋊C₈	1	1	yes	characteristic
140	C₂×C₄	#11719936	1	1	yes	characteristic
141	C₂×C₄	C₂²⋊#6157	1	1	yes	characteristic
142	C₂×C₄	C₂²⋊#6157	1	1	yes	characteristic
143	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
144	C₈	—	2	2	no	—
145	C₈	—	53	1	no	—
146	C₈	—	53	1	no	—
147	C₈	—	53	1	no	—
148	C₈	—	53	1	no	—
149	C₂²	—	4	1	no	—
150	C₂²	—	106	2	no	—
151	C₂²	—	212	2	no	—
152	C₂²	#13133274	1	1	yes	characteristic
153	C₂²	#13133274	1	1	yes	characteristic
154	C₂²	C₂×#11719936	1	1	yes	characteristic
155	C₄	—	106	2	no	—
156	C₄	#13133274	1	1	yes	characteristic
157	C₄	#13133274	1	1	yes	characteristic
158	C₄	C₂×C₂²⋊#6157	1	1	yes	characteristic
159	C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
160	C₂	—	2	1	no	—
161	C₂	—	2	1	no	—
162	C₂	—	2	1	no	—
163	C₂	—	212	2	no	—
164	C₂	C₂×#13133274	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×C₂²⋊#6157
commutator	C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄	C₂²×#6157
Fitting	C₂×M₁₆×C₅₃	C₄
socle	C₂×C₅₃	C₂×C₂²⋊C₈
hypercenter	C₂×M₁₆	#6157
hypocenter	C₅₃	#2117
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#13133291	C₂	semidirect
2. C₂×#13133291	C₂	semidirect
3. #16811522	C₂	semidirect
8. #13133291	C₂²	semidirect
10. C₂×#11719910	C₂²	semidirect
87. C₅₃	#2117	semidirect