Finite Group Databasebeta

Subgroups — #20290781

Order 6784.

186 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 60 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	#16811630	C₂	1	1	yes	maximal, characteristic
2	#16811588	C₂	1	1	yes	maximal, characteristic
3	#16811499	C₂	1	1	yes	maximal, characteristic
4	#16811493	C₂	1	1	yes	maximal, characteristic
5	#16811493	C₂	1	1	yes	maximal, characteristic
6	(C₂×#6157)⋊C₈	C₂	1	1	yes	maximal, characteristic
7	#16810658	C₂	1	1	yes	maximal, characteristic
8	C₂×#11719936	C₂²	1	1	yes	characteristic
9	C₂×#11719936	C₂²	1	1	yes	characteristic
10	C₂×D₅₃⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×#6157	C₂²	1	1	yes	characteristic
12	#13133284	—	2	1	no	—
13	C₄⋊#88324	—	2	1	no	—
14	#13133272	—	2	1	no	—
15	C₄×#88324	—	2	1	no	—
16	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
17	#13133175	C₂²	1	1	yes	characteristic
18	#13133175	C₂²	1	1	yes	characteristic
19	#13133150	C₄	1	1	yes	characteristic, complemented
20	#13133150	C₄	1	1	yes	characteristic, complemented
21	#13133149	C₄	1	1	yes	characteristic, complemented
22	#106×C₅₃	C₄	1	1	yes	characteristic, complemented
23	C₂²×#6157	—	2	1	no	—
24	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
25	C₂×#88324	—	2	1	no	—
26	C₂×#88324	—	2	1	no	—
27	C₂×#88324	D₄	1	1	yes	characteristic
28	C₂×#88324	D₄	1	1	yes	characteristic
29	C₄×#6157	—	2	1	no	—
30	C₄×#6157	—	2	1	no	—
31	#11719936	—	2	1	no	—
32	#11719936	—	2	1	no	—
33	#11719936	—	2	1	no	—
34	#11719936	Q₈	1	2	yes	—
35	D₅₃⋊C₈	—	2	1	no	—
36	D₅₃⋊C₈	—	2	1	no	—
37	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
40	#11719921	—	2	1	no	—
41	#11719921	—	2	1	no	—
42	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
43	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
44	C₄×Dic₅₃	—	2	1	no	—
45	C₄×Dic₅₃	—	2	1	no	—
46	C₂×#88322	C₂×C₄	1	1	yes	characteristic
47	C₈×D₅₃	—	2	1	no	—
48	C₈×D₅₃	—	2	1	no	—
49	C₂²×D₅₃	#41	1	1	yes	characteristic
50	C₂×#6157	—	2	2	no	—
51	C₂×#6157	—	4	1	no	—
52	C₂×#6157	—	4	1	no	—
53	C₂×C₄×C₅₃	—	2	1	no	—
54	C₂×C₄×C₅₃	—	2	1	no	—
55	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×Dic₅₃	—	2	1	no	—
57	C₂×Dic₅₃	—	2	1	no	—
58	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
59	C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
60	C₄×D₅₃	#41	1	1	yes	characteristic
61	C₄×D₅₃	#41	1	1	yes	characteristic
62	C₄×D₅₃	#41	1	1	yes	characteristic
63	#88324	—	2	2	no	—
64	#88324	—	2	1	no	—
65	#88324	—	2	1	no	—
66	#88324	—	4	1	no	—
67	C₈×C₅₃	—	2	1	no	—
68	#88322	—	2	1	no	—
69	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
70	C₂×D₅₃	—	2	1	no	—
71	C₂×D₅₃	—	2	1	no	—
72	C₂×D₅₃	#125	1	1	yes	characteristic
73	C₂×D₅₃	#126	1	1	yes	characteristic
74	#6157	—	4	2	no	—
75	C₄×C₅₃	—	4	1	no	—
76	C₄×C₅₃	—	4	1	no	—
77	C₄×C₅₃	#117	1	1	yes	characteristic
78	C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
79	Dic₅₃	—	4	1	no	—
80	Dic₅₃	—	4	1	no	—
81	Dic₅₃	C₂²⋊Q₈	1	1	yes	characteristic
82	Dic₅₃	#123	1	1	yes	characteristic
83	#1922	—	53	1	no	maximal, Sylow
84	C₂×C₅₃	#387	1	1	yes	characteristic
85	C₂×C₅₃	#443	1	1	yes	characteristic
86	C₂×C₅₃	#444	1	1	yes	characteristic
87	D₅₃	—	2	1	no	—
88	D₅₃	—	2	1	no	—
89	#427	—	53	1	no	—
90	#424	—	53	1	no	—
91	#405	—	53	1	no	—
92	#341	—	53	1	no	—
93	#339	—	53	1	no	—
94	#339	—	53	1	no	—
95	#336	—	53	1	no	—
96	C₅₃	#1922	1	1	yes	characteristic, complemented, Sylow
97	C₂×M₁₆	—	53	1	no	—
98	C₂×M₁₆	—	53	1	no	—
99	C₂²×C₈	—	53	1	no	—
100	C₂²×C₈	—	53	1	no	—
101	#117	—	53	1	no	—
102	#117	—	53	1	no	—
103	C₂×C₄²	—	53	1	no	—
104	#107	—	53	1	no	—
105	#107	—	53	1	no	—
106	#106	—	53	1	no	—
107	#106	#6157	1	1	yes	characteristic, complemented
108	C₄⋊C₈	—	106	1	no	—
109	C₄⋊C₈	—	106	1	no	—
110	#97	—	106	1	no	—
111	C₄×C₈	—	106	1	no	—
112	C₂²×C₄	—	53	1	no	—
113	C₂²×C₄	—	106	1	no	—
114	M₁₆	—	53	2	no	—
115	M₁₆	—	106	1	no	—
116	M₁₆	—	106	1	no	—
117	M₁₆	—	106	1	no	—
118	C₂×C₈	—	53	1	no	—
119	C₂×C₈	—	53	1	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	—	106	1	no	—
126	C₂×C₈	—	106	1	no	—
127	C₂×C₈	C₂×#6157	1	1	yes	characteristic
128	C₄⋊C₄	—	53	1	no	—
129	C₄⋊C₄	—	53	1	no	—
130	C₄⋊C₄	C₂×#6157	1	1	yes	characteristic
131	C₄⋊C₄	C₂×#6157	1	1	yes	characteristic
132	C₂²⋊C₄	—	106	1	no	—
133	C₂²⋊C₄	—	106	1	no	—
134	C₄²	—	106	1	no	—
135	C₄²	—	106	1	no	—
136	C₄²	—	106	1	no	—
137	C₄²	—	106	1	no	—
138	C₂³	—	53	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	53	1	no	—
142	C₂×C₄	—	53	1	no	—
143	C₂×C₄	—	53	1	no	—
144	C₂×C₄	—	53	1	no	—
145	C₂×C₄	—	53	1	no	—
146	C₂×C₄	—	106	2	no	—
147	C₂×C₄	—	106	1	no	—
148	C₂×C₄	—	106	1	no	—
149	C₂×C₄	—	212	1	no	—
150	C₂×C₄	—	212	1	no	—
151	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
152	C₈	—	2	1	no	—
153	C₈	—	106	1	no	—
154	C₈	—	106	1	no	—
155	C₈	—	106	2	no	—
156	C₈	—	106	1	no	—
157	C₈	—	212	1	no	—
158	C₂²	—	53	1	no	—
159	C₂²	—	53	1	no	—
160	C₂²	—	106	1	no	—
161	C₂²	—	106	1	no	—
162	C₂²	D₄×#6157	1	1	yes	characteristic
163	C₄	—	4	1	no	—
164	C₄	—	4	1	no	—
165	C₄	—	53	1	no	—
166	C₄	—	53	1	no	—
167	C₄	—	212	2	no	—
168	C₄	—	212	1	no	—
169	C₄	—	212	1	no	—
170	C₄	#13133296	1	1	yes	characteristic
171	C₄	Q₈×#6157	1	1	yes	characteristic
172	C₂	—	106	1	no	—
173	C₂	—	106	1	no	—
174	C₂	#16811572	1	1	yes	characteristic
175	C₂	#16811666	1	1	yes	characteristic
176	C₂	#16811668	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×#6157
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6157
Fitting	#106×C₅₃	C₄
socle	C₂²×C₅₃	C₄×D₄
hypercenter	#106	#6157
hypocenter	C₅₃	#1922
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

N	H ≅ G/N	type
19. #13133150	C₄	semidirect
20. #13133150	C₄	semidirect
21. #13133149	C₄	semidirect
22. #106×C₅₃	C₄	semidirect
96. C₅₃	#1922	semidirect
107. #106	#6157	semidirect