Finite Group Databasebeta

Subgroups — #20290757

Order 6784.

312 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 66 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	#16811703	C₂	1	1	yes	maximal, characteristic
2	#16811616	C₂	1	2	yes	maximal, complemented
3	#16811604	C₂	1	1	yes	maximal, characteristic
4	#16811490	C₂	1	2	yes	maximal, complemented
5	#16811198	C₂	1	1	yes	maximal, characteristic
6	#13133305	—	2	4	no	—
7	D₅₃⋊M₁₆	—	2	2	no	—
8	D₅₃⋊M₁₆	C₂²	1	2	yes	—
9	#13133296	C₂²	1	2	yes	—
10	#13133280	—	2	4	no	—
11	#13133276	—	2	2	no	—
12	C₂×#11719949	C₂²	1	1	yes	characteristic
13	C₂×Q₈×D₅₃	C₂²	1	1	yes	characteristic
14	#13133215	C₄	1	1	yes	characteristic, complemented
15	#13133206	C₄	1	1	yes	characteristic, complemented
16	#13133205	C₄	1	1	yes	characteristic, complemented
17	#13133202	—	2	2	no	—
18	#13133195	—	2	2	no	—
19	#13133174	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic, complemented
21	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
23	#11719949	—	2	2	no	—
24	#11719949	—	2	2	no	—
25	Q₈×D₅₃	—	2	2	no	—
26	Q₈×D₅₃	—	2	2	no	—
27	C₂×D₂₁₂	C₂×C₄	1	1	yes	characteristic
28	C₂×D₂₁₂	D₄	1	2	yes	—
29	C₂×C₄×D₅₃	—	2	1	no	—
30	C₂×C₄×D₅₃	D₄	1	2	yes	—
31	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
32	C₂×Dic₁₀₆	—	2	1	no	—
33	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
34	C₂²⋊#6157	—	2	2	no	—
35	#11719940	—	2	2	no	—
36	C₄⋊#6157	—	2	2	no	—
37	C₄×#6157	—	2	2	no	—
38	#11719936	—	2	4	no	—
39	#11719936	—	2	2	no	—
40	D₅₃⋊C₈	—	2	4	no	—
41	D₅₃⋊C₈	D₄	1	4	yes	—
42	C₄⋊C₄×C₅₃	—	2	2	no	—
43	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
44	#11719921	—	2	2	no	—
45	#11719921	—	2	2	no	—
46	#11719921	—	2	1	no	—
47	#11719919	—	2	2	no	—
48	#11719919	—	2	1	no	—
49	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂²×D₅₃	C₂²⋊C₄	1	2	yes	—
51	C₂²×D₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂×#6157	—	4	2	no	—
53	Q₈×C₅₃	—	2	2	no	—
54	Q₈×C₅₃	—	2	2	no	—
55	C₂×C₄×C₅₃	—	2	1	no	—
56	C₂×C₄×C₅₃	—	2	1	no	—
57	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
58	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×Dic₅₃	—	2	1	no	—
60	C₂×Dic₅₃	—	2	1	no	—
61	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
62	D₂₁₂	—	2	4	no	—
63	D₂₁₂	—	2	2	no	—
64	C₄×D₅₃	—	2	2	no	—
65	C₄×D₅₃	—	2	2	no	—
66	C₄×D₅₃	—	4	2	no	—
67	C₄×D₅₃	C₂×D₄	1	2	yes	—
68	C₄×D₅₃	#41	1	2	yes	—
69	Dic₁₀₆	—	2	2	no	—
70	Dic₁₀₆	—	4	1	no	—
71	Dic₁₀₆	—	4	1	no	—
72	#88324	—	2	4	no	—
73	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×D₅₃	—	2	2	no	—
75	C₂×D₅₃	—	4	1	no	—
76	C₂×D₅₃	—	4	2	no	—
77	C₂×D₅₃	#121	1	2	yes	—
78	#6157	—	8	2	no	—
79	C₄×C₅₃	—	2	2	no	—
80	C₄×C₅₃	—	4	1	no	—
81	C₄×C₅₃	—	4	1	no	—
82	C₄×C₅₃	C₄×D₄	1	2	yes	—
83	Dic₅₃	—	4	1	no	—
84	Dic₅₃	—	4	1	no	—
85	Dic₅₃	#120	1	1	yes	characteristic
86	Dic₅₃	#123	1	1	yes	characteristic
87	#1909	—	53	1	no	maximal, Sylow
88	C₂×C₅₃	—	2	1	no	—
89	C₂×C₅₃	#386	1	1	yes	characteristic
90	D₅₃	—	2	2	no	—
91	D₅₃	—	4	2	no	—
92	#568	—	53	1	no	—
93	#536	—	53	1	no	—
94	#419	—	53	2	no	—
95	#410	—	53	1	no	—
96	H(3,ℤ/4)	—	53	2	no	—
97	C₅₃	#1909	1	1	yes	characteristic, complemented, Sylow
98	C₂×#41	—	53	1	no	—
99	C₂²×Q₈	—	53	1	no	—
100	#131	—	106	4	no	—
101	C₂×M₁₆	—	53	2	no	—
102	C₂×M₁₆	—	106	2	no	—
103	C₄⋊Q₈	—	53	1	no	—
104	C₄⋊Q₈	#6157	1	1	yes	characteristic, complemented
105	#124	—	53	1	no	—
106	#124	—	53	1	no	—
107	#123	—	106	2	no	—
108	C₂²⋊Q₈	—	106	2	no	—
109	#117	—	53	1	no	—
110	#117	—	53	2	no	—
111	#103	—	106	4	no	—
112	#99	—	106	2	no	—
113	#41	—	106	2	no	—
114	#41	—	106	2	no	—
115	C₂×Q₈	—	53	1	no	—
116	C₂×Q₈	—	106	1	no	—
117	C₂×Q₈	—	106	2	no	—
118	C₂×Q₈	—	106	2	no	—
119	C₂×Q₈	C₂×#6157	1	1	yes	characteristic
120	C₂×Q₈	C₂×#6157	1	1	yes	characteristic
121	C₂×D₄	—	53	2	no	—
122	C₂×D₄	—	53	1	no	—
123	C₂²×C₄	—	53	1	no	—
124	C₂²×C₄	—	53	2	no	—
125	C₂²×C₄	—	106	1	no	—
126	M₁₆	—	106	2	no	—
127	M₁₆	—	106	2	no	—
128	M₁₆	—	106	4	no	—
129	C₂×C₈	—	53	4	no	—
130	C₂×C₈	—	106	4	no	—
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	—	106	1	no	—
133	C₄⋊C₄	—	106	2	no	—
134	C₄⋊C₄	—	106	2	no	—
135	C₂²⋊C₄	—	106	1	no	—
136	C₂²⋊C₄	—	106	2	no	—
137	C₂²⋊C₄	—	106	2	no	—
138	C₂²⋊C₄	—	106	2	no	—
139	C₄²	—	53	1	no	—
140	C₄²	—	106	2	no	—
141	C₄²	C₂×#6157	1	1	yes	characteristic
142	C₂³	—	53	1	no	—
143	C₂³	—	53	2	no	—
144	Q₈	—	2	2	no	—
145	Q₈	—	2	2	no	—
146	Q₈	—	106	2	no	—
147	Q₈	—	212	1	no	—
148	Q₈	—	212	1	no	—
149	D₄	—	106	4	no	—
150	D₄	—	106	2	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	53	1	no	—
154	C₂×C₄	—	53	2	no	—
155	C₂×C₄	—	53	2	no	—
156	C₂×C₄	—	106	2	no	—
157	C₂×C₄	—	106	2	no	—
158	C₂×C₄	—	106	1	no	—
159	C₂×C₄	—	106	1	no	—
160	C₂×C₄	—	212	2	no	—
161	C₂×C₄	—	212	2	no	—
162	C₂×C₄	C₂²⋊#6157	1	2	yes	—
163	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
164	C₈	—	106	4	no	—
165	C₂²	—	53	2	no	—
166	C₂²	—	106	2	no	—
167	C₂²	—	212	1	no	—
168	C₂²	—	212	2	no	—
169	C₂²	C₂×C₂²⋊#6157	1	1	yes	characteristic
170	C₄	—	2	2	no	—
171	C₄	—	4	1	no	—
172	C₄	—	4	1	no	—
173	C₄	—	53	1	no	—
174	C₄	—	53	1	no	—
175	C₄	—	424	2	no	—
176	C₄	—	212	1	no	—
177	C₄	—	212	1	no	—
178	C₄	D₄×#6157	1	2	yes	—
179	C₂	—	2	1	no	—
180	C₂	—	106	2	no	—
181	C₂	—	212	2	no	—
182	C₂	#16811571	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16811571
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6157
Fitting	C₄⋊Q₈×C₅₃	C₄
socle	C₂×C₅₃	#386
hypercenter	C₄⋊Q₈	#6157
hypocenter	C₅₃	#1909
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #16811616	C₂	semidirect
4. #16811490	C₂	semidirect
14. #13133215	C₄	semidirect
15. #13133206	C₄	semidirect
16. #13133205	C₄	semidirect
20. C₄⋊Q₈×C₅₃	C₄	semidirect
97. C₅₃	#1909	semidirect
104. C₄⋊Q₈	#6157	semidirect