Finite Group Databasebeta

Subgroups — #20290995

Order 6784.

270 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 52 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	#16811604	C₂	1	2	yes	maximal, complemented
2	#16811528	C₂	1	2	yes	maximal, complemented
3	#16811527	C₂	1	2	yes	maximal, complemented
4	#16811191	C₂	1	1	yes	maximal, characteristic
5	#13133296	—	2	2	no	—
6	#13133276	—	2	2	no	—
7	#13133276	C₂²	1	4	yes	complemented
8	C₂×#11719949	C₂²	1	1	yes	characteristic
9	C₂×D₄×D₅₃	C₄	1	1	yes	characteristic, complemented
10	#13133214	C₂²	1	1	yes	characteristic
11	#13133213	C₂²	1	1	yes	characteristic
12	#13133206	C₄	1	1	yes	characteristic, complemented
13	#13133190	—	2	2	no	—
14	#13133187	—	2	2	no	—
15	#13133174	C₄	1	1	yes	characteristic, complemented
16	#124×C₅₃	C₄	1	1	yes	characteristic, complemented
17	C₂³×D₅₃	—	2	1	no	—
18	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₂²⋊D₅₃	—	2	2	no	—
21	C₂×C₂²⋊D₅₃	—	2	1	no	—
22	#11719949	—	4	2	no	—
23	D₄×D₅₃	—	4	2	no	—
24	C₂×D₂₁₂	—	2	2	no	—
25	C₂×D₂₁₂	C₂×C₄	1	1	yes	characteristic
26	C₂×D₂₁₂	D₄	1	2	yes	—
27	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
28	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×D₅₃	D₄	1	2	yes	—
30	C₂²⋊#6157	—	2	4	no	—
31	C₄⋊#6157	—	4	2	no	—
32	C₄×#6157	—	4	2	no	—
33	C₂²⋊C₄×C₅₃	—	2	2	no	—
34	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
35	#11719921	—	2	1	no	—
36	#11719921	—	2	2	no	—
37	#11719919	—	2	1	no	—
38	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₅₃	—	2	1	no	—
40	C₂²×D₅₃	—	2	1	no	—
41	C₂²×D₅₃	—	4	2	no	—
42	C₂²×D₅₃	—	8	2	no	—
43	C₂²×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂²×D₅₃	C₂×D₄	1	2	yes	—
45	C₂×#6157	—	4	4	no	—
46	Q₈×C₅₃	—	4	1	no	—
47	D₄×C₅₃	—	4	1	no	—
48	C₂×C₄×C₅₃	—	2	1	no	—
49	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
50	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂²⋊D₅₃	—	4	2	no	—
52	C₂²⋊D₅₃	—	8	2	no	—
53	C₂×Dic₅₃	—	2	1	no	—
54	C₂×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
55	D₂₁₂	—	4	2	no	—
56	D₂₁₂	—	4	1	no	—
57	D₂₁₂	—	4	1	no	—
58	D₂₁₂	—	8	2	no	—
59	C₄×D₅₃	—	2	2	no	—
60	C₄×D₅₃	—	2	2	no	—
61	C₄×D₅₃	—	2	2	no	—
62	C₂²×C₅₃	—	4	2	no	—
63	C₂²×C₅₃	—	4	1	no	—
64	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂×D₅₃	—	2	2	no	—
66	C₂×D₅₃	—	2	1	no	—
67	C₂×D₅₃	—	4	2	no	—
68	C₂×D₅₃	—	4	1	no	—
69	C₂×D₅₃	—	4	2	no	—
70	C₂×D₅₃	—	4	1	no	—
71	C₂×D₅₃	—	8	4	no	—
72	#6157	—	8	4	no	—
73	C₄×C₅₃	—	2	1	no	—
74	C₄×C₅₃	—	2	2	no	—
75	C₄×C₅₃	—	4	1	no	—
76	Dic₅₃	—	4	1	no	—
77	Dic₅₃	#99	1	2	yes	—
78	#2128	—	53	1	no	maximal, Sylow
79	C₂×C₅₃	—	2	1	no	—
80	C₂×C₅₃	—	8	1	no	—
81	C₂×C₅₃	C₂×#99	1	1	yes	characteristic
82	D₅₃	—	4	2	no	—
83	D₅₃	—	4	1	no	—
84	D₅₃	—	8	1	no	—
85	#535	—	53	1	no	—
86	#410	—	53	2	no	—
87	#353	—	53	2	no	—
88	#353	—	53	2	no	—
89	C₅₃	#2128	1	1	yes	characteristic, complemented, Sylow
90	C₂×#41	—	53	1	no	—
91	C₂²×D₄	—	53	1	no	—
92	#127	—	53	1	no	—
93	#127	—	53	1	no	—
94	#124	—	53	1	no	—
95	#124	#6157	1	1	yes	characteristic, complemented
96	#121	—	106	2	no	—
97	#120	—	106	2	no	—
98	#117	—	53	1	no	—
99	#117	—	106	2	no	—
100	#99	—	53	4	no	—
101	#99	—	106	2	no	—
102	C₂⁴	—	106	1	no	—
103	#41	—	212	2	no	—
104	C₂×Q₈	C₂×#6157	1	1	yes	characteristic
105	C₂×D₄	—	53	2	no	—
106	C₂×D₄	—	53	1	no	—
107	C₂×D₄	—	53	1	no	—
108	C₂×D₄	—	106	2	no	—
109	C₂×D₄	—	106	1	no	—
110	C₂×D₄	—	106	2	no	—
111	C₂×D₄	—	212	2	no	—
112	C₂×D₄	C₂×#6157	1	1	yes	characteristic
113	C₂²×C₄	—	53	2	no	—
114	C₂²×C₄	—	53	1	no	—
115	C₄⋊C₄	—	106	1	no	—
116	C₄⋊C₄	—	212	2	no	—
117	C₂²⋊C₄	—	2	2	no	—
118	C₂²⋊C₄	—	106	1	no	—
119	C₂²⋊C₄	—	106	2	no	—
120	C₂²⋊C₄	—	106	4	no	—
121	C₄²	—	53	1	no	—
122	C₄²	—	212	2	no	—
123	C₄²	C₂×#6157	1	1	yes	characteristic
124	C₂³	—	2	1	no	—
125	C₂³	—	53	1	no	—
126	C₂³	—	53	2	no	—
127	C₂³	—	106	1	no	—
128	C₂³	—	424	2	no	—
129	C₂³	—	212	2	no	—
130	Q₈	—	4	1	no	—
131	D₄	—	4	1	no	—
132	D₄	—	424	2	no	—
133	D₄	—	424	2	no	—
134	D₄	—	212	2	no	—
135	D₄	—	212	1	no	—
136	D₄	—	212	1	no	—
137	D₄	—	212	2	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	53	1	no	—
140	C₂×C₄	—	106	2	no	—
141	C₂×C₄	—	106	2	no	—
142	C₂×C₄	—	106	2	no	—
143	C₂×C₄	—	106	1	no	—
144	C₂×C₄	—	212	4	no	—
145	C₂×C₄	C₂²⋊#6157	1	2	yes	—
146	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
147	C₂²	—	4	2	no	—
148	C₂²	—	4	1	no	—
149	C₂²	—	106	2	no	—
150	C₂²	—	106	1	no	—
151	C₂²	—	424	4	no	—
152	C₂²	—	212	2	no	—
153	C₂²	—	212	2	no	—
154	C₂²	—	212	1	no	—
155	C₂²	—	212	1	no	—
156	C₂²	C₂×C₂²⋊#6157	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	—	2	1	no	—
159	C₄	—	4	1	no	—
160	C₄	—	53	2	no	—
161	C₄	—	424	4	no	—
162	C₄	—	212	1	no	—
163	C₂	—	2	1	no	—
164	C₂	—	8	1	no	—
165	C₂	—	424	1	no	—
166	C₂	—	212	2	no	—
167	C₂	—	212	1	no	—
168	C₂	D₅₃⋊#99	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₅₃⋊#99
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6157
Fitting	#124×C₅₃	C₄
socle	C₂×C₅₃	C₂×#99
hypercenter	#124	#6157
hypocenter	C₅₃	#2128
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ #124

Complements

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

N	H ≅ G/N	type
1. #16811604	C₂	semidirect
2. #16811528	C₂	semidirect
3. #16811527	C₂	semidirect
7. #13133276	C₂²	semidirect
9. C₂×D₄×D₅₃	C₄	semidirect
12. #13133206	C₄	semidirect
15. #13133174	C₄	semidirect
16. #124×C₅₃	C₄	semidirect
89. C₅₃	#2128	semidirect
95. #124	#6157	semidirect