Finite Group Databasebeta

Subgroups — #20276113

Order 6784.

216 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 65 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	#16810672	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810671	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810671	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#13133134	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133134	C₂	1	1	yes	maximal, characteristic
6	#16810381	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#100×C₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#11719917	C₂²	1	1	yes	characteristic
9	C₂×#11719917	C₂²	1	1	yes	characteristic
10	C₂×#11719917	C₂²	1	1	yes	characteristic
11	C₂×#11719917	C₂²	1	1	yes	characteristic
12	#13133153	—	2	2	no	—
13	#13133152	—	2	2	no	—
14	#13133152	—	2	2	no	—
15	#13133134	—	2	2	no	—
16	#13133134	—	2	2	no	—
17	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
20	#100×C₅₃	—	2	2	no	—
21	C₂⁴×C₅₃	Q₈	1	2	yes	—
22	C₂×D₄×C₅₃	—	2	1	no	—
23	C₂×D₄×C₅₃	—	2	1	no	—
24	C₂×D₄×C₅₃	—	2	1	no	—
25	C₂×D₄×C₅₃	—	2	1	no	—
26	C₂×D₄×C₅₃	—	2	1	no	—
27	C₂×D₄×C₅₃	—	2	1	no	—
28	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
29	M₁₆×C₅₃	—	2	1	no	—
30	M₁₆×C₅₃	—	2	1	no	—
31	M₁₆×C₅₃	—	2	1	no	—
32	M₁₆×C₅₃	D₄	1	2	yes	—
33	C₂×C₈×C₅₃	—	2	1	no	—
34	C₂×C₈×C₅₃	—	2	1	no	—
35	#11719917	—	2	1	no	—
36	#11719917	—	2	1	no	—
37	#11719917	—	2	1	no	—
38	#11719917	—	2	1	no	—
39	#11719917	—	2	1	no	—
40	#11719917	—	2	1	no	—
41	#11719917	D₄	1	2	yes	—
42	#11719917	D₄	1	2	yes	—
43	C₂×#88322	—	2	1	no	—
44	C₂×#88322	—	2	1	no	—
45	C₂×#88322	—	2	1	no	—
46	C₂×#88322	—	2	1	no	—
47	C₂³×C₅₃	—	2	2	no	—
48	C₂³×C₅₃	—	2	2	no	—
49	C₂³×C₅₃	—	2	2	no	—
50	C₂³×C₅₃	—	8	2	no	—
51	C₂³×C₅₃	C₂×Q₈	1	1	yes	characteristic
52	D₄×C₅₃	—	4	1	no	—
53	D₄×C₅₃	—	4	1	no	—
54	D₄×C₅₃	—	4	1	no	—
55	D₄×C₅₃	—	4	1	no	—
56	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
60	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
61	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
62	C₈×C₅₃	—	2	2	no	—
63	C₈×C₅₃	—	4	1	no	—
64	#88322	—	2	2	no	—
65	#88322	—	2	2	no	—
66	#88322	—	4	1	no	—
67	#88322	—	4	1	no	—
68	C₂²×C₅₃	—	4	1	no	—
69	C₂²×C₅₃	—	4	2	no	—
70	C₂²×C₅₃	—	8	2	no	—
71	C₂²×C₅₃	—	8	2	no	—
72	C₂²×C₅₃	—	8	2	no	—
73	C₂²×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
74	C₂²×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂²×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
76	C₄×C₅₃	#121	1	1	yes	characteristic
77	C₄×C₅₃	#121	1	1	yes	characteristic
78	C₄×C₅₃	#121	1	1	yes	characteristic
79	C₄×C₅₃	#126	1	1	yes	characteristic
80	#2069	—	53	1	no	maximal, Sylow
81	C₂×C₅₃	—	2	1	no	—
82	C₂×C₅₃	—	2	1	no	—
83	C₂×C₅₃	—	2	1	no	—
84	C₂×C₅₃	—	8	2	no	—
85	C₂×C₅₃	#396	1	1	yes	characteristic
86	#430	—	53	1	no	—
87	#430	—	53	1	no	—
88	#430	—	53	1	no	—
89	C₂×#100	—	53	1	no	—
90	C₂×#100	—	53	1	no	—
91	C₂×#100	D₅₃	1	1	yes	characteristic
92	#338	—	53	1	no	—
93	C₅₃	#2069	1	1	yes	characteristic, complemented, Sylow
94	C₂²×D₄	C₂×D₅₃	1	1	yes	characteristic
95	C₂×M₁₆	—	53	1	no	—
96	C₂×M₁₆	—	53	1	no	—
97	C₂×M₁₆	—	53	1	no	—
98	C₂×M₁₆	—	53	1	no	—
99	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
100	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
101	#108	—	106	2	no	—
102	#108	—	106	2	no	—
103	#108	—	106	2	no	—
104	#100	—	2	2	no	—
105	#100	—	106	2	no	—
106	#100	—	106	2	no	—
107	C₂⁴	Dic₁₀₆	1	2	yes	—
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	—	2	1	no	—
110	C₂×D₄	—	2	1	no	—
111	C₂×D₄	—	2	1	no	—
112	C₂×D₄	—	2	1	no	—
113	C₂×D₄	—	2	1	no	—
114	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
115	M₁₆	—	2	1	no	—
116	M₁₆	—	2	1	no	—
117	M₁₆	—	2	1	no	—
118	M₁₆	—	53	2	no	—
119	M₁₆	—	53	2	no	—
120	M₁₆	—	106	1	no	—
121	M₁₆	—	106	1	no	—
122	M₁₆	—	106	1	no	—
123	M₁₆	—	106	1	no	—
124	M₁₆	—	106	1	no	—
125	M₁₆	—	106	1	no	—
126	M₁₆	C₂²⋊D₅₃	1	2	yes	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	106	1	no	—
130	C₂×C₈	—	106	1	no	—
131	C₂×C₈	—	106	1	no	—
132	C₂×C₈	—	106	1	no	—
133	C₂³	—	2	2	no	—
134	C₂³	—	2	2	no	—
135	C₂³	—	2	2	no	—
136	C₂³	—	8	2	no	—
137	C₂³	C₂×Dic₁₀₆	1	1	yes	characteristic
138	D₄	—	4	1	no	—
139	D₄	—	4	1	no	—
140	D₄	—	4	1	no	—
141	D₄	—	4	1	no	—
142	C₂×C₄	#11719945	1	1	yes	characteristic
143	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
144	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
145	C₂×C₄	#11719947	1	1	yes	characteristic
146	C₂×C₄	#11719947	1	1	yes	characteristic
147	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
148	C₈	—	2	2	no	—
149	C₈	—	4	1	no	—
150	C₈	—	106	2	no	—
151	C₈	—	106	2	no	—
152	C₈	—	212	1	no	—
153	C₈	—	212	1	no	—
154	C₂²	—	4	1	no	—
155	C₂²	—	4	2	no	—
156	C₂²	—	8	2	no	—
157	C₂²	—	8	2	no	—
158	C₂²	—	8	2	no	—
159	C₂²	#13133194	1	1	yes	characteristic
160	C₂²	#13133194	1	1	yes	characteristic
161	C₂²	#13133196	1	1	yes	characteristic
162	C₄	#13133188	1	1	yes	characteristic
163	C₄	#13133189	1	1	yes	characteristic
164	C₄	#13133190	1	1	yes	characteristic
165	C₄	#13133210	1	1	yes	characteristic
166	C₂	—	2	1	no	—
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	—	8	2	no	—
170	C₂	#16810546	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16810546
commutator	C₂²×C₄×C₅₃	C₂³
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×#100×C₅₃	C₂
socle	C₂×C₅₃	#396
hypercenter	C₂×#100	D₅₃
hypocenter	C₅₃	#2069
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#100

Complements

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

N	H ≅ G/N	type
1. #16810672	C₂	semidirect
2. #16810671	C₂	semidirect
3. #16810671	C₂	semidirect
6. #16810381	C₂	semidirect
93. C₅₃	#2069	semidirect