Finite Group Databasebeta

Subgroups — #20290874

Order 6784.

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

normal subgroups only

This group has 284 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	#16811633	C₂	1	1	yes	maximal, characteristic, complemented
2	#16811622	C₂	1	2	yes	maximal, complemented
3	#16811604	C₂	1	2	yes	maximal, complemented
4	#16811586	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#13133240	C₂	1	1	yes	maximal, characteristic
6	D₅₃⋊M₁₆	C₂²	1	1	yes	characteristic
7	#13133296	C₂²	1	2	yes	—
8	#13133296	C₂²	1	1	yes	characteristic
9	#13133286	—	2	1	no	—
10	#13133285	—	2	1	no	—
11	#13133282	—	2	4	no	—
12	#13133276	—	2	4	no	—
13	#13133271	—	2	1	no	—
14	C₈×#6157	—	2	1	no	—
15	C₂×#11719949	C₂²	1	2	yes	—
16	#13133240	—	2	1	no	—
17	#13133240	—	2	1	no	—
18	#13133240	—	2	1	no	—
19	#13133240	—	2	1	no	—
20	C₂×#11719924	C₄	1	2	yes	complemented
21	C₂×D₄₂₄	C₄	1	1	yes	characteristic, complemented
22	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
23	C₂×Q₁₆×C₅₃	C₄	1	1	yes	characteristic, complemented
24	C₂×Q₈×C₅₃	C₂×C₄	1	2	yes	—
25	#11719949	—	2	2	no	—
26	#11719949	—	2	2	no	—
27	#11719949	—	2	2	no	—
28	#11719949	—	2	2	no	—
29	C₂×D₂₁₂	—	2	2	no	—
30	C₂×D₂₁₂	C₂×C₄	1	2	yes	—
31	C₂×C₄×D₅₃	—	2	2	no	—
32	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
33	C₂²⋊#6157	—	2	2	no	—
34	C₂²⋊#6157	D₄	1	2	yes	—
35	#11719940	—	2	1	no	—
36	C₄⋊#6157	—	2	2	no	—
37	C₄⋊#6157	—	2	1	no	—
38	C₄×#6157	—	2	1	no	—
39	C₄×#6157	—	2	2	no	—
40	#11719936	D₄	1	2	yes	—
41	D₅₃⋊C₈	—	2	1	no	—
42	Q₁₆×C₅₃	—	2	1	no	—
43	Q₁₆×C₅₃	—	2	1	no	—
44	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
45	#11719924	—	2	2	no	—
46	#11719924	—	2	2	no	—
47	C₂×#88322	C₂×C₄	1	1	yes	characteristic
48	D₄₂₄	—	2	1	no	—
49	D₄₂₄	—	2	1	no	—
50	C₈×D₅₃	D₄	1	1	yes	characteristic
51	C₈×D₅₃	D₄	1	1	yes	characteristic
52	C₈×D₅₃	D₄	1	1	yes	characteristic
53	C₈×D₅₃	D₄	1	1	yes	characteristic
54	C₂²×D₅₃	—	2	2	no	—
55	C₂²×D₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂×#6157	—	2	2	no	—
57	C₂×#6157	—	4	2	no	—
58	Q₈×C₅₃	—	2	2	no	—
59	Q₈×C₅₃	—	2	2	no	—
60	C₂×C₄×C₅₃	—	2	2	no	—
61	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×Dic₅₃	#41	1	1	yes	characteristic
63	D₂₁₂	—	2	2	no	—
64	D₂₁₂	—	2	2	no	—
65	D₂₁₂	—	4	2	no	—
66	D₂₁₂	—	4	2	no	—
67	C₄×D₅₃	—	4	2	no	—
68	C₄×D₅₃	—	4	2	no	—
69	C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
70	C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
71	C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
72	C₄×D₅₃	#41	1	1	yes	characteristic
73	#88324	—	2	2	no	—
74	C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
76	#88322	C₂²⋊C₄	1	1	yes	characteristic
77	#88322	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
79	C₂×D₅₃	—	4	1	no	—
80	C₂×D₅₃	—	4	2	no	—
81	C₂×D₅₃	—	8	2	no	—
82	C₂×D₅₃	#121	1	1	yes	characteristic
83	C₂×D₅₃	#127	1	1	yes	characteristic
84	#6157	—	4	2	no	—
85	#6157	—	8	2	no	—
86	C₄×C₅₃	—	4	2	no	—
87	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
89	Dic₅₃	#121	1	1	yes	characteristic
90	Dic₅₃	#124	1	1	yes	characteristic
91	#1976	—	53	1	no	maximal, Sylow
92	C₂×C₅₃	—	2	1	no	—
93	C₂×C₅₃	#390	1	1	yes	characteristic
94	D₅₃	—	2	1	no	—
95	D₅₃	—	2	1	no	—
96	D₅₃	—	8	2	no	—
97	C₂×#135	—	53	1	no	—
98	#427	—	53	1	no	—
99	#421	—	53	2	no	—
100	#410	—	53	2	no	—
101	#405	—	53	1	no	—
102	C₅₃	#1976	1	1	yes	characteristic, complemented, Sylow
103	C₂×#41	—	53	2	no	—
104	#135	—	106	1	no	—
105	#135	—	106	1	no	—
106	#135	—	106	1	no	—
107	#135	—	106	1	no	—
108	C₂×Q₁₆	#6157	1	1	yes	characteristic, complemented
109	C₂×SD₁₆	—	53	2	no	—
110	C₂×D₈	—	53	1	no	—
111	C₂×M₁₆	—	53	1	no	—
112	C₂²×C₈	—	53	1	no	—
113	#117	—	53	1	no	—
114	#117	—	53	2	no	—
115	#107	—	106	1	no	—
116	#106	—	106	1	no	—
117	#104	—	106	4	no	—
118	#99	—	106	4	no	—
119	#97	—	106	1	no	—
120	C₄×C₈	—	106	1	no	—
121	#41	—	106	2	no	—
122	#41	—	106	2	no	—
123	#41	—	106	2	no	—
124	#41	—	106	2	no	—
125	C₂×Q₈	C₂×#6157	1	2	yes	—
126	C₂×D₄	—	53	2	no	—
127	C₂×D₄	—	106	2	no	—
128	C₂²×C₄	—	53	1	no	—
129	C₂²×C₄	—	106	2	no	—
130	Q₁₆	—	2	1	no	—
131	Q₁₆	—	2	1	no	—
132	SD₁₆	—	106	2	no	—
133	SD₁₆	—	106	2	no	—
134	D₈	—	106	1	no	—
135	D₈	—	106	1	no	—
136	M₁₆	—	53	2	no	—
137	M₁₆	—	106	1	no	—
138	C₂×C₈	—	53	1	no	—
139	C₂×C₈	—	53	1	no	—
140	C₂×C₈	—	53	1	no	—
141	C₂×C₈	—	53	1	no	—
142	C₂×C₈	—	53	1	no	—
143	C₂×C₈	—	106	1	no	—
144	C₂×C₈	C₂×#6157	1	1	yes	characteristic
145	C₄⋊C₄	—	106	1	no	—
146	C₄⋊C₄	—	106	2	no	—
147	C₂²⋊C₄	—	53	2	no	—
148	C₂²⋊C₄	—	106	2	no	—
149	C₄²	—	106	1	no	—
150	C₄²	—	106	2	no	—
151	C₂³	—	53	1	no	—
152	C₂³	—	106	2	no	—
153	Q₈	—	2	2	no	—
154	Q₈	—	2	2	no	—
155	D₄	—	106	2	no	—
156	D₄	—	106	2	no	—
157	D₄	—	212	2	no	—
158	D₄	—	212	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	53	1	no	—
161	C₂×C₄	—	53	1	no	—
162	C₂×C₄	—	53	1	no	—
163	C₂×C₄	—	53	1	no	—
164	C₂×C₄	—	53	1	no	—
165	C₂×C₄	—	106	2	no	—
166	C₂×C₄	—	212	2	no	—
167	C₂×C₄	—	212	2	no	—
168	C₂×C₄	—	212	2	no	—
169	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
170	C₈	—	53	1	no	—
171	C₈	—	53	1	no	—
172	C₈	—	106	2	no	—
173	C₈	C₂²⋊#6157	1	1	yes	characteristic
174	C₈	C₂²⋊#6157	1	1	yes	characteristic
175	C₂²	—	53	1	no	—
176	C₂²	—	53	1	no	—
177	C₂²	—	424	2	no	—
178	C₂²	—	212	1	no	—
179	C₂²	—	212	2	no	—
180	C₂²	D₄×#6157	1	1	yes	characteristic
181	C₄	—	4	2	no	—
182	C₄	—	53	1	no	—
183	C₄	—	53	1	no	—
184	C₄	—	424	2	no	—
185	C₄	—	212	2	no	—
186	C₄	C₂×C₂²⋊#6157	1	1	yes	characteristic
187	C₄	D₄×#6157	1	1	yes	characteristic
188	C₂	—	2	1	no	—
189	C₂	—	106	1	no	—
190	C₂	—	106	1	no	—
191	C₂	—	424	2	no	—
192	C₂	#16811575	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16811575
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6157
Fitting	C₂×Q₁₆×C₅₃	C₄
socle	C₂×C₅₃	#390
hypercenter	C₂×Q₁₆	#6157
hypocenter	C₅₃	#1976
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
1. #16811633	C₂	semidirect
2. #16811622	C₂	semidirect
3. #16811604	C₂	semidirect
4. #16811586	C₂	semidirect
20. C₂×#11719924	C₄	semidirect
21. C₂×D₄₂₄	C₄	semidirect
23. C₂×Q₁₆×C₅₃	C₄	semidirect
102. C₅₃	#1976	semidirect
108. C₂×Q₁₆	#6157	semidirect