Finite Group Databasebeta

Subgroups — #20290989

Order 6784.

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

normal subgroups only

This group has 258 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	#16811529	C₂	1	2	yes	maximal, complemented
3	#16811528	C₂	1	2	yes	maximal, complemented
4	#16811174	C₂	1	1	yes	maximal, characteristic
5	#13133296	—	2	2	no	—
6	#13133276	—	2	2	no	—
7	#13133276	C₂²	1	4	yes	—
8	C₂×#11719949	C₄	1	1	yes	characteristic, complemented
9	C₂×#11719949	C₂²	1	1	yes	characteristic
10	#13133214	C₂²	1	1	yes	characteristic
11	#13133206	C₄	1	1	yes	characteristic, complemented
12	#13133206	C₂²	1	1	yes	characteristic
13	#13133192	—	2	2	no	—
14	#13133179	—	2	2	no	—
15	C₄²×D₅₃	C₄	1	1	yes	characteristic, complemented
16	C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic, complemented
17	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
19	#11719949	—	4	1	no	—
20	#11719949	—	4	1	no	—
21	#11719949	—	4	1	no	—
22	#11719949	—	4	1	no	—
23	C₂×D₂₁₂	—	2	1	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	—	2	1	no	—
29	C₂×C₄×D₅₃	—	2	1	no	—
30	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×D₅₃	D₄	1	2	yes	—
32	C₂²⋊#6157	—	2	4	no	—
33	C₄⋊#6157	—	4	2	no	—
34	C₄×#6157	—	4	2	no	—
35	C₄⋊C₄×C₅₃	—	2	2	no	—
36	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
37	#11719921	—	2	2	no	—
38	#11719921	—	2	2	no	—
39	C₄×Dic₅₃	—	2	1	no	—
40	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×D₅₃	—	2	1	no	—
42	C₂²×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×D₅₃	C₂×D₄	1	2	yes	—
44	C₂×#6157	—	4	4	no	—
45	Q₈×C₅₃	—	4	1	no	—
46	Q₈×C₅₃	—	4	1	no	—
47	C₂×C₄×C₅₃	—	2	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	C₂×Dic₅₃	—	2	1	no	—
52	C₂×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
53	D₂₁₂	—	4	2	no	—
54	D₂₁₂	—	4	1	no	—
55	D₂₁₂	—	4	1	no	—
56	D₂₁₂	—	4	4	no	—
57	D₂₁₂	—	4	2	no	—
58	C₄×D₅₃	—	2	2	no	—
59	C₄×D₅₃	—	2	2	no	—
60	C₄×D₅₃	—	2	1	no	—
61	C₄×D₅₃	—	2	1	no	—
62	C₄×D₅₃	—	4	2	no	—
63	C₄×D₅₃	—	4	1	no	—
64	C₄×D₅₃	—	4	1	no	—
65	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×D₅₃	—	2	1	no	—
67	C₂×D₅₃	—	2	2	no	—
68	C₂×D₅₃	—	4	1	no	—
69	C₂×D₅₃	—	4	2	no	—
70	C₂×D₅₃	—	4	2	no	—
71	C₂×D₅₃	—	4	1	no	—
72	#6157	—	8	4	no	—
73	C₄×C₅₃	—	2	2	no	—
74	C₄×C₅₃	—	2	1	no	—
75	C₄×C₅₃	—	2	2	no	—
76	C₄×C₅₃	—	4	1	no	—
77	Dic₅₃	—	4	1	no	—
78	Dic₅₃	#99	1	1	yes	characteristic
79	Dic₅₃	#99	1	1	yes	characteristic
80	#2126	—	53	1	no	maximal, Sylow
81	C₂×C₅₃	—	2	1	no	—
82	C₂×C₅₃	C₂×#99	1	1	yes	characteristic
83	D₅₃	—	4	2	no	—
84	D₅₃	—	4	1	no	—
85	D₅₃	—	8	1	no	—
86	#532	—	53	1	no	—
87	#410	—	53	2	no	—
88	#354	—	53	2	no	—
89	#353	—	53	2	no	—
90	C₅₃	#2126	1	1	yes	characteristic, complemented, Sylow
91	C₂×#41	—	53	1	no	—
92	C₂×#41	—	53	1	no	—
93	C₄⋊Q₈	#6157	1	1	yes	characteristic, complemented
94	#127	—	53	1	no	—
95	#124	—	53	1	no	—
96	#124	—	53	1	no	—
97	#121	—	106	2	no	—
98	C₄×D₄	—	106	2	no	—
99	#117	—	106	2	no	—
100	C₂×C₄²	—	53	1	no	—
101	#99	—	53	4	no	—
102	#99	—	106	2	no	—
103	#41	—	212	1	no	—
104	#41	—	212	1	no	—
105	#41	—	212	1	no	—
106	#41	—	212	1	no	—
107	C₂×Q₈	C₂×#6157	1	1	yes	characteristic
108	C₂×Q₈	C₂×#6157	1	1	yes	characteristic
109	C₂×D₄	—	53	2	no	—
110	C₂×D₄	—	53	1	no	—
111	C₂×D₄	—	53	1	no	—
112	C₂×D₄	—	106	2	no	—
113	C₂×D₄	—	106	1	no	—
114	C₂²×C₄	—	53	2	no	—
115	C₂²×C₄	—	53	1	no	—
116	C₂²×C₄	—	106	1	no	—
117	C₂²×C₄	—	106	1	no	—
118	C₄⋊C₄	—	2	2	no	—
119	C₄⋊C₄	—	212	2	no	—
120	C₂²⋊C₄	—	106	2	no	—
121	C₂²⋊C₄	—	106	2	no	—
122	C₂²⋊C₄	—	106	4	no	—
123	C₄²	—	53	1	no	—
124	C₄²	—	106	1	no	—
125	C₄²	—	212	2	no	—
126	C₄²	C₂×#6157	1	1	yes	characteristic
127	C₂³	—	53	1	no	—
128	C₂³	—	53	2	no	—
129	C₂³	—	106	1	no	—
130	Q₈	—	4	1	no	—
131	Q₈	—	4	1	no	—
132	D₄	—	212	2	no	—
133	D₄	—	212	1	no	—
134	D₄	—	212	1	no	—
135	D₄	—	212	4	no	—
136	D₄	—	212	2	no	—
137	C₂×C₄	—	2	1	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	1	no	—
143	C₂×C₄	—	106	1	no	—
144	C₂×C₄	—	106	1	no	—
145	C₂×C₄	—	212	2	no	—
146	C₂×C₄	—	212	1	no	—
147	C₂×C₄	—	212	1	no	—
148	C₂×C₄	—	212	4	no	—
149	C₂×C₄	C₂²⋊#6157	1	2	yes	—
150	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
151	C₂²	—	106	2	no	—
152	C₂²	—	106	1	no	—
153	C₂²	—	212	2	no	—
154	C₂²	—	212	1	no	—
155	C₂²	—	212	2	no	—
156	C₂²	—	212	1	no	—
157	C₂²	C₂×C₂²⋊#6157	1	1	yes	characteristic
158	C₄	—	2	2	no	—
159	C₄	—	2	2	no	—
160	C₄	—	2	1	no	—
161	C₄	—	4	1	no	—
162	C₄	—	53	1	no	—
163	C₄	—	53	1	no	—
164	C₄	—	424	4	no	—
165	C₄	—	212	1	no	—
166	C₂	—	2	1	no	—
167	C₂	—	424	1	no	—
168	C₂	—	212	2	no	—
169	C₂	—	212	1	no	—
170	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	C₄⋊Q₈×C₅₃	C₄
socle	C₂×C₅₃	C₂×#99
hypercenter	C₄⋊Q₈	#6157
hypocenter	C₅₃	#2126
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #16811604	C₂	semidirect
2. #16811529	C₂	semidirect
3. #16811528	C₂	semidirect
8. C₂×#11719949	C₄	semidirect
11. #13133206	C₄	semidirect
15. C₄²×D₅₃	C₄	semidirect
16. C₄⋊Q₈×C₅₃	C₄	semidirect
90. C₅₃	#2126	semidirect
93. C₄⋊Q₈	#6157	semidirect