Subgroups — C₂×#16811180 #20288006

Order 6784.

1168 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811180	C₂	1	16	yes	maximal, direct product factor
2	C₂×#13133217	C₂	1	2	yes	maximal, complemented
3	C₂×#13133209	C₂	1	2	yes	maximal, complemented
4	C₂×#13133196	C₂	1	4	yes	maximal
5	C₂×C₄×Dic₁₀₆	C₂	1	4	yes	maximal, complemented
6	C₂×#13133176	C₂	1	2	yes	maximal
7	C₂²×C₄²×C₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₀₆	C₂²	1	4	yes	—
9	#13133217	C₂²	1	16	yes	complemented
10	#13133209	C₂²	1	16	yes	complemented
11	#13133196	C₂²	1	32	yes	—
12	C₄×Dic₁₀₆	C₂²	1	32	yes	complemented
13	#13133176	C₂²	1	16	yes	—
14	C₂×C₄⋊Dic₅₃	C₂²	1	4	yes	—
15	C₂×C₄⋊Dic₅₃	C₂²	1	4	yes	—
16	C₂×#11719919	C₂²	1	8	yes	—
17	C₂×C₂²⋊Dic₅₃	C₂²	1	4	yes	—
18	C₂×C₄×Dic₅₃	C₂²	1	4	yes	—
19	C₂³×C₄×C₅₃	C₂²	1	2	yes	—
20	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
22	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₄²×C₅₃	C₂²	1	8	yes	—
25	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	—	2	8	no	—
27	C₂²×C₄×C₅₃	Q₈	1	8	yes	complemented
28	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
29	C₂²×C₄×C₅₃	C₂³	1	8	yes	—
30	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
33	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
34	C₂²×Dic₅₃	C₂³	1	8	yes	—
35	C₂×Dic₁₀₆	—	2	16	no	—
36	C₂×Dic₁₀₆	C₂³	1	16	yes	—
37	C₄²×C₅₃	C₂³	1	8	yes	—
38	C₄²×C₅₃	C₂³	1	4	yes	—
39	C₄²×C₅₃	C₂³	1	4	yes	—
40	C₂²⋊Dic₅₃	C₂³	1	16	yes	—
41	C₄⋊Dic₅₃	C₂³	1	16	yes	—
42	C₄⋊Dic₅₃	C₂³	1	16	yes	—
43	#11719919	C₂³	1	32	yes	—
44	C₄×Dic₅₃	C₂³	1	16	yes	—
45	C₂³×C₅₃	—	2	4	no	—
46	C₂³×C₅₃	C₂×Q₈	1	4	yes	—
47	C₂³×C₅₃	C₂⁴	1	2	yes	—
48	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
49	C₂×C₄×C₅₃	—	2	4	no	—
50	C₂×C₄×C₅₃	—	2	16	no	—
51	C₂×C₄×C₅₃	C₂×Q₈	1	4	yes	—
52	C₂×C₄×C₅₃	C₂×Q₈	1	4	yes	—
53	C₂×C₄×C₅₃	C₂×Q₈	1	16	yes	complemented
54	C₂×C₄×C₅₃	#41	1	8	yes	—
55	C₂×C₄×C₅₃	C₂⁴	1	4	yes	—
56	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
57	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
58	C₂×C₄×C₅₃	C₂⁴	1	4	yes	—
59	C₂×Dic₅₃	—	2	16	no	—
60	C₂×Dic₅₃	C₂⁴	1	16	yes	—
61	Dic₁₀₆	—	2	32	no	—
62	C₂²×C₅₃	—	2	8	no	—
63	C₂²×C₅₃	—	2	4	no	—
64	C₂²×C₅₃	C₂²×Q₈	1	4	yes	—
65	C₂²×C₅₃	C₂²×Q₈	1	2	yes	—
66	C₂²×C₅₃	C₂×#41	1	4	yes	—
67	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
68	C₄×C₅₃	—	2	4	no	—
69	C₄×C₅₃	C₂²×Q₈	1	4	yes	—
70	C₄×C₅₃	C₂²×Q₈	1	4	yes	—
71	C₄×C₅₃	C₂×#41	1	8	yes	—
72	Dic₅₃	—	2	16	no	—
73	C₂×#533	—	53	1	no	maximal, Sylow
74	C₂×C₅₃	—	2	4	no	—
75	C₂×C₅₃	#533	1	4	yes	complemented
76	C₂×C₅₃	C₂³×Q₈	1	1	yes	characteristic
77	C₂×C₅₃	C₂²×#41	1	2	yes	—
78	#533	—	53	16	no	—
79	C₂×C₄⋊Q₈	—	53	2	no	—
80	C₂×#125	—	53	2	no	—
81	C₂×C₂²⋊Q₈	—	53	4	no	—
82	C₂×C₄×Q₈	—	53	4	no	—
83	C₂×#117	—	53	2	no	—
84	C₂²×C₄²	D₅₃	1	1	yes	characteristic
85	C₅₃	C₂×#533	1	1	yes	characteristic, complemented, Sylow
86	C₂²×Q₈	—	53	4	no	—
87	C₂³×C₄	C₂×D₅₃	1	2	yes	—
88	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
89	C₄⋊Q₈	—	53	16	no	—
90	#125	—	53	16	no	—
91	C₂²⋊Q₈	—	53	32	no	—
92	C₄×Q₈	—	53	32	no	—
93	#117	—	53	16	no	—
94	C₂×C₄⋊C₄	—	53	4	no	—
95	C₂×C₄⋊C₄	—	53	8	no	—
96	C₂×C₄⋊C₄	—	53	4	no	—
97	C₂×C₂²⋊C₄	—	53	4	no	—
98	C₂×C₄²	—	53	4	no	—
99	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
100	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
101	C₂×C₄²	C₂×D₅₃	1	8	yes	—
102	C₂×C₄²	C₂×D₅₃	1	2	yes	—
103	C₂⁴	C₂²×D₅₃	1	1	yes	characteristic
104	C₂×Q₈	—	53	16	no	—
105	C₂×Q₈	—	106	16	no	—
106	C₂²×C₄	—	2	8	no	—
107	C₂²×C₄	—	53	8	no	—
108	C₂²×C₄	Dic₁₀₆	1	8	yes	complemented
109	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
110	C₂²×C₄	C₂²×D₅₃	1	8	yes	—
111	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
112	C₂²×C₄	C₂²×D₅₃	1	4	yes	—
113	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
114	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
115	C₄⋊C₄	—	53	16	no	—
116	C₄⋊C₄	—	53	32	no	—
117	C₄⋊C₄	—	53	16	no	—
118	C₂²⋊C₄	—	53	16	no	—
119	C₄²	—	53	16	no	—
120	C₄²	C₂²×D₅₃	1	8	yes	—
121	C₄²	C₂²×D₅₃	1	4	yes	—
122	C₄²	C₂²×D₅₃	1	4	yes	—
123	C₂³	—	2	4	no	—
124	C₂³	C₂×Dic₁₀₆	1	4	yes	—
125	C₂³	C₂³×D₅₃	1	2	yes	—
126	C₂³	C₂³×D₅₃	1	1	yes	characteristic
127	Q₈	—	106	32	no	—
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	2	16	no	—
130	C₂×C₄	—	53	16	no	—
131	C₂×C₄	—	106	16	no	—
132	C₂×C₄	C₂×Dic₁₀₆	1	4	yes	—
133	C₂×C₄	C₂×Dic₁₀₆	1	4	yes	—
134	C₂×C₄	C₂×Dic₁₀₆	1	16	yes	complemented
135	C₂×C₄	#11719945	1	8	yes	—
136	C₂×C₄	C₂³×D₅₃	1	4	yes	—
137	C₂×C₄	C₂³×D₅₃	1	2	yes	—
138	C₂×C₄	C₂³×D₅₃	1	2	yes	—
139	C₂×C₄	C₂³×D₅₃	1	4	yes	—
140	C₂²	—	2	8	no	—
141	C₂²	—	2	4	no	—
142	C₂²	C₂²×Dic₁₀₆	1	4	yes	—
143	C₂²	C₂²×Dic₁₀₆	1	2	yes	—
144	C₂²	C₂×#11719945	1	4	yes	—
145	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
146	C₄	—	2	4	no	—
147	C₄	—	106	16	no	—
148	C₄	C₂²×Dic₁₀₆	1	4	yes	—
149	C₄	C₂²×Dic₁₀₆	1	4	yes	—
150	C₄	C₂×#11719945	1	8	yes	—
151	C₂	—	2	4	no	—
152	C₂	#16811180	1	4	yes	direct product factor
153	C₂	C₂³×Dic₁₀₆	1	1	yes	characteristic
154	C₂	C₂²×#11719945	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂⁵
Frattini	C₂²	C₂⁴×D₅₃
Fitting	C₂²×C₄²×C₅₃	C₂
socle	C₂³×C₅₃	C₂⁴
hypercenter	C₂²×C₄²	D₅₃
hypocenter	C₅₃	C₂×#533
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
1. #16811180	C₂	direct (N × H)
2. C₂×#13133217	C₂	semidirect
3. C₂×#13133209	C₂	semidirect
5. C₂×C₄×Dic₁₀₆	C₂	semidirect
9. #13133217	C₂²	semidirect
10. #13133209	C₂²	semidirect
12. C₄×Dic₁₀₆	C₂²	semidirect
27. C₂²×C₄×C₅₃	Q₈	semidirect
53. C₂×C₄×C₅₃	C₂×Q₈	semidirect
75. C₂×C₅₃	#533	semidirect
85. C₅₃	C₂×#533	semidirect
108. C₂²×C₄	Dic₁₀₆	semidirect
134. C₂×C₄	C₂×Dic₁₀₆	semidirect
152. C₂	#16811180	direct (N × H)

Subgroups — C2×#16811180 #20288006

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16811180 #20288006