Subgroups — C₂×#16810381 #20274418

Order 6784.

372 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#11719917	C₂	1	2	yes	maximal
2	#16810381	C₂	1	4	yes	maximal, direct product factor
3	C₂²×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#11719917	—	2	2	no	—
5	C₂×#11719917	—	2	2	no	—
6	C₂×#11719917	—	2	2	no	—
7	C₂×#11719917	—	2	2	no	—
8	C₂×#11719917	C₄	1	4	yes	—
9	C₂×#11719917	C₂²	1	4	yes	—
10	C₂²×#88322	C₄	1	4	yes	—
11	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×M₁₆×C₅₃	—	2	2	no	—
13	C₂×M₁₆×C₅₃	—	2	2	no	—
14	C₂×M₁₆×C₅₃	C₄	1	2	yes	—
15	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
16	C₂²×C₈×C₅₃	C₄	1	2	yes	—
17	C₂⁴×C₅₃	Q₈	1	1	yes	characteristic
18	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₅₃	D₄	1	2	yes	—
25	C₂²×C₄×C₅₃	D₄	1	2	yes	—
26	C₂²×C₄×C₅₃	Q₈	1	1	yes	characteristic
27	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
28	M₁₆×C₅₃	—	2	4	no	—
29	M₁₆×C₅₃	—	2	4	no	—
30	C₂×C₈×C₅₃	—	2	2	no	—
31	C₂×C₈×C₅₃	—	2	2	no	—
32	C₂×C₈×C₅₃	C₂×C₄	1	4	yes	—
33	#11719917	—	2	4	no	—
34	#11719917	—	2	4	no	—
35	#11719917	—	2	4	no	—
36	#11719917	—	2	4	no	—
37	C₂×#88322	—	2	4	no	—
38	C₂×#88322	—	2	4	no	—
39	C₂×#88322	C₂×C₄	1	8	yes	—
40	C₂³×C₅₃	—	4	2	no	—
41	C₂³×C₅₃	C₄⋊C₄	1	2	yes	—
42	C₂³×C₅₃	C₄⋊C₄	1	2	yes	—
43	C₂³×C₅₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂³×C₅₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂³×C₅₃	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₅₃	C₄²	1	2	yes	—
47	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
48	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
50	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
51	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
52	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
53	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
56	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
57	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
58	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
61	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
64	C₈×C₅₃	—	2	4	no	—
65	#88322	—	2	8	no	—
66	C₂²×C₅₃	—	2	4	no	—
67	C₂²×C₅₃	—	2	2	no	—
68	C₂²×C₅₃	—	4	2	no	—
69	C₂²×C₅₃	—	4	1	no	—
70	C₂²×C₅₃	—	4	1	no	—
71	C₂²×C₅₃	#95	1	2	yes	—
72	C₂²×C₅₃	#95	1	1	yes	characteristic
73	C₂²×C₅₃	#95	1	1	yes	characteristic
74	C₂²×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
75	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₄×C₅₃	#95	1	2	yes	—
77	C₄×C₅₃	#95	1	1	yes	characteristic
78	C₄×C₅₃	#95	1	1	yes	characteristic
79	C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
80	C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
81	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×#338	—	53	1	no	maximal, Sylow
83	C₂×C₅₃	—	2	2	no	—
84	C₂×C₅₃	—	2	1	no	—
85	C₂×C₅₃	—	2	2	no	—
86	C₂×C₅₃	—	2	1	no	—
87	C₂×C₅₃	#338	1	2	yes	complemented
88	C₂×C₅₃	C₂×#95	1	1	yes	characteristic
89	C₂²×M₁₆	—	53	2	no	—
90	C₂²×M₁₆	D₅₃	1	1	yes	characteristic
91	#338	—	53	4	no	—
92	C₅₃	C₂×#338	1	1	yes	characteristic, complemented, Sylow
93	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
94	C₂×M₁₆	—	2	2	no	—
95	C₂×M₁₆	—	2	2	no	—
96	C₂×M₁₆	—	53	4	no	—
97	C₂×M₁₆	—	53	4	no	—
98	C₂×M₁₆	—	106	2	no	—
99	C₂×M₁₆	—	106	2	no	—
100	C₂×M₁₆	—	106	2	no	—
101	C₂×M₁₆	—	106	2	no	—
102	C₂×M₁₆	Dic₅₃	1	2	yes	—
103	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
104	C₂²×C₈	—	53	4	no	—
105	C₂²×C₈	Dic₅₃	1	2	yes	—
106	C₂⁴	Dic₁₀₆	1	1	yes	characteristic
107	C₂²×C₄	Dic₁₀₆	1	1	yes	characteristic
108	C₂²×C₄	C₄×D₅₃	1	2	yes	—
109	C₂²×C₄	C₄×D₅₃	1	2	yes	—
110	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
111	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
112	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
113	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
114	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
115	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
116	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
117	M₁₆	—	2	4	no	—
118	M₁₆	—	2	4	no	—
119	M₁₆	—	106	4	no	—
120	M₁₆	—	106	4	no	—
121	M₁₆	—	106	4	no	—
122	M₁₆	—	106	4	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	53	8	no	—
126	C₂×C₈	—	106	4	no	—
127	C₂×C₈	—	106	4	no	—
128	C₂×C₈	C₂×Dic₅₃	1	4	yes	—
129	C₂³	—	4	2	no	—
130	C₂³	#11719919	1	2	yes	—
131	C₂³	#11719919	1	2	yes	—
132	C₂³	C₄⋊Dic₅₃	1	1	yes	characteristic
133	C₂³	C₄⋊Dic₅₃	1	1	yes	characteristic
134	C₂³	C₂×Dic₁₀₆	1	1	yes	characteristic
135	C₂×C₄	C₄×Dic₅₃	1	1	yes	characteristic
136	C₂×C₄	C₄×Dic₅₃	1	2	yes	—
137	C₂×C₄	C₄×Dic₅₃	1	1	yes	characteristic
138	C₂×C₄	#11719919	1	2	yes	—
139	C₂×C₄	#11719919	1	2	yes	—
140	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
141	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
142	C₂×C₄	#11719921	1	2	yes	—
143	C₂×C₄	#11719921	1	2	yes	—
144	C₂×C₄	#11719921	1	2	yes	—
145	C₂×C₄	#11719921	1	2	yes	—
146	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
147	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
148	C₂×C₄	C₂²⋊Dic₅₃	1	2	yes	—
149	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
150	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
151	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
152	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
153	C₈	—	2	4	no	—
154	C₈	—	106	8	no	—
155	C₂²	—	2	4	no	—
156	C₂²	—	2	2	no	—
157	C₂²	—	4	1	no	—
158	C₂²	—	4	2	no	—
159	C₂²	—	4	1	no	—
160	C₂²	#13133124	1	2	yes	—
161	C₂²	#13133124	1	1	yes	characteristic
162	C₂²	#13133124	1	1	yes	characteristic
163	C₂²	C₂×#11719919	1	2	yes	—
164	C₂²	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
165	C₄	#13133124	1	2	yes	—
166	C₄	#13133124	1	1	yes	characteristic
167	C₄	#13133124	1	1	yes	characteristic
168	C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
169	C₄	C₂×#11719921	1	2	yes	—
170	C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
171	C₂	—	2	2	no	—
172	C₂	—	2	1	no	—
173	C₂	—	2	2	no	—
174	C₂	—	2	1	no	—
175	C₂	#16810381	1	2	yes	direct product factor
176	C₂	C₂×#13133124	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄×Dic₅₃
commutator	C₄×C₅₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂²×M₁₆×C₅₃	C₂
socle	C₂²×C₅₃	#95
hypercenter	C₂²×M₁₆	D₅₃
hypocenter	C₅₃	C₂×#338
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
2. #16810381	C₂	direct (N × H)
87. C₂×C₅₃	#338	semidirect
92. C₅₃	C₂×#338	semidirect
175. C₂	#16810381	direct (N × H)

Subgroups — C2×#16810381 #20274418

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16810381 #20274418