Subgroups — C₂×#20283796 #28249031

Order 13568.

2234 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283796	C₂	1	16	yes	maximal, direct product factor
2	C₂×Q₈×#88322	C₂	1	8	yes	maximal, complemented
3	C₂×#16810676	C₂	1	6	yes	maximal
4	C₂²×C₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	Q₈×#88322	C₂²	1	64	yes	complemented
6	#16810676	C₂²	1	48	yes	—
7	C₂×C₄⋊#88322	C₂²	1	12	yes	—
8	C₂×C₂²⋊#88322	C₂²	1	4	yes	—
9	C₂×C₄×#88322	C₂²	1	12	yes	—
10	C₂³×Q₈×C₅₃	C₄	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₅₃	C₄	1	4	yes	—
12	C₂×C₄×Q₈×C₅₃	C₄	1	8	yes	—
13	C₂×C₄×Q₈×C₅₃	C₂²	1	8	yes	—
14	C₂×C₄×Q₈×C₅₃	C₂²	1	4	yes	—
15	C₂²×C₄⋊C₄×C₅₃	C₄	1	3	yes	—
16	C₂²×C₄²×C₅₃	C₂²	1	3	yes	—
17	C₂²×#88322	C₂³	1	8	yes	—
18	C₄⋊#88322	C₂³	1	48	yes	—
19	C₂²⋊#88322	C₂³	1	16	yes	—
20	C₄×#88322	C₂³	1	48	yes	—
21	C₂²×Q₈×C₅₃	C₈	1	16	yes	complemented
22	C₂²×Q₈×C₅₃	C₂×C₄	1	4	yes	—
23	C₂²×Q₈×C₅₃	C₂×C₄	1	8	yes	—
24	C₂³×C₄×C₅₃	C₂×C₄	1	3	yes	—
25	C₂³×C₄×C₅₃	C₂×C₄	1	3	yes	—
26	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
27	C₄×Q₈×C₅₃	C₂×C₄	1	32	yes	—
28	C₄×Q₈×C₅₃	C₂×C₄	1	16	yes	—
29	C₄×Q₈×C₅₃	C₂³	1	16	yes	—
30	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	12	yes	—
31	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	24	yes	—
32	C₂×C₄²×C₅₃	C₂×C₄	1	6	yes	—
33	C₂×C₄²×C₅₃	C₂×C₄	1	12	yes	—
34	C₂×C₄²×C₅₃	C₂³	1	6	yes	—
35	C₂×C₄²×C₅₃	C₂³	1	12	yes	—
36	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×Q₈×C₅₃	C₂×C₈	1	32	yes	—
38	C₂×Q₈×C₅₃	C₂×C₈	1	64	yes	complemented
39	C₂×Q₈×C₅₃	C₂²×C₄	1	16	yes	—
40	C₂²×C₄×C₅₃	—	2	4	no	—
41	C₂²×C₄×C₅₃	—	4	6	no	—
42	C₂²×C₄×C₅₃	C₂×C₈	1	24	yes	—
43	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
45	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
46	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
47	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
48	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
49	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
50	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
51	C₄⋊C₄×C₅₃	C₂²×C₄	1	48	yes	—
52	C₄²×C₅₃	C₂²×C₄	1	24	yes	—
53	C₄²×C₅₃	C₂²×C₄	1	12	yes	—
54	C₄²×C₅₃	C₂⁴	1	12	yes	—
55	C₂×#88322	—	2	16	no	—
56	C₂×#88322	C₂⁴	1	16	yes	—
57	C₂³×C₅₃	—	2	2	no	—
58	C₂³×C₅₃	—	2	2	no	—
59	C₂³×C₅₃	C₂²×C₈	1	4	yes	—
60	C₂³×C₅₃	C₂³×C₄	1	2	yes	—
61	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
62	Q₈×C₅₃	C₂²×C₈	1	64	yes	—
63	C₂×C₄×C₅₃	—	2	2	no	—
64	C₂×C₄×C₅₃	—	2	12	no	—
65	C₂×C₄×C₅₃	—	2	4	no	—
66	C₂×C₄×C₅₃	—	2	4	no	—
67	C₂×C₄×C₅₃	—	4	12	no	—
68	C₂×C₄×C₅₃	C₂²×C₈	1	24	yes	—
69	C₂×C₄×C₅₃	C₂²×C₈	1	48	yes	—
70	C₂×C₄×C₅₃	C₂³×C₄	1	12	yes	—
71	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
72	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂³×C₄	1	12	yes	—
74	C₂×C₄×C₅₃	2₋¹⁺⁴	1	4	yes	—
75	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
76	#88322	—	2	16	no	—
77	C₂×#2982	—	53	1	no	maximal, Sylow
78	C₂²×C₅₃	—	2	2	no	—
79	C₂²×C₅₃	—	2	8	no	—
80	C₂²×C₅₃	—	2	2	no	—
81	C₂²×C₅₃	#519	1	2	yes	—
82	C₂²×C₅₃	C₂³×C₈	1	4	yes	—
83	C₂²×C₅₃	C₂³×C₈	1	2	yes	—
84	C₂²×C₅₃	#568	1	2	yes	—
85	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
86	C₄×C₅₃	—	2	2	no	—
87	C₄×C₅₃	—	2	12	no	—
88	C₄×C₅₃	#519	1	2	yes	—
89	C₄×C₅₃	C₂³×C₈	1	24	yes	—
90	C₄×C₅₃	C₂×2₋¹⁺⁴	1	2	yes	—
91	C₂²×C₄×Q₈	D₅₃	1	1	yes	characteristic
92	#2982	—	53	16	no	—
93	C₂×C₈×Q₈	—	53	8	no	—
94	C₂×#431	—	53	6	no	—
95	C₂×C₅₃	—	2	4	no	—
96	C₂×C₅₃	#2982	1	4	yes	complemented
97	C₂×C₅₃	C₂×#519	1	1	yes	characteristic
98	C₂×C₅₃	C₂⁴×C₈	1	1	yes	characteristic
99	C₂×C₅₃	C₂×#568	1	1	yes	characteristic
100	C₂³×Q₈	Dic₅₃	1	1	yes	characteristic
101	C₂×C₄×Q₈	Dic₅₃	1	4	yes	—
102	C₂×C₄×Q₈	Dic₅₃	1	8	yes	—
103	C₂×C₄×Q₈	C₂×D₅₃	1	4	yes	—
104	C₂×C₄×Q₈	C₂×D₅₃	1	8	yes	—
105	C₂²×C₄⋊C₄	Dic₅₃	1	3	yes	—
106	C₂²×C₄²	C₂×D₅₃	1	3	yes	—
107	C₈×Q₈	—	53	64	no	—
108	#431	—	53	48	no	—
109	C₂×C₄⋊C₈	—	53	12	no	—
110	C₂×C₂²⋊C₈	—	53	4	no	—
111	C₂×C₄×C₈	—	53	12	no	—
112	C₅₃	C₂×#2982	1	1	yes	characteristic, complemented, Sylow
113	C₂²×Q₈	#88322	1	16	yes	complemented
114	C₂²×Q₈	C₂×Dic₅₃	1	4	yes	—
115	C₂²×Q₈	C₂×Dic₅₃	1	8	yes	—
116	C₂³×C₄	C₂×Dic₅₃	1	3	yes	—
117	C₂³×C₄	C₂×Dic₅₃	1	3	yes	—
118	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
119	C₂²×C₈	—	53	8	no	—
120	C₄×Q₈	C₂×Dic₅₃	1	16	yes	—
121	C₄×Q₈	C₂×Dic₅₃	1	32	yes	—
122	C₄×Q₈	C₂²×D₅₃	1	16	yes	—
123	C₂×C₄⋊C₄	C₂×Dic₅₃	1	12	yes	—
124	C₂×C₄⋊C₄	C₂×Dic₅₃	1	24	yes	—
125	C₂×C₄²	C₂×Dic₅₃	1	6	yes	—
126	C₂×C₄²	C₂×Dic₅₃	1	12	yes	—
127	C₂×C₄²	C₂²×D₅₃	1	12	yes	—
128	C₂×C₄²	C₂²×D₅₃	1	6	yes	—
129	C₄⋊C₈	—	53	48	no	—
130	C₂²⋊C₈	—	53	16	no	—
131	C₄×C₈	—	53	48	no	—
132	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
133	C₂×Q₈	C₂×#88322	1	32	yes	—
134	C₂×Q₈	C₂×#88322	1	64	yes	complemented
135	C₂×Q₈	C₂²×Dic₅₃	1	16	yes	—
136	C₂²×C₄	—	2	4	no	—
137	C₂²×C₄	—	4	6	no	—
138	C₂²×C₄	C₂×#88322	1	24	yes	—
139	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
140	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
141	C₂²×C₄	C₂²×Dic₅₃	1	6	yes	—
142	C₂²×C₄	C₂²×Dic₅₃	1	12	yes	—
143	C₂²×C₄	C₂²×Dic₅₃	1	6	yes	—
144	C₂²×C₄	C₂²×Dic₅₃	1	12	yes	—
145	C₂²×C₄	C₂³×D₅₃	1	2	yes	—
146	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
147	C₂×C₈	—	53	16	no	—
148	C₂×C₈	—	106	16	no	—
149	C₄⋊C₄	C₂²×Dic₅₃	1	48	yes	—
150	C₄²	C₂²×Dic₅₃	1	24	yes	—
151	C₄²	C₂²×Dic₅₃	1	12	yes	—
152	C₄²	C₂³×D₅₃	1	12	yes	—
153	C₂³	—	2	2	no	—
154	C₂³	—	2	2	no	—
155	C₂³	C₂²×#88322	1	4	yes	—
156	C₂³	C₂³×Dic₅₃	1	2	yes	—
157	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
158	Q₈	C₂²×#88322	1	64	yes	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	12	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	4	12	no	—
164	C₂×C₄	C₂²×#88322	1	24	yes	—
165	C₂×C₄	C₂²×#88322	1	48	yes	—
166	C₂×C₄	C₂³×Dic₅₃	1	2	yes	—
167	C₂×C₄	C₂³×Dic₅₃	1	12	yes	—
168	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
169	C₂×C₄	C₂³×Dic₅₃	1	12	yes	—
170	C₂×C₄	#13133265	1	4	yes	—
171	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
172	C₈	—	106	16	no	—
173	C₂²	—	2	2	no	—
174	C₂²	—	2	8	no	—
175	C₂²	—	2	2	no	—
176	C₂²	#16811101	1	2	yes	—
177	C₂²	C₂³×#88322	1	4	yes	—
178	C₂²	C₂³×#88322	1	2	yes	—
179	C₂²	#16811390	1	2	yes	—
180	C₂²	C₂⁴×Dic₅₃	1	1	yes	characteristic
181	C₄	—	2	12	no	—
182	C₄	—	2	2	no	—
183	C₄	#16811101	1	2	yes	—
184	C₄	C₂³×#88322	1	24	yes	—
185	C₄	C₂×#13133265	1	2	yes	—
186	C₂	—	2	4	no	—
187	C₂	#20283796	1	4	yes	direct product factor
188	C₂	C₂×#16811101	1	1	yes	characteristic
189	C₂	C₂⁴×#88322	1	1	yes	characteristic
190	C₂	C₂×#16811390	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂³×D₅₃
commutator	C₂×C₅₃	C₂⁴×C₈
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂²×C₄×Q₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂³×C₄
prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂×C₄×C₅₃	C₂⁵
hypercenter	C₂²×C₄×Q₈	D₅₃
hypocenter	C₅₃	C₂×#2982
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #20283796	C₂	direct (N × H)
2. C₂×Q₈×#88322	C₂	semidirect
5. Q₈×#88322	C₂²	semidirect
21. C₂²×Q₈×C₅₃	C₈	semidirect
38. C₂×Q₈×C₅₃	C₂×C₈	semidirect
96. C₂×C₅₃	#2982	semidirect
112. C₅₃	C₂×#2982	semidirect
113. C₂²×Q₈	#88322	semidirect
134. C₂×Q₈	C₂×#88322	semidirect
187. C₂	#20283796	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20283796	2
C₂×Q₈×#88322	2
C₂×#16810676	2
C₂²×C₄×Q₈×C₅₃	2
C₂×#2982	53

Subgroups — C2×#20283796 #28249031

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20283796 #28249031