Subgroups — C₂×#20291608 #28329955

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	#20291608	C₂	1	16	yes	maximal, direct product factor
2	C₂×Q₈×#88324	C₂	1	8	yes	maximal, complemented
3	C₂×#16811639	C₂	1	6	yes	maximal
4	C₂²×Q₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
5	Q₈×#88324	C₂²	1	64	yes	complemented
6	#16811639	C₂²	1	48	yes	—
7	C₂×C₄⋊#88324	C₂²	1	12	yes	—
8	C₂×C₂²⋊#88324	C₂²	1	4	yes	—
9	C₂×C₄×#88324	C₂²	1	12	yes	—
10	C₂×Q₈×Dic₅₃	C₄	1	4	yes	—
11	C₂×Q₈×Dic₅₃	C₄	1	8	yes	—
12	C₂×Q₈×Dic₅₃	C₂²	1	8	yes	—
13	C₂×Q₈×Dic₅₃	C₂²	1	4	yes	—
14	C₂²×C₄⋊Dic₅₃	C₄	1	3	yes	—
15	C₂²×C₄×Dic₅₃	C₂²	1	3	yes	—
16	C₂³×Q₈×C₅₃	C₄	1	1	yes	characteristic
17	C₂²×#88324	C₂³	1	8	yes	—
18	C₄⋊#88324	C₂³	1	48	yes	—
19	C₂²⋊#88324	C₂³	1	16	yes	—
20	C₄×#88324	C₂³	1	48	yes	—
21	C₂³×Dic₅₃	C₂×C₄	1	3	yes	—
22	C₂³×Dic₅₃	C₂³	1	1	yes	characteristic
23	Q₈×Dic₅₃	C₂×C₄	1	32	yes	—
24	Q₈×Dic₅₃	C₂×C₄	1	16	yes	—
25	Q₈×Dic₅₃	C₂³	1	16	yes	—
26	C₂×C₄⋊Dic₅₃	C₂×C₄	1	12	yes	—
27	C₂×C₄⋊Dic₅₃	C₂×C₄	1	24	yes	—
28	C₂×C₄×Dic₅₃	C₂×C₄	1	6	yes	—
29	C₂×C₄×Dic₅₃	C₂×C₄	1	12	yes	—
30	C₂×C₄×Dic₅₃	C₂³	1	6	yes	—
31	C₂×C₄×Dic₅₃	C₂³	1	12	yes	—
32	C₂²×Q₈×C₅₃	C₈	1	16	yes	complemented
33	C₂²×Q₈×C₅₃	C₂×C₄	1	4	yes	—
34	C₂²×Q₈×C₅₃	C₂×C₄	1	8	yes	—
35	C₂³×C₄×C₅₃	C₂×C₄	1	3	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	C₂×C₈	1	24	yes	—
41	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
42	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
43	C₂²×Dic₅₃	—	2	4	no	—
44	C₂²×Dic₅₃	—	4	6	no	—
45	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×Dic₅₃	C₂²×C₄	1	2	yes	—
47	C₂²×Dic₅₃	C₂²×C₄	1	6	yes	—
48	C₂²×Dic₅₃	C₂²×C₄	1	12	yes	—
49	C₂²×Dic₅₃	C₂⁴	1	2	yes	—
50	C₂²×Dic₅₃	C₂⁴	1	1	yes	characteristic
51	C₂×#88324	—	2	16	no	—
52	C₂×#88324	C₂⁴	1	16	yes	—
53	C₄⋊Dic₅₃	C₂²×C₄	1	48	yes	—
54	C₄×Dic₅₃	C₂²×C₄	1	24	yes	—
55	C₄×Dic₅₃	C₂²×C₄	1	12	yes	—
56	C₄×Dic₅₃	C₂⁴	1	12	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	C₂²×C₈	1	24	yes	—
64	C₂×C₄×C₅₃	C₂²×C₈	1	48	yes	—
65	C₂×C₄×C₅₃	C₂³×C₄	1	12	yes	—
66	C₂×Dic₅₃	—	2	2	no	—
67	C₂×Dic₅₃	—	2	12	no	—
68	C₂×Dic₅₃	—	2	4	no	—
69	C₂×Dic₅₃	—	2	4	no	—
70	C₂×Dic₅₃	—	4	12	no	—
71	C₂×Dic₅₃	C₂³×C₄	1	12	yes	—
72	C₂×Dic₅₃	C₂³×C₄	1	2	yes	—
73	C₂×Dic₅₃	C₂³×C₄	1	1	yes	characteristic
74	C₂×Dic₅₃	2₋¹⁺⁴	1	4	yes	—
75	C₂×Dic₅₃	C₂⁵	1	1	yes	characteristic
76	#88324	—	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	C₂³×C₈	1	24	yes	—
87	Dic₅₃	—	2	2	no	—
88	Dic₅₃	—	2	12	no	—
89	Dic₅₃	#519	1	2	yes	—
90	Dic₅₃	C₂×2₋¹⁺⁴	1	2	yes	—
91	C₂²×C₄×Q₈	—	53	1	no	—
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	#6157	1	1	yes	characteristic
101	C₂×C₄×Q₈	—	53	4	no	—
102	C₂×C₄×Q₈	—	53	8	no	—
103	C₂×C₄×Q₈	—	53	4	no	—
104	C₂×C₄×Q₈	—	53	8	no	—
105	C₂²×C₄⋊C₄	—	53	3	no	—
106	C₂²×C₄²	—	53	3	no	—
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	#88324	1	16	yes	complemented
114	C₂²×Q₈	C₂×#6157	1	4	yes	—
115	C₂²×Q₈	C₂×#6157	1	8	yes	—
116	C₂³×C₄	—	53	1	no	—
117	C₂³×C₄	—	53	3	no	—
118	C₂³×C₄	C₂×#6157	1	3	yes	—
119	C₂²×C₈	—	53	8	no	—
120	C₄×Q₈	—	53	16	no	—
121	C₄×Q₈	—	53	32	no	—
122	C₄×Q₈	—	53	16	no	—
123	C₂×C₄⋊C₄	—	53	12	no	—
124	C₂×C₄⋊C₄	—	53	24	no	—
125	C₂×C₄²	—	53	12	no	—
126	C₂×C₄²	—	53	6	no	—
127	C₂×C₄²	—	53	6	no	—
128	C₂×C₄²	—	53	12	no	—
129	C₄⋊C₈	—	53	48	no	—
130	C₂²⋊C₈	—	53	16	no	—
131	C₄×C₈	—	53	48	no	—
132	C₂⁴	C₂²×#6157	1	1	yes	characteristic
133	C₂×Q₈	C₂×#88324	1	32	yes	—
134	C₂×Q₈	C₂×#88324	1	64	yes	complemented
135	C₂×Q₈	C₂²×#6157	1	16	yes	—
136	C₂²×C₄	—	53	1	no	—
137	C₂²×C₄	—	53	2	no	—
138	C₂²×C₄	—	53	2	no	—
139	C₂²×C₄	—	53	1	no	—
140	C₂²×C₄	—	53	6	no	—
141	C₂²×C₄	—	53	12	no	—
142	C₂²×C₄	—	106	4	no	—
143	C₂²×C₄	—	212	6	no	—
144	C₂²×C₄	C₂×#88324	1	24	yes	—
145	C₂²×C₄	C₂²×#6157	1	12	yes	—
146	C₂²×C₄	C₂²×#6157	1	6	yes	—
147	C₂×C₈	—	53	16	no	—
148	C₂×C₈	—	106	16	no	—
149	C₄⋊C₄	—	53	48	no	—
150	C₄²	—	53	12	no	—
151	C₄²	—	53	12	no	—
152	C₄²	—	53	24	no	—
153	C₂³	—	2	2	no	—
154	C₂³	—	2	2	no	—
155	C₂³	C₂²×#88324	1	4	yes	—
156	C₂³	C₂³×#6157	1	2	yes	—
157	C₂³	C₂³×#6157	1	1	yes	characteristic
158	Q₈	C₂²×#88324	1	64	yes	—
159	C₂×C₄	—	53	4	no	—
160	C₂×C₄	—	53	1	no	—
161	C₂×C₄	—	53	1	no	—
162	C₂×C₄	—	53	2	no	—
163	C₂×C₄	—	53	12	no	—
164	C₂×C₄	—	106	4	no	—
165	C₂×C₄	—	106	4	no	—
166	C₂×C₄	—	106	2	no	—
167	C₂×C₄	—	106	12	no	—
168	C₂×C₄	—	212	12	no	—
169	C₂×C₄	C₂²×#88324	1	24	yes	—
170	C₂×C₄	C₂²×#88324	1	48	yes	—
171	C₂×C₄	C₂³×#6157	1	12	yes	—
172	C₈	—	106	16	no	—
173	C₂²	—	2	8	no	—
174	C₂²	—	2	2	no	—
175	C₂²	—	2	2	no	—
176	C₂²	#16811691	1	2	yes	—
177	C₂²	C₂³×#88324	1	4	yes	—
178	C₂²	C₂³×#88324	1	2	yes	—
179	C₂²	#16811703	1	2	yes	—
180	C₂²	C₂⁴×#6157	1	1	yes	characteristic
181	C₄	—	53	2	no	—
182	C₄	—	53	2	no	—
183	C₄	—	106	2	no	—
184	C₄	—	106	12	no	—
185	C₄	C₂³×#88324	1	24	yes	—
186	C₂	—	2	4	no	—
187	C₂	#20291608	1	4	yes	direct product factor
188	C₂	C₂×#16811691	1	1	yes	characteristic
189	C₂	C₂⁴×#88324	1	1	yes	characteristic
190	C₂	C₂×#16811703	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×#6157
commutator	C₂×C₅₃	C₂⁴×C₈
Frattini	C₂²	C₂⁴×#6157
Fitting	C₂³×Q₈×C₅₃	C₄
socle	C₂³×C₅₃	C₂³×C₄
prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂×Dic₅₃	C₂⁵
hypercenter	C₂³×Q₈	#6157
hypocenter	C₅₃	C₂×#2982
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂³×Q₈

Complements

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

N	H ≅ G/N	type
1. #20291608	C₂	direct (N × H)
2. C₂×Q₈×#88324	C₂	semidirect
5. Q₈×#88324	C₂²	semidirect
32. 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	#88324	semidirect
134. C₂×Q₈	C₂×#88324	semidirect
187. C₂	#20291608	direct (N × H)

Maximal subgroups

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

subgroup	index
#20291608	2
C₂×Q₈×#88324	2
C₂×#16811639	2
C₂²×Q₈×Dic₅₃	2
C₂×#2982	53

Subgroups — C2×#20291608 #28329955

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20291608 #28329955