Subgroups — C₂×#20291129 #28328419

Order 13568.

582 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291129	C₂	1	8	yes	maximal, direct product factor
2	C₂×#16811544	C₂	1	2	yes	maximal
3	C₂×C₄⋊#11719910	C₂	1	1	yes	maximal, characteristic
4	C₂×#16811509	C₂	1	2	yes	maximal
5	C₂×C₄×#11719910	C₂	1	1	yes	maximal, characteristic
6	C₂×#16810747	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11719910	C₂²	1	1	yes	characteristic
8	C₂²×#11719910	C₂²	1	2	yes	—
9	C₂²×#11719910	C₂²	1	1	yes	characteristic
10	#16811544	C₂²	1	8	yes	—
11	C₄⋊#11719910	C₂²	1	4	yes	—
12	#16811509	C₂²	1	8	yes	—
13	C₄×#11719910	C₂²	1	4	yes	—
14	C₂×C₄×Dic₁₀₆	C₄	1	1	yes	characteristic
15	#16810747	C₄	1	4	yes	—
16	#16810747	C₂²	1	4	yes	—
17	C₂×#13133146	C₄	1	2	yes	—
18	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
20	C₂×C₄⋊C₈×C₅₃	C₄	1	1	yes	characteristic
21	C₂×#11719910	—	2	2	no	—
22	C₂×#11719910	—	2	4	no	—
23	C₂×#11719910	Q₈	1	4	yes	complemented
24	C₂×#11719910	C₂³	1	4	yes	—
25	C₂×#11719910	C₂³	1	2	yes	—
26	C₂×#11719910	C₂³	1	2	yes	—
27	C₂²×Dic₁₀₆	C₈	1	1	yes	characteristic
28	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
29	C₂²×#88322	C₂³	1	1	yes	characteristic
30	C₄×Dic₁₀₆	C₈	1	4	yes	—
31	C₄×Dic₁₀₆	C₂×C₄	1	4	yes	—
32	C₂×C₄⋊Dic₅₃	C₈	1	1	yes	characteristic
33	C₂×#11719919	C₈	1	2	yes	—
34	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
35	#13133146	C₂×C₄	1	8	yes	—
36	C₄×#88322	C₂×C₄	1	2	yes	—
37	C₄×#88322	C₂³	1	2	yes	—
38	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
39	C₈×Dic₅₃	C₂³	1	4	yes	—
40	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
43	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
47	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
48	C₂×Dic₁₀₆	C₁₆	1	8	yes	complemented
49	C₂×Dic₁₀₆	C₂×C₈	1	4	yes	—
50	C₂×C₈×C₅₃	—	2	4	no	—
51	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
52	C₄²×C₅₃	C₂×C₈	1	2	yes	—
53	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
54	C₄⋊Dic₅₃	C₂×C₈	1	4	yes	—
55	#11719919	C₂×C₈	1	8	yes	—
56	C₄×Dic₅₃	C₂×C₈	1	4	yes	—
57	C₄×Dic₅₃	C₂²×C₄	1	4	yes	—
58	C₂×#88322	—	2	2	no	—
59	C₂×#88322	C₂²×C₄	1	2	yes	—
60	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
61	C₂×#88322	C₂×Q₈	1	2	yes	—
62	C₂×#88322	#41	1	2	yes	—
63	C₂×#88322	C₂⁴	1	1	yes	characteristic
64	#11719910	—	2	4	no	—
65	#11719910	—	2	2	no	—
66	#11719910	C₂×Q₈	1	4	yes	complemented
67	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
68	C₂×C₄×C₅₃	—	2	2	no	—
69	C₂×C₄×C₅₃	C₂×C₁₆	1	4	yes	—
70	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
71	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
72	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
74	C₂×C₄×C₅₃	#131	1	2	yes	—
75	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
76	C₂×Dic₅₃	—	2	4	no	—
77	C₂×Dic₅₃	C₂×C₁₆	1	8	yes	—
78	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
79	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
80	Dic₁₀₆	C₂×C₁₆	1	16	yes	complemented
81	C₈×C₅₃	—	2	4	no	—
82	#88322	—	2	2	no	—
83	#88322	C₄×Q₈	1	2	yes	—
84	#88322	C₂²×Q₈	1	1	yes	characteristic
85	#88322	C₂×#41	1	1	yes	characteristic
86	C₂×Q₈×C₁₆	—	53	1	no	maximal, Sylow
87	C₂²×C₅₃	C₈×Q₈	1	2	yes	—
88	C₂²×C₅₃	C₂²×C₁₆	1	2	yes	—
89	C₂²×C₅₃	#504	1	2	yes	—
90	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
91	C₄×C₅₃	—	2	2	no	—
92	C₄×C₅₃	C₈×Q₈	1	2	yes	—
93	C₄×C₅₃	C₂²×C₁₆	1	4	yes	—
94	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
95	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
96	Dic₅₃	—	2	4	no	—
97	Dic₅₃	C₂²×C₁₆	1	8	yes	—
98	C₂×C₈×Q₈	—	53	1	no	—
99	Q₈×C₁₆	—	53	8	no	—
100	C₂×C₄⋊C₁₆	—	53	1	no	—
101	C₂×C₄⋊C₁₆	—	53	2	no	—
102	C₂×C₄×C₁₆	—	53	1	no	—
103	C₂×C₄×C₁₆	—	53	2	no	—
104	C₂×C₅₃	Q₈×C₁₆	1	4	yes	complemented
105	C₂×C₅₃	C₂×C₈×Q₈	1	1	yes	characteristic
106	C₂×C₅₃	C₂³×C₁₆	1	1	yes	characteristic
107	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
108	C₂×C₄×Q₈	—	53	1	no	—
109	C₂²×C₁₆	—	53	1	no	—
110	C₂²×C₁₆	—	53	1	no	—
111	C₂²×C₁₆	—	53	2	no	—
112	C₈×Q₈	—	53	4	no	—
113	C₈×Q₈	—	53	4	no	—
114	C₂×C₄⋊C₈	—	53	2	no	—
115	C₂×C₄⋊C₈	#6157	1	1	yes	characteristic
116	C₂×C₄×C₈	—	53	2	no	—
117	C₂×C₄×C₈	—	53	1	no	—
118	C₄⋊C₁₆	—	53	4	no	—
119	C₄⋊C₁₆	—	53	8	no	—
120	C₄×C₁₆	—	53	4	no	—
121	C₄×C₁₆	—	53	8	no	—
122	C₅₃	C₂×Q₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
123	C₂²×Q₈	—	53	1	no	—
124	C₂²×C₈	—	53	1	no	—
125	C₂²×C₈	—	53	1	no	—
126	C₂²×C₈	C₂×#6157	1	2	yes	—
127	C₄×Q₈	—	53	4	no	—
128	C₄×Q₈	—	53	4	no	—
129	C₂×C₄⋊C₄	—	53	2	no	—
130	C₂×C₄⋊C₄	—	53	1	no	—
131	C₂×C₄²	—	53	2	no	—
132	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
133	C₂×C₁₆	—	53	2	no	—
134	C₂×C₁₆	—	53	4	no	—
135	C₂×C₁₆	—	53	2	no	—
136	C₂×C₁₆	—	53	4	no	—
137	C₂×C₁₆	—	106	2	no	—
138	C₂×C₁₆	—	106	4	no	—
139	C₄⋊C₈	—	53	8	no	—
140	C₄⋊C₈	C₂×#6157	1	4	yes	—
141	C₄×C₈	—	53	4	no	—
142	C₄×C₈	—	53	4	no	—
143	C₄×C₈	—	53	2	no	—
144	C₄×C₈	—	53	2	no	—
145	C₂×Q₈	—	53	8	no	—
146	C₂×Q₈	—	53	4	no	—
147	C₂²×C₄	—	53	2	no	—
148	C₂²×C₄	—	53	2	no	—
149	C₂²×C₄	D₅₃⋊C₈	1	1	yes	characteristic
150	C₂²×C₄	D₅₃⋊C₈	1	1	yes	characteristic
151	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	—	53	2	no	—
154	C₂×C₈	—	53	2	no	—
155	C₂×C₈	—	53	1	no	—
156	C₂×C₈	—	53	1	no	—
157	C₂×C₈	—	53	2	no	—
158	C₂×C₈	—	106	2	no	—
159	C₂×C₈	C₂²×#6157	1	4	yes	—
160	C₄⋊C₄	—	53	8	no	—
161	C₄⋊C₄	—	53	4	no	—
162	C₄²	—	53	4	no	—
163	C₄²	—	53	4	no	—
164	C₄²	D₅₃⋊C₈	1	2	yes	—
165	C₄²	C₂²×#6157	1	2	yes	—
166	C₁₆	—	53	4	no	—
167	C₁₆	—	106	2	no	—
168	C₁₆	—	106	4	no	—
169	C₂³	C₂×D₅₃⋊C₈	1	1	yes	characteristic
170	Q₈	—	53	16	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	53	8	no	—
173	C₂×C₄	—	53	4	no	—
174	C₂×C₄	—	53	4	no	—
175	C₂×C₄	—	106	4	no	—
176	C₂×C₄	D₅₃⋊C₁₆	1	4	yes	—
177	C₂×C₄	Q₈×#6157	1	2	yes	—
178	C₂×C₄	C₂×D₅₃⋊C₈	1	2	yes	—
179	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
180	C₂×C₄	C₂×D₅₃⋊C₈	1	2	yes	—
181	C₂×C₄	#13133304	1	2	yes	—
182	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
183	C₈	—	2	4	no	—
184	C₈	—	53	2	no	—
185	C₈	—	53	1	no	—
186	C₈	—	53	1	no	—
187	C₈	—	106	2	no	—
188	C₂²	#16811672	1	2	yes	—
189	C₂²	C₂×D₅₃⋊C₁₆	1	2	yes	—
190	C₂²	#16811682	1	2	yes	—
191	C₂²	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
192	C₄	—	2	2	no	—
193	C₄	—	53	8	no	—
194	C₄	—	106	4	no	—
195	C₄	#16811672	1	2	yes	—
196	C₄	C₂×D₅₃⋊C₁₆	1	4	yes	—
197	C₄	C₂×Q₈×#6157	1	1	yes	characteristic
198	C₄	C₂×#13133304	1	1	yes	characteristic
199	C₂	#20291129	1	4	yes	direct product factor
200	C₂	C₂×#16811672	1	1	yes	characteristic
201	C₂	C₂²×D₅₃⋊C₁₆	1	1	yes	characteristic
202	C₂	C₂×#16811682	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×#6157
commutator	C₂×C₅₃	C₂³×C₁₆
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₂×C₄⋊C₈×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂×#88322	C₂⁴
hypercenter	C₂×C₄⋊C₈	#6157
hypocenter	C₅₃	C₂×Q₈×C₁₆
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. #20291129	C₂	direct (N × H)
23. C₂×#11719910	Q₈	semidirect
48. C₂×Dic₁₀₆	C₁₆	semidirect
66. #11719910	C₂×Q₈	semidirect
80. Dic₁₀₆	C₂×C₁₆	semidirect
104. C₂×C₅₃	Q₈×C₁₆	semidirect
122. C₅₃	C₂×Q₈×C₁₆	semidirect
199. C₂	#20291129	direct (N × H)

Maximal subgroups

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

subgroup	index
#20291129	2
C₂×#16811544	2
C₂×C₄⋊#11719910	2
C₂×#16811509	2
C₂×C₄×#11719910	2
C₂×#16810747	2
C₂×Q₈×C₁₆	53

Subgroups — C2×#20291129 #28328419

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20291129 #28328419