Subgroups — C₂×C₄²×Dic₁₀₆ #27542288

Order 13568.

3966 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Dic₁₀₆	C₂	1	3	yes	maximal
2	C₄²×Dic₁₀₆	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄×#11719919	C₂	1	6	yes	maximal
4	C₂×C₄×C₄⋊Dic₅₃	C₂	1	3	yes	maximal
5	C₂×C₄²×Dic₅₃	C₂	1	2	yes	maximal
6	C₂×C₄³×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₁₀₆	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₀₆	C₄	1	48	yes	direct product factor
9	C₂×C₄×Dic₁₀₆	C₂²	1	24	yes	—
10	C₂²×C₄⋊Dic₅₃	C₂²	1	3	yes	—
11	C₂²×#11719919	C₂²	1	6	yes	—
12	C₂²×C₄×Dic₅₃	C₂²	1	6	yes	—
13	C₂²×C₄×Dic₅₃	C₂²	1	6	yes	—
14	C₄×#11719919	C₂²	1	48	yes	—
15	C₄×C₄⋊Dic₅₃	C₂²	1	24	yes	—
16	C₂×#13133124	C₂²	1	6	yes	—
17	C₄²×Dic₅₃	C₂²	1	16	yes	—
18	C₂²×C₄²×C₅₃	C₂²	1	3	yes	—
19	C₂²×C₄²×C₅₃	C₂²	1	3	yes	—
20	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
21	C₄³×C₅₃	C₂²	1	8	yes	—
22	C₂²×Dic₁₀₆	C₂×C₄	1	24	yes	—
23	C₂²×Dic₁₀₆	C₂³	1	4	yes	—
24	C₂³×Dic₅₃	C₂³	1	6	yes	—
25	C₂³×Dic₅₃	C₂³	1	2	yes	—
26	C₄×Dic₁₀₆	C₂×C₄	1	192	yes	direct product factor
27	C₂×C₄⋊Dic₅₃	C₂×C₄	1	24	yes	—
28	C₂×C₄⋊Dic₅₃	C₂³	1	12	yes	—
29	C₂×#11719919	C₂×C₄	1	48	yes	—
30	C₂×#11719919	C₂³	1	24	yes	—
31	C₂×C₄×Dic₅₃	—	2	24	no	—
32	C₂×C₄×Dic₅₃	C₂×C₄	1	48	yes	—
33	C₂×C₄×Dic₅₃	C₂³	1	24	yes	—
34	C₂×C₄×Dic₅₃	C₂³	1	24	yes	—
35	#13133124	C₂³	1	24	yes	—
36	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
37	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
38	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
39	C₂×C₄²×C₅₃	—	2	12	no	—
40	C₂×C₄²×C₅₃	C₂×C₄	1	24	yes	—
41	C₂×C₄²×C₅₃	Q₈	1	8	yes	complemented
42	C₂×C₄²×C₅₃	C₂³	1	12	yes	—
43	C₂×C₄²×C₅₃	C₂³	1	12	yes	—
44	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
45	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	—	2	12	no	—
47	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
48	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
49	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
50	C₂²×C₄×C₅₃	C₂×Q₈	1	12	yes	—
51	C₂²×C₄×C₅₃	#41	1	12	yes	—
52	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
53	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
54	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
55	C₂²×Dic₅₃	—	2	24	no	—
56	C₂²×Dic₅₃	C₂²×C₄	1	24	yes	—
57	C₂²×Dic₅₃	C₂²×C₄	1	24	yes	—
58	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
59	C₂²×Dic₅₃	C₂⁴	1	4	yes	—
60	C₂×Dic₁₀₆	C₄²	1	64	yes	direct product factor
61	C₂×Dic₁₀₆	C₂²×C₄	1	48	yes	—
62	C₄²×C₅₃	—	2	24	no	—
63	C₄²×C₅₃	C₂²×C₄	1	48	yes	—
64	C₄²×C₅₃	C₂×Q₈	1	16	yes	complemented
65	C₄⋊Dic₅₃	C₂²×C₄	1	48	yes	—
66	#11719919	C₂²×C₄	1	96	yes	—
67	C₄×Dic₅₃	—	2	48	no	—
68	C₄×Dic₅₃	C₂²×C₄	1	96	yes	—
69	C₂³×C₅₃	C₂³×C₄	1	6	yes	—
70	C₂³×C₅₃	C₂²×Q₈	1	2	yes	—
71	C₂³×C₅₃	C₂×#41	1	6	yes	—
72	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
73	C₂×C₄×C₅₃	—	2	24	no	—
74	C₂×C₄×C₅₃	—	2	12	no	—
75	C₂×C₄×C₅₃	C₂×C₄²	1	16	yes	—
76	C₂×C₄×C₅₃	C₄×Q₈	1	48	yes	complemented
77	C₂×C₄×C₅₃	C₂³×C₄	1	12	yes	—
78	C₂×C₄×C₅₃	C₂³×C₄	1	12	yes	—
79	C₂×C₄×C₅₃	C₂³×C₄	1	12	yes	—
80	C₂×C₄×C₅₃	C₂²×Q₈	1	12	yes	—
81	C₂×C₄×C₅₃	C₂×#41	1	12	yes	—
82	C₂×Dic₅₃	—	2	48	no	—
83	C₂×Dic₅₃	—	2	24	no	—
84	C₂×Dic₅₃	C₂×C₄²	1	32	yes	—
85	C₂×Dic₅₃	C₂³×C₄	1	24	yes	—
86	C₂×Dic₅₃	C₂³×C₄	1	24	yes	—
87	Dic₁₀₆	C₂×C₄²	1	64	yes	direct product factor
88	C₂×C₄²×Q₈	—	53	1	no	maximal, Sylow
89	C₂²×C₅₃	C₂²×C₄²	1	4	yes	—
90	C₂²×C₅₃	C₂×C₄×Q₈	1	12	yes	—
91	C₂²×C₅₃	C₄×#41	1	12	yes	—
92	C₂²×C₅₃	C₂⁴×C₄	1	3	yes	—
93	C₂²×C₅₃	C₂³×Q₈	1	1	yes	characteristic
94	C₂²×C₅₃	C₂²×#41	1	3	yes	—
95	C₄×C₅₃	—	2	12	no	—
96	C₄×C₅₃	C₂²×C₄²	1	8	yes	—
97	C₄×C₅₃	C₂×C₄×Q₈	1	24	yes	complemented
98	Dic₅₃	—	2	24	no	—
99	Dic₅₃	C₂²×C₄²	1	16	yes	—
100	C₂²×C₄×Q₈	—	53	3	no	—
101	C₄²×Q₈	—	53	16	no	—
102	C₂×C₄×C₄⋊C₄	—	53	3	no	—
103	C₂×C₄×C₄⋊C₄	—	53	6	no	—
104	C₂×C₄³	—	53	2	no	—
105	C₂×C₄³	D₅₃	1	1	yes	characteristic
106	C₂×C₅₃	C₄²×Q₈	1	8	yes	complemented
107	C₂×C₅₃	C₂³×C₄²	1	1	yes	characteristic
108	C₂×C₅₃	C₂²×C₄×Q₈	1	3	yes	—
109	C₂×C₅₃	C₂×C₄×#41	1	3	yes	—
110	C₂³×Q₈	—	53	1	no	—
111	C₂×C₄×Q₈	—	53	48	no	—
112	C₂×C₄×Q₈	—	53	24	no	—
113	C₂²×C₄⋊C₄	—	53	6	no	—
114	C₂²×C₄⋊C₄	—	53	3	no	—
115	C₂²×C₄²	—	53	6	no	—
116	C₂²×C₄²	—	53	6	no	—
117	C₂²×C₄²	C₂×D₅₃	1	3	yes	—
118	C₂²×C₄²	C₂×D₅₃	1	3	yes	—
119	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
120	C₄×C₄⋊C₄	—	53	24	no	—
121	C₄×C₄⋊C₄	—	53	48	no	—
122	C₂×#95	—	53	6	no	—
123	C₄³	—	53	16	no	—
124	C₄³	C₂×D₅₃	1	8	yes	—
125	C₅₃	C₂×C₄²×Q₈	1	1	yes	characteristic, complemented, Sylow
126	C₂²×Q₈	—	53	24	no	—
127	C₂²×Q₈	—	53	4	no	—
128	C₂³×C₄	—	53	2	no	—
129	C₂³×C₄	—	53	6	no	—
130	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
131	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
132	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
133	C₄×Q₈	—	53	192	no	—
134	C₂×C₄⋊C₄	—	53	48	no	—
135	C₂×C₄⋊C₄	—	53	24	no	—
136	C₂×C₄⋊C₄	—	53	24	no	—
137	C₂×C₄⋊C₄	—	53	12	no	—
138	C₂×C₄²	—	2	12	no	—
139	C₂×C₄²	—	53	48	no	—
140	C₂×C₄²	—	53	24	no	—
141	C₂×C₄²	—	53	24	no	—
142	C₂×C₄²	—	106	24	no	—
143	C₂×C₄²	Dic₁₀₆	1	8	yes	direct product factor
144	C₂×C₄²	C₄×D₅₃	1	24	yes	—
145	C₂×C₄²	C₂²×D₅₃	1	12	yes	—
146	C₂×C₄²	C₂²×D₅₃	1	4	yes	—
147	C₂×C₄²	C₂²×D₅₃	1	12	yes	—
148	#95	—	53	24	no	—
149	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
150	C₂×Q₈	—	53	64	no	—
151	C₂×Q₈	—	53	48	no	—
152	C₂²×C₄	—	2	12	no	—
153	C₂²×C₄	—	53	24	no	—
154	C₂²×C₄	—	53	4	no	—
155	C₂²×C₄	—	53	24	no	—
156	C₂²×C₄	—	53	12	no	—
157	C₂²×C₄	—	106	24	no	—
158	C₂²×C₄	C₂×Dic₁₀₆	1	12	yes	—
159	C₂²×C₄	C₂×C₄×D₅₃	1	12	yes	—
160	C₂²×C₄	C₂×C₄×D₅₃	1	12	yes	—
161	C₂²×C₄	C₂×C₄×D₅₃	1	12	yes	—
162	C₂²×C₄	#11719945	1	12	yes	—
163	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
164	C₂²×C₄	C₂³×D₅₃	1	2	yes	—
165	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
166	C₄⋊C₄	—	53	96	no	—
167	C₄⋊C₄	—	53	48	no	—
168	C₄²	—	2	24	no	—
169	C₄²	—	53	96	no	—
170	C₄²	—	106	48	no	—
171	C₄²	C₂×Dic₁₀₆	1	16	yes	direct product factor
172	C₄²	C₂×C₄×D₅₃	1	48	yes	—
173	C₂³	C₂²×C₄×D₅₃	1	6	yes	—
174	C₂³	C₂²×Dic₁₀₆	1	2	yes	—
175	C₂³	C₂×#11719945	1	6	yes	—
176	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
177	Q₈	—	53	64	no	—
178	C₂×C₄	—	2	24	no	—
179	C₂×C₄	—	2	12	no	—
180	C₂×C₄	—	53	32	no	—
181	C₂×C₄	—	53	24	no	—
182	C₂×C₄	—	53	24	no	—
183	C₂×C₄	—	106	48	no	—
184	C₂×C₄	—	106	24	no	—
185	C₂×C₄	C₄²×D₅₃	1	16	yes	—
186	C₂×C₄	C₄×Dic₁₀₆	1	48	yes	direct product factor
187	C₂×C₄	C₂²×C₄×D₅₃	1	12	yes	—
188	C₂×C₄	C₂²×C₄×D₅₃	1	12	yes	—
189	C₂×C₄	C₂²×C₄×D₅₃	1	12	yes	—
190	C₂×C₄	C₂²×Dic₁₀₆	1	12	yes	—
191	C₂×C₄	C₂×#11719945	1	12	yes	—
192	C₂²	C₂×C₄²×D₅₃	1	4	yes	—
193	C₂²	C₂×C₄×Dic₁₀₆	1	12	yes	—
194	C₂²	C₄×#11719945	1	12	yes	—
195	C₂²	C₂³×C₄×D₅₃	1	3	yes	—
196	C₂²	C₂³×Dic₁₀₆	1	1	yes	characteristic
197	C₂²	C₂²×#11719945	1	3	yes	—
198	C₄	—	2	12	no	—
199	C₄	—	53	16	no	—
200	C₄	—	106	24	no	—
201	C₄	C₂×C₄²×D₅₃	1	8	yes	—
202	C₄	C₂×C₄×Dic₁₀₆	1	24	yes	direct product factor
203	C₂	C₄²×Dic₁₀₆	1	8	yes	direct product factor
204	C₂	C₂²×C₄²×D₅₃	1	1	yes	characteristic
205	C₂	C₂²×C₄×Dic₁₀₆	1	3	yes	—
206	C₂	C₂×C₄×#11719945	1	3	yes	—

	subgroup	G/H
center	C₂²×C₄²	C₂×D₅₃
commutator	C₂×C₅₃	C₂³×C₄²
Frattini	C₂³	C₂⁴×D₅₃
Fitting	C₂×C₄³×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	C₂⁴
cosocle	C₂³×C₅₃	C₂⁵
hypercenter	C₂×C₄³	D₅₃
hypocenter	C₅₃	C₂×C₄²×Q₈
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄² ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
2. C₄²×Dic₁₀₆	C₂	direct (N × H)
8. C₂×C₄×Dic₁₀₆	C₄	direct (N × H)
26. C₄×Dic₁₀₆	C₂×C₄	direct (N × H)
41. C₂×C₄²×C₅₃	Q₈	semidirect
60. C₂×Dic₁₀₆	C₄²	direct (N × H)
64. C₄²×C₅₃	C₂×Q₈	semidirect
76. C₂×C₄×C₅₃	C₄×Q₈	semidirect
87. Dic₁₀₆	C₂×C₄²	direct (N × H)
97. C₄×C₅₃	C₂×C₄×Q₈	semidirect
106. C₂×C₅₃	C₄²×Q₈	semidirect
125. C₅₃	C₂×C₄²×Q₈	semidirect
143. C₂×C₄²	Dic₁₀₆	direct (N × H)
171. C₄²	C₂×Dic₁₀₆	direct (N × H)
186. C₂×C₄	C₄×Dic₁₀₆	direct (N × H)
202. C₄	C₂×C₄×Dic₁₀₆	direct (N × H)
203. C₂	C₄²×Dic₁₀₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×Dic₁₀₆	2
C₄²×Dic₁₀₆	2
C₂×C₄×#11719919	2
C₂×C₄×C₄⋊Dic₅₃	2
C₂×C₄²×Dic₅₃	2
C₂×C₄³×C₅₃	2
C₂×C₄²×Q₈	53

Subgroups — C2×C42×Dic106 #27542288

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄²×Dic₁₀₆ #27542288