Subgroups — C₂²×Q₈×Dic₁₀₆ #28291159

Order 13568.

7846 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×Dic₁₀₆	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#13133217	C₂	1	3	yes	maximal
3	C₂²×#13133216	C₂	1	6	yes	maximal
4	C₂²×Q₈×Dic₅₃	C₂	1	2	yes	maximal
5	C₂²×C₄×Dic₁₀₆	C₂	1	3	yes	maximal
6	C₂²×C₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₁₀₆	C₂²	1	6	yes	—
8	C₂³×Dic₁₀₆	C₂²	1	1	yes	characteristic
9	Q₈×Dic₁₀₆	C₂²	1	256	yes	direct product factor
10	C₂×#13133217	C₂²	1	72	yes	—
11	C₂×#13133216	C₂²	1	144	yes	—
12	C₂×Q₈×Dic₅₃	C₂²	1	48	yes	—
13	C₂×C₄×Dic₁₀₆	C₂²	1	72	yes	—
14	C₂²×C₄⋊Dic₅₃	C₂²	1	3	yes	—
15	C₂²×C₄⋊Dic₅₃	C₂²	1	6	yes	—
16	C₂²×#11719919	C₂²	1	6	yes	—
17	C₂²×C₄×Dic₅₃	C₂²	1	6	yes	—
18	C₂³×Q₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×Q₈×C₅₃	C₂²	1	24	yes	—
20	C₂²×C₄⋊C₄×C₅₃	C₂²	1	3	yes	—
21	C₂²×C₄²×C₅₃	C₂²	1	3	yes	—
22	C₂²×Dic₁₀₆	—	4	24	no	—
23	C₂²×Dic₁₀₆	Q₈	1	16	yes	direct product factor
24	C₂²×Dic₁₀₆	C₂³	1	72	yes	—
25	C₂²×Dic₁₀₆	C₂³	1	12	yes	—
26	C₂³×Dic₅₃	C₂³	1	6	yes	—
27	C₂³×Dic₅₃	C₂³	1	2	yes	—
28	#13133217	C₂³	1	192	yes	—
29	#13133216	C₂³	1	384	yes	—
30	Q₈×Dic₅₃	C₂³	1	128	yes	—
31	C₄×Dic₁₀₆	C₂³	1	192	yes	—
32	C₂×C₄⋊Dic₅₃	C₂³	1	36	yes	—
33	C₂×C₄⋊Dic₅₃	C₂³	1	72	yes	—
34	C₂×#11719919	C₂³	1	72	yes	—
35	C₂×C₄×Dic₅₃	C₂³	1	72	yes	—
36	C₂²×Q₈×C₅₃	Q₈	1	16	yes	complemented
37	C₂²×Q₈×C₅₃	C₂³	1	12	yes	—
38	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
39	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
40	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
41	C₄×Q₈×C₅₃	C₂³	1	64	yes	—
42	C₂×C₄⋊C₄×C₅₃	C₂³	1	36	yes	—
43	C₂×C₄²×C₅₃	C₂³	1	36	yes	—
44	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
45	C₂×Q₈×C₅₃	C₂×Q₈	1	96	yes	complemented
46	C₂×Q₈×C₅₃	C₂⁴	1	16	yes	—
47	C₂²×C₄×C₅₃	—	2	12	no	—
48	C₂²×C₄×C₅₃	C₂×Q₈	1	24	yes	—
49	C₂²×C₄×C₅₃	C₂×Q₈	1	8	yes	—
50	C₂²×C₄×C₅₃	C₂⁴	1	18	yes	—
51	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
52	C₂²×C₄×C₅₃	C₂⁴	1	18	yes	—
53	C₂²×Dic₅₃	—	2	24	no	—
54	C₂²×Dic₅₃	C₂×Q₈	1	16	yes	—
55	C₂²×Dic₅₃	C₂⁴	1	36	yes	—
56	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
57	C₂×Dic₁₀₆	—	4	144	no	—
58	C₂×Dic₁₀₆	C₂×Q₈	1	96	yes	direct product factor
59	C₂×Dic₁₀₆	C₂⁴	1	96	yes	—
60	C₂×Dic₁₀₆	C₂⁴	1	16	yes	—
61	C₄⋊C₄×C₅₃	C₂⁴	1	48	yes	—
62	C₄²×C₅₃	C₂⁴	1	48	yes	—
63	C₄⋊Dic₅₃	C₂⁴	1	48	yes	—
64	C₄⋊Dic₅₃	C₂⁴	1	96	yes	—
65	#11719919	C₂⁴	1	96	yes	—
66	C₄×Dic₅₃	C₂⁴	1	96	yes	—
67	C₂³×C₅₃	C₂²×Q₈	1	4	yes	—
68	C₂³×C₅₃	C₂²×Q₈	1	4	yes	—
69	C₂³×C₅₃	H(5,𝔽₂)	1	4	yes	—
70	C₂³×C₅₃	C₂⁵	1	3	yes	—
71	Q₈×C₅₃	C₂²×Q₈	1	64	yes	complemented
72	C₂×C₄×C₅₃	—	2	36	no	—
73	C₂×C₄×C₅₃	C₂²×Q₈	1	24	yes	—
74	C₂×C₄×C₅₃	C₂²×Q₈	1	72	yes	—
75	C₂×C₄×C₅₃	C₂⁵	1	12	yes	—
76	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
77	C₂×C₄×C₅₃	C₂⁵	1	12	yes	—
78	C₂×Dic₅₃	—	2	72	no	—
79	C₂×Dic₅₃	C₂²×Q₈	1	48	yes	—
80	C₂×Dic₅₃	C₂⁵	1	24	yes	—
81	C₂×Dic₅₃	C₂⁵	1	8	yes	—
82	Dic₁₀₆	—	4	96	no	—
83	Dic₁₀₆	C₂²×Q₈	1	64	yes	direct product factor
84	C₂²×Q₈²	—	53	1	no	maximal, Sylow
85	C₂²×C₅₃	Q₈²	1	16	yes	complemented
86	C₂²×C₅₃	C₂³×Q₈	1	6	yes	—
87	C₂²×C₅₃	C₂³×Q₈	1	6	yes	—
88	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	6	yes	—
89	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
90	C₄×C₅₃	—	2	12	no	—
91	C₄×C₅₃	C₂³×Q₈	1	8	yes	—
92	C₄×C₅₃	C₂³×Q₈	1	24	yes	—
93	Dic₅₃	—	2	24	no	—
94	Dic₅₃	C₂³×Q₈	1	16	yes	—
95	C₂×Q₈²	—	53	48	no	—
96	C₂²×C₄⋊Q₈	—	53	3	no	—
97	C₂²×C₄⋊Q₈	—	53	6	no	—
98	C₂²×C₄×Q₈	—	53	3	no	—
99	C₂²×C₄×Q₈	—	53	2	no	—
100	C₂²×C₄×Q₈	D₅₃	1	1	yes	characteristic
101	C₂×C₅₃	C₂×Q₈²	1	12	yes	complemented
102	C₂×C₅₃	C₂⁴×Q₈	1	1	yes	characteristic
103	C₂×C₅₃	C₂⁴×Q₈	1	1	yes	characteristic
104	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
105	C₂³×Q₈	—	53	1	no	—
106	C₂³×Q₈	—	53	6	no	—
107	C₂³×Q₈	C₂×D₅₃	1	1	yes	characteristic
108	Q₈²	—	53	256	no	—
109	C₂×C₄⋊Q₈	—	53	144	no	—
110	C₂×C₄⋊Q₈	—	53	72	no	—
111	C₂×C₄×Q₈	—	53	48	no	—
112	C₂×C₄×Q₈	—	53	72	no	—
113	C₂×C₄×Q₈	C₂×D₅₃	1	24	yes	—
114	C₂²×C₄⋊C₄	—	53	6	no	—
115	C₂²×C₄⋊C₄	—	53	6	no	—
116	C₂²×C₄⋊C₄	—	53	3	no	—
117	C₂²×C₄⋊C₄	C₂×D₅₃	1	3	yes	—
118	C₂²×C₄²	—	53	6	no	—
119	C₂²×C₄²	C₂×D₅₃	1	3	yes	—
120	C₅₃	C₂²×Q₈²	1	1	yes	characteristic, complemented, Sylow
121	C₂²×Q₈	—	53	16	no	—
122	C₂²×Q₈	—	53	12	no	—
123	C₂²×Q₈	—	53	72	no	—
124	C₂²×Q₈	—	212	24	no	—
125	C₂²×Q₈	Dic₁₀₆	1	16	yes	direct product factor
126	C₂²×Q₈	C₂²×D₅₃	1	12	yes	—
127	C₂³×C₄	—	53	2	no	—
128	C₂³×C₄	—	53	6	no	—
129	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
130	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
131	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
132	C₄⋊Q₈	—	53	384	no	—
133	C₄⋊Q₈	—	53	192	no	—
134	C₄×Q₈	—	53	128	no	—
135	C₄×Q₈	—	53	192	no	—
136	C₄×Q₈	C₂²×D₅₃	1	64	yes	—
137	C₂×C₄⋊C₄	—	53	72	no	—
138	C₂×C₄⋊C₄	—	53	36	no	—
139	C₂×C₄⋊C₄	—	53	72	no	—
140	C₂×C₄⋊C₄	C₂²×D₅₃	1	36	yes	—
141	C₂×C₄²	—	53	72	no	—
142	C₂×C₄²	C₂²×D₅₃	1	36	yes	—
143	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
144	C₂×Q₈	—	53	96	no	—
145	C₂×Q₈	—	53	16	no	—
146	C₂×Q₈	—	53	96	no	—
147	C₂×Q₈	—	212	144	no	—
148	C₂×Q₈	C₂×Dic₁₀₆	1	96	yes	direct product factor
149	C₂×Q₈	C₂³×D₅₃	1	16	yes	—
150	C₂²×C₄	—	2	12	no	—
151	C₂²×C₄	—	53	16	no	—
152	C₂²×C₄	—	53	12	no	—
153	C₂²×C₄	—	53	36	no	—
154	C₂²×C₄	—	106	24	no	—
155	C₂²×C₄	C₂×Dic₁₀₆	1	24	yes	—
156	C₂²×C₄	Q₈×D₅₃	1	8	yes	—
157	C₂²×C₄	C₂³×D₅₃	1	18	yes	—
158	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
159	C₂²×C₄	C₂³×D₅₃	1	18	yes	—
160	C₄⋊C₄	—	53	96	no	—
161	C₄⋊C₄	—	53	48	no	—
162	C₄⋊C₄	—	53	96	no	—
163	C₄⋊C₄	C₂³×D₅₃	1	48	yes	—
164	C₄²	—	53	96	no	—
165	C₄²	C₂³×D₅₃	1	48	yes	—
166	C₂³	C₂×Q₈×D₅₃	1	4	yes	—
167	C₂³	C₂²×Dic₁₀₆	1	4	yes	—
168	C₂³	#13133263	1	4	yes	—
169	C₂³	C₂⁴×D₅₃	1	3	yes	—
170	Q₈	—	53	64	no	—
171	Q₈	—	212	96	no	—
172	Q₈	C₂²×Dic₁₀₆	1	64	yes	direct product factor
173	C₂×C₄	—	2	36	no	—
174	C₂×C₄	—	53	48	no	—
175	C₂×C₄	—	53	8	no	—
176	C₂×C₄	—	53	24	no	—
177	C₂×C₄	—	106	72	no	—
178	C₂×C₄	C₂×Q₈×D₅₃	1	24	yes	—
179	C₂×C₄	C₂²×Dic₁₀₆	1	72	yes	—
180	C₂×C₄	C₂⁴×D₅₃	1	12	yes	—
181	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
182	C₂×C₄	C₂⁴×D₅₃	1	12	yes	—
183	C₂²	Q₈×Dic₁₀₆	1	16	yes	direct product factor
184	C₂²	C₂²×Q₈×D₅₃	1	6	yes	—
185	C₂²	C₂³×Dic₁₀₆	1	6	yes	—
186	C₂²	C₂×#13133263	1	6	yes	—
187	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
188	C₄	—	2	12	no	—
189	C₄	—	53	16	no	—
190	C₄	—	106	24	no	—
191	C₄	C₂²×Q₈×D₅₃	1	8	yes	—
192	C₄	C₂³×Dic₁₀₆	1	24	yes	—
193	C₂	C₂×Q₈×Dic₁₀₆	1	12	yes	direct product factor
194	C₂	C₂³×Q₈×D₅₃	1	1	yes	characteristic
195	C₂	C₂⁴×Dic₁₀₆	1	1	yes	characteristic
196	C₂	C₂²×#13133263	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂²×C₄×Q₈×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	C₂⁴
hypercenter	C₂²×C₄×Q₈	D₅₃
hypocenter	C₅₃	C₂²×Q₈²
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×Dic₁₀₆	C₂	direct (N × H)
9. Q₈×Dic₁₀₆	C₂²	direct (N × H)
23. C₂²×Dic₁₀₆	Q₈	direct (N × H)
36. C₂²×Q₈×C₅₃	Q₈	semidirect
45. C₂×Q₈×C₅₃	C₂×Q₈	semidirect
58. C₂×Dic₁₀₆	C₂×Q₈	direct (N × H)
71. Q₈×C₅₃	C₂²×Q₈	semidirect
83. Dic₁₀₆	C₂²×Q₈	direct (N × H)
85. C₂²×C₅₃	Q₈²	semidirect
101. C₂×C₅₃	C₂×Q₈²	semidirect
120. C₅₃	C₂²×Q₈²	semidirect
125. C₂²×Q₈	Dic₁₀₆	direct (N × H)
148. C₂×Q₈	C₂×Dic₁₀₆	direct (N × H)
172. Q₈	C₂²×Dic₁₀₆	direct (N × H)
183. C₂²	Q₈×Dic₁₀₆	direct (N × H)
193. C₂	C₂×Q₈×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
C₂×Q₈×Dic₁₀₆	2
C₂²×#13133217	2
C₂²×#13133216	2
C₂²×Q₈×Dic₅₃	2
C₂²×C₄×Dic₁₀₆	2
C₂²×C₄×Q₈×C₅₃	2
C₂²×Q₈²	53

Subgroups — C22×Q8×Dic106 #28291159

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×Q₈×Dic₁₀₆ #28291159