Subgroups — Q₈×#13133156 #27537602

Order 13568.

530 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	C₈×Q₈×D₅₃	C₂	1	1	yes	maximal, characteristic
2	#20276623	C₂	1	3	yes	maximal, complemented
3	#20276620	C₂	1	3	yes	maximal, complemented
4	Q₈×#11719908	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276469	C₂	1	3	yes	maximal
6	C₄×#13133156	C₂	1	3	yes	maximal
7	Q₈×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×Q₈×D₅₃	C₄	1	1	yes	characteristic
9	C₂×#13133156	C₂²	1	3	yes	—
10	C₂×#13133156	C₂²	1	1	yes	characteristic
11	Q₈×#88322	C₄	1	1	yes	characteristic
12	C₈×Dic₁₀₆	C₂²	1	3	yes	—
13	#16810747	C₄	1	3	yes	—
14	C₄⋊C₈×D₅₃	C₄	1	3	yes	—
15	C₄×C₈×D₅₃	C₂²	1	3	yes	—
16	C₄⋊#11719908	C₂²	1	3	yes	—
17	#16810441	C₂²	1	6	yes	—
18	#16810398	C₂²	1	3	yes	—
19	#16810398	C₂²	1	3	yes	—
20	C₄×#11719908	C₂²	1	3	yes	—
21	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₅₃×C₄⋊C₁₆	C₂²	1	3	yes	—
23	C₄×C₁₆×C₅₃	C₂²	1	3	yes	—
24	C₂×Q₈×D₅₃	C₈	1	1	yes	characteristic
25	C₂×C₈×D₅₃	C₂×C₄	1	3	yes	—
26	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
27	Q₈×Dic₅₃	C₈	1	1	yes	characteristic
28	C₄×Dic₁₀₆	C₂×C₄	1	3	yes	—
29	#13133182	C₈	1	3	yes	—
30	C₄⋊C₄×D₅₃	C₈	1	3	yes	—
31	C₄²×D₅₃	C₂×C₄	1	3	yes	—
32	#13133156	—	2	6	no	—
33	#13133156	Q₈	1	4	yes	direct product factor
34	C₂×#11719908	C₂³	1	3	yes	—
35	C₂×#11719908	C₂³	1	1	yes	characteristic
36	C₄⋊#88322	C₂×C₄	1	3	yes	—
37	#13133146	C₂×C₄	1	6	yes	—
38	C₄×#88322	C₂×C₄	1	3	yes	—
39	C₈×Dic₅₃	C₂×C₄	1	3	yes	—
40	C₈×Dic₅₃	C₂³	1	3	yes	—
41	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₁₆×C₅₃	C₂³	1	3	yes	—
43	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₅₃	C₂×C₄	1	3	yes	—
45	C₄×C₈×C₅₃	C₂³	1	3	yes	—
46	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
47	Q₈×D₅₃	—	2	4	no	—
48	C₂×C₄×D₅₃	C₂×C₈	1	3	yes	—
49	C₂×C₄×D₅₃	C₂×C₈	1	3	yes	—
50	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×Dic₁₀₆	C₂×C₈	1	3	yes	—
52	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
53	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
54	C₄⋊C₄×C₅₃	C₂×C₈	1	3	yes	—
55	C₄²×C₅₃	C₂²×C₄	1	3	yes	—
56	C₄⋊Dic₅₃	C₂×C₈	1	3	yes	—
57	#11719919	C₂×C₈	1	6	yes	—
58	C₄×Dic₅₃	C₂×C₈	1	3	yes	—
59	C₄×Dic₅₃	C₂×C₈	1	3	yes	—
60	C₄×Dic₅₃	C₂²×C₄	1	3	yes	—
61	C₂×#88322	C₂²×C₄	1	3	yes	—
62	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
63	C₈×D₅₃	—	2	6	no	—
64	C₈×D₅₃	C₂×Q₈	1	2	yes	—
65	C₈×D₅₃	#41	1	2	yes	—
66	C₁₆×C₅₃	—	2	3	no	—
67	C₁₆×C₅₃	C₂×Q₈	1	2	yes	complemented
68	#11719908	—	2	3	no	—
69	#11719908	C₂×Q₈	1	2	yes	complemented
70	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
71	Q₈×C₅₃	M₃₂	1	4	yes	complemented
72	C₂×C₄×C₅₃	C₂²×C₈	1	3	yes	—
73	C₂×C₄×C₅₃	C₂²×C₈	1	3	yes	—
74	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
75	C₂×Dic₅₃	C₂²×C₈	1	3	yes	—
76	C₂×Dic₅₃	C₂²×C₈	1	3	yes	—
77	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
78	C₄×D₅₃	—	2	6	no	—
79	C₄×D₅₃	—	4	3	no	—
80	C₄×D₅₃	C₄×Q₈	1	2	yes	—
81	C₄×D₅₃	#131	1	2	yes	—
82	Dic₁₀₆	—	2	6	no	—
83	C₈×C₅₃	—	2	3	no	—
84	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
85	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
86	#88322	—	2	3	no	—
87	#88322	C₄×Q₈	1	2	yes	—
88	Q₈×M₃₂	—	53	1	no	maximal, Sylow
89	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
90	C₂×D₅₃	—	2	1	no	—
91	C₂×D₅₃	—	2	1	no	—
92	C₂×D₅₃	C₈×Q₈	1	2	yes	—
93	C₄×C₅₃	—	2	3	no	—
94	C₄×C₅₃	C₂×M₃₂	1	6	yes	—
95	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
96	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
97	Dic₅₃	—	2	3	no	—
98	Dic₅₃	—	2	3	no	—
99	Dic₅₃	C₈×Q₈	1	2	yes	—
100	C₂×C₈×Q₈	—	53	1	no	—
101	#2183	—	53	3	no	—
102	#2183	—	53	3	no	—
103	Q₈×C₁₆	—	53	1	no	—
104	Q₈×C₁₆	D₅₃	1	1	yes	characteristic, complemented
105	#2150	—	53	3	no	—
106	C₄×M₃₂	—	53	3	no	—
107	C₂×C₅₃	C₂×C₈×Q₈	1	1	yes	characteristic
108	C₂×C₅₃	C₂²×M₃₂	1	1	yes	characteristic
109	C₂×C₅₃	#3407	1	1	yes	characteristic
110	D₅₃	—	2	2	no	—
111	C₂×C₄×Q₈	—	53	1	no	—
112	C₂×M₃₂	—	53	1	no	—
113	C₂×M₃₂	—	53	3	no	—
114	C₈×Q₈	—	53	1	no	—
115	C₈×Q₈	—	53	3	no	—
116	C₈×Q₈	—	53	3	no	—
117	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
118	C₂×C₄⋊C₈	—	53	3	no	—
119	C₂×C₄×C₈	—	53	3	no	—
120	C₄⋊C₁₆	—	53	6	no	—
121	C₄⋊C₁₆	—	53	3	no	—
122	C₄⋊C₁₆	C₂×D₅₃	1	3	yes	—
123	#346	—	53	3	no	—
124	#346	—	53	3	no	—
125	C₄×C₁₆	—	53	3	no	—
126	C₄×C₁₆	C₂×D₅₃	1	3	yes	—
127	C₅₃	Q₈×M₃₂	1	1	yes	characteristic, complemented, Sylow
128	C₂²×Q₈	—	53	1	no	—
129	C₂²×C₈	—	53	1	no	—
130	C₂²×C₈	—	53	3	no	—
131	C₄×Q₈	—	53	1	no	—
132	C₄×Q₈	—	53	3	no	—
133	C₄×Q₈	—	53	3	no	—
134	C₄×Q₈	C₄×D₅₃	1	1	yes	characteristic
135	C₂×C₄⋊C₄	—	53	3	no	—
136	C₂×C₄²	—	53	3	no	—
137	M₃₂	—	53	4	no	—
138	M₃₂	—	106	6	no	—
139	C₂×C₁₆	—	53	1	no	—
140	C₂×C₁₆	—	53	3	no	—
141	C₂×C₁₆	C₂²×D₅₃	1	3	yes	—
142	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
143	C₄⋊C₈	—	53	6	no	—
144	C₄⋊C₈	—	53	3	no	—
145	C₄⋊C₈	C₄×D₅₃	1	3	yes	—
146	C₄×C₈	—	53	3	no	—
147	C₄×C₈	—	53	3	no	—
148	C₄×C₈	—	53	3	no	—
149	C₄×C₈	C₂²×D₅₃	1	3	yes	—
150	C₂×Q₈	—	53	3	no	—
151	C₂×Q₈	—	106	4	no	—
152	C₂×Q₈	C₈×D₅₃	1	1	yes	characteristic
153	C₂²×C₄	—	53	1	no	—
154	C₂²×C₄	—	53	3	no	—
155	C₂²×C₄	—	53	3	no	—
156	C₂×C₈	—	53	1	no	—
157	C₂×C₈	—	53	2	no	—
158	C₂×C₈	—	53	2	no	—
159	C₂×C₈	—	53	3	no	—
160	C₂×C₈	—	106	6	no	—
161	C₂×C₈	C₂×C₄×D₅₃	1	3	yes	—
162	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
163	C₄⋊C₄	—	53	6	no	—
164	C₄⋊C₄	—	53	3	no	—
165	C₄⋊C₄	C₈×D₅₃	1	3	yes	—
166	C₄²	—	53	3	no	—
167	C₄²	—	53	3	no	—
168	C₄²	—	53	3	no	—
169	C₄²	C₂×C₄×D₅₃	1	3	yes	—
170	C₁₆	—	2	3	no	—
171	C₁₆	—	53	2	no	—
172	C₁₆	—	106	3	no	—
173	C₁₆	Q₈×D₅₃	1	2	yes	complemented
174	C₂³	—	53	1	no	—
175	Q₈	—	106	6	no	—
176	Q₈	#13133156	1	4	yes	direct product factor
177	C₂×C₄	—	53	1	no	—
178	C₂×C₄	—	53	2	no	—
179	C₂×C₄	—	53	2	no	—
180	C₂×C₄	—	53	3	no	—
181	C₂×C₄	—	53	3	no	—
182	C₂×C₄	—	106	6	no	—
183	C₂×C₄	—	212	3	no	—
184	C₂×C₄	C₂×C₈×D₅₃	1	3	yes	—
185	C₂×C₄	C₂×C₈×D₅₃	1	3	yes	—
186	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
187	C₈	—	2	3	no	—
188	C₈	—	53	2	no	—
189	C₈	—	106	3	no	—
190	C₈	C₂×Q₈×D₅₃	1	1	yes	characteristic
191	C₈	#41×D₅₃	1	1	yes	characteristic
192	C₂²	—	53	2	no	—
193	C₂²	—	106	1	no	—
194	C₂²	—	106	1	no	—
195	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
196	C₄	—	2	3	no	—
197	C₄	—	53	2	no	—
198	C₄	—	106	3	no	—
199	C₄	—	106	3	no	—
200	C₄	C₂×#13133156	1	6	yes	—
201	C₄	C₄×Q₈×D₅₃	1	1	yes	characteristic
202	C₄	#131×D₅₃	1	1	yes	characteristic
203	C₂	—	106	2	no	—
204	C₂	C₈×Q₈×D₅₃	1	1	yes	characteristic
205	C₂	C₂²×#13133156	1	1	yes	characteristic
206	C₂	#20287816	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	Q₈×C₁₆×C₅₃	C₂
prime radical	C₂²×D₅₃	C₂²×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	Q₈×C₁₆	D₅₃
hypocenter	C₅₃	Q₈×M₃₂
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ Q₈×C₁₆

Complements

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

N	H ≅ G/N	type
2. #20276623	C₂	semidirect
3. #20276620	C₂	semidirect
4. Q₈×#11719908	C₂	semidirect
7. Q₈×C₁₆×C₅₃	C₂	semidirect
33. #13133156	Q₈	direct (N × H)
67. C₁₆×C₅₃	C₂×Q₈	semidirect
69. #11719908	C₂×Q₈	semidirect
71. Q₈×C₅₃	M₃₂	semidirect
104. Q₈×C₁₆	D₅₃	semidirect
127. C₅₃	Q₈×M₃₂	semidirect
173. C₁₆	Q₈×D₅₃	semidirect
176. Q₈	#13133156	direct (N × H)

Maximal subgroups

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

subgroup	index
C₈×Q₈×D₅₃	2
#20276623	2
#20276620	2
Q₈×#11719908	2
#20276469	2
C₄×#13133156	2
Q₈×C₁₆×C₅₃	2
Q₈×M₃₂	53

Subgroups — Q8×#13133156 #27537602

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#13133156 #27537602