Subgroups — Q₈×#108×C₅₃ #27230244

Order 13568.

658 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	Q₈×M₁₆×C₅₃	C₂	1	2	yes	maximal
2	C₂×C₈×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1950	C₂	1	6	yes	maximal
4	C₅₃×#1848	C₂	1	3	yes	maximal
5	C₄×#108×C₅₃	C₂	1	3	yes	maximal
6	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₅₃×#446	—	2	6	no	—
8	C₈×Q₈×C₅₃	—	2	2	no	—
9	C₈×Q₈×C₅₃	C₄	1	6	yes	—
10	C₈×Q₈×C₅₃	C₄	1	2	yes	—
11	C₂×#108×C₅₃	C₂²	1	3	yes	—
12	C₂×#108×C₅₃	C₂²	1	6	yes	—
13	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#423	C₂²	1	6	yes	—
15	C₂×C₄⋊C₈×C₅₃	C₂²	1	3	yes	—
16	C₄×M₁₆×C₅₃	C₂²	1	6	yes	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	3	yes	—
18	C₅₃×#341	C₂²	1	6	yes	—
19	C₂²×Q₈×C₅₃	Q₈	1	1	yes	characteristic
20	C₂×M₁₆×C₅₃	C₂³	1	6	yes	—
21	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
22	C₂²×C₈×C₅₃	C₂³	1	3	yes	—
23	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
24	C₄×Q₈×C₅₃	C₂×C₄	1	3	yes	—
25	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
27	C₄×Q₈×C₅₃	D₄	1	3	yes	—
28	C₂×C₄⋊C₄×C₅₃	Q₈	1	3	yes	—
29	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
30	#108×C₅₃	—	2	6	no	—
31	#108×C₅₃	—	4	6	no	—
32	#108×C₅₃	Q₈	1	4	yes	direct product factor
33	C₄⋊C₈×C₅₃	—	2	6	no	—
34	C₄⋊C₈×C₅₃	—	2	6	no	—
35	C₄⋊C₈×C₅₃	C₂×C₄	1	6	yes	—
36	C₄⋊C₈×C₅₃	C₂×C₄	1	6	yes	—
37	#97×C₅₃	—	2	6	no	—
38	C₄×C₈×C₅₃	—	2	6	no	—
39	C₄×C₈×C₅₃	C₂×C₄	1	6	yes	—
40	C₄×C₈×C₅₃	C₂×C₄	1	6	yes	—
41	C₂×Q₈×C₅₃	—	2	4	no	—
42	C₂×Q₈×C₅₃	C₄⋊C₄	1	3	yes	—
43	C₂×Q₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂×Q₈	1	3	yes	—
45	C₂²×C₄×C₅₃	C₂×Q₈	1	3	yes	—
46	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
47	M₁₆×C₅₃	—	2	2	no	—
48	M₁₆×C₅₃	—	2	6	no	—
49	M₁₆×C₅₃	—	2	6	no	—
50	M₁₆×C₅₃	C₂×Q₈	1	4	yes	—
51	C₂×C₈×C₅₃	—	2	2	no	—
52	C₂×C₈×C₅₃	—	2	3	no	—
53	C₂×C₈×C₅₃	—	2	3	no	—
54	C₂×C₈×C₅₃	—	2	6	no	—
55	C₂×C₈×C₅₃	C₂²×C₄	1	6	yes	—
56	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
57	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
58	C₂×C₈×C₅₃	#41	1	2	yes	—
59	C₄⋊C₄×C₅₃	C₄⋊C₄	1	6	yes	—
60	C₄⋊C₄×C₅₃	C₄⋊C₄	1	3	yes	—
61	C₄⋊C₄×C₅₃	C₄⋊C₄	1	3	yes	—
62	C₄²×C₅₃	C₂²×C₄	1	3	yes	—
63	C₄²×C₅₃	C₂²×C₄	1	3	yes	—
64	C₄²×C₅₃	C₂×D₄	1	3	yes	—
65	C₄²×C₅₃	C₂×D₄	1	3	yes	—
66	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
67	Q₈×C₅₃	—	2	6	no	—
68	Q₈×C₅₃	#108	1	4	yes	direct product factor
69	C₂×C₄×C₅₃	—	2	6	no	—
70	C₂×C₄×C₅₃	—	4	3	no	—
71	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
72	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
73	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
74	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
75	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
79	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
80	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
81	C₈×C₅₃	—	2	6	no	—
82	C₈×C₅₃	—	2	4	no	—
83	C₈×C₅₃	—	4	6	no	—
84	C₈×C₅₃	C₄×Q₈	1	4	yes	—
85	Q₈×#108	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂²×C₅₃	—	2	1	no	—
87	C₂²×C₅₃	—	2	1	no	—
88	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₅₃	#557	1	1	yes	characteristic
90	C₂²×C₅₃	Q₈²	1	1	yes	characteristic
91	C₄×C₅₃	—	2	3	no	—
92	C₄×C₅₃	—	2	3	no	—
93	C₄×C₅₃	—	2	3	no	—
94	C₄×C₅₃	C₂×#108	1	6	yes	—
95	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
96	C₄×C₅₃	#520	1	1	yes	characteristic
97	C₄×C₅₃	D₄×Q₈	1	1	yes	characteristic
98	C₄×C₅₃	#550	1	1	yes	characteristic
99	Q₈×M₁₆	C₂×C₅₃	1	2	yes	—
100	C₂×C₈×Q₈	C₂×C₅₃	1	1	yes	characteristic
101	#1950	C₂×C₅₃	1	6	yes	—
102	#1848	C₂×C₅₃	1	3	yes	—
103	C₄×#108	C₂×C₅₃	1	3	yes	—
104	C₂×C₅₃	—	2	1	no	—
105	C₂×C₅₃	—	2	1	no	—
106	C₂×C₅₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₅₃	C₂²×#108	1	1	yes	characteristic
108	C₂×C₅₃	#2916	1	1	yes	characteristic
109	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
110	#446	—	2	6	no	—
111	C₈×Q₈	—	2	2	no	—
112	C₈×Q₈	C₄×C₅₃	1	6	yes	—
113	C₈×Q₈	C₄×C₅₃	1	2	yes	—
114	C₂×#108	C₂²×C₅₃	1	3	yes	—
115	C₂×#108	C₂²×C₅₃	1	6	yes	—
116	C₂×#108	C₂²×C₅₃	1	1	yes	characteristic
117	#423	C₂²×C₅₃	1	6	yes	—
118	C₂×C₄⋊C₈	C₂²×C₅₃	1	3	yes	—
119	C₄×M₁₆	C₂²×C₅₃	1	6	yes	—
120	C₂×C₄×C₈	C₂²×C₅₃	1	3	yes	—
121	#341	C₂²×C₅₃	1	6	yes	—
122	C₅₃	Q₈×#108	1	1	yes	characteristic, direct product factor, Sylow
123	C₂²×Q₈	Q₈×C₅₃	1	1	yes	characteristic
124	C₂×M₁₆	C₂³×C₅₃	1	6	yes	—
125	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
126	C₂²×C₈	C₂³×C₅₃	1	3	yes	—
127	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
128	C₄×Q₈	C₂×C₄×C₅₃	1	3	yes	—
129	C₄×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
130	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
131	C₄×Q₈	D₄×C₅₃	1	3	yes	—
132	C₂×C₄⋊C₄	Q₈×C₅₃	1	3	yes	—
133	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
134	#108	—	2	6	no	—
135	#108	—	4	6	no	—
136	#108	Q₈×C₅₃	1	4	yes	direct product factor
137	C₄⋊C₈	—	2	6	no	—
138	C₄⋊C₈	—	2	6	no	—
139	C₄⋊C₈	C₂×C₄×C₅₃	1	6	yes	—
140	C₄⋊C₈	C₂×C₄×C₅₃	1	6	yes	—
141	#97	—	2	6	no	—
142	C₄×C₈	—	2	6	no	—
143	C₄×C₈	C₂×C₄×C₅₃	1	6	yes	—
144	C₄×C₈	C₂×C₄×C₅₃	1	6	yes	—
145	C₂×Q₈	—	2	4	no	—
146	C₂×Q₈	C₄⋊C₄×C₅₃	1	3	yes	—
147	C₂×Q₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
148	C₂²×C₄	C₂×Q₈×C₅₃	1	3	yes	—
149	C₂²×C₄	C₂×Q₈×C₅₃	1	3	yes	—
150	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
151	M₁₆	—	2	2	no	—
152	M₁₆	—	2	6	no	—
153	M₁₆	—	2	6	no	—
154	M₁₆	C₂×Q₈×C₅₃	1	4	yes	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	3	no	—
157	C₂×C₈	—	2	3	no	—
158	C₂×C₈	—	2	6	no	—
159	C₂×C₈	C₂²×C₄×C₅₃	1	6	yes	—
160	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
161	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
162	C₂×C₈	#41×C₅₃	1	2	yes	—
163	C₄⋊C₄	C₄⋊C₄×C₅₃	1	6	yes	—
164	C₄⋊C₄	C₄⋊C₄×C₅₃	1	3	yes	—
165	C₄⋊C₄	C₄⋊C₄×C₅₃	1	3	yes	—
166	C₄²	C₂²×C₄×C₅₃	1	3	yes	—
167	C₄²	C₂²×C₄×C₅₃	1	3	yes	—
168	C₄²	C₂×D₄×C₅₃	1	3	yes	—
169	C₄²	C₂×D₄×C₅₃	1	3	yes	—
170	C₂³	C₂²×Q₈×C₅₃	1	1	yes	characteristic
171	Q₈	—	2	6	no	—
172	Q₈	#108×C₅₃	1	4	yes	direct product factor
173	C₂×C₄	—	2	6	no	—
174	C₂×C₄	—	4	3	no	—
175	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	3	yes	—
176	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	3	yes	—
177	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	3	yes	—
178	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	3	yes	—
179	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
180	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
181	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
182	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
183	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
184	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
185	C₈	—	2	6	no	—
186	C₈	—	2	4	no	—
187	C₈	—	4	6	no	—
188	C₈	C₄×Q₈×C₅₃	1	4	yes	—
189	C₂²	—	2	1	no	—
190	C₂²	—	2	1	no	—
191	C₂²	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
192	C₂²	C₅₃×#557	1	1	yes	characteristic
193	C₂²	Q₈²×C₅₃	1	1	yes	characteristic
194	C₄	—	2	3	no	—
195	C₄	—	2	3	no	—
196	C₄	—	2	3	no	—
197	C₄	C₂×#108×C₅₃	1	6	yes	—
198	C₄	C₂×C₄×Q₈×C₅₃	1	1	yes	characteristic
199	C₄	C₅₃×#520	1	1	yes	characteristic
200	C₄	D₄×Q₈×C₅₃	1	1	yes	characteristic
201	C₄	C₅₃×#550	1	1	yes	characteristic
202	C₂	—	2	1	no	—
203	C₂	—	2	1	no	—
204	C₂	Q₈×C₄⋊C₄×C₅₃	1	1	yes	characteristic
205	C₂	C₂²×#108×C₅₃	1	1	yes	characteristic
206	C₂	C₅₃×#2916	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂³×C₅₃	C₂²×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄×Q₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
32. #108×C₅₃	Q₈	direct (N × H)
68. Q₈×C₅₃	#108	direct (N × H)
85. Q₈×#108	C₅₃	direct (N × H)
122. C₅₃	Q₈×#108	direct (N × H)
136. #108	Q₈×C₅₃	direct (N × H)
172. Q₈	#108×C₅₃	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×M₁₆×C₅₃	2
C₂×C₈×Q₈×C₅₃	2
C₅₃×#1950	2
C₅₃×#1848	2
C₄×#108×C₅₃	2
Q₈×#108	53

Subgroups — Q8×#108×C53 #27230244

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#108×C₅₃ #27230244