Subgroups — Q₈×#97×C₅₃ #27228034

Order 13568.

954 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Q₈×C₅₃	C₂	1	2	yes	maximal
2	C₄²×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1770	C₂	1	6	yes	maximal
4	C₅₃×#1767	C₂	1	3	yes	maximal
5	C₄×#97×C₅₃	C₂	1	3	yes	maximal
6	C₂×C₄×Q₈×C₅₃	C₄	1	2	yes	—
7	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₈×Q₈×C₅₃	C₄	1	16	yes	complemented
9	C₂×C₄⋊C₈×C₅₃	C₂²	1	6	yes	—
10	C₂×#97×C₅₃	C₂²	1	3	yes	—
11	C₂×#97×C₅₃	C₂²	1	6	yes	—
12	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₅₃	C₂²	1	6	yes	—
14	C₄×C₄⋊C₄×C₅₃	C₄	1	6	yes	—
15	C₄×C₄⋊C₄×C₅₃	C₂²	1	3	yes	—
16	C₄³×C₅₃	C₂²	1	3	yes	—
17	C₅₃×#336	C₂²	1	6	yes	—
18	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₅₃	C₂³	1	6	yes	—
20	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
21	C₄×Q₈×C₅₃	—	2	8	no	—
22	C₄×Q₈×C₅₃	C₂×C₄	1	8	yes	—
23	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	6	yes	—
24	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	3	yes	—
25	C₂×C₄²×C₅₃	C₂×C₄	1	6	yes	—
26	C₂×C₄²×C₅₃	C₂×C₄	1	6	yes	—
27	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
28	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
29	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₅₃	C₂×C₄	1	24	yes	—
31	#97×C₅₃	—	2	12	no	—
32	#97×C₅₃	—	2	6	no	—
33	#97×C₅₃	Q₈	1	4	yes	direct product factor
34	C₄×C₈×C₅₃	C₂×C₄	1	24	yes	—
35	#95×C₅₃	C₂×C₄	1	6	yes	—
36	C₂×Q₈×C₅₃	C₄²	1	4	yes	—
37	C₂×Q₈×C₅₃	M₁₆	1	8	yes	—
38	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
39	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
40	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
41	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
42	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
43	C₂×C₈×C₅₃	—	2	6	no	—
44	C₂×C₈×C₅₃	—	2	6	no	—
45	C₂×C₈×C₅₃	C₂²×C₄	1	12	yes	—
46	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
47	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
48	C₂×C₈×C₅₃	#41	1	4	yes	—
49	C₄⋊C₄×C₅₃	—	2	12	no	—
50	C₄⋊C₄×C₅₃	C₄²	1	12	yes	—
51	C₄²×C₅₃	—	2	12	no	—
52	C₄²×C₅₃	—	2	6	no	—
53	C₄²×C₅₃	—	4	6	no	—
54	C₄²×C₅₃	C₂²×C₄	1	12	yes	—
55	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
56	C₄²×C₅₃	#41	1	2	yes	—
57	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
58	Q₈×C₅₃	#97	1	16	yes	direct product factor
59	C₂×C₄×C₅₃	—	2	2	no	—
60	C₂×C₄×C₅₃	—	2	6	no	—
61	C₂×C₄×C₅₃	—	2	2	no	—
62	C₂×C₄×C₅₃	—	2	6	no	—
63	C₂×C₄×C₅₃	—	2	6	no	—
64	C₂×C₄×C₅₃	—	2	6	no	—
65	C₂×C₄×C₅₃	C₂×C₄²	1	6	yes	—
66	C₂×C₄×C₅₃	C₂×C₄²	1	6	yes	—
67	C₂×C₄×C₅₃	C₄×Q₈	1	4	yes	—
68	C₂×C₄×C₅₃	C₂×M₁₆	1	12	yes	—
69	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
70	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
72	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
73	C₈×C₅₃	—	2	12	no	—
74	C₈×C₅₃	C₄×Q₈	1	8	yes	complemented
75	Q₈×#97	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
77	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
78	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
79	C₂²×C₅₃	C₂²×M₁₆	1	2	yes	—
80	C₂²×C₅₃	#568	1	2	yes	—
81	C₄×C₅₃	—	2	6	no	—
82	C₄×C₅₃	—	2	4	no	—
83	C₄×C₅₃	—	4	6	no	—
84	C₄×C₅₃	C₂×#97	1	12	yes	—
85	C₄×C₅₃	C₂×C₄×Q₈	1	2	yes	—
86	C₄×C₅₃	C₄×#41	1	2	yes	—
87	C₂×C₈×Q₈	C₂×C₅₃	1	2	yes	—
88	C₄²×Q₈	C₂×C₅₃	1	1	yes	characteristic
89	#1770	C₂×C₅₃	1	6	yes	—
90	#1767	C₂×C₅₃	1	3	yes	—
91	C₄×#97	C₂×C₅₃	1	3	yes	—
92	C₂×C₅₃	C₄²×Q₈	1	1	yes	characteristic
93	C₂×C₅₃	C₂²×#97	1	1	yes	characteristic
94	C₂×C₅₃	#2875	1	1	yes	characteristic
95	C₂×C₅₃	Q₈×M₁₆	1	2	yes	—
96	C₂×C₅₃	#2971	1	2	yes	—
97	C₂×C₄×Q₈	C₄×C₅₃	1	2	yes	—
98	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
99	C₈×Q₈	C₄×C₅₃	1	16	yes	complemented
100	C₂×C₄⋊C₈	C₂²×C₅₃	1	6	yes	—
101	C₂×#97	C₂²×C₅₃	1	3	yes	—
102	C₂×#97	C₂²×C₅₃	1	6	yes	—
103	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
104	C₂×C₄×C₈	C₂²×C₅₃	1	6	yes	—
105	C₄×C₄⋊C₄	C₄×C₅₃	1	6	yes	—
106	C₄×C₄⋊C₄	C₂²×C₅₃	1	3	yes	—
107	C₄³	C₂²×C₅₃	1	3	yes	—
108	#336	C₂²×C₅₃	1	6	yes	—
109	C₅₃	Q₈×#97	1	1	yes	characteristic, direct product factor, Sylow
110	C₂²×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
111	C₂²×C₈	C₂³×C₅₃	1	6	yes	—
112	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
113	C₄×Q₈	—	2	8	no	—
114	C₄×Q₈	C₂×C₄×C₅₃	1	8	yes	—
115	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	6	yes	—
116	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	3	yes	—
117	C₂×C₄²	C₂×C₄×C₅₃	1	6	yes	—
118	C₂×C₄²	C₂×C₄×C₅₃	1	6	yes	—
119	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
120	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
121	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
122	C₄⋊C₈	C₂×C₄×C₅₃	1	24	yes	—
123	#97	—	2	12	no	—
124	#97	—	2	6	no	—
125	#97	Q₈×C₅₃	1	4	yes	direct product factor
126	C₄×C₈	C₂×C₄×C₅₃	1	24	yes	—
127	#95	C₂×C₄×C₅₃	1	6	yes	—
128	C₂×Q₈	C₄²×C₅₃	1	4	yes	—
129	C₂×Q₈	M₁₆×C₅₃	1	8	yes	—
130	C₂²×C₄	C₂²×C₄×C₅₃	1	6	yes	—
131	C₂²×C₄	C₂²×C₄×C₅₃	1	3	yes	—
132	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
133	C₂²×C₄	C₂²×C₄×C₅₃	1	3	yes	—
134	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
135	C₂×C₈	—	2	6	no	—
136	C₂×C₈	—	2	6	no	—
137	C₂×C₈	C₂²×C₄×C₅₃	1	12	yes	—
138	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
139	C₂×C₈	C₂×Q₈×C₅₃	1	4	yes	—
140	C₂×C₈	#41×C₅₃	1	4	yes	—
141	C₄⋊C₄	—	2	12	no	—
142	C₄⋊C₄	C₄²×C₅₃	1	12	yes	—
143	C₄²	—	2	12	no	—
144	C₄²	—	2	6	no	—
145	C₄²	—	4	6	no	—
146	C₄²	C₂²×C₄×C₅₃	1	12	yes	—
147	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
148	C₄²	#41×C₅₃	1	2	yes	—
149	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
150	Q₈	#97×C₅₃	1	16	yes	direct product factor
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	6	no	—
154	C₂×C₄	—	2	6	no	—
155	C₂×C₄	—	2	6	no	—
156	C₂×C₄	—	2	6	no	—
157	C₂×C₄	C₂×C₄²×C₅₃	1	6	yes	—
158	C₂×C₄	C₂×C₄²×C₅₃	1	6	yes	—
159	C₂×C₄	C₄×Q₈×C₅₃	1	4	yes	—
160	C₂×C₄	C₂×M₁₆×C₅₃	1	12	yes	—
161	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
162	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
163	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
164	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
165	C₈	—	2	12	no	—
166	C₈	C₄×Q₈×C₅₃	1	8	yes	complemented
167	C₂²	C₂²×C₄²×C₅₃	1	1	yes	characteristic
168	C₂²	C₂×C₄×Q₈×C₅₃	1	1	yes	characteristic
169	C₂²	C₄×#41×C₅₃	1	1	yes	characteristic
170	C₂²	C₂²×M₁₆×C₅₃	1	2	yes	—
171	C₂²	C₅₃×#568	1	2	yes	—
172	C₄	—	2	4	no	—
173	C₄	—	2	6	no	—
174	C₄	—	4	6	no	—
175	C₄	C₂×#97×C₅₃	1	12	yes	—
176	C₄	C₂×C₄×Q₈×C₅₃	1	2	yes	—
177	C₄	C₄×#41×C₅₃	1	2	yes	—
178	C₂	C₄²×Q₈×C₅₃	1	1	yes	characteristic
179	C₂	C₂²×#97×C₅₃	1	1	yes	characteristic
180	C₂	C₅₃×#2875	1	1	yes	characteristic
181	C₂	Q₈×M₁₆×C₅₃	1	2	yes	—
182	C₂	C₅₃×#2971	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂⁴
commutator	C₂²	C₂²×C₄²×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle, prime radical	C₂³×C₅₃	C₂³×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₈×Q₈×C₅₃	C₄	semidirect
33. #97×C₅₃	Q₈	direct (N × H)
58. Q₈×C₅₃	#97	direct (N × H)
74. C₈×C₅₃	C₄×Q₈	semidirect
75. Q₈×#97	C₅₃	direct (N × H)
99. C₈×Q₈	C₄×C₅₃	semidirect
109. C₅₃	Q₈×#97	direct (N × H)
125. #97	Q₈×C₅₃	direct (N × H)
150. Q₈	#97×C₅₃	direct (N × H)
166. C₈	C₄×Q₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Q₈×C₅₃	2
C₄²×Q₈×C₅₃	2
C₅₃×#1770	2
C₅₃×#1767	2
C₄×#97×C₅₃	2
Q₈×#97	53

Subgroups — Q8×#97×C53 #27228034

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#97×C₅₃ #27228034