Subgroups — #27383156

Order 13568.

914 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×#88322	C₂	1	2	yes	maximal
2	#20274584	C₂	1	6	yes	maximal
3	#20274568	C₂	1	3	yes	maximal
4	#20274461	C₂	1	3	yes	maximal
5	C₄²×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	Q₈×#88322	C₄	1	16	yes	complemented
7	C₂×C₄⋊#88322	C₂²	1	6	yes	—
8	C₂×#13133128	C₂²	1	6	yes	—
9	C₂×C₄×#88322	C₂²	1	6	yes	—
10	#16810378	C₂²	1	1	yes	characteristic
11	#16810378	C₂²	1	3	yes	—
12	#16810378	C₂²	1	6	yes	—
13	C₂×C₄×Q₈×C₅₃	C₄	1	2	yes	—
14	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₅₃	C₄	1	6	yes	—
16	C₄×C₄⋊C₄×C₅₃	C₂²	1	3	yes	—
17	C₄³×C₅₃	C₂²	1	3	yes	—
18	C₂²×#88322	C₂³	1	2	yes	—
19	C₂²×#88322	C₂³	1	6	yes	—
20	C₄⋊#88322	C₂×C₄	1	24	yes	—
21	#13133128	—	4	6	no	—
22	C₄×#88322	C₂×C₄	1	24	yes	—
23	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₄×Q₈×C₅₃	—	2	8	no	—
25	C₄×Q₈×C₅₃	C₂×C₄	1	8	yes	—
26	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	6	yes	—
27	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	3	yes	—
28	C₂×C₄²×C₅₃	C₂×C₄	1	6	yes	—
29	C₂×C₄²×C₅₃	C₂×C₄	1	6	yes	—
30	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
31	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
33	#95×C₅₃	C₂×C₄	1	6	yes	—
34	C₂×Q₈×C₅₃	C₄²	1	4	yes	—
35	C₂×Q₈×C₅₃	M₁₆	1	8	yes	—
36	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
37	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
38	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
39	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
40	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
41	C₄⋊C₄×C₅₃	—	2	12	no	—
42	C₄⋊C₄×C₅₃	C₄²	1	12	yes	—
43	C₄²×C₅₃	—	2	12	no	—
44	C₄²×C₅₃	—	2	12	no	—
45	C₄²×C₅₃	—	2	3	no	—
46	C₄²×C₅₃	—	2	2	no	—
47	C₄²×C₅₃	—	2	3	no	—
48	C₄²×C₅₃	C₂²×C₄	1	12	yes	—
49	C₂×#88322	—	2	6	no	—
50	C₂×#88322	—	2	2	no	—
51	C₂×#88322	—	2	2	no	—
52	C₂×#88322	—	2	6	no	—
53	C₂×#88322	C₂²×C₄	1	4	yes	—
54	C₂×#88322	C₂²×C₄	1	12	yes	—
55	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
56	Q₈×C₅₃	#97	1	16	yes	complemented
57	C₂×C₄×C₅₃	—	2	2	no	—
58	C₂×C₄×C₅₃	—	2	2	no	—
59	C₂×C₄×C₅₃	—	2	6	no	—
60	C₂×C₄×C₅₃	—	2	6	no	—
61	C₂×C₄×C₅₃	—	2	6	no	—
62	C₂×C₄×C₅₃	—	2	6	no	—
63	C₂×C₄×C₅₃	C₂×C₄²	1	6	yes	—
64	C₂×C₄×C₅₃	C₂×C₄²	1	6	yes	—
65	C₂×C₄×C₅₃	C₂×M₁₆	1	12	yes	—
66	C₂×C₄×C₅₃	#131	1	4	yes	—
67	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
68	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
69	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
70	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
71	#88322	—	2	4	no	—
72	#88322	—	2	12	no	—
73	#20572	—	53	1	no	maximal, Sylow
74	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
75	C₂²×C₅₃	#519	1	1	yes	characteristic
76	C₂²×C₅₃	#520	1	1	yes	characteristic
77	C₂²×C₅₃	C₂²×M₁₆	1	2	yes	—
78	C₂²×C₅₃	C₂×#131	1	2	yes	—
79	C₄×C₅₃	—	2	6	no	—
80	C₄×C₅₃	—	2	4	no	—
81	C₄×C₅₃	—	4	6	no	—
82	C₄×C₅₃	C₂×#97	1	12	yes	—
83	C₄×C₅₃	#519	1	2	yes	—
84	C₄×C₅₃	#520	1	2	yes	—
85	C₂×C₈×Q₈	—	53	2	no	—
86	C₄²×Q₈	D₅₃	1	1	yes	characteristic
87	#1770	—	53	6	no	—
88	#1767	—	53	3	no	—
89	#1744	—	53	3	no	—
90	C₂×C₅₃	#2276	1	1	yes	characteristic
91	C₂×C₅₃	C₂²×#97	1	1	yes	characteristic
92	C₂×C₅₃	C₄×#131	1	1	yes	characteristic
93	C₂×C₅₃	#2984	1	2	yes	—
94	C₂×C₅₃	#2991	1	2	yes	—
95	C₂×C₄×Q₈	Dic₅₃	1	2	yes	—
96	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
97	C₈×Q₈	—	53	16	no	—
98	C₂×C₄⋊C₈	—	53	6	no	—
99	C₂×#97	—	53	6	no	—
100	C₂×C₄×C₈	—	53	6	no	—
101	C₄×C₄⋊C₄	Dic₅₃	1	6	yes	—
102	C₄×C₄⋊C₄	C₂×D₅₃	1	3	yes	—
103	C₄³	C₂×D₅₃	1	3	yes	—
104	#336	—	53	1	no	—
105	#336	—	53	3	no	—
106	#336	—	53	6	no	—
107	C₅₃	#20572	1	1	yes	characteristic, complemented, Sylow
108	C₂²×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
109	C₂²×C₈	—	53	2	no	—
110	C₂²×C₈	—	53	6	no	—
111	C₄×Q₈	—	2	8	no	—
112	C₄×Q₈	C₄×D₅₃	1	8	yes	—
113	C₂×C₄⋊C₄	C₂×Dic₅₃	1	6	yes	—
114	C₂×C₄⋊C₄	C₂×Dic₅₃	1	3	yes	—
115	C₂×C₄²	C₂×Dic₅₃	1	6	yes	—
116	C₂×C₄²	C₂×Dic₅₃	1	6	yes	—
117	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
118	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
119	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
120	C₄⋊C₈	—	53	24	no	—
121	#97	—	212	6	no	—
122	C₄×C₈	—	53	24	no	—
123	#95	C₂×Dic₅₃	1	6	yes	—
124	C₂×Q₈	#11719917	1	8	yes	—
125	C₂×Q₈	C₄×Dic₅₃	1	4	yes	—
126	C₂²×C₄	C₂²×Dic₅₃	1	6	yes	—
127	C₂²×C₄	C₂²×Dic₅₃	1	3	yes	—
128	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
129	C₂²×C₄	C₂²×Dic₅₃	1	3	yes	—
130	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
131	C₂×C₈	—	53	4	no	—
132	C₂×C₈	—	53	12	no	—
133	C₂×C₈	—	106	6	no	—
134	C₂×C₈	—	106	2	no	—
135	C₂×C₈	—	106	2	no	—
136	C₂×C₈	—	106	6	no	—
137	C₄⋊C₄	—	2	12	no	—
138	C₄⋊C₄	C₄×Dic₅₃	1	12	yes	—
139	C₄²	—	2	12	no	—
140	C₄²	—	2	12	no	—
141	C₄²	—	2	3	no	—
142	C₄²	—	2	3	no	—
143	C₄²	—	2	2	no	—
144	C₄²	C₂×C₄×D₅₃	1	12	yes	—
145	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
146	Q₈	#13133128	1	16	yes	complemented
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	6	no	—
150	C₂×C₄	—	2	6	no	—
151	C₂×C₄	—	2	6	no	—
152	C₂×C₄	—	2	6	no	—
153	C₂×C₄	C₂×C₄×Dic₅₃	1	6	yes	—
154	C₂×C₄	C₂×C₄×Dic₅₃	1	6	yes	—
155	C₂×C₄	C₂×#11719917	1	12	yes	—
156	C₂×C₄	#13133224	1	4	yes	—
157	C₂×C₄	C₂²×C₄×D₅₃	1	2	yes	—
158	C₂×C₄	#13133263	1	1	yes	characteristic
159	C₂×C₄	#13133264	1	1	yes	characteristic
160	C₂×C₄	#13133265	1	2	yes	—
161	C₈	—	106	4	no	—
162	C₈	—	106	12	no	—
163	C₂²	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
164	C₂²	#16811101	1	1	yes	characteristic
165	C₂²	#16811106	1	1	yes	characteristic
166	C₂²	C₂²×#11719917	1	2	yes	—
167	C₂²	C₂×#13133224	1	2	yes	—
168	C₄	—	2	6	no	—
169	C₄	—	2	4	no	—
170	C₄	—	4	6	no	—
171	C₄	C₂×#13133128	1	12	yes	—
172	C₄	#16811099	1	2	yes	—
173	C₄	#16811105	1	2	yes	—
174	C₂	#20277080	1	1	yes	characteristic
175	C₂	C₂²×#13133128	1	1	yes	characteristic
176	C₂	C₄×#13133224	1	1	yes	characteristic
177	C₂	#20283804	1	2	yes	—
178	C₂	#20283847	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂²×C₄²
Fitting	C₄²×Q₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂³×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₄²×Q₈	D₅₃
hypocenter	C₅₃	#20572
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×Q₈

Complements

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

N	H ≅ G/N	type
6. Q₈×#88322	C₄	semidirect
56. Q₈×C₅₃	#97	semidirect
107. C₅₃	#20572	semidirect
146. Q₈	#13133128	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×#88322	2
#20274584	2
#20274568	2
#20274461	2
C₄²×Q₈×C₅₃	2
#20572	53