Subgroups — #27537587

Order 13568.

506 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276624	C₂	1	4	yes	maximal, complemented
2	Q₈×#11719908	C₂	1	4	yes	maximal, complemented
3	#20276474	C₂	1	2	yes	maximal
4	#20276471	C₂	1	1	yes	maximal, characteristic
5	#20276312	C₂	1	2	yes	maximal
6	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#2960	C₂	1	1	yes	maximal, characteristic
8	C₂×#13133157	C₂²	1	2	yes	—
9	C₂²×#11719908	C₂²	1	1	yes	characteristic
10	C₂²×#11719908	C₂²	1	1	yes	characteristic
11	C₄⋊#11719908	C₂²	1	4	yes	—
12	C₄⋊#11719908	C₂²	1	8	yes	—
13	#16810399	C₂²	1	4	yes	—
14	C₄×#11719908	C₂²	1	2	yes	—
15	C₄×#11719908	C₂²	1	2	yes	—
16	C₄×#11719908	C₂²	1	4	yes	—
17	C₅₃×#533	C₄	1	1	yes	characteristic
18	C₅₃×#446	C₄	1	4	yes	—
19	C₈×Q₈×C₅₃	C₂²	1	4	yes	—
20	C₅₃×#424	C₄	1	2	yes	—
21	C₅₃×#423	C₄	1	1	yes	characteristic
22	C₅₃×#405	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	#13133157	—	2	4	no	—
25	C₂×#11719908	—	2	2	no	—
26	C₂×#11719908	Q₈	1	4	yes	complemented
27	C₂×#11719908	C₂³	1	2	yes	—
28	C₂×#11719908	C₂³	1	4	yes	—
29	C₂×#11719908	C₂³	1	1	yes	characteristic
30	C₂×#11719908	C₂³	1	1	yes	characteristic
31	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
34	C₄⋊Q₈×C₅₃	C₈	1	2	yes	—
35	#125×C₅₃	C₈	1	2	yes	—
36	C₂²⋊Q₈×C₅₃	C₈	1	4	yes	—
37	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	—
38	#117×C₅₃	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
41	C₄⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
42	#97×C₅₃	C₂×C₄	1	4	yes	—
43	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₅₃	C₂³	1	2	yes	—
45	C₄×C₈×C₅₃	C₂³	1	4	yes	—
46	C₂×Q₈×C₅₃	C₂×C₈	1	4	yes	—
47	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
49	M₁₆×C₅₃	—	2	4	no	—
50	C₂×C₈×C₅₃	—	2	2	no	—
51	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
52	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
53	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
55	C₂×C₈×C₅₃	#41	1	2	yes	—
56	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
57	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
58	C₄⋊C₄×C₅₃	C₂×C₈	1	8	yes	—
59	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
60	C₂²⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
61	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
62	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
63	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
64	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
65	#11719908	—	2	2	no	—
66	#11719908	—	2	4	no	—
67	#11719908	C₂×Q₈	1	2	yes	—
68	#11719908	C₂×Q₈	1	2	yes	—
69	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
70	Q₈×C₅₃	—	2	8	no	—
71	C₂×C₄×C₅₃	—	2	4	no	—
72	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
73	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
74	C₂×C₄×C₅₃	C₂²×C₈	1	8	yes	—
75	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
77	C₂×C₄×C₅₃	#131	1	2	yes	—
78	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
79	C₈×C₅₃	—	2	2	no	—
80	C₈×C₅₃	—	2	4	no	—
81	C₈×C₅₃	C₄×Q₈	1	2	yes	—
82	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
83	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
84	#33474	—	53	1	no	maximal, Sylow
85	C₂²×C₅₃	—	2	2	no	—
86	C₂²×C₅₃	C₈×Q₈	1	2	yes	—
87	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
88	C₄×C₅₃	—	2	2	no	—
89	C₄×C₅₃	—	2	8	no	—
90	C₄×C₅₃	C₈×Q₈	1	2	yes	—
91	C₄×C₅₃	#504	1	4	yes	—
92	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
93	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
94	#2960	D₅₃	1	1	yes	characteristic
95	#2183	—	53	4	no	—
96	Q₈×C₁₆	—	53	4	no	—
97	#2151	—	53	2	no	—
98	#2150	—	53	1	no	—
99	#2108	—	53	2	no	—
100	C₂×C₄×C₁₆	—	53	1	no	—
101	C₂×C₅₃	—	2	2	no	—
102	C₂×C₅₃	C₂×C₈×Q₈	1	1	yes	characteristic
103	C₂×C₅₃	C₂×#504	1	2	yes	—
104	#533	Dic₅₃	1	1	yes	characteristic
105	C₂×M₃₂	—	53	2	no	—
106	C₂²×C₁₆	—	53	1	no	—
107	C₂²×C₁₆	—	53	1	no	—
108	#446	Dic₅₃	1	4	yes	—
109	C₈×Q₈	C₂×D₅₃	1	4	yes	—
110	#424	Dic₅₃	1	2	yes	—
111	#423	Dic₅₃	1	1	yes	characteristic
112	#405	C₂×D₅₃	1	2	yes	—
113	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
114	C₄⋊C₁₆	—	53	4	no	—
115	C₄⋊C₁₆	—	53	8	no	—
116	#346	—	53	4	no	—
117	C₄×C₁₆	—	53	2	no	—
118	C₄×C₁₆	—	53	2	no	—
119	C₄×C₁₆	—	53	4	no	—
120	C₅₃	#33474	1	1	yes	characteristic, complemented, Sylow
121	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
122	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
123	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
124	C₄⋊Q₈	#88322	1	2	yes	—
125	#125	#88322	1	2	yes	—
126	C₂²⋊Q₈	#88322	1	4	yes	—
127	C₄×Q₈	C₂×Dic₅₃	1	4	yes	—
128	#117	C₂×Dic₅₃	1	2	yes	—
129	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
130	M₃₂	—	106	4	no	—
131	C₂×C₁₆	—	53	4	no	—
132	C₂×C₁₆	—	53	1	no	—
133	C₂×C₁₆	—	53	1	no	—
134	C₂×C₁₆	—	53	2	no	—
135	C₂×C₁₆	—	53	4	no	—
136	C₂×C₁₆	—	106	2	no	—
137	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
138	C₄⋊C₈	C₂×Dic₅₃	1	8	yes	—
139	#97	C₂×Dic₅₃	1	4	yes	—
140	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
141	C₄×C₈	C₂²×D₅₃	1	2	yes	—
142	C₄×C₈	C₂²×D₅₃	1	4	yes	—
143	C₂×Q₈	C₂×#88322	1	4	yes	—
144	C₂²×C₄	C₂×#88322	1	2	yes	—
145	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
146	M₁₆	—	2	4	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	Q₈×D₅₃	1	2	yes	—
149	C₂×C₈	#11719949	1	2	yes	—
150	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
151	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
152	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
153	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
154	C₄⋊C₄	C₂×#88322	1	4	yes	—
155	C₄⋊C₄	C₂×#88322	1	8	yes	—
156	C₄⋊C₄	C₂×#88322	1	4	yes	—
157	C₂²⋊C₄	C₂×#88322	1	4	yes	—
158	C₄²	C₂×#88322	1	1	yes	characteristic
159	C₄²	C₂×#88322	1	1	yes	characteristic
160	C₄²	C₂²×Dic₅₃	1	2	yes	—
161	C₄²	C₂²×Dic₅₃	1	4	yes	—
162	C₁₆	—	53	2	no	—
163	C₁₆	—	53	2	no	—
164	C₁₆	—	106	2	no	—
165	C₁₆	—	106	4	no	—
166	C₂³	C₂²×#88322	1	1	yes	characteristic
167	Q₈	—	2	8	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	Q₈×Dic₅₃	1	2	yes	—
170	C₂×C₄	C₂²×#88322	1	2	yes	—
171	C₂×C₄	C₂²×#88322	1	8	yes	—
172	C₂×C₄	C₂²×#88322	1	2	yes	—
173	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
174	C₂×C₄	#13133224	1	2	yes	—
175	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
176	C₈	—	2	2	no	—
177	C₈	—	2	4	no	—
178	C₈	Q₈×Dic₅₃	1	2	yes	—
179	C₈	C₂×Q₈×D₅₃	1	1	yes	characteristic
180	C₈	C₂×#11719949	1	1	yes	characteristic
181	C₂²	—	2	2	no	—
182	C₂²	Q₈×#88322	1	2	yes	—
183	C₂²	C₂³×#88322	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	2	8	no	—
186	C₄	Q₈×#88322	1	2	yes	—
187	C₄	#16811040	1	4	yes	—
188	C₄	C₂×Q₈×Dic₅₃	1	1	yes	characteristic
189	C₄	C₂×#13133224	1	1	yes	characteristic
190	C₂	—	2	2	no	—
191	C₂	C₂×Q₈×#88322	1	1	yes	characteristic
192	C₂	C₂×#16811040	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#2960	C₂
prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2960	D₅₃
hypocenter	C₅₃	#33474
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #2960

Complements

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

N	H ≅ G/N	type
1. #20276624	C₂	semidirect
2. Q₈×#11719908	C₂	semidirect
26. C₂×#11719908	Q₈	semidirect
120. C₅₃	#33474	semidirect

Maximal subgroups

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

subgroup	index
#20276624	2
Q₈×#11719908	2
#20276474	2
#20276471	2
#20276312	2
C₂×C₄×#11719908	2
C₅₃×#2960	2
#33474	53