Subgroups — #27537784

Order 13568.

532 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#11719908	C₂	1	2	yes	maximal, complemented
2	#20276547	C₂	1	6	yes	maximal, complemented
3	#20276514	C₂	1	3	yes	maximal, complemented
4	#20276471	C₂	1	3	yes	maximal, complemented
5	C₈×#41×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133157	C₂²	1	6	yes	—
7	C₄⋊#11719908	C₂²	1	1	yes	characteristic
8	C₄⋊#11719908	C₂²	1	3	yes	—
9	C₄⋊#11719908	C₂²	1	6	yes	—
10	C₂²⋊#11719908	C₂²	1	6	yes	—
11	C₄×#11719908	C₂²	1	6	yes	—
12	C₄×#41×C₅₃	C₄	1	1	yes	characteristic
13	C₈×Q₈×C₅₃	C₄	1	1	yes	characteristic
14	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₈×D₄×C₅₃	C₄	1	3	yes	—
16	C₈×D₄×C₅₃	C₂²	1	3	yes	—
17	C₅₃×#431	C₄	1	3	yes	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	3	yes	—
19	#13133157	—	4	6	no	—
20	C₂×#11719908	C₂³	1	6	yes	—
21	C₂×#11719908	C₂³	1	2	yes	—
22	C₂×#41×C₅₃	C₈	1	1	yes	characteristic
23	C₂²×C₈×C₅₃	C₂×C₄	1	3	yes	—
24	C₂²×C₈×C₅₃	C₂³	1	3	yes	—
25	C₄×Q₈×C₅₃	C₈	1	1	yes	characteristic
26	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₄×D₄×C₅₃	C₈	1	3	yes	—
28	C₄×D₄×C₅₃	C₂×C₄	1	3	yes	—
29	#117×C₅₃	C₈	1	3	yes	—
30	C₂×C₄²×C₅₃	C₂×C₄	1	3	yes	—
31	C₄⋊C₈×C₅₃	C₂×C₄	1	3	yes	—
32	C₄⋊C₈×C₅₃	C₂×C₄	1	3	yes	—
33	C₂²⋊C₈×C₅₃	C₂×C₄	1	6	yes	—
34	C₄×C₈×C₅₃	C₂×C₄	1	3	yes	—
35	C₄×C₈×C₅₃	C₂×C₄	1	3	yes	—
36	C₄×C₈×C₅₃	C₂³	1	3	yes	—
37	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
38	#41×C₅₃	—	2	4	no	—
39	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
40	C₂×D₄×C₅₃	C₂×C₈	1	3	yes	—
41	C₂²×C₄×C₅₃	C₂×C₈	1	3	yes	—
42	C₂²×C₄×C₅₃	C₂×C₈	1	3	yes	—
43	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
44	C₂×C₈×C₅₃	—	2	3	no	—
45	C₂×C₈×C₅₃	—	2	3	no	—
46	C₂×C₈×C₅₃	—	4	3	no	—
47	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
49	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
50	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
51	C₄⋊C₄×C₅₃	C₂×C₈	1	3	yes	—
52	C₄⋊C₄×C₅₃	C₂×C₈	1	3	yes	—
53	C₂²⋊C₄×C₅₃	C₂×C₈	1	6	yes	—
54	C₄²×C₅₃	C₂×C₈	1	3	yes	—
55	C₄²×C₅₃	C₂×C₈	1	3	yes	—
56	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₅₃	C₂²×C₄	1	3	yes	—
58	#11719908	—	2	6	no	—
59	#11719908	—	2	2	no	—
60	C₂³×C₅₃	C₂²×C₈	1	3	yes	—
61	Q₈×C₅₃	M₃₂	1	4	yes	complemented
62	D₄×C₅₃	—	2	6	no	—
63	C₂×C₄×C₅₃	—	2	3	no	—
64	C₂×C₄×C₅₃	—	2	3	no	—
65	C₂×C₄×C₅₃	—	2	6	no	—
66	C₂×C₄×C₅₃	—	2	6	no	—
67	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂²×C₈	1	3	yes	—
69	C₂×C₄×C₅₃	C₂²×C₈	1	3	yes	—
70	C₂×C₄×C₅₃	C₂²×C₈	1	3	yes	—
71	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
73	C₈×C₅₃	—	2	3	no	—
74	C₈×C₅₃	—	2	1	no	—
75	C₈×C₅₃	—	2	3	no	—
76	C₈×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
77	C₈×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
78	#33505	—	53	1	no	maximal, Sylow
79	C₂²×C₅₃	—	2	3	no	—
80	C₂²×C₅₃	—	2	3	no	—
81	C₂²×C₅₃	—	2	3	no	—
82	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
83	C₄×C₅₃	—	2	1	no	—
84	C₄×C₅₃	—	2	3	no	—
85	C₄×C₅₃	—	2	3	no	—
86	C₄×C₅₃	C₂×M₃₂	1	6	yes	—
87	C₄×C₅₃	#504	1	2	yes	—
88	C₄×C₅₃	#520	1	1	yes	characteristic
89	C₄×C₅₃	#568	1	1	yes	characteristic
90	C₈×#41	D₅₃	1	1	yes	characteristic
91	Q₈×C₁₆	—	53	2	no	—
92	#2169	—	53	6	no	—
93	#2162	—	53	3	no	—
94	#2150	—	53	3	no	—
95	C₂×C₅₃	—	4	3	no	—
96	C₂×C₅₃	#2988	1	1	yes	characteristic
97	C₂×C₅₃	C₂²×M₃₂	1	1	yes	characteristic
98	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
99	C₄×#41	Dic₅₃	1	1	yes	characteristic
100	C₂×M₃₂	—	53	6	no	—
101	C₈×Q₈	Dic₅₃	1	1	yes	characteristic
102	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
103	C₈×D₄	Dic₅₃	1	3	yes	—
104	C₈×D₄	C₂×D₅₃	1	3	yes	—
105	#431	Dic₅₃	1	3	yes	—
106	C₂×C₄×C₈	C₂×D₅₃	1	3	yes	—
107	C₄⋊C₁₆	—	53	3	no	—
108	C₄⋊C₁₆	—	53	6	no	—
109	C₄⋊C₁₆	—	53	1	no	—
110	C₂²⋊C₁₆	—	53	6	no	—
111	C₄×C₁₆	—	53	6	no	—
112	C₅₃	#33505	1	1	yes	characteristic, complemented, Sylow
113	C₂×#41	#88322	1	1	yes	characteristic
114	C₂²×C₈	C₂×Dic₅₃	1	3	yes	—
115	C₂²×C₈	C₂²×D₅₃	1	3	yes	—
116	C₄×Q₈	#88322	1	1	yes	characteristic
117	C₄×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
118	C₄×D₄	#88322	1	3	yes	—
119	C₄×D₄	C₂×Dic₅₃	1	3	yes	—
120	#117	#88322	1	3	yes	—
121	C₂×C₄²	C₂×Dic₅₃	1	3	yes	—
122	M₃₂	—	212	6	no	—
123	C₂×C₁₆	—	53	2	no	—
124	C₂×C₁₆	—	53	6	no	—
125	C₄⋊C₈	C₂×Dic₅₃	1	3	yes	—
126	C₄⋊C₈	C₂×Dic₅₃	1	3	yes	—
127	C₂²⋊C₈	C₂×Dic₅₃	1	6	yes	—
128	C₄×C₈	C₂×Dic₅₃	1	3	yes	—
129	C₄×C₈	C₂×Dic₅₃	1	3	yes	—
130	C₄×C₈	C₂²×D₅₃	1	3	yes	—
131	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
132	#41	—	2	4	no	—
133	C₂×Q₈	C₂×#88322	1	1	yes	characteristic
134	C₂×D₄	C₂×#88322	1	3	yes	—
135	C₂²×C₄	C₂×#88322	1	3	yes	—
136	C₂²×C₄	C₂×#88322	1	3	yes	—
137	C₂²×C₄	C₂²×Dic₅₃	1	3	yes	—
138	C₂×C₈	—	2	3	no	—
139	C₂×C₈	—	2	3	no	—
140	C₂×C₈	—	4	3	no	—
141	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
142	C₂×C₈	C₂²×Dic₅₃	1	3	yes	—
143	C₂×C₈	C₂²×Dic₅₃	1	3	yes	—
144	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
145	C₄⋊C₄	C₂×#88322	1	3	yes	—
146	C₄⋊C₄	C₂×#88322	1	3	yes	—
147	C₂²⋊C₄	C₂×#88322	1	6	yes	—
148	C₄²	C₂×#88322	1	3	yes	—
149	C₄²	C₂×#88322	1	3	yes	—
150	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
151	C₄²	C₂²×Dic₅₃	1	3	yes	—
152	C₁₆	—	106	6	no	—
153	C₁₆	—	106	2	no	—
154	C₂³	C₂²×#88322	1	3	yes	—
155	Q₈	#13133157	1	4	yes	complemented
156	D₄	—	2	6	no	—
157	C₂×C₄	—	2	6	no	—
158	C₂×C₄	—	2	6	no	—
159	C₂×C₄	—	2	3	no	—
160	C₂×C₄	—	2	3	no	—
161	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
162	C₂×C₄	C₂²×#88322	1	3	yes	—
163	C₂×C₄	C₂²×#88322	1	3	yes	—
164	C₂×C₄	C₂²×#88322	1	3	yes	—
165	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
166	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
167	C₈	—	2	3	no	—
168	C₈	—	2	1	no	—
169	C₈	—	2	3	no	—
170	C₈	#13133263	1	1	yes	characteristic
171	C₈	#13133264	1	1	yes	characteristic
172	C₂²	—	2	3	no	—
173	C₂²	—	2	3	no	—
174	C₂²	—	2	3	no	—
175	C₂²	C₂³×#88322	1	1	yes	characteristic
176	C₄	—	2	1	no	—
177	C₄	—	2	3	no	—
178	C₄	—	2	3	no	—
179	C₄	C₂×#13133157	1	6	yes	—
180	C₄	#16811040	1	2	yes	—
181	C₄	#16811106	1	1	yes	characteristic
182	C₄	#16811390	1	1	yes	characteristic
183	C₂	—	4	3	no	—
184	C₂	#20283826	1	1	yes	characteristic
185	C₂	C₂²×#13133157	1	1	yes	characteristic
186	C₂	C₂×#16811040	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₈×#41×C₅₃	C₂
prime radical	C₂×#41×C₅₃	C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₈×#41	D₅₃
hypocenter	C₅₃	#33505
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
1. Q₈×#11719908	C₂	semidirect
2. #20276547	C₂	semidirect
3. #20276514	C₂	semidirect
4. #20276471	C₂	semidirect
61. Q₈×C₅₃	M₃₂	semidirect
112. C₅₃	#33505	semidirect
155. Q₈	#13133157	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#11719908	2
#20276547	2
#20276514	2
#20276471	2
C₈×#41×C₅₃	2
#33505	53