Subgroups — #27537672

Order 13568.

548 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	Q₈×#11719908	C₂	1	2	yes	maximal, complemented
2	#20276547	C₂	1	6	yes	maximal, complemented
3	#20276518	C₂	1	3	yes	maximal, complemented
4	C₄×#13133157	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	6	yes	—
8	C₂²⋊#11719908	C₂²	1	6	yes	—
9	#16810399	C₂²	1	3	yes	—
10	#16810399	C₂²	1	1	yes	characteristic
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	—	2	12	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	6	no	—
45	C₂×C₈×C₅₃	—	2	6	no	—
46	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
47	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
48	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
50	C₄⋊C₄×C₅₃	C₂×C₈	1	3	yes	—
51	C₄⋊C₄×C₅₃	C₂×C₈	1	3	yes	—
52	C₂²⋊C₄×C₅₃	C₂×C₈	1	6	yes	—
53	C₄²×C₅₃	C₂×C₈	1	3	yes	—
54	C₄²×C₅₃	C₂×C₈	1	3	yes	—
55	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₅₃	C₂²×C₄	1	3	yes	—
57	#11719908	—	2	6	no	—
58	#11719908	#41	1	4	yes	complemented
59	C₂³×C₅₃	C₂²×C₈	1	3	yes	—
60	Q₈×C₅₃	M₃₂	1	4	yes	complemented
61	D₄×C₅₃	—	2	6	no	—
62	C₂×C₄×C₅₃	—	2	3	no	—
63	C₂×C₄×C₅₃	—	2	3	no	—
64	C₂×C₄×C₅₃	—	2	6	no	—
65	C₂×C₄×C₅₃	—	4	3	no	—
66	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
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	C₂³×C₄	1	1	yes	characteristic
72	C₈×C₅₃	—	2	3	no	—
73	C₈×C₅₃	—	2	3	no	—
74	C₈×C₅₃	#131	1	2	yes	—
75	C₈×C₅₃	C₂×#41	1	2	yes	—
76	#33485	—	53	1	no	maximal, Sylow
77	C₂²×C₅₃	—	2	3	no	—
78	C₂²×C₅₃	—	2	3	no	—
79	C₂²×C₅₃	—	2	3	no	—
80	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
81	C₄×C₅₃	—	2	1	no	—
82	C₄×C₅₃	—	2	1	no	—
83	C₄×C₅₃	—	2	3	no	—
84	C₄×C₅₃	—	2	3	no	—
85	C₄×C₅₃	C₂×M₃₂	1	6	yes	—
86	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
87	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
88	C₈×#41	D₅₃	1	1	yes	characteristic
89	Q₈×C₁₆	—	53	2	no	—
90	#2169	—	53	6	no	—
91	#2163	—	53	3	no	—
92	C₄×M₃₂	—	53	3	no	—
93	C₂×C₅₃	—	4	3	no	—
94	C₂×C₅₃	C₈×#41	1	1	yes	characteristic
95	C₂×C₅₃	C₂²×M₃₂	1	1	yes	characteristic
96	C₂×C₅₃	#3407	1	1	yes	characteristic
97	C₄×#41	Dic₅₃	1	1	yes	characteristic
98	C₂×M₃₂	—	53	6	no	—
99	C₈×Q₈	Dic₅₃	1	1	yes	characteristic
100	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
101	C₈×D₄	Dic₅₃	1	3	yes	—
102	C₈×D₄	C₂×D₅₃	1	3	yes	—
103	#431	Dic₅₃	1	3	yes	—
104	C₂×C₄×C₈	C₂×D₅₃	1	3	yes	—
105	C₄⋊C₁₆	—	53	6	no	—
106	C₂²⋊C₁₆	—	53	6	no	—
107	#346	—	53	1	no	—
108	#346	—	53	3	no	—
109	C₄×C₁₆	—	53	6	no	—
110	C₅₃	#33485	1	1	yes	characteristic, complemented, Sylow
111	C₂×#41	#88322	1	1	yes	characteristic
112	C₂²×C₈	C₂×Dic₅₃	1	3	yes	—
113	C₂²×C₈	C₂²×D₅₃	1	3	yes	—
114	C₄×Q₈	#88322	1	1	yes	characteristic
115	C₄×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
116	C₄×D₄	#88322	1	3	yes	—
117	C₄×D₄	C₂×Dic₅₃	1	3	yes	—
118	#117	#88322	1	3	yes	—
119	C₂×C₄²	C₂×Dic₅₃	1	3	yes	—
120	M₃₂	—	106	12	no	—
121	C₂×C₁₆	—	53	2	no	—
122	C₂×C₁₆	—	53	6	no	—
123	C₄⋊C₈	C₂×Dic₅₃	1	3	yes	—
124	C₄⋊C₈	C₂×Dic₅₃	1	3	yes	—
125	C₂²⋊C₈	C₂×Dic₅₃	1	6	yes	—
126	C₄×C₈	C₂×Dic₅₃	1	3	yes	—
127	C₄×C₈	C₂×Dic₅₃	1	3	yes	—
128	C₄×C₈	C₂²×D₅₃	1	3	yes	—
129	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
130	#41	—	2	4	no	—
131	C₂×Q₈	C₂×#88322	1	1	yes	characteristic
132	C₂×D₄	C₂×#88322	1	3	yes	—
133	C₂²×C₄	C₂×#88322	1	3	yes	—
134	C₂²×C₄	C₂×#88322	1	3	yes	—
135	C₂²×C₄	C₂²×Dic₅₃	1	3	yes	—
136	C₂×C₈	—	2	6	no	—
137	C₂×C₈	—	2	6	no	—
138	C₂×C₈	C₂²×Dic₅₃	1	3	yes	—
139	C₂×C₈	C₂²×Dic₅₃	1	3	yes	—
140	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
141	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
142	C₄⋊C₄	C₂×#88322	1	3	yes	—
143	C₄⋊C₄	C₂×#88322	1	3	yes	—
144	C₂²⋊C₄	C₂×#88322	1	6	yes	—
145	C₄²	C₂×#88322	1	3	yes	—
146	C₄²	C₂×#88322	1	3	yes	—
147	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
148	C₄²	C₂²×Dic₅₃	1	3	yes	—
149	C₁₆	—	53	4	no	—
150	C₁₆	—	106	6	no	—
151	C₂³	C₂²×#88322	1	3	yes	—
152	Q₈	#13133157	1	4	yes	complemented
153	D₄	—	2	6	no	—
154	C₂×C₄	—	2	6	no	—
155	C₂×C₄	—	2	3	no	—
156	C₂×C₄	—	2	3	no	—
157	C₂×C₄	—	4	3	no	—
158	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
159	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
160	C₂×C₄	C₂²×#88322	1	3	yes	—
161	C₂×C₄	C₂²×#88322	1	3	yes	—
162	C₂×C₄	C₂²×#88322	1	3	yes	—
163	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
164	C₈	—	2	3	no	—
165	C₈	—	2	3	no	—
166	C₈	#13133224	1	2	yes	—
167	C₈	#41×D₅₃	1	2	yes	—
168	C₂²	—	2	3	no	—
169	C₂²	—	2	3	no	—
170	C₂²	—	2	3	no	—
171	C₂²	C₂³×#88322	1	1	yes	characteristic
172	C₄	—	2	3	no	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	2	3	no	—
176	C₄	C₂×#13133157	1	6	yes	—
177	C₄	#41×Dic₅₃	1	1	yes	characteristic
178	C₄	C₂×#13133224	1	1	yes	characteristic
179	C₂	—	4	3	no	—
180	C₂	#41×#88322	1	1	yes	characteristic
181	C₂	C₂²×#13133157	1	1	yes	characteristic
182	C₂	#20287818	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	#33485
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. #20276518	C₂	semidirect
4. C₄×#13133157	C₂	semidirect
58. #11719908	#41	semidirect
60. Q₈×C₅₃	M₃₂	semidirect
110. C₅₃	#33485	semidirect
152. 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
#20276518	2
C₄×#13133157	2
C₈×#41×C₅₃	2
#33485	53