Subgroups — #131×#88324 #28313762

Order 13568.

922 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×#88324	C₂	1	2	yes	maximal
2	#20289918	C₂	1	3	yes	maximal, complemented
3	M₁₆×#88324	C₂	1	6	yes	maximal, complemented
4	C₂×C₈×#88324	C₂	1	3	yes	maximal, complemented
5	#131×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	Q₈×#88324	C₄	1	4	yes	—
7	D₄×#88324	C₄	1	12	yes	—
8	#16811639	C₂²	1	6	yes	—
9	C₂×C₄×#88324	C₂²	1	6	yes	—
10	#16811464	C₂²	1	6	yes	—
11	C₈×#88324	C₂²	1	1	yes	characteristic
12	C₈×#88324	C₂²	1	6	yes	—
13	C₈×#88324	C₂²	1	3	yes	—
14	C₂×#13133224	C₄	1	1	yes	characteristic
15	#41×Dic₅₃	C₂²	1	1	yes	characteristic
16	#16810581	C₄	1	3	yes	—
17	#16810578	C₄	1	3	yes	—
18	M₁₆×Dic₅₃	C₂²	1	3	yes	—
19	C₂×C₈×Dic₅₃	C₂²	1	3	yes	—
20	C₂×#131×C₅₃	C₄	1	1	yes	characteristic
21	C₂²×#88324	C₂×C₄	1	12	yes	—
22	C₄⋊#88324	C₂×C₄	1	12	yes	—
23	C₂²⋊#88324	C₂×C₄	1	12	yes	—
24	C₄×#88324	C₂×C₄	1	12	yes	—
25	C₄×#88324	C₂³	1	2	yes	—
26	C₄×#88324	C₂³	1	6	yes	—
27	#13133224	C₈	1	8	yes	complemented
28	C₂×#11719917	C₂×C₄	1	3	yes	—
29	C₂²×#88322	C₂×C₄	1	3	yes	—
30	Q₈×Dic₅₃	C₂×C₄	1	2	yes	—
31	D₄×Dic₅₃	C₂×C₄	1	6	yes	—
32	#13133176	C₂×C₄	1	3	yes	—
33	C₂×C₄×Dic₅₃	C₂³	1	3	yes	—
34	#13133127	C₂×C₄	1	6	yes	—
35	C₈×Dic₅₃	C₂×C₄	1	3	yes	—
36	C₈×Dic₅₃	C₂×C₄	1	3	yes	—
37	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
38	C₈×Dic₅₃	C₂³	1	3	yes	—
39	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
40	#131×C₅₃	C₈	1	8	yes	complemented
41	C₂×M₁₆×C₅₃	C₂×C₄	1	3	yes	—
42	C₂²×C₈×C₅₃	C₂×C₄	1	3	yes	—
43	#41×C₅₃	C₂×C₈	1	4	yes	—
44	#41×C₅₃	C₂×C₈	1	4	yes	—
45	C₂×Q₈×C₅₃	C₄²	1	1	yes	characteristic
46	C₂×D₄×C₅₃	C₄²	1	3	yes	—
47	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
48	C₂²×Dic₅₃	C₂²×C₄	1	6	yes	—
49	C₂×#88324	—	2	24	no	—
50	C₂×#88324	C₂²×C₄	1	12	yes	—
51	C₂×#88324	C₂²×C₄	1	4	yes	—
52	M₁₆×C₅₃	C₂×C₈	1	12	yes	complemented
53	C₂×C₈×C₅₃	C₂×C₈	1	12	yes	complemented
54	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
55	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²⋊Dic₅₃	C₄²	1	6	yes	—
57	C₄⋊Dic₅₃	C₄²	1	6	yes	—
58	C₄×Dic₅₃	C₂²×C₄	1	3	yes	—
59	C₄×Dic₅₃	C₂²×C₄	1	6	yes	—
60	C₄×Dic₅₃	C₂⁴	1	1	yes	characteristic
61	#11719917	C₂×C₈	1	12	yes	complemented
62	C₂×#88322	C₂×C₈	1	12	yes	complemented
63	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
64	C₂×#88322	C₂²×C₄	1	3	yes	—
65	C₂³×C₅₃	C₂×C₄²	1	3	yes	—
66	Q₈×C₅₃	C₄×C₈	1	4	yes	—
67	D₄×C₅₃	C₄×C₈	1	12	yes	—
68	C₂×C₄×C₅₃	C₂×C₄²	1	3	yes	—
69	C₂×C₄×C₅₃	C₂²×C₈	1	6	yes	—
70	C₂×C₄×C₅₃	C₂²×C₈	1	6	yes	—
71	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
72	C₂×Dic₅₃	—	2	12	no	—
73	C₂×Dic₅₃	C₂×C₄²	1	6	yes	—
74	C₂×Dic₅₃	C₂³×C₄	1	2	yes	—
75	#88324	—	2	12	no	—
76	#88324	#131	1	8	yes	direct product factor
77	C₈×C₅₃	C₂²×C₈	1	2	yes	—
78	C₈×C₅₃	C₂²×C₈	1	6	yes	—
79	#88322	C₂²×C₈	1	2	yes	—
80	#88322	C₂²×C₈	1	6	yes	—
81	C₈×#131	—	53	1	no	maximal, Sylow
82	C₂²×C₅₃	—	2	6	no	—
83	C₂²×C₅₃	C₂×C₄×C₈	1	6	yes	—
84	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
85	C₄×C₅₃	C₂×C₄×C₈	1	6	yes	—
86	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
87	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
88	Dic₅₃	—	2	6	no	—
89	Dic₅₃	C₂×#131	1	4	yes	—
90	C₈×#41	—	53	2	no	—
91	C₄×#131	—	53	1	no	—
92	#1450	—	53	3	no	—
93	C₈×M₁₆	—	53	6	no	—
94	C₂×C₈²	—	53	3	no	—
95	C₂×C₅₃	—	2	6	no	—
96	C₂×C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic
97	C₂×C₅₃	C₄×#131	1	2	yes	—
98	C₂×#131	—	53	1	no	—
99	C₂×#131	#6157	1	1	yes	characteristic
100	C₄×#41	—	53	1	no	—
101	C₈×Q₈	—	53	4	no	—
102	C₈×D₄	—	53	12	no	—
103	#431	—	53	6	no	—
104	#405	—	53	3	no	—
105	#405	—	53	3	no	—
106	C₄×M₁₆	—	53	3	no	—
107	C₂×C₄×C₈	—	53	6	no	—
108	C₂×C₄×C₈	—	53	3	no	—
109	#322	—	53	6	no	—
110	C₈²	—	53	1	no	—
111	C₈²	—	53	6	no	—
112	C₈²	—	53	3	no	—
113	C₅₃	C₈×#131	1	1	yes	characteristic, complemented, Sylow
114	C₂×#41	C₂×#6157	1	1	yes	characteristic
115	#131	—	53	8	no	—
116	#131	#88324	1	8	yes	direct product factor
117	C₂×M₁₆	—	53	3	no	—
118	C₂×M₁₆	C₂×#6157	1	3	yes	—
119	C₂²×C₈	—	53	3	no	—
120	C₂²×C₈	—	53	12	no	—
121	C₂²×C₈	C₂×#6157	1	3	yes	—
122	C₄×Q₈	—	53	2	no	—
123	C₄×D₄	—	53	6	no	—
124	#117	—	53	3	no	—
125	C₂×C₄²	—	53	3	no	—
126	C₄⋊C₈	—	53	12	no	—
127	C₂²⋊C₈	—	53	12	no	—
128	#97	—	53	6	no	—
129	C₄×C₈	—	53	2	no	—
130	C₄×C₈	—	53	6	no	—
131	C₄×C₈	—	53	12	no	—
132	C₄×C₈	—	53	3	no	—
133	C₄×C₈	—	53	3	no	—
134	C₄×C₈	—	53	1	no	—
135	C₄×C₈	—	53	3	no	—
136	#41	D₅₃⋊C₈	1	4	yes	—
137	#41	C₂×#88324	1	4	yes	—
138	C₂×Q₈	C₄×#6157	1	1	yes	characteristic
139	C₂×D₄	C₄×#6157	1	3	yes	—
140	C₂²×C₄	—	53	6	no	—
141	C₂²×C₄	C₂²×#6157	1	3	yes	—
142	M₁₆	—	53	12	no	—
143	M₁₆	C₂×#88324	1	12	yes	complemented
144	C₂×C₈	—	53	4	no	—
145	C₂×C₈	—	53	12	no	—
146	C₂×C₈	—	53	1	no	—
147	C₂×C₈	—	53	3	no	—
148	C₂×C₈	—	53	12	no	—
149	C₂×C₈	—	106	24	no	—
150	C₂×C₈	C₂×#88324	1	12	yes	complemented
151	C₂×C₈	C₂²×#6157	1	3	yes	—
152	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
153	C₄⋊C₄	—	53	6	no	—
154	C₂²⋊C₄	—	53	6	no	—
155	C₄²	—	53	6	no	—
156	C₄²	—	53	1	no	—
157	C₄²	—	53	3	no	—
158	C₂³	C₂×C₄×#6157	1	3	yes	—
159	Q₈	C₄×#88324	1	4	yes	—
160	D₄	C₄×#88324	1	12	yes	—
161	C₂×C₄	—	53	2	no	—
162	C₂×C₄	—	53	6	no	—
163	C₂×C₄	—	106	12	no	—
164	C₂×C₄	C₂×C₄×#6157	1	3	yes	—
165	C₂×C₄	C₂²×#88324	1	6	yes	—
166	C₂×C₄	C₂×D₅₃⋊C₈	1	6	yes	—
167	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
168	C₈	—	53	8	no	—
169	C₈	—	53	2	no	—
170	C₈	—	53	6	no	—
171	C₈	—	106	12	no	—
172	C₈	C₂²×#88324	1	2	yes	—
173	C₈	C₂²×#88324	1	6	yes	—
174	C₂²	—	2	6	no	—
175	C₂²	C₂×C₄×#88324	1	6	yes	—
176	C₂²	C₂²×C₄×#6157	1	1	yes	characteristic
177	C₄	—	53	4	no	—
178	C₄	—	106	6	no	—
179	C₄	C₂×C₄×#88324	1	6	yes	—
180	C₄	C₂³×#88324	1	1	yes	characteristic
181	C₄	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
182	C₂	—	2	6	no	—
183	C₂	C₂²×C₄×#88324	1	1	yes	characteristic
184	C₂	#131×#6157	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²×#6157
commutator	C₂×C₅₃	C₂²×C₄×C₈
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₂×#131×C₅₃	C₄
socle	C₂²×C₅₃	C₂²×C₄²
prime radical	C₂×#41×C₅₃	C₂×C₄
cosocle	C₄×Dic₅₃	C₂⁴
hypercenter	C₂×#131	#6157
hypocenter	C₅₃	C₈×#131
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
2. #20289918	C₂	semidirect
3. M₁₆×#88324	C₂	semidirect
4. C₂×C₈×#88324	C₂	semidirect
27. #13133224	C₈	semidirect
40. #131×C₅₃	C₈	semidirect
52. M₁₆×C₅₃	C₂×C₈	semidirect
53. C₂×C₈×C₅₃	C₂×C₈	semidirect
61. #11719917	C₂×C₈	semidirect
62. C₂×#88322	C₂×C₈	semidirect
76. #88324	#131	direct (N × H)
113. C₅₃	C₈×#131	semidirect
116. #131	#88324	direct (N × H)
143. M₁₆	C₂×#88324	semidirect
150. C₂×C₈	C₂×#88324	semidirect

Maximal subgroups

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

subgroup	index
#41×#88324	2
#20289918	2
M₁₆×#88324	2
C₂×C₈×#88324	2
#131×Dic₅₃	2
C₈×#131	53