Subgroups — #28302030

Order 13568.

502 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊#88324	C₂	1	2	yes	maximal
2	C₂×#16811464	C₂	1	2	yes	maximal
3	#20289537	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#88324	C₄	1	4	yes	—
6	C₂×C₄×#88324	C₂²	1	4	yes	—
7	#16811489	C₄	1	4	yes	—
8	#16811464	—	2	4	no	—
9	C₂×C₄×#88322	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
14	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
16	C₂²×#88324	C₂×C₄	1	8	yes	—
17	C₄⋊#88324	—	2	8	no	—
18	C₄×#88324	—	2	4	no	—
19	C₄×#88324	—	2	4	no	—
20	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
21	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
24	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
25	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
26	C₄×#88322	C₈	1	4	yes	complemented
27	C₈×Dic₅₃	—	2	2	no	—
28	C₈×Dic₅₃	—	2	2	no	—
29	C₈×Dic₅₃	—	2	2	no	—
30	C₈×Dic₅₃	D₄	1	2	yes	—
31	C₈×Dic₅₃	Q₈	1	2	yes	—
32	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₅₃	C₈	1	4	yes	complemented
36	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₂²×Dic₅₃	C₄²	1	2	yes	—
40	C₂²×Dic₅₃	C₂²×C₄	1	2	yes	—
41	C₂×#88324	—	2	8	no	—
42	C₂×#88324	—	2	8	no	—
43	C₂×#88324	—	2	8	no	—
44	C₂×C₈×C₅₃	—	2	1	no	—
45	C₂×C₈×C₅₃	—	2	1	no	—
46	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
47	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
48	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
49	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
50	C₄²×C₅₃	C₂×C₈	1	2	yes	—
51	C₄²×C₅₃	C₂×C₈	1	2	yes	—
52	C₄×Dic₅₃	—	2	1	no	—
53	C₄×Dic₅₃	—	2	1	no	—
54	C₄×Dic₅₃	—	2	4	no	—
55	C₄×Dic₅₃	M₁₆	1	8	yes	—
56	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
57	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
58	C₄×Dic₅₃	#41	1	2	yes	—
59	C₂×#88322	—	2	1	no	—
60	C₂×#88322	—	2	1	no	—
61	C₂×#88322	C₄⋊C₄	1	2	yes	—
62	C₂×#88322	C₂×C₈	1	2	yes	—
63	C₂×#88322	C₂×C₈	1	2	yes	—
64	C₂×#88322	M₁₆	1	2	yes	—
65	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
66	C₂×C₄×C₅₃	—	2	2	no	—
67	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
68	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
69	C₂×C₄×C₅₃	#97	1	4	yes	—
70	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
76	C₂×Dic₅₃	—	2	2	no	—
77	C₂×Dic₅₃	—	2	2	no	—
78	C₂×Dic₅₃	#97	1	4	yes	—
79	C₂×Dic₅₃	C₄×D₄	1	2	yes	—
80	C₂×Dic₅₃	C₄×Q₈	1	2	yes	—
81	C₂×Dic₅₃	C₂×M₁₆	1	4	yes	—
82	C₂×Dic₅₃	#131	1	4	yes	—
83	#88324	—	4	8	no	—
84	C₈×C₅₃	—	2	2	no	—
85	C₈×C₅₃	C₄⋊C₈	1	4	yes	complemented
86	#88322	—	2	2	no	—
87	#88322	C₄⋊C₈	1	4	yes	complemented
88	#8260	—	53	1	no	maximal, Sylow
89	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
90	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
91	C₂²×C₅₃	C₂×#97	1	2	yes	—
92	C₂²×C₅₃	C₄×M₁₆	1	1	yes	characteristic
93	C₂²×C₅₃	#405	1	2	yes	—
94	C₄×C₅₃	—	2	2	no	—
95	C₄×C₅₃	#322	1	4	yes	complemented
96	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
97	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
98	C₄×C₅₃	#431	1	1	yes	characteristic
99	C₄×C₅₃	#431	1	1	yes	characteristic
100	Dic₅₃	—	2	4	no	—
101	Dic₅₃	#436	1	4	yes	—
102	Dic₅₃	#446	1	4	yes	—
103	C₄×C₄⋊C₈	—	53	2	no	—
104	C₄²×C₈	—	53	1	no	—
105	C₂×#322	—	53	2	no	—
106	#1273	—	53	2	no	—
107	C₂×C₅₃	C₂×#322	1	1	yes	characteristic
108	C₂×C₅₃	#1450	1	1	yes	characteristic
109	C₂×C₅₃	#1451	1	2	yes	—
110	C₂×C₅₃	C₄×C₄⋊C₈	1	1	yes	characteristic
111	C₂×C₅₃	#1770	1	2	yes	—
112	C₂×C₄⋊C₈	—	53	4	no	—
113	C₂×C₄×C₈	—	53	1	no	—
114	C₂×C₄×C₈	—	53	1	no	—
115	C₂×C₄×C₈	—	53	4	no	—
116	C₂×C₄×C₈	—	53	1	no	—
117	C₂×C₄×C₈	—	53	1	no	—
118	C₂×C₄×C₈	—	53	1	no	—
119	C₂×C₄×C₈	#6157	1	1	yes	characteristic
120	C₄³	—	53	1	no	—
121	#336	—	53	4	no	—
122	#322	—	106	4	no	—
123	C₅₃	#8260	1	1	yes	characteristic, complemented, Sylow
124	C₂²×C₈	—	53	1	no	—
125	C₂²×C₈	—	53	1	no	—
126	C₂²×C₈	—	53	8	no	—
127	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
128	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
129	C₂×C₄²	—	53	2	no	—
130	C₂×C₄²	—	53	1	no	—
131	C₂×C₄²	—	53	2	no	—
132	C₂×C₄²	—	53	1	no	—
133	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
134	C₄⋊C₈	—	106	8	no	—
135	C₄×C₈	—	53	2	no	—
136	C₄×C₈	—	53	2	no	—
137	C₄×C₈	—	53	4	no	—
138	C₄×C₈	—	106	2	no	—
139	C₄×C₈	—	106	4	no	—
140	C₄×C₈	—	106	4	no	—
141	C₄×C₈	—	106	2	no	—
142	C₄×C₈	—	106	2	no	—
143	C₄×C₈	#88324	1	4	yes	complemented
144	C₂²×C₄	—	53	2	no	—
145	C₂²×C₄	—	53	2	no	—
146	C₂²×C₄	C₄×#6157	1	1	yes	characteristic
147	C₂²×C₄	C₄×#6157	1	1	yes	characteristic
148	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	2	1	no	—
151	C₂×C₈	—	53	2	no	—
152	C₂×C₈	—	53	2	no	—
153	C₂×C₈	—	53	2	no	—
154	C₂×C₈	—	53	2	no	—
155	C₂×C₈	—	106	1	no	—
156	C₂×C₈	—	106	1	no	—
157	C₂×C₈	—	106	8	no	—
158	C₂×C₈	—	106	8	no	—
159	C₂×C₈	—	106	8	no	—
160	C₂×C₈	#11719936	1	2	yes	—
161	C₂×C₈	C₄⋊#6157	1	2	yes	—
162	C₂×C₈	C₂×#88324	1	2	yes	—
163	C₂×C₈	C₂×#88324	1	2	yes	—
164	C₄²	—	53	8	no	—
165	C₄²	—	53	1	no	—
166	C₄²	—	53	1	no	—
167	C₄²	—	53	2	no	—
168	C₄²	—	106	4	no	—
169	C₄²	—	106	1	no	—
170	C₄²	—	106	1	no	—
171	C₄²	D₅₃⋊C₈	1	2	yes	—
172	C₄²	C₂×#88324	1	2	yes	—
173	C₂³	C₂×C₄×#6157	1	1	yes	characteristic
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	53	2	no	—
176	C₂×C₄	—	53	2	no	—
177	C₂×C₄	—	53	4	no	—
178	C₂×C₄	—	53	4	no	—
179	C₂×C₄	—	53	4	no	—
180	C₂×C₄	—	106	2	no	—
181	C₂×C₄	—	106	2	no	—
182	C₂×C₄	C₄×#88324	1	2	yes	—
183	C₂×C₄	C₄×#88324	1	2	yes	—
184	C₂×C₄	#13133271	1	4	yes	—
185	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
186	C₂×C₄	#13133296	1	1	yes	characteristic
187	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
188	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
189	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
190	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
191	C₈	—	2	2	no	—
192	C₈	—	53	4	no	—
193	C₈	—	106	2	no	—
194	C₈	—	212	8	no	—
195	C₈	C₄⋊#88324	1	4	yes	complemented
196	C₂²	C₄×C₄⋊#6157	1	1	yes	characteristic
197	C₂²	C₂×C₄×#88324	1	1	yes	characteristic
198	C₂²	C₂×#13133271	1	2	yes	—
199	C₂²	C₄×#11719936	1	1	yes	characteristic
200	C₂²	#16811586	1	2	yes	—
201	C₄	—	2	2	no	—
202	C₄	—	53	4	no	—
203	C₄	—	53	4	no	—
204	C₄	—	106	4	no	—
205	C₄	#16811464	1	4	yes	complemented
206	C₄	C₂×C₄⋊#88324	1	1	yes	characteristic
207	C₄	D₅₃⋊(C₄⋊C₈)	1	1	yes	characteristic
208	C₄	#16811639	1	1	yes	characteristic
209	C₄	#16811642	1	1	yes	characteristic
210	C₂	C₂×#16811464	1	1	yes	characteristic
211	C₂	#20289918	1	1	yes	characteristic
212	C₂	#20289925	1	2	yes	—
213	C₂	C₄×C₄⋊#88324	1	1	yes	characteristic
214	C₂	#20290509	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#6157
commutator	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₂×C₄×C₈×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	C₂×C₄²
cosocle	C₂×C₄×Dic₅₃	C₂³
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	#8260
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
26. C₄×#88322	C₈	semidirect
35. C₄×C₈×C₅₃	C₈	semidirect
85. C₈×C₅₃	C₄⋊C₈	semidirect
87. #88322	C₄⋊C₈	semidirect
95. C₄×C₅₃	#322	semidirect
123. C₅₃	#8260	semidirect
143. C₄×C₈	#88324	semidirect
195. C₈	C₄⋊#88324	semidirect
205. C₄	#16811464	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊#88324	2
C₂×#16811464	2
#20289537	2
C₄×C₈×Dic₅₃	2
#8260	53