Subgroups — #28302040

Order 13568.

494 conjugacy classes of subgroups.

The subgroup lattice has 236 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	1	yes	maximal, characteristic
3	C₂×C₈×#88324	C₂	1	1	yes	maximal, characteristic
4	#20289537	C₂	1	2	yes	maximal
5	#97×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88324	C₄	1	4	yes	—
7	C₂×C₄×#88324	C₂²	1	2	yes	—
8	C₂×C₄×#88324	C₂²	1	2	yes	—
9	#16811489	C₄	1	4	yes	—
10	#16811464	—	2	2	no	—
11	C₈×#88324	—	2	2	no	—
12	C₂×#13133128	C₄	1	1	yes	characteristic
13	C₂×#13133127	C₄	1	1	yes	characteristic
14	C₂×#13133127	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
17	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
18	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
19	C₂²×#88324	C₂×C₄	1	4	yes	—
20	C₂²×#88324	C₂×C₄	1	4	yes	—
21	C₄⋊#88324	—	2	8	no	—
22	C₄×#88324	—	2	2	no	—
23	C₄×#88324	—	2	2	no	—
24	C₄×#88324	—	2	2	no	—
25	C₄×#88324	—	2	2	no	—
26	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
27	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
30	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
31	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
32	#13133128	C₈	1	4	yes	complemented
33	#13133127	—	2	2	no	—
34	#13133127	—	2	2	no	—
35	C₈×Dic₅₃	—	2	2	no	—
36	C₈×Dic₅₃	D₄	1	2	yes	—
37	C₈×Dic₅₃	Q₈	1	2	yes	—
38	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
41	#97×C₅₃	C₈	1	4	yes	complemented
42	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
43	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
45	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
46	C₂²×Dic₅₃	C₂²×C₄	1	2	yes	—
47	C₂×#88324	—	2	4	no	—
48	C₂×#88324	—	2	4	no	—
49	C₂×#88324	—	2	4	no	—
50	C₂×#88324	—	2	4	no	—
51	C₂×#88324	—	2	4	no	—
52	C₂×#88324	M₁₆	1	8	yes	—
53	C₂×C₈×C₅₃	—	2	1	no	—
54	C₂×C₈×C₅₃	—	2	1	no	—
55	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
56	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
57	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
58	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
59	C₄²×C₅₃	C₂×C₈	1	2	yes	—
60	C₄²×C₅₃	C₂×C₈	1	2	yes	—
61	C₄×Dic₅₃	—	2	1	no	—
62	C₄×Dic₅₃	—	2	1	no	—
63	C₄×Dic₅₃	—	2	4	no	—
64	C₄×Dic₅₃	—	2	4	no	—
65	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
66	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
67	C₄×Dic₅₃	#41	1	2	yes	—
68	C₂×#88322	—	2	1	no	—
69	C₂×#88322	—	2	1	no	—
70	C₂×#88322	C₄⋊C₄	1	2	yes	—
71	C₂×#88322	C₂×C₈	1	2	yes	—
72	C₂×#88322	C₂×C₈	1	2	yes	—
73	C₂×#88322	M₁₆	1	2	yes	—
74	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
75	C₂×C₄×C₅₃	—	2	2	no	—
76	C₂×C₄×C₅₃	—	2	2	no	—
77	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
78	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
84	C₂×Dic₅₃	—	2	2	no	—
85	C₂×Dic₅₃	—	2	1	no	—
86	C₂×Dic₅₃	—	2	1	no	—
87	C₂×Dic₅₃	—	2	2	no	—
88	C₂×Dic₅₃	C₄×D₄	1	2	yes	—
89	C₂×Dic₅₃	C₄×Q₈	1	2	yes	—
90	C₂×Dic₅₃	C₂×M₁₆	1	4	yes	—
91	C₂×Dic₅₃	#131	1	4	yes	—
92	#88324	—	2	8	no	—
93	#88324	—	4	4	no	—
94	C₈×C₅₃	—	2	2	no	—
95	C₈×C₅₃	C₄⋊C₈	1	4	yes	complemented
96	#88322	—	2	2	no	—
97	#88322	C₄⋊C₈	1	4	yes	complemented
98	#8262	—	53	1	no	maximal, Sylow
99	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
100	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
101	C₂²×C₅₃	C₄×M₁₆	1	2	yes	—
102	C₂²×C₅₃	C₄×M₁₆	1	1	yes	characteristic
103	C₂²×C₅₃	#405	1	2	yes	—
104	C₄×C₅₃	—	2	4	no	—
105	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
106	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
107	C₄×C₅₃	#431	1	1	yes	characteristic
108	C₄×C₅₃	#431	1	1	yes	characteristic
109	Dic₅₃	—	4	1	no	—
110	Dic₅₃	—	4	1	no	—
111	Dic₅₃	#435	1	4	yes	—
112	Dic₅₃	#446	1	4	yes	—
113	C₄×C₄⋊C₈	—	53	2	no	—
114	C₄×#97	—	53	1	no	—
115	C₂×#322	—	53	1	no	—
116	C₂×C₈²	—	53	1	no	—
117	#1273	—	53	2	no	—
118	C₂×C₅₃	C₈×M₁₆	1	2	yes	—
119	C₂×C₅₃	#1453	1	1	yes	characteristic
120	C₂×C₅₃	#1455	1	1	yes	characteristic
121	C₂×C₅₃	C₄×C₄⋊C₈	1	1	yes	characteristic
122	C₂×C₅₃	#1771	1	2	yes	—
123	C₂×C₄⋊C₈	—	53	4	no	—
124	C₂×#97	—	53	1	no	—
125	C₂×#97	—	53	1	no	—
126	C₂×#97	—	53	1	no	—
127	C₂×#97	#6157	1	1	yes	characteristic
128	C₂×C₄×C₈	—	53	1	no	—
129	C₂×C₄×C₈	—	53	2	no	—
130	C₂×C₄×C₈	—	53	2	no	—
131	C₂×C₄×C₈	—	53	1	no	—
132	C₄³	—	53	1	no	—
133	#336	—	53	4	no	—
134	#322	—	106	2	no	—
135	C₈²	—	106	2	no	—
136	C₅₃	#8262	1	1	yes	characteristic, complemented, Sylow
137	C₂²×C₈	—	53	1	no	—
138	C₂²×C₈	—	53	4	no	—
139	C₂²×C₈	—	53	1	no	—
140	C₂²×C₈	—	53	4	no	—
141	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
142	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
143	C₂×C₄²	—	53	1	no	—
144	C₂×C₄²	—	53	2	no	—
145	C₂×C₄²	—	53	2	no	—
146	C₂×C₄²	—	53	1	no	—
147	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
148	C₄⋊C₈	—	106	8	no	—
149	#97	—	53	4	no	—
150	#97	—	106	2	no	—
151	#97	—	106	2	no	—
152	#97	#88324	1	4	yes	complemented
153	C₄×C₈	—	53	2	no	—
154	C₄×C₈	—	53	2	no	—
155	C₄×C₈	—	106	2	no	—
156	C₄×C₈	—	106	2	no	—
157	C₄×C₈	—	106	2	no	—
158	C₄×C₈	—	106	2	no	—
159	C₄×C₈	—	106	2	no	—
160	C₂²×C₄	—	53	2	no	—
161	C₂²×C₄	—	53	1	no	—
162	C₂²×C₄	—	53	1	no	—
163	C₂²×C₄	C₄×#6157	1	2	yes	—
164	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	53	2	no	—
168	C₂×C₈	—	53	2	no	—
169	C₂×C₈	—	53	2	no	—
170	C₂×C₈	—	53	8	no	—
171	C₂×C₈	—	53	2	no	—
172	C₂×C₈	—	106	4	no	—
173	C₂×C₈	—	106	4	no	—
174	C₂×C₈	—	106	1	no	—
175	C₂×C₈	—	106	1	no	—
176	C₂×C₈	—	106	4	no	—
177	C₂×C₈	—	106	4	no	—
178	C₂×C₈	—	106	4	no	—
179	C₂×C₈	#11719936	1	2	yes	—
180	C₂×C₈	C₄⋊#6157	1	2	yes	—
181	C₂×C₈	C₂×#88324	1	2	yes	—
182	C₂×C₈	C₂×#88324	1	2	yes	—
183	C₄²	—	53	1	no	—
184	C₄²	—	53	1	no	—
185	C₄²	—	53	2	no	—
186	C₄²	—	106	4	no	—
187	C₄²	—	106	4	no	—
188	C₄²	—	106	1	no	—
189	C₄²	—	106	1	no	—
190	C₄²	D₅₃⋊C₈	1	2	yes	—
191	C₄²	C₂×#88324	1	2	yes	—
192	C₂³	C₂×C₄×#6157	1	1	yes	characteristic
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	53	2	no	—
196	C₂×C₄	—	53	2	no	—
197	C₂×C₄	—	53	4	no	—
198	C₂×C₄	—	53	4	no	—
199	C₂×C₄	—	106	2	no	—
200	C₂×C₄	—	106	1	no	—
201	C₂×C₄	—	106	1	no	—
202	C₂×C₄	—	106	2	no	—
203	C₂×C₄	C₄×#88324	1	4	yes	—
204	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
205	C₂×C₄	#13133296	1	1	yes	characteristic
206	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
207	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
208	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
209	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
210	C₈	—	2	2	no	—
211	C₈	—	53	4	no	—
212	C₈	—	106	8	no	—
213	C₈	—	106	2	no	—
214	C₈	—	212	4	no	—
215	C₈	C₄⋊#88324	1	4	yes	complemented
216	C₂²	C₄×C₄⋊#6157	1	1	yes	characteristic
217	C₂²	C₂×C₄×#88324	1	1	yes	characteristic
218	C₂²	M₁₆×#6157	1	2	yes	—
219	C₂²	C₄×#11719936	1	1	yes	characteristic
220	C₂²	#16811589	1	2	yes	—
221	C₄	—	2	4	no	—
222	C₄	—	53	4	no	—
223	C₄	—	53	4	no	—
224	C₄	—	212	1	no	—
225	C₄	—	212	1	no	—
226	C₄	C₂×C₄⋊#88324	1	1	yes	characteristic
227	C₄	D₅₃⋊(C₄⋊C₈)	1	1	yes	characteristic
228	C₄	#16811639	1	1	yes	characteristic
229	C₄	#16811642	1	1	yes	characteristic
230	C₂	M₁₆×#88324	1	2	yes	—
231	C₂	#20289935	1	1	yes	characteristic
232	C₂	#20289939	1	1	yes	characteristic
233	C₂	C₄×C₄⋊#88324	1	1	yes	characteristic
234	C₂	#20290513	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#6157
commutator	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₂×#97×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	C₂×C₄²
cosocle	C₂×C₄×Dic₅₃	C₂³
hypercenter	C₂×#97	#6157
hypocenter	C₅₃	#8262
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
32. #13133128	C₈	semidirect
41. #97×C₅₃	C₈	semidirect
95. C₈×C₅₃	C₄⋊C₈	semidirect
97. #88322	C₄⋊C₈	semidirect
136. C₅₃	#8262	semidirect
152. #97	#88324	semidirect
215. C₈	C₄⋊#88324	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
C₂×C₈×#88324	2
#20289537	2
#97×Dic₅₃	2
#8262	53