Subgroups — #28302046

Order 13568.

470 conjugacy classes of subgroups.

The subgroup lattice has 234 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×#88324⋊C₈	C₂	1	2	yes	maximal
3	#20289537	C₂	1	1	yes	maximal, characteristic
4	#20289537	C₂	1	1	yes	maximal, characteristic
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	#88324⋊C₈	—	2	4	no	—
11	C₂×#13133128	C₄	1	1	yes	characteristic
12	C₂×#13133127	C₄	1	1	yes	characteristic
13	C₂×#13133127	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
16	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
17	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
18	C₂²×#88324	C₂×C₄	1	4	yes	—
19	C₂²×#88324	C₂×C₄	1	4	yes	—
20	C₄⋊#88324	—	2	8	no	—
21	C₄×#88324	—	2	2	no	—
22	C₄×#88324	—	2	2	no	—
23	C₄×#88324	—	2	2	no	—
24	C₄×#88324	—	2	2	no	—
25	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
26	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
29	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
30	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
31	#13133128	—	2	2	no	—
32	#13133127	—	2	2	no	—
33	#13133127	—	2	2	no	—
34	C₈×Dic₅₃	—	2	2	no	—
35	C₈×Dic₅₃	D₄	1	2	yes	—
36	C₈×Dic₅₃	Q₈	1	2	yes	—
37	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
40	#97×C₅₃	—	2	2	no	—
41	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
42	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
44	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
45	C₂²×Dic₅₃	C₂²×C₄	1	2	yes	—
46	C₂×#88324	—	2	4	no	—
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	M₁₆	1	8	yes	—
52	C₂×C₈×C₅₃	—	2	1	no	—
53	C₂×C₈×C₅₃	—	2	1	no	—
54	C₂×C₈×C₅₃	—	2	1	no	—
55	C₂×C₈×C₅₃	—	2	1	no	—
56	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
57	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
58	C₄²×C₅₃	M₁₆	1	2	yes	—
59	C₄²×C₅₃	M₁₆	1	2	yes	—
60	C₄×Dic₅₃	—	2	1	no	—
61	C₄×Dic₅₃	—	2	1	no	—
62	C₄×Dic₅₃	—	2	4	no	—
63	C₄×Dic₅₃	—	2	4	no	—
64	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
65	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
66	C₄×Dic₅₃	#41	1	2	yes	—
67	C₂×#88322	—	2	1	no	—
68	C₂×#88322	—	2	1	no	—
69	C₂×#88322	—	2	1	no	—
70	C₂×#88322	—	2	1	no	—
71	C₂×#88322	C₄⋊C₄	1	2	yes	—
72	C₂×#88322	M₁₆	1	2	yes	—
73	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
74	C₂×C₄×C₅₃	—	2	2	no	—
75	C₂×C₄×C₅₃	—	2	2	no	—
76	C₂×C₄×C₅₃	#97	1	4	yes	—
77	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
83	C₂×Dic₅₃	—	2	2	no	—
84	C₂×Dic₅₃	—	2	1	no	—
85	C₂×Dic₅₃	—	2	1	no	—
86	C₂×Dic₅₃	—	2	2	no	—
87	C₂×Dic₅₃	C₄×D₄	1	2	yes	—
88	C₂×Dic₅₃	C₄×Q₈	1	2	yes	—
89	C₂×Dic₅₃	C₂×M₁₆	1	4	yes	—
90	C₂×Dic₅₃	#131	1	4	yes	—
91	#88324	—	2	8	no	—
92	#88324	—	4	4	no	—
93	C₈×C₅₃	—	2	2	no	—
94	C₈×C₅₃	—	2	2	no	—
95	#88322	—	2	2	no	—
96	#88322	—	2	2	no	—
97	#8263	—	53	1	no	maximal, Sylow
98	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
99	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
100	C₂²×C₅₃	C₄×M₁₆	1	2	yes	—
101	C₂²×C₅₃	C₄×M₁₆	1	1	yes	characteristic
102	C₂²×C₅₃	#405	1	2	yes	—
103	C₄×C₅₃	—	2	4	no	—
104	C₄×C₅₃	#423	1	1	yes	characteristic
105	C₄×C₅₃	#423	1	1	yes	characteristic
106	C₄×C₅₃	#432	1	1	yes	characteristic
107	C₄×C₅₃	#432	1	1	yes	characteristic
108	Dic₅₃	—	4	1	no	—
109	Dic₅₃	—	4	1	no	—
110	Dic₅₃	#435	1	4	yes	—
111	Dic₅₃	#446	1	4	yes	—
112	C₄×C₄⋊C₈	—	53	2	no	—
113	C₄×#97	—	53	1	no	—
114	C₂×#322	—	53	2	no	—
115	#1273	—	53	1	no	—
116	#1273	—	53	1	no	—
117	C₂×C₅₃	#1451	1	2	yes	—
118	C₂×C₅₃	#1453	1	1	yes	characteristic
119	C₂×C₅₃	#1455	1	1	yes	characteristic
120	C₂×C₅₃	#1767	1	1	yes	characteristic
121	C₂×C₅₃	#1771	1	2	yes	—
122	C₂×C₄⋊C₈	—	53	4	no	—
123	C₂×#97	—	53	1	no	—
124	C₂×#97	—	53	1	no	—
125	C₂×#97	—	53	1	no	—
126	C₂×#97	#6157	1	1	yes	characteristic
127	C₂×C₄×C₈	—	53	1	no	—
128	C₂×C₄×C₈	—	53	2	no	—
129	C₂×C₄×C₈	—	53	2	no	—
130	C₂×C₄×C₈	—	53	1	no	—
131	C₄³	—	53	1	no	—
132	#336	—	53	4	no	—
133	#322	—	106	4	no	—
134	C₅₃	#8263	1	1	yes	characteristic, complemented, Sylow
135	C₂²×C₈	—	53	4	no	—
136	C₂²×C₈	—	53	1	no	—
137	C₂²×C₈	—	53	1	no	—
138	C₂²×C₈	—	53	4	no	—
139	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
140	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
141	C₂×C₄²	—	53	1	no	—
142	C₂×C₄²	—	53	2	no	—
143	C₂×C₄²	—	53	2	no	—
144	C₂×C₄²	—	53	1	no	—
145	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
146	C₄⋊C₈	—	106	8	no	—
147	#97	—	2	2	no	—
148	#97	—	106	2	no	—
149	#97	—	106	2	no	—
150	#97	—	106	2	no	—
151	C₄×C₈	—	53	2	no	—
152	C₄×C₈	—	53	2	no	—
153	C₄×C₈	—	106	2	no	—
154	C₄×C₈	—	106	2	no	—
155	C₄×C₈	—	106	2	no	—
156	C₄×C₈	—	106	2	no	—
157	C₄×C₈	—	106	2	no	—
158	C₂²×C₄	—	53	2	no	—
159	C₂²×C₄	—	53	1	no	—
160	C₂²×C₄	—	53	1	no	—
161	C₂²×C₄	C₄×#6157	1	2	yes	—
162	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	53	8	no	—
168	C₂×C₈	—	53	2	no	—
169	C₂×C₈	—	53	2	no	—
170	C₂×C₈	—	106	4	no	—
171	C₂×C₈	—	106	4	no	—
172	C₂×C₈	—	106	1	no	—
173	C₂×C₈	—	106	1	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	C₄²	—	53	1	no	—
182	C₄²	—	53	1	no	—
183	C₄²	—	53	2	no	—
184	C₄²	—	106	4	no	—
185	C₄²	—	106	4	no	—
186	C₄²	—	106	1	no	—
187	C₄²	—	106	1	no	—
188	C₄²	#11719936	1	2	yes	—
189	C₄²	#11719940	1	2	yes	—
190	C₂³	C₂×C₄×#6157	1	1	yes	characteristic
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	53	2	no	—
194	C₂×C₄	—	53	2	no	—
195	C₂×C₄	—	53	4	no	—
196	C₂×C₄	—	53	4	no	—
197	C₂×C₄	—	106	2	no	—
198	C₂×C₄	—	106	1	no	—
199	C₂×C₄	—	106	1	no	—
200	C₂×C₄	—	106	2	no	—
201	C₂×C₄	#13133272	1	4	yes	—
202	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
203	C₂×C₄	#13133296	1	1	yes	characteristic
204	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
205	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
206	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
207	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
208	C₈	—	2	2	no	—
209	C₈	—	2	2	no	—
210	C₈	—	106	8	no	—
211	C₈	—	106	2	no	—
212	C₈	—	106	2	no	—
213	C₈	—	212	4	no	—
214	C₂²	C₄×C₄⋊#6157	1	1	yes	characteristic
215	C₂²	C₂×#13133272	1	1	yes	characteristic
216	C₂²	M₁₆×#6157	1	2	yes	—
217	C₂²	C₄×#11719936	1	1	yes	characteristic
218	C₂²	#16811589	1	2	yes	—
219	C₄	—	2	4	no	—
220	C₄	—	53	4	no	—
221	C₄	—	53	4	no	—
222	C₄	—	212	1	no	—
223	C₄	—	212	1	no	—
224	C₄	#16811626	1	1	yes	characteristic
225	C₄	#16811627	1	1	yes	characteristic
226	C₄	#16811644	1	1	yes	characteristic
227	C₄	#16811645	1	1	yes	characteristic
228	C₂	#20289922	1	2	yes	—
229	C₂	#20289935	1	1	yes	characteristic
230	C₂	#20289939	1	1	yes	characteristic
231	C₂	#20290501	1	1	yes	characteristic
232	C₂	#20290513	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#6157
commutator, socle, prime radical	C₂³×C₅₃	C₂×C₄²
Fitting	C₂×#97×C₅₃	C₄
cosocle	C₂×C₄×Dic₅₃	C₂³
hypercenter	C₂×#97	#6157
hypocenter	C₅₃	#8263
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
134. C₅₃	#8263	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×#88324⋊C₈	2
#20289537	2
#97×Dic₅₃	2
#8263	53