Subgroups — #28310458

Order 13568.

614 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811518	C₂	1	2	yes	maximal
2	C₂×#16811509	C₂	1	2	yes	maximal, complemented
3	#20289753	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719910	C₂²	1	4	yes	—
6	#16811518	C₄	1	8	yes	complemented
7	#16811509	—	2	4	no	—
8	C₂²×C₈×D₅₃	C₄	1	1	yes	characteristic
9	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₄²×D₅₃	C₄	1	1	yes	characteristic
11	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
12	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
14	C₄×C₈×D₅₃	—	2	4	no	—
15	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
16	C₂×#11719910	—	2	4	no	—
17	C₂×#11719910	—	2	4	no	—
18	C₂×#11719910	C₂×C₄	1	8	yes	complemented
19	C₂²×C₄×D₅₃	C₈	1	2	yes	—
20	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
22	C₂²×#88322	C₂³	1	1	yes	characteristic
23	C₂×C₈×D₅₃	—	2	2	no	—
24	C₂×C₈×D₅₃	—	2	2	no	—
25	C₂×C₈×D₅₃	—	2	2	no	—
26	C₂×C₈×D₅₃	—	2	2	no	—
27	C₂×C₈×D₅₃	C₈	1	4	yes	—
28	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
29	C₂×C₄×Dic₅₃	C₈	1	2	yes	—
30	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
31	C₄²×D₅₃	—	2	2	no	—
32	C₄²×D₅₃	—	2	2	no	—
33	C₄×#88322	—	2	2	no	—
34	C₈×Dic₅₃	—	2	2	no	—
35	C₈×Dic₅₃	D₄	1	4	yes	—
36	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₅₃	—	2	2	no	—
40	C₂³×D₅₃	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
42	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
44	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
45	C₂×C₄×D₅₃	—	2	1	no	—
46	C₂×C₄×D₅₃	—	2	1	no	—
47	C₂×C₄×D₅₃	—	2	1	no	—
48	C₂×C₄×D₅₃	—	2	1	no	—
49	C₂×C₄×D₅₃	—	2	4	no	—
50	C₂×C₄×D₅₃	—	2	4	no	—
51	C₂×C₄×D₅₃	C₁₆	1	8	yes	complemented
52	C₂×C₄×D₅₃	C₄²	1	2	yes	—
53	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
54	C₂×C₈×C₅₃	—	2	1	no	—
55	C₂×C₈×C₅₃	—	2	1	no	—
56	C₂×C₈×C₅₃	C₄²	1	2	yes	—
57	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
58	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
59	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
60	C₄²×C₅₃	—	2	1	no	—
61	C₄²×C₅₃	—	2	1	no	—
62	C₄×Dic₅₃	—	2	4	no	—
63	C₄×Dic₅₃	C₂²⋊C₄	1	2	yes	—
64	C₄×Dic₅₃	M₁₆	1	2	yes	—
65	C₂×#88322	—	2	1	no	—
66	C₂×#88322	—	2	1	no	—
67	C₂×#88322	C₂×C₈	1	2	yes	—
68	C₂×#88322	C₂²×C₄	1	2	yes	—
69	C₂×#88322	C₂×D₄	1	2	yes	—
70	C₂×#88322	#41	1	2	yes	—
71	C₈×D₅₃	—	2	8	no	—
72	C₈×D₅₃	—	2	8	no	—
73	#11719910	—	2	8	no	—
74	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
75	C₂²×D₅₃	—	2	1	no	—
76	C₂²×D₅₃	—	2	1	no	—
77	C₂²×D₅₃	—	2	2	no	—
78	C₂²×D₅₃	C₄×C₈	1	2	yes	—
79	C₂²×D₅₃	C₂×C₁₆	1	4	yes	—
80	C₂×C₄×C₅₃	—	2	2	no	—
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	C₂×C₁₆	1	4	yes	—
83	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
89	C₂×Dic₅₃	—	2	1	no	—
90	C₂×Dic₅₃	—	2	1	no	—
91	C₂×Dic₅₃	C₄×C₈	1	2	yes	—
92	C₂×Dic₅₃	C₂²⋊C₈	1	4	yes	—
93	C₂×Dic₅₃	C₂×C₁₆	1	4	yes	—
94	C₂×Dic₅₃	M₃₂	1	4	yes	—
95	C₄×D₅₃	—	2	2	no	—
96	C₄×D₅₃	—	2	2	no	—
97	C₄×D₅₃	—	2	2	no	—
98	C₄×D₅₃	—	2	2	no	—
99	C₄×D₅₃	—	2	16	no	—
100	C₈×C₅₃	—	2	2	no	—
101	C₈×C₅₃	C₂²⋊C₈	1	4	yes	—
102	#88322	—	2	2	no	—
103	#88322	C₄×D₄	1	4	yes	—
104	C₄×C₂²⋊C₁₆	—	53	1	no	maximal, Sylow
105	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
106	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
107	C₂²×C₅₃	#431	1	1	yes	characteristic
108	C₂²×C₅₃	C₂²×C₁₆	1	2	yes	—
109	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
110	C₂×D₅₃	—	2	2	no	—
111	C₂×D₅₃	—	2	4	no	—
112	C₂×D₅₃	—	2	4	no	—
113	C₂×D₅₃	—	2	2	no	—
114	C₂×D₅₃	C₄×C₁₆	1	4	yes	complemented
115	C₄×C₅₃	—	2	4	no	—
116	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
117	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
118	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
119	C₄×C₅₃	#431	1	1	yes	characteristic
120	Dic₅₃	—	2	2	no	—
121	Dic₅₃	C₂²⋊C₁₆	1	8	yes	complemented
122	C₂²×C₄×C₈	—	53	1	no	—
123	C₂×C₂²⋊C₁₆	—	53	2	no	—
124	C₂×C₄×C₁₆	—	53	2	no	—
125	#1374	—	53	2	no	—
126	C₂×C₅₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
127	C₂×C₅₃	C₂×C₄×C₁₆	1	1	yes	characteristic
128	C₂×C₅₃	C₄×M₃₂	1	1	yes	characteristic
129	C₂×C₅₃	C₂×C₂²⋊C₁₆	1	2	yes	—
130	C₂×C₅₃	#2162	1	2	yes	—
131	D₅₃	—	2	4	no	—
132	C₂³×C₈	—	53	1	no	—
133	C₂³×C₈	—	53	1	no	—
134	C₂²×C₄²	—	53	1	no	—
135	C₂²×C₁₆	—	53	4	no	—
136	C₂×C₄×C₈	—	53	1	no	—
137	C₂×C₄×C₈	—	53	1	no	—
138	C₂×C₄×C₈	—	53	1	no	—
139	C₂×C₄×C₈	—	106	4	no	—
140	C₂×C₄×C₈	#6157	1	1	yes	characteristic
141	C₂²⋊C₁₆	—	53	8	no	—
142	C₄×C₁₆	—	106	4	no	—
143	C₅₃	C₄×C₂²⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
144	C₂³×C₄	—	53	1	no	—
145	C₂³×C₄	—	53	2	no	—
146	C₂²×C₈	—	53	1	no	—
147	C₂²×C₈	—	53	1	no	—
148	C₂²×C₈	—	53	4	no	—
149	C₂²×C₈	—	53	4	no	—
150	C₂²×C₈	—	106	2	no	—
151	C₂²×C₈	—	106	2	no	—
152	C₂²×C₈	—	106	2	no	—
153	C₂²×C₈	—	106	2	no	—
154	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
155	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
156	C₂×C₄²	—	53	1	no	—
157	C₂×C₄²	—	53	2	no	—
158	C₂×C₄²	—	106	2	no	—
159	C₂×C₄²	—	106	2	no	—
160	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
161	C₂×C₁₆	—	53	8	no	—
162	C₂×C₁₆	—	106	4	no	—
163	C₂×C₁₆	—	106	4	no	—
164	C₄×C₈	—	2	2	no	—
165	C₄×C₈	—	53	4	no	—
166	C₄×C₈	—	106	2	no	—
167	C₄×C₈	—	106	2	no	—
168	C₂⁴	—	53	1	no	—
169	C₂²×C₄	—	53	2	no	—
170	C₂²×C₄	—	53	1	no	—
171	C₂²×C₄	—	53	8	no	—
172	C₂²×C₄	—	53	2	no	—
173	C₂²×C₄	—	53	2	no	—
174	C₂²×C₄	—	106	4	no	—
175	C₂²×C₄	—	106	4	no	—
176	C₂²×C₄	—	106	1	no	—
177	C₂²×C₄	—	106	1	no	—
178	C₂²×C₄	—	106	1	no	—
179	C₂²×C₄	—	106	1	no	—
180	C₂²×C₄	D₅₃⋊C₈	1	2	yes	—
181	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
182	C₂×C₈	—	2	1	no	—
183	C₂×C₈	—	2	1	no	—
184	C₂×C₈	—	53	2	no	—
185	C₂×C₈	—	53	2	no	—
186	C₂×C₈	—	53	2	no	—
187	C₂×C₈	—	53	2	no	—
188	C₂×C₈	—	106	8	no	—
189	C₂×C₈	—	106	1	no	—
190	C₂×C₈	—	106	1	no	—
191	C₂×C₈	—	106	8	no	—
192	C₂×C₈	D₅₃⋊C₈	1	2	yes	—
193	C₂×C₈	#11719936	1	2	yes	—
194	C₂×C₈	C₄×#6157	1	2	yes	—
195	C₂×C₈	C₂²⋊#6157	1	2	yes	—
196	C₄²	—	2	1	no	—
197	C₄²	—	2	1	no	—
198	C₄²	—	53	2	no	—
199	C₄²	—	53	2	no	—
200	C₄²	—	106	4	no	—
201	C₁₆	—	106	8	no	—
202	C₂³	—	53	2	no	—
203	C₂³	—	53	4	no	—
204	C₂³	—	106	1	no	—
205	C₂³	—	106	1	no	—
206	C₂³	—	106	2	no	—
207	C₂³	C₂×D₅₃⋊C₈	1	1	yes	characteristic
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	53	4	no	—
211	C₂×C₄	—	53	4	no	—
212	C₂×C₄	—	53	4	no	—
213	C₂×C₄	—	53	2	no	—
214	C₂×C₄	—	106	16	no	—
215	C₂×C₄	—	106	1	no	—
216	C₂×C₄	—	106	1	no	—
217	C₂×C₄	—	106	2	no	—
218	C₂×C₄	—	106	2	no	—
219	C₂×C₄	—	106	2	no	—
220	C₂×C₄	—	106	2	no	—
221	C₂×C₄	D₅₃⋊C₁₆	1	4	yes	—
222	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
223	C₂×C₄	C₂×C₂²⋊#6157	1	1	yes	characteristic
224	C₂×C₄	#13133296	1	1	yes	characteristic
225	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
226	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
227	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
228	C₈	—	2	2	no	—
229	C₈	—	53	4	no	—
230	C₈	—	106	2	no	—
231	C₈	#13133274	1	4	yes	—
232	C₂²	—	53	4	no	—
233	C₂²	—	106	4	no	—
234	C₂²	—	106	2	no	—
235	C₂²	—	106	4	no	—
236	C₂²	—	106	2	no	—
237	C₂²	C₄×D₅₃⋊C₈	1	1	yes	characteristic
238	C₂²	D₅₃⋊(C₂²⋊C₈)	1	1	yes	characteristic
239	C₂²	#16811642	1	1	yes	characteristic
240	C₂²	C₂×D₅₃⋊C₁₆	1	2	yes	—
241	C₂²	D₅₃⋊M₃₂	1	2	yes	—
242	C₄	—	2	4	no	—
243	C₄	—	53	8	no	—
244	C₄	—	106	2	no	—
245	C₄	C₄×C₂²⋊#6157	1	1	yes	characteristic
246	C₄	C₄×#11719936	1	1	yes	characteristic
247	C₄	C₂×#13133274	1	1	yes	characteristic
248	C₄	#16811641	1	1	yes	characteristic
249	C₂	—	106	4	no	—
250	C₂	C₄×#13133274	1	1	yes	characteristic
251	C₂	C₄×D₅₃⋊C₁₆	1	1	yes	characteristic
252	C₂	#20290922	1	1	yes	characteristic
253	C₂	D₅₃⋊(C₂²⋊C₁₆)	1	2	yes	—
254	C₂	#20291075	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂×#6157
commutator	C₂×C₅₃	C₂×C₄×C₁₆
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₂×C₄×C₈×C₅₃	C₄
socle	C₂³×C₅₃	C₂²×C₈
prime radical	C₂³×D₅₃	C₂×C₈
cosocle	C₂²×#88322	C₂³
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	C₄×C₂²⋊C₁₆
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#16811509	C₂	semidirect
3. #20289753	C₂	semidirect
6. #16811518	C₄	semidirect
18. C₂×#11719910	C₂×C₄	semidirect
51. C₂×C₄×D₅₃	C₁₆	semidirect
114. C₂×D₅₃	C₄×C₁₆	semidirect
121. Dic₅₃	C₂²⋊C₁₆	semidirect
143. C₅₃	C₄×C₂²⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811518	2
C₂×#16811509	2
#20289753	2
C₂×C₄×C₈×D₅₃	2
C₄×C₂²⋊C₁₆	53