Subgroups — #28328228

Order 13568.

682 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291076	C₂	1	4	yes	maximal, complemented
2	#20291072	C₂	1	4	yes	maximal, complemented
3	#20291047	C₂	1	2	yes	maximal, complemented
4	#20290942	C₂	1	2	yes	maximal
5	C₄×#13133291	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×#88322	C₂	1	1	yes	maximal, characteristic
7	C₂×#13133291	C₂²	1	4	yes	—
8	C₄⋊#11719910	C₂²	1	8	yes	complemented
9	C₂²⋊#11719910	C₂²	1	8	yes	—
10	#16811511	C₂²	1	4	yes	complemented
11	C₄×#11719910	C₂²	1	4	yes	complemented
12	C₂³×#88322	C₄	1	1	yes	characteristic
13	C₂³×#88322	C₂²	1	1	yes	characteristic
14	C₂×C₄×#88322	C₄	1	2	yes	—
15	C₂×C₄×#88322	C₄	1	1	yes	characteristic
16	C₂×C₄×#88322	C₄	1	2	yes	—
17	C₂×C₄×#88322	C₄	1	1	yes	characteristic
18	C₂×C₄×#88322	C₂²	1	2	yes	—
19	C₂×C₄×#88322	C₂²	1	2	yes	—
20	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
21	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
22	C₂²×C₄²×C₅₃	C₄	1	1	yes	characteristic
23	#13133291	—	2	8	no	—
24	C₂×#11719910	C₂³	1	8	yes	—
25	C₂²×#88322	—	2	4	no	—
26	C₂²×#88322	—	2	4	no	—
27	C₂²×#88322	C₂×C₄	1	2	yes	—
28	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
29	C₂²×#88322	C₂×C₄	1	2	yes	—
30	C₂²×#88322	C₂×C₄	1	2	yes	—
31	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
32	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
33	C₂²×#88322	C₂³	1	2	yes	—
34	C₂²×#88322	C₂³	1	1	yes	characteristic
35	C₄×#88322	C₂×C₄	1	2	yes	—
36	C₄×#88322	C₂×C₄	1	2	yes	—
37	C₄×#88322	C₂×C₄	1	2	yes	—
38	C₄×#88322	C₂×C₄	1	2	yes	—
39	C₄×#88322	C₂×C₄	1	1	yes	characteristic
40	C₄×#88322	C₂×C₄	1	2	yes	—
41	C₄×#88322	C₂×C₄	1	1	yes	characteristic
42	C₄×#88322	C₂³	1	2	yes	—
43	C₄×#88322	C₂³	1	1	yes	characteristic
44	C₄×#88322	C₂³	1	1	yes	characteristic
45	C₂³×C₄×C₅₃	C₈	1	2	yes	—
46	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₅₃	C₈	1	2	yes	—
48	C₂×C₄²×C₅₃	C₈	1	2	yes	—
49	C₂×C₄²×C₅₃	C₈	1	2	yes	—
50	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
54	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	—	2	8	no	—
56	C₂²×C₄×C₅₃	—	2	2	no	—
57	C₂²×C₄×C₅₃	—	2	2	no	—
58	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
60	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
61	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
62	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
63	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
64	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
65	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
66	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₄²×C₅₃	C₂×C₈	1	2	yes	—
69	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
70	C₄²×C₅₃	C₂×C₈	1	2	yes	—
71	C₄²×C₅₃	C₂×C₈	1	2	yes	—
72	C₄²×C₅₃	C₂×C₈	1	2	yes	—
73	C₄²×C₅₃	C₂×C₈	1	2	yes	—
74	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
75	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
77	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×#88322	—	2	2	no	—
79	C₂×#88322	—	2	2	no	—
80	C₂×#88322	—	2	4	no	—
81	C₂×#88322	—	2	4	no	—
82	C₂×#88322	—	2	4	no	—
83	C₂×#88322	—	2	4	no	—
84	C₂×#88322	M₁₆	1	4	yes	—
85	C₂×#88322	C₂²×C₄	1	2	yes	—
86	C₂×#88322	C₂²×C₄	1	2	yes	—
87	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
88	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
89	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
90	C₂×#88322	#41	1	4	yes	—
91	C₂×#88322	C₂⁴	1	1	yes	characteristic
92	#11719910	—	2	8	no	—
93	C₂³×C₅₃	—	2	2	no	—
94	C₂³×C₅₃	—	2	4	no	—
95	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
96	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
97	C₂×C₄×C₅₃	—	2	2	no	—
98	C₂×C₄×C₅₃	—	2	4	no	—
99	C₂×C₄×C₅₃	—	2	2	no	—
100	C₂×C₄×C₅₃	—	2	2	no	—
101	C₂×C₄×C₅₃	—	2	4	no	—
102	C₂×C₄×C₅₃	—	2	4	no	—
103	C₂×C₄×C₅₃	—	2	4	no	—
104	C₂×C₄×C₅₃	—	2	4	no	—
105	C₂×C₄×C₅₃	—	2	4	no	—
106	C₂×C₄×C₅₃	M₃₂	1	8	yes	complemented
107	C₂×C₄×C₅₃	#117	1	2	yes	—
108	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
109	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
110	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
111	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
113	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
114	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
115	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
116	#88322	—	2	2	no	—
117	#88322	—	2	2	no	—
118	#88322	#117	1	2	yes	—
119	#88322	C₂×M₁₆	1	2	yes	—
120	#88322	C₂×M₁₆	1	2	yes	—
121	#88322	C₂×#41	1	2	yes	—
122	#33418	—	53	1	no	maximal, Sylow
123	C₂²×C₅₃	—	2	8	no	—
124	C₂²×C₅₃	—	2	2	no	—
125	C₂²×C₅₃	—	2	2	no	—
126	C₂²×C₅₃	—	2	2	no	—
127	C₂²×C₅₃	#431	1	2	yes	—
128	C₂²×C₅₃	C₂×M₃₂	1	4	yes	—
129	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
130	C₄×C₅₃	—	2	2	no	—
131	C₄×C₅₃	—	2	2	no	—
132	C₄×C₅₃	—	2	2	no	—
133	C₄×C₅₃	#431	1	2	yes	—
134	C₄×C₅₃	C₂×M₃₂	1	4	yes	—
135	C₄×C₅₃	C₂×M₃₂	1	4	yes	—
136	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
137	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
138	C₂²×C₄×C₈	—	53	1	no	—
139	#2163	—	53	4	no	—
140	#2162	—	53	4	no	—
141	#2150	—	53	2	no	—
142	#2112	—	53	2	no	—
143	C₄×M₃₂	—	53	2	no	—
144	C₂×C₅₃	—	2	2	no	—
145	C₂×C₅₃	—	2	4	no	—
146	C₂×C₅₃	C₂×#431	1	1	yes	characteristic
147	C₂×C₅₃	C₂²×M₃₂	1	2	yes	—
148	C₂³×C₈	—	53	1	no	—
149	C₂³×C₈	—	53	1	no	—
150	C₂²×C₄²	#6157	1	1	yes	characteristic
151	C₂×M₃₂	—	53	4	no	—
152	C₂×C₄×C₈	—	53	1	no	—
153	C₂×C₄×C₈	—	53	1	no	—
154	C₂×C₄×C₈	—	53	2	no	—
155	C₂×C₄×C₈	—	53	2	no	—
156	C₂×C₄×C₈	—	53	1	no	—
157	C₂×C₄×C₈	—	53	1	no	—
158	C₂×C₄×C₈	—	53	2	no	—
159	C₂×C₄×C₈	—	53	2	no	—
160	C₄⋊C₁₆	—	53	8	no	—
161	C₂²⋊C₁₆	—	53	8	no	—
162	#346	—	53	4	no	—
163	C₄×C₁₆	—	53	4	no	—
164	C₅₃	#33418	1	1	yes	characteristic, complemented, Sylow
165	C₂³×C₄	#88324	1	2	yes	—
166	C₂³×C₄	C₂×#6157	1	1	yes	characteristic
167	C₂²×C₈	—	53	1	no	—
168	C₂²×C₈	—	53	1	no	—
169	C₂²×C₈	—	53	2	no	—
170	C₂²×C₈	—	53	2	no	—
171	C₂²×C₈	—	53	2	no	—
172	C₂²×C₈	—	53	2	no	—
173	C₂²×C₈	—	53	1	no	—
174	C₂²×C₈	—	53	1	no	—
175	C₂²×C₈	—	106	4	no	—
176	C₂²×C₈	—	106	4	no	—
177	C₂×C₄²	#88324	1	2	yes	—
178	C₂×C₄²	#88324	1	2	yes	—
179	C₂×C₄²	#88324	1	2	yes	—
180	C₂×C₄²	C₂×#6157	1	2	yes	—
181	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
182	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
183	C₂×C₄²	C₂×#6157	1	2	yes	—
184	M₃₂	—	106	8	no	—
185	C₂×C₁₆	—	53	8	no	—
186	C₄×C₈	—	53	1	no	—
187	C₄×C₈	—	53	2	no	—
188	C₄×C₈	—	53	2	no	—
189	C₄×C₈	—	53	2	no	—
190	C₄×C₈	—	53	2	no	—
191	C₄×C₈	—	53	1	no	—
192	C₄×C₈	—	53	2	no	—
193	C₄×C₈	—	53	1	no	—
194	C₄×C₈	—	53	2	no	—
195	C₄×C₈	—	53	1	no	—
196	C₂⁴	C₂×#88324	1	1	yes	characteristic
197	C₂²×C₄	—	2	8	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	C₂×#88324	1	1	yes	characteristic
201	C₂²×C₄	C₂×#88324	1	2	yes	—
202	C₂²×C₄	C₂×#88324	1	2	yes	—
203	C₂²×C₄	C₂×#88324	1	2	yes	—
204	C₂²×C₄	C₂×#88324	1	2	yes	—
205	C₂²×C₄	C₂×#88324	1	2	yes	—
206	C₂²×C₄	C₂×#88324	1	2	yes	—
207	C₂²×C₄	C₂×#88324	1	2	yes	—
208	C₂²×C₄	C₂²×#6157	1	2	yes	—
209	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
210	C₂×C₈	—	53	4	no	—
211	C₂×C₈	—	53	4	no	—
212	C₂×C₈	—	53	1	no	—
213	C₂×C₈	—	53	1	no	—
214	C₂×C₈	—	53	1	no	—
215	C₂×C₈	—	53	1	no	—
216	C₂×C₈	—	53	2	no	—
217	C₂×C₈	—	53	2	no	—
218	C₂×C₈	—	106	4	no	—
219	C₂×C₈	—	106	4	no	—
220	C₂×C₈	—	106	4	no	—
221	C₂×C₈	—	106	4	no	—
222	C₂×C₈	—	106	2	no	—
223	C₂×C₈	—	106	2	no	—
224	C₄²	C₂×#88324	1	2	yes	—
225	C₄²	C₂×#88324	1	2	yes	—
226	C₄²	C₂×#88324	1	2	yes	—
227	C₄²	C₂×#88324	1	2	yes	—
228	C₄²	C₂×#88324	1	1	yes	characteristic
229	C₄²	C₂×#88324	1	2	yes	—
230	C₄²	C₂×#88324	1	1	yes	characteristic
231	C₄²	C₂²×#6157	1	1	yes	characteristic
232	C₄²	C₂²×#6157	1	2	yes	—
233	C₄²	C₂²×#6157	1	1	yes	characteristic
234	C₁₆	—	106	8	no	—
235	C₂³	—	2	2	no	—
236	C₂³	—	2	4	no	—
237	C₂³	C₂²×#88324	1	1	yes	characteristic
238	C₂³	C₂²×#88324	1	2	yes	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	#13133291	1	8	yes	complemented
249	C₂×C₄	#13133296	1	2	yes	—
250	C₂×C₄	C₂²×#88324	1	2	yes	—
251	C₂×C₄	C₂²×#88324	1	2	yes	—
252	C₂×C₄	C₂²×#88324	1	2	yes	—
253	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
254	C₂×C₄	C₂²×#88324	1	2	yes	—
255	C₂×C₄	C₂²×#88324	1	2	yes	—
256	C₂×C₄	D₅₃⋊M₁₆	1	2	yes	—
257	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
258	C₈	—	53	2	no	—
259	C₈	—	53	2	no	—
260	C₈	—	53	2	no	—
261	C₈	—	53	2	no	—
262	C₈	—	106	2	no	—
263	C₈	—	106	2	no	—
264	C₂²	—	2	8	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	#16811639	1	2	yes	—
269	C₂²	C₂×#13133291	1	4	yes	—
270	C₂²	C₂³×#88324	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	2	2	no	—
274	C₄	#16811639	1	2	yes	—
275	C₄	C₂×#13133291	1	4	yes	—
276	C₄	C₂×#13133291	1	4	yes	—
277	C₄	C₂×#13133296	1	1	yes	characteristic
278	C₄	C₂×D₅₃⋊M₁₆	1	1	yes	characteristic
279	C₂	—	2	2	no	—
280	C₂	—	2	4	no	—
281	C₂	C₂×#16811639	1	1	yes	characteristic
282	C₂	C₂²×#13133291	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂²×#6157
commutator, socle	C₂²×C₅₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₂²×C₄²×C₅₃	C₄
prime radical	C₂⁴×C₅₃	C₂×C₈
cosocle	C₂×#88322	C₂⁴
hypercenter	C₂²×C₄²	#6157
hypocenter	C₅₃	#33418
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
1. #20291076	C₂	semidirect
2. #20291072	C₂	semidirect
3. #20291047	C₂	semidirect
5. C₄×#13133291	C₂	semidirect
8. C₄⋊#11719910	C₂²	semidirect
10. #16811511	C₂²	semidirect
11. C₄×#11719910	C₂²	semidirect
106. C₂×C₄×C₅₃	M₃₂	semidirect
164. C₅₃	#33418	semidirect
248. C₂×C₄	#13133291	semidirect

Maximal subgroups

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

subgroup	index
#20291076	2
#20291072	2
#20291047	2
#20290942	2
C₄×#13133291	2
C₂²×C₄×#88322	2
#33418	53