Subgroups — #28328279

Order 13568.

524 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291088	C₂	1	4	yes	maximal, complemented
2	#20291084	C₂	1	4	yes	maximal, complemented
3	#20291058	C₂	1	2	yes	maximal, complemented
4	C₂×#16811519	C₂	1	2	yes	maximal
5	#20290936	C₂	1	2	yes	maximal, complemented
6	#20283134	C₂	1	1	yes	maximal, characteristic
7	C₂×#13133291	C₂²	1	4	yes	—
8	#16811546	C₂²	1	8	yes	complemented
9	#16811519	C₂²	1	4	yes	—
10	#16811519	C₂²	1	4	yes	—
11	#16811515	C₂²	1	8	yes	complemented
12	C₂²×#11719917	C₄	1	1	yes	characteristic
13	C₂²×#11719917	C₂²	1	1	yes	characteristic
14	#16810579	C₄	1	2	yes	—
15	#16810579	C₄	1	1	yes	characteristic
16	#16810579	C₄	1	1	yes	characteristic
17	#16810579	C₂²	1	1	yes	characteristic
18	#16810579	C₂²	1	2	yes	—
19	#16810579	C₂²	1	1	yes	characteristic
20	C₄×#11719917	C₄	1	2	yes	—
21	C₄×#11719917	C₂²	1	2	yes	—
22	C₂×#117×C₅₃	C₄	1	1	yes	characteristic
23	#13133291	—	2	4	no	—
24	#13133291	C₂³	1	8	yes	—
25	C₂×#11719910	—	2	4	no	—
26	C₂×#11719917	—	2	4	no	—
27	C₂×#11719917	—	2	4	no	—
28	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
29	C₂×#11719917	C₂×C₄	1	2	yes	—
30	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
31	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
32	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
33	C₂×#11719917	C₂³	1	1	yes	characteristic
34	C₂×#11719917	C₂³	1	1	yes	characteristic
35	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
36	C₂²×#88322	C₂×C₄	1	2	yes	—
37	C₂²×#88322	C₂³	1	1	yes	characteristic
38	#13133128	C₂×C₄	1	2	yes	—
39	#13133128	C₂×C₄	1	2	yes	—
40	#13133128	C₂×C₄	1	2	yes	—
41	#13133128	C₂³	1	2	yes	—
42	C₄×#88322	C₂×C₄	1	2	yes	—
43	C₄×#88322	C₂×C₄	1	2	yes	—
44	C₄×#88322	C₂×C₄	1	2	yes	—
45	C₄×#88322	C₂³	1	2	yes	—
46	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
47	#117×C₅₃	—	2	1	no	—
48	#117×C₅₃	—	2	1	no	—
49	#117×C₅₃	C₂×C₄	1	2	yes	—
50	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
51	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
53	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
54	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
55	C₂⁴×C₅₃	M₁₆	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	—	2	2	no	—
57	C₂²×C₄×C₅₃	—	2	2	no	—
58	C₂²×C₄×C₅₃	—	2	2	no	—
59	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₄⋊C₄×C₅₃	—	2	2	no	—
66	C₄⋊C₄×C₅₃	—	2	1	no	—
67	C₄⋊C₄×C₅₃	—	2	1	no	—
68	C₂²⋊C₄×C₅₃	—	2	2	no	—
69	C₂²⋊C₄×C₅₃	—	2	1	no	—
70	C₂²⋊C₄×C₅₃	—	2	1	no	—
71	C₄²×C₅₃	—	2	2	no	—
72	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
73	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
74	#11719917	—	2	2	no	—
75	#11719917	—	2	2	no	—
76	#11719917	—	2	2	no	—
77	#11719917	—	2	4	no	—
78	#11719917	—	2	2	no	—
79	#11719917	—	2	2	no	—
80	#11719917	M₁₆	1	4	yes	—
81	C₂×#88322	—	2	2	no	—
82	C₂×#88322	—	2	4	no	—
83	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
84	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
85	C₂×#88322	C₂²×C₄	1	2	yes	—
86	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
87	C₂×#88322	C₂²×C₄	1	2	yes	—
88	C₂×#88322	#41	1	4	yes	—
89	C₂×#88322	C₂⁴	1	1	yes	characteristic
90	#11719910	—	2	8	no	—
91	C₂³×C₅₃	—	2	2	no	—
92	C₂³×C₅₃	—	8	1	no	—
93	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
94	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
95	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
96	C₂×C₄×C₅₃	—	2	2	no	—
97	C₂×C₄×C₅₃	—	2	1	no	—
98	C₂×C₄×C₅₃	—	2	1	no	—
99	C₂×C₄×C₅₃	—	2	1	no	—
100	C₂×C₄×C₅₃	—	2	1	no	—
101	C₂×C₄×C₅₃	—	2	2	no	—
102	C₂×C₄×C₅₃	—	2	2	no	—
103	C₂×C₄×C₅₃	—	2	2	no	—
104	C₂×C₄×C₅₃	—	2	2	no	—
105	C₂×C₄×C₅₃	—	8	2	no	—
106	C₂×C₄×C₅₃	#117	1	2	yes	—
107	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
111	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
112	#88322	—	2	1	no	—
113	#88322	—	2	1	no	—
114	#88322	—	2	2	no	—
115	#88322	#117	1	2	yes	—
116	#88322	C₂×M₁₆	1	4	yes	—
117	#88322	C₂×#41	1	2	yes	—
118	#33427	—	53	1	no	maximal, Sylow
119	C₂²×C₅₃	—	2	1	no	—
120	C₂²×C₅₃	—	2	1	no	—
121	C₂²×C₅₃	—	4	1	no	—
122	C₂²×C₅₃	—	4	1	no	—
123	C₂²×C₅₃	—	4	1	no	—
124	C₂²×C₅₃	—	8	2	no	—
125	C₂²×C₅₃	#432	1	2	yes	—
126	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
127	C₄×C₅₃	—	2	1	no	—
128	C₄×C₅₃	—	2	1	no	—
129	C₄×C₅₃	—	4	2	no	—
130	C₄×C₅₃	—	4	2	no	—
131	C₄×C₅₃	#432	1	2	yes	—
132	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
133	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
134	#2873	—	53	1	no	—
135	#2166	—	53	4	no	—
136	#2165	—	53	4	no	—
137	#2153	—	53	2	no	—
138	C₂×#349	—	53	2	no	—
139	#2110	—	53	2	no	—
140	C₂×C₅₃	—	2	1	no	—
141	C₂×C₅₃	—	2	2	no	—
142	C₂×C₅₃	—	4	1	no	—
143	C₂×C₅₃	—	4	1	no	—
144	C₂×C₅₃	C₂×#432	1	1	yes	characteristic
145	C₂²×M₁₆	—	53	1	no	—
146	C₂²×M₁₆	—	53	1	no	—
147	C₂×#117	#6157	1	1	yes	characteristic
148	C₂×M₃₂	—	53	4	no	—
149	#405	—	53	1	no	—
150	#405	—	53	2	no	—
151	#405	—	53	1	no	—
152	#405	—	53	2	no	—
153	#405	—	53	1	no	—
154	#405	—	53	1	no	—
155	C₄×M₁₆	—	53	2	no	—
156	C₄×M₁₆	—	53	2	no	—
157	#364	—	53	8	no	—
158	#349	—	53	4	no	—
159	#349	—	53	4	no	—
160	#347	—	53	8	no	—
161	C₅₃	#33427	1	1	yes	characteristic, complemented, Sylow
162	C₂³×C₄	C₂×#6157	1	1	yes	characteristic
163	C₂×M₁₆	—	53	1	no	—
164	C₂×M₁₆	—	53	1	no	—
165	C₂×M₁₆	—	53	1	no	—
166	C₂×M₁₆	—	53	1	no	—
167	C₂×M₁₆	—	53	2	no	—
168	C₂×M₁₆	—	53	1	no	—
169	C₂×M₁₆	—	53	1	no	—
170	C₂×M₁₆	—	106	4	no	—
171	C₂×M₁₆	—	106	4	no	—
172	C₂²×C₈	—	53	1	no	—
173	C₂²×C₈	—	53	2	no	—
174	C₂²×C₈	—	53	1	no	—
175	#117	—	2	1	no	—
176	#117	—	2	1	no	—
177	#117	C₂×#6157	1	1	yes	characteristic
178	#117	C₂×#6157	1	2	yes	—
179	#117	C₂×#6157	1	1	yes	characteristic
180	C₂×C₄⋊C₄	—	2	1	no	—
181	C₂×C₂²⋊C₄	—	2	1	no	—
182	C₂×C₄²	C₂×#6157	1	2	yes	—
183	M₃₂	—	53	8	no	—
184	M₃₂	—	106	4	no	—
185	C₂×C₁₆	—	106	4	no	—
186	#97	—	53	2	no	—
187	#97	—	53	2	no	—
188	#97	—	53	2	no	—
189	#97	—	53	2	no	—
190	C₄×C₈	—	53	2	no	—
191	C₄×C₈	—	53	2	no	—
192	C₄×C₈	—	53	2	no	—
193	C₄×C₈	—	53	2	no	—
194	C₂⁴	#11719940	1	1	yes	characteristic
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	#11719940	1	1	yes	characteristic
199	C₂²×C₄	#11719940	1	1	yes	characteristic
200	C₂²×C₄	#11719940	1	1	yes	characteristic
201	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
202	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
203	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
204	M₁₆	—	53	4	no	—
205	M₁₆	—	106	2	no	—
206	M₁₆	—	106	2	no	—
207	M₁₆	—	106	2	no	—
208	M₁₆	—	106	2	no	—
209	M₁₆	—	106	4	no	—
210	M₁₆	—	106	2	no	—
211	C₂×C₈	—	53	4	no	—
212	C₂×C₈	—	53	1	no	—
213	C₂×C₈	—	53	1	no	—
214	C₂×C₈	—	53	2	no	—
215	C₂×C₈	—	53	1	no	—
216	C₂×C₈	—	53	1	no	—
217	C₂×C₈	—	53	2	no	—
218	C₂×C₈	—	106	4	no	—
219	C₂×C₈	—	106	2	no	—
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	2	1	no	—
222	C₄⋊C₄	—	2	1	no	—
223	C₂²⋊C₄	—	2	2	no	—
224	C₂²⋊C₄	—	2	1	no	—
225	C₂²⋊C₄	—	2	1	no	—
226	C₄²	—	2	2	no	—
227	C₄²	C₂²×#6157	1	2	yes	—
228	C₄²	C₂²×#6157	1	2	yes	—
229	C₁₆	—	106	8	no	—
230	C₂³	—	2	2	no	—
231	C₂³	—	8	1	no	—
232	C₂³	C₂×#11719940	1	1	yes	characteristic
233	C₂³	C₂×#11719940	1	1	yes	characteristic
234	C₂³	C₂×#11719940	1	1	yes	characteristic
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	8	2	no	—
245	C₂×C₄	#13133296	1	2	yes	—
246	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
247	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
248	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
249	C₂×C₄	D₅₃⋊M₁₆	1	2	yes	—
250	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
251	C₈	—	53	2	no	—
252	C₈	—	53	2	no	—
253	C₈	—	53	4	no	—
254	C₈	—	106	1	no	—
255	C₈	—	106	1	no	—
256	C₈	—	106	2	no	—
257	C₂²	—	2	1	no	—
258	C₂²	—	2	1	no	—
259	C₂²	—	4	1	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	4	1	no	—
262	C₂²	—	8	2	no	—
263	C₂²	#16811643	1	2	yes	—
264	C₂²	C₂²×#11719940	1	1	yes	characteristic
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	4	2	no	—
268	C₄	—	4	2	no	—
269	C₄	#16811643	1	2	yes	—
270	C₄	C₂×#13133296	1	1	yes	characteristic
271	C₄	C₂×D₅₃⋊M₁₆	1	1	yes	characteristic
272	C₂	—	2	1	no	—
273	C₂	—	2	2	no	—
274	C₂	—	4	1	no	—
275	C₂	—	4	1	no	—
276	C₂	C₂×#16811643	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#13133296
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₂×#117×C₅₃	C₄
socle	C₂×C₅₃	C₂×#432
prime radical	C₂⁴×C₅₃	M₁₆
cosocle	C₂×#88322	C₂⁴
hypercenter	C₂×#117	#6157
hypocenter	C₅₃	#33427
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ #117 ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
1. #20291088	C₂	semidirect
2. #20291084	C₂	semidirect
3. #20291058	C₂	semidirect
5. #20290936	C₂	semidirect
8. #16811546	C₂²	semidirect
11. #16811515	C₂²	semidirect
161. C₅₃	#33427	semidirect

Maximal subgroups

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

subgroup	index
#20291088	2
#20291084	2
#20291058	2
C₂×#16811519	2
#20290936	2
#20283134	2
#33427	53