Subgroups — #28306200

Order 13568.

528 conjugacy classes of subgroups.

The subgroup lattice has 288 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	1	yes	maximal, characteristic
2	C₄×C₄⋊#88324	C₂	1	1	yes	maximal, characteristic
3	#20289551	C₂	1	4	yes	maximal
4	C₄⋊Q₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#88324	—	2	1	no	—
6	C₂×C₄⋊#88324	C₂²	1	2	yes	—
7	C₂×C₄×#88324	—	2	1	no	—
8	C₂×C₄×#88324	C₂²	1	2	yes	—
9	#16811489	—	2	1	no	—
10	#16811489	—	2	1	no	—
11	C₂×Q₈×Dic₅₃	—	2	1	no	—
12	#16810530	—	2	1	no	—
13	#16810522	—	2	1	no	—
14	#16810522	C₄	1	2	yes	—
15	#16810493	C₄	1	1	yes	characteristic
16	C₄⋊C₄×Dic₅₃	—	2	1	no	—
17	C₄⋊C₄×Dic₅₃	C₂²	1	2	yes	—
18	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
20	C₂²×#88324	—	2	2	no	—
21	C₂²×#88324	—	2	2	no	—
22	C₄⋊#88324	—	2	4	no	—
23	C₄⋊#88324	—	2	2	no	—
24	C₄⋊#88324	Q₈	1	4	yes	complemented
25	C₄×#88324	—	2	4	no	—
26	C₄×#88324	—	4	2	no	—
27	Q₈×Dic₅₃	—	4	4	no	—
28	C₂×C₄⋊Dic₅₃	—	2	1	no	—
29	C₂×C₄⋊Dic₅₃	—	2	1	no	—
30	C₂×C₄⋊Dic₅₃	—	2	2	no	—
31	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
32	C₂×C₄×Dic₅₃	—	2	2	no	—
33	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
35	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
36	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
37	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
38	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
39	#13133124	—	2	2	no	—
40	#13133124	—	2	2	no	—
41	C₂²×Q₈×C₅₃	—	2	1	no	—
42	C₄⋊Q₈×C₅₃	—	2	2	no	—
43	C₄⋊Q₈×C₅₃	C₈	1	4	yes	complemented
44	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
45	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂×Q₈×C₅₃	—	2	4	no	—
48	C₂×Q₈×C₅₃	—	4	2	no	—
49	C₂×Q₈×C₅₃	—	4	2	no	—
50	C₂²×C₄×C₅₃	—	2	2	no	—
51	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×Dic₅₃	—	2	2	no	—
55	C₂²×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×Dic₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×Dic₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂×#88324	—	2	4	no	—
60	C₂×#88324	—	2	4	no	—
61	C₂×#88324	—	4	2	no	—
62	C₂×#88324	—	4	2	no	—
63	C₂×#88324	—	4	2	no	—
64	C₂×#88324	—	4	2	no	—
65	C₄⋊C₄×C₅₃	—	2	4	no	—
66	C₄⋊C₄×C₅₃	—	2	2	no	—
67	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
68	C₄²×C₅₃	C₂×C₈	1	2	yes	—
69	C₄²×C₅₃	M₁₆	1	2	yes	—
70	C₄⋊Dic₅₃	—	2	4	no	—
71	C₄⋊Dic₅₃	—	4	4	no	—
72	C₄⋊Dic₅₃	—	4	2	no	—
73	C₄⋊Dic₅₃	—	4	2	no	—
74	C₄×Dic₅₃	—	2	2	no	—
75	C₄×Dic₅₃	—	2	2	no	—
76	C₄×Dic₅₃	—	2	2	no	—
77	C₄×Dic₅₃	—	4	1	no	—
78	C₄×Dic₅₃	—	4	4	no	—
79	C₄×Dic₅₃	SD₁₆	1	2	yes	—
80	C₄×Dic₅₃	Q₁₆	1	2	yes	—
81	C₄×Dic₅₃	C₂×Q₈	1	2	yes	—
82	C₄×Dic₅₃	#41	1	2	yes	—
83	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	Q₈×C₅₃	—	4	8	no	—
85	C₂×C₄×C₅₃	—	2	1	no	—
86	C₂×C₄×C₅₃	—	2	1	no	—
87	C₂×C₄×C₅₃	—	2	4	no	—
88	C₂×C₄×C₅₃	—	4	2	no	—
89	C₂×C₄×C₅₃	—	4	2	no	—
90	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
91	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
92	C₂×C₄×C₅₃	#103	1	2	yes	—
93	C₂×C₄×C₅₃	#104	1	2	yes	—
94	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
95	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
97	C₂×C₄×C₅₃	#131	1	2	yes	—
98	C₂×Dic₅₃	—	2	1	no	—
99	C₂×Dic₅₃	—	2	1	no	—
100	C₂×Dic₅₃	—	2	1	no	—
101	C₂×Dic₅₃	—	2	1	no	—
102	C₂×Dic₅₃	—	2	1	no	—
103	C₂×Dic₅₃	—	2	1	no	—
104	C₂×Dic₅₃	—	2	1	no	—
105	C₂×Dic₅₃	—	4	2	no	—
106	C₂×Dic₅₃	—	4	2	no	—
107	C₂×Dic₅₃	—	4	2	no	—
108	C₂×Dic₅₃	#103	1	2	yes	—
109	C₂×Dic₅₃	C₂²⋊Q₈	1	2	yes	—
110	C₂×Dic₅₃	#124	1	1	yes	characteristic
111	C₂×Dic₅₃	C₄⋊Q₈	1	1	yes	characteristic
112	C₂×Dic₅₃	C₂×SD₁₆	1	1	yes	characteristic
113	C₂×Dic₅₃	C₂×Q₁₆	1	1	yes	characteristic
114	C₂×Dic₅₃	AGL(1,ℤ/8)	1	2	yes	—
115	#88324	—	4	4	no	—
116	#88324	—	4	4	no	—
117	#9864	—	53	1	no	maximal, Sylow
118	C₂²×C₅₃	#391	1	1	yes	characteristic
119	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
120	C₂²×C₅₃	#408	1	1	yes	characteristic
121	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
122	C₂²×C₅₃	#418	1	1	yes	characteristic
123	C₂²×C₅₃	C₂×#104	1	1	yes	characteristic
124	C₂²×C₅₃	#421	1	1	yes	characteristic
125	C₄×C₅₃	—	2	2	no	—
126	C₄×C₅₃	—	4	4	no	—
127	C₄×C₅₃	Q₈⋊C₈	1	4	yes	—
128	C₄×C₅₃	C₈×Q₈	1	2	yes	—
129	C₄×C₅₃	#446	1	2	yes	—
130	Dic₅₃	—	2	1	no	—
131	Dic₅₃	—	2	1	no	—
132	Dic₅₃	—	2	2	no	—
133	Dic₅₃	—	4	1	no	—
134	Dic₅₃	—	8	2	no	—
135	Dic₅₃	#476	1	2	yes	—
136	Dic₅₃	#477	1	2	yes	—
137	C₄×C₄⋊Q₈	—	53	1	no	—
138	C₄×C₄⋊C₈	—	53	1	no	—
139	C₄×C₄⋊C₈	—	53	1	no	—
140	#1279	—	53	4	no	—
141	C₂×C₅₃	C₂×Q₈⋊C₈	1	1	yes	characteristic
142	C₂×C₅₃	#1478	1	1	yes	characteristic
143	C₂×C₅₃	#1481	1	1	yes	characteristic
144	C₂×C₅₃	#1484	1	1	yes	characteristic
145	C₂×C₅₃	#1984	1	1	yes	characteristic
146	C₂×C₅₃	#1986	1	1	yes	characteristic
147	C₂×C₅₃	#1987	1	1	yes	characteristic
148	C₂×C₄⋊Q₈	#6157	1	1	yes	characteristic
149	C₂×C₄×Q₈	—	106	1	no	—
150	C₂×C₄⋊C₈	—	53	2	no	—
151	C₂×C₄⋊C₈	—	106	1	no	—
152	C₂×C₄×C₈	—	53	2	no	—
153	C₂×C₄×C₈	—	106	1	no	—
154	#391	—	106	1	no	—
155	#389	—	53	2	no	—
156	#389	—	106	1	no	—
157	#384	—	53	1	no	—
158	C₄×C₄⋊C₄	—	53	2	no	—
159	C₄×C₄⋊C₄	—	106	1	no	—
160	C₄³	—	53	1	no	—
161	#336	—	106	1	no	—
162	#336	—	106	1	no	—
163	C₅₃	#9864	1	1	yes	characteristic, complemented, Sylow
164	C₂²×Q₈	—	2	1	no	—
165	C₂²×C₈	—	106	2	no	—
166	C₂²×C₈	—	106	2	no	—
167	C₄⋊Q₈	—	2	2	no	—
168	C₄⋊Q₈	#88324	1	4	yes	complemented
169	C₄×Q₈	—	212	4	no	—
170	C₂×C₄⋊C₄	—	2	1	no	—
171	C₂×C₄⋊C₄	—	53	2	no	—
172	C₂×C₄⋊C₄	—	106	2	no	—
173	C₂×C₄⋊C₄	—	106	1	no	—
174	C₂×C₄⋊C₄	—	106	1	no	—
175	C₂×C₄⋊C₄	C₂×#6157	1	2	yes	—
176	C₂×C₄²	—	53	1	no	—
177	C₂×C₄²	—	53	1	no	—
178	C₂×C₄²	—	53	1	no	—
179	C₂×C₄²	—	53	1	no	—
180	C₂×C₄²	—	53	1	no	—
181	C₂×C₄²	—	53	1	no	—
182	C₂×C₄²	—	106	2	no	—
183	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
184	C₄⋊C₈	—	53	4	no	—
185	C₄⋊C₈	—	106	2	no	—
186	C₄⋊C₈	—	106	4	no	—
187	C₄×C₈	—	106	4	no	—
188	C₄×C₈	—	212	2	no	—
189	#95	—	106	2	no	—
190	#95	—	106	2	no	—
191	C₂×Q₈	—	2	4	no	—
192	C₂×Q₈	—	4	2	no	—
193	C₂×Q₈	—	4	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	53	1	no	—
196	C₂²×C₄	—	53	1	no	—
197	C₂²×C₄	—	53	1	no	—
198	C₂²×C₄	—	53	1	no	—
199	C₂²×C₄	—	106	2	no	—
200	C₂²×C₄	C₂²⋊#6157	1	1	yes	characteristic
201	C₂²×C₄	C₂²⋊#6157	1	1	yes	characteristic
202	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
203	C₂×C₈	—	106	4	no	—
204	C₂×C₈	—	106	4	no	—
205	C₂×C₈	—	212	2	no	—
206	C₂×C₈	—	212	2	no	—
207	C₂×C₈	—	212	2	no	—
208	C₂×C₈	—	212	2	no	—
209	C₄⋊C₄	—	2	4	no	—
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	106	4	no	—
212	C₄⋊C₄	—	212	2	no	—
213	C₄⋊C₄	—	212	4	no	—
214	C₄⋊C₄	—	212	2	no	—
215	C₄⋊C₄	C₂×#88324	1	4	yes	—
216	C₄²	—	53	2	no	—
217	C₄²	—	53	2	no	—
218	C₄²	—	53	2	no	—
219	C₄²	—	53	2	no	—
220	C₄²	—	106	2	no	—
221	C₄²	—	106	2	no	—
222	C₄²	—	106	2	no	—
223	C₄²	—	212	1	no	—
224	C₄²	—	212	4	no	—
225	C₄²	C₂×#88324	1	2	yes	—
226	C₄²	#11719940	1	2	yes	—
227	C₂³	C₂×C₂²⋊#6157	1	1	yes	characteristic
228	Q₈	—	4	8	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	53	2	no	—
235	C₂×C₄	—	53	1	no	—
236	C₂×C₄	—	53	1	no	—
237	C₂×C₄	—	53	2	no	—
238	C₂×C₄	—	53	1	no	—
239	C₂×C₄	—	53	1	no	—
240	C₂×C₄	—	53	2	no	—
241	C₂×C₄	—	106	1	no	—
242	C₂×C₄	—	106	1	no	—
243	C₂×C₄	—	106	1	no	—
244	C₂×C₄	—	106	1	no	—
245	C₂×C₄	—	106	1	no	—
246	C₂×C₄	—	106	1	no	—
247	C₂×C₄	—	106	1	no	—
248	C₂×C₄	—	212	2	no	—
249	C₂×C₄	—	212	2	no	—
250	C₂×C₄	—	212	2	no	—
251	C₂×C₄	C₂²⋊#88324	1	2	yes	—
252	C₂×C₄	C₂²⋊#88324	1	2	yes	—
253	C₂×C₄	#13133280	1	2	yes	—
254	C₂×C₄	#13133282	1	2	yes	—
255	C₂×C₄	Q₈×#6157	1	2	yes	—
256	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
257	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
258	C₂×C₄	#13133305	1	2	yes	—
259	C₈	—	212	4	no	—
260	C₈	—	212	4	no	—
261	C₂²	(C₂×#6157)⋊Q₈	1	1	yes	characteristic
262	C₂²	C₂×C₂²⋊#88324	1	1	yes	characteristic
263	C₂²	#16811598	1	1	yes	characteristic
264	C₂²	C₂×#13133280	1	1	yes	characteristic
265	C₂²	#16811615	1	1	yes	characteristic
266	C₂²	C₂×#13133282	1	1	yes	characteristic
267	C₂²	#16811621	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	4	4	no	—
270	C₄	—	53	2	no	—
271	C₄	—	53	2	no	—
272	C₄	—	106	1	no	—
273	C₄	—	106	1	no	—
274	C₄	—	106	2	no	—
275	C₄	—	424	2	no	—
276	C₄	—	212	1	no	—
277	C₄	Q₈⋊#88324	1	4	yes	—
278	C₄	Q₈×#88324	1	2	yes	—
279	C₄	#16811674	1	2	yes	—
280	C₂	C₂×Q₈⋊#88324	1	1	yes	characteristic
281	C₂	#20290014	1	1	yes	characteristic
282	C₂	#20290025	1	1	yes	characteristic
283	C₂	#20290036	1	1	yes	characteristic
284	C₂	#20290884	1	1	yes	characteristic
285	C₂	#20290890	1	1	yes	characteristic
286	C₂	#20290893	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊#6157
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₂×C₄⋊Q₈×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄×Dic₅₃	C₂³
hypercenter	C₂×C₄⋊Q₈	#6157
hypocenter	C₅₃	#9864
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
24. C₄⋊#88324	Q₈	semidirect
43. C₄⋊Q₈×C₅₃	C₈	semidirect
163. C₅₃	#9864	semidirect
168. C₄⋊Q₈	#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
#20289551	2
C₄⋊Q₈×Dic₅₃	2
#9864	53