Subgroups — #28305748

Order 13568.

502 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20290884	C₂	1	2	yes	maximal
2	C₂×#16811486	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊#88324	C₂	1	2	yes	maximal
4	C₂×C₈×#88324	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88324	C₂²	1	2	yes	—
7	C₂×C₄×#88324	C₂²	1	1	yes	characteristic
8	C₂×C₄×#88324	C₂²	1	1	yes	characteristic
9	#16811489	—	2	4	no	—
10	#16811486	—	2	2	no	—
11	Q₈⋊#88324	—	2	4	no	—
12	C₈×#88324	—	2	2	no	—
13	C₂×Q₈×Dic₅₃	C₂²	1	2	yes	—
14	Q₁₆×Dic₅₃	—	2	2	no	—
15	Q₁₆×Dic₅₃	—	2	2	no	—
16	C₂×#13133151	C₄	1	1	yes	characteristic
17	C₂×#13133141	C₄	1	2	yes	—
18	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
19	C₂²×Q₁₆×C₅₃	C₄	1	1	yes	characteristic
20	C₂²×#88324	—	2	1	no	—
21	C₂²×#88324	—	2	2	no	—
22	C₂²×#88324	D₄	1	2	yes	—
23	C₄⋊#88324	—	2	2	no	—
24	C₄⋊#88324	—	2	2	no	—
25	C₄×#88324	—	2	1	no	—
26	C₄×#88324	—	2	1	no	—
27	C₄×#88324	—	2	1	no	—
28	C₄×#88324	D₄	1	2	yes	—
29	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
30	Q₈×Dic₅₃	—	2	2	no	—
31	Q₈×Dic₅₃	—	2	2	no	—
32	Q₈×Dic₅₃	—	2	2	no	—
33	Q₈×Dic₅₃	—	2	2	no	—
34	C₂×C₄⋊Dic₅₃	—	2	2	no	—
35	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
36	C₂×C₄×Dic₅₃	—	2	2	no	—
37	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
38	#13133151	—	2	2	no	—
39	#13133141	—	2	4	no	—
40	C₈×Dic₅₃	D₄	1	2	yes	—
41	C₈×Dic₅₃	D₄	1	2	yes	—
42	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
43	C₂×Q₁₆×C₅₃	—	2	2	no	—
44	C₂×Q₁₆×C₅₃	—	2	2	no	—
45	C₂×Q₁₆×C₅₃	C₈	1	4	yes	complemented
46	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂×Q₈×C₅₃	—	2	2	no	—
48	C₂×Q₈×C₅₃	—	2	2	no	—
49	C₂×Q₈×C₅₃	—	2	2	no	—
50	C₂×Q₈×C₅₃	—	2	2	no	—
51	C₂×Q₈×C₅₃	—	2	2	no	—
52	C₂×Q₈×C₅₃	C₂×C₈	1	4	yes	—
53	C₂²×C₄×C₅₃	—	2	2	no	—
54	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×Dic₅₃	—	2	2	no	—
56	C₂²×Dic₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×Dic₅₃	#41	1	1	yes	characteristic
58	C₂×#88324	—	2	4	no	—
59	C₂×#88324	—	2	2	no	—
60	C₂×#88324	—	2	2	no	—
61	C₂×#88324	—	2	2	no	—
62	C₂×#88324	—	4	2	no	—
63	C₂×#88324	—	4	4	no	—
64	C₂×#88324	Q₁₆	1	4	yes	complemented
65	Q₁₆×C₅₃	—	2	8	no	—
66	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
67	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
68	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
69	C₄⋊Dic₅₃	—	2	2	no	—
70	C₄⋊Dic₅₃	—	2	2	no	—
71	C₄⋊Dic₅₃	—	4	2	no	—
72	C₄⋊Dic₅₃	—	4	2	no	—
73	C₄×Dic₅₃	—	4	2	no	—
74	C₄×Dic₅₃	—	4	2	no	—
75	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
76	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
77	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
78	C₄×Dic₅₃	#41	1	1	yes	characteristic
79	C₂×#88322	—	2	1	no	—
80	C₂×#88322	—	2	1	no	—
81	C₂×#88322	C₂²⋊C₄	1	2	yes	—
82	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
83	Q₈×C₅₃	—	2	4	no	—
84	Q₈×C₅₃	—	2	4	no	—
85	Q₈×C₅₃	—	4	2	no	—
86	Q₈×C₅₃	—	4	2	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	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
94	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
95	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
96	C₂×Dic₅₃	—	2	2	no	—
97	C₂×Dic₅₃	—	4	2	no	—
98	C₂×Dic₅₃	—	4	4	no	—
99	C₂×Dic₅₃	#121	1	1	yes	characteristic
100	C₂×Dic₅₃	#121	1	1	yes	characteristic
101	C₂×Dic₅₃	#124	1	1	yes	characteristic
102	C₂×Dic₅₃	#127	1	1	yes	characteristic
103	C₂×Dic₅₃	C₂×Q₁₆	1	2	yes	—
104	C₂×Dic₅₃	#135	1	2	yes	—
105	#88324	—	2	4	no	—
106	#88324	—	2	4	no	—
107	#88324	—	8	2	no	—
108	C₈×C₅₃	C₂²⋊C₈	1	4	yes	—
109	#88322	—	2	2	no	—
110	#9656	—	53	1	no	maximal, Sylow
111	C₂²×C₅₃	#390	1	1	yes	characteristic
112	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
113	C₂²×C₅₃	#436	1	1	yes	characteristic
114	C₂²×C₅₃	C₄×Q₁₆	1	2	yes	—
115	C₂²×C₅₃	#443	1	2	yes	—
116	C₄×C₅₃	—	4	4	no	—
117	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
118	C₄×C₅₃	#408	1	1	yes	characteristic
119	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
120	C₄×C₅₃	#436	1	1	yes	characteristic
121	Dic₅₃	—	2	1	no	—
122	Dic₅₃	—	2	1	no	—
123	Dic₅₃	—	8	2	no	—
124	Dic₅₃	#467	1	2	yes	—
125	Dic₅₃	#494	1	1	yes	characteristic
126	Dic₅₃	#495	1	1	yes	characteristic
127	C₂×C₄×Q₁₆	—	53	1	no	—
128	#1984	—	53	2	no	—
129	C₂×#335	—	53	1	no	—
130	C₂×Q₈⋊C₈	—	53	2	no	—
131	C₂×C₈²	—	53	1	no	—
132	C₂×C₅₃	C₈×Q₁₆	1	2	yes	—
133	C₂×C₅₃	#1591	1	2	yes	—
134	C₂×C₅₃	#1954	1	1	yes	characteristic
135	C₂×C₅₃	#1969	1	1	yes	characteristic
136	C₂×C₅₃	#1974	1	1	yes	characteristic
137	C₂²×Q₁₆	#6157	1	1	yes	characteristic
138	C₂×C₄×Q₈	—	53	2	no	—
139	C₄×Q₁₆	—	106	2	no	—
140	C₄×Q₁₆	—	106	2	no	—
141	C₂×#107	—	53	1	no	—
142	C₂×C₄⋊C₈	—	53	2	no	—
143	C₂×#103	—	53	2	no	—
144	C₂×C₄×C₈	—	53	1	no	—
145	C₂×C₄×C₈	—	53	1	no	—
146	C₂×C₄×C₈	—	53	1	no	—
147	#336	—	106	4	no	—
148	#335	—	106	2	no	—
149	Q₈⋊C₈	—	106	4	no	—
150	C₈²	—	106	2	no	—
151	C₅₃	#9656	1	1	yes	characteristic, complemented, Sylow
152	C₂²×Q₈	C₂×#6157	1	2	yes	—
153	C₂×Q₁₆	—	2	2	no	—
154	C₂×Q₁₆	—	2	2	no	—
155	C₂×Q₁₆	#88324	1	4	yes	complemented
156	C₂²×C₈	—	53	2	no	—
157	C₂²×C₈	—	53	1	no	—
158	C₂²×C₈	—	106	1	no	—
159	C₂²×C₈	—	106	2	no	—
160	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
161	C₄×Q₈	—	106	2	no	—
162	C₄×Q₈	—	106	2	no	—
163	C₄×Q₈	—	106	2	no	—
164	C₄×Q₈	—	106	2	no	—
165	C₂×C₄⋊C₄	—	53	2	no	—
166	C₂×C₄⋊C₄	—	106	2	no	—
167	C₂×C₄²	—	53	1	no	—
168	C₂×C₄²	—	106	2	no	—
169	#107	—	106	2	no	—
170	C₄⋊C₈	—	106	2	no	—
171	C₄⋊C₈	—	106	2	no	—
172	#103	—	106	4	no	—
173	C₄×C₈	—	53	2	no	—
174	C₄×C₈	—	53	2	no	—
175	C₄×C₈	—	53	2	no	—
176	C₄×C₈	—	106	1	no	—
177	C₄×C₈	—	106	1	no	—
178	C₄×C₈	—	106	1	no	—
179	C₂×Q₈	—	2	2	no	—
180	C₂×Q₈	—	2	2	no	—
181	C₂×Q₈	—	2	2	no	—
182	C₂×Q₈	—	2	2	no	—
183	C₂×Q₈	—	2	2	no	—
184	C₂×Q₈	C₂×#88324	1	4	yes	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	53	1	no	—
187	C₂²×C₄	—	53	1	no	—
188	C₂²×C₄	—	106	2	no	—
189	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
190	Q₁₆	—	2	8	no	—
191	C₂×C₈	—	53	4	no	—
192	C₂×C₈	—	53	2	no	—
193	C₂×C₈	—	106	2	no	—
194	C₂×C₈	—	106	2	no	—
195	C₂×C₈	—	106	2	no	—
196	C₂×C₈	—	106	4	no	—
197	C₂×C₈	—	106	1	no	—
198	C₂×C₈	—	106	1	no	—
199	C₂×C₈	—	212	2	no	—
200	C₂×C₈	—	212	4	no	—
201	C₂×C₈	C₂×#88324	1	2	yes	—
202	C₂×C₈	#11719940	1	2	yes	—
203	C₂×C₈	C₂²⋊#6157	1	2	yes	—
204	C₄⋊C₄	—	106	2	no	—
205	C₄⋊C₄	—	106	2	no	—
206	C₄⋊C₄	—	212	2	no	—
207	C₄⋊C₄	—	212	2	no	—
208	C₄²	—	53	1	no	—
209	C₄²	—	53	1	no	—
210	C₄²	—	53	1	no	—
211	C₄²	—	53	1	no	—
212	C₄²	—	212	2	no	—
213	C₄²	—	212	2	no	—
214	C₂³	D₄×#6157	1	1	yes	characteristic
215	Q₈	—	2	4	no	—
216	Q₈	—	2	4	no	—
217	Q₈	—	4	2	no	—
218	Q₈	—	4	2	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	53	1	no	—
223	C₂×C₄	—	53	1	no	—
224	C₂×C₄	—	53	2	no	—
225	C₂×C₄	—	53	2	no	—
226	C₂×C₄	—	53	1	no	—
227	C₂×C₄	—	53	1	no	—
228	C₂×C₄	—	106	2	no	—
229	C₂×C₄	—	212	2	no	—
230	C₂×C₄	—	212	4	no	—
231	C₂×C₄	C₂×C₂²⋊#6157	1	1	yes	characteristic
232	C₂×C₄	D₄×#6157	1	1	yes	characteristic
233	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
234	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
235	C₂×C₄	#13133304	1	1	yes	characteristic
236	C₂×C₄	#13133304	1	1	yes	characteristic
237	C₈	—	106	4	no	—
238	C₈	—	106	4	no	—
239	C₈	—	106	2	no	—
240	C₈	—	424	2	no	—
241	C₈	C₂²⋊#88324	1	4	yes	—
242	C₂²	#16811575	1	1	yes	characteristic
243	C₂²	D₄×#88324	1	1	yes	characteristic
244	C₂²	#16811658	1	1	yes	characteristic
245	C₂²	Q₁₆×#6157	1	2	yes	—
246	C₂²	#16811667	1	2	yes	—
247	C₄	—	4	4	no	—
248	C₄	—	53	2	no	—
249	C₄	—	53	1	no	—
250	C₄	—	53	1	no	—
251	C₄	—	106	1	no	—
252	C₄	—	106	1	no	—
253	C₄	—	424	2	no	—
254	C₄	C₂×C₂²⋊#88324	1	1	yes	characteristic
255	C₄	#16811597	1	1	yes	characteristic
256	C₄	D₄×#88324	1	1	yes	characteristic
257	C₄	#16811658	1	1	yes	characteristic
258	C₂	Q₁₆×#88324	1	2	yes	—
259	C₂	#20290355	1	2	yes	—
260	C₂	#20290836	1	1	yes	characteristic
261	C₂	#20290861	1	1	yes	characteristic
262	C₂	#20290866	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄×#6157
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₂²×Q₁₆×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₂×C₄×Dic₅₃	C₂³
hypercenter	C₂²×Q₁₆	#6157
hypocenter	C₅₃	#9656
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

Complements

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

N	H ≅ G/N	type
45. C₂×Q₁₆×C₅₃	C₈	semidirect
64. C₂×#88324	Q₁₆	semidirect
151. C₅₃	#9656	semidirect
155. C₂×Q₁₆	#88324	semidirect

Maximal subgroups

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

subgroup	index
#20290884	2
C₂×#16811486	2
C₂×Q₈⋊#88324	2
C₂×C₈×#88324	2
C₂×Q₁₆×Dic₅₃	2
#9656	53