Subgroups — #28308520

Order 13568.

380 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811545	C₂	1	1	yes	maximal, characteristic, complemented
2	#20289918	C₂	1	1	yes	maximal, characteristic, complemented
3	#20289679	C₂	1	4	yes	maximal, complemented
4	C₂×#16810740	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133291	—	2	1	no	—
6	#16811639	—	2	1	no	—
7	#16811545	—	2	1	no	—
8	#16811545	C₂²	1	2	yes	complemented
9	#16811464	C₂²	1	2	yes	complemented
10	C₈×#88324	—	2	1	no	—
11	C₂×#13133224	—	2	1	no	—
12	#16810740	—	2	1	no	—
13	#16810740	—	2	1	no	—
14	#16810740	C₄	1	2	yes	—
15	#16810740	C₂²	1	2	yes	—
16	C₂×#13133153	C₄	1	1	yes	characteristic
17	C₂×#13133144	—	2	1	no	—
18	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
19	C₂×#135×C₅₃	C₄	1	1	yes	characteristic
20	#13133291	—	2	2	no	—
21	#13133291	—	4	1	no	—
22	C₂×#11719910	—	4	1	no	—
23	C₄⋊#88324	—	2	2	no	—
24	C₂²⋊#88324	—	2	2	no	—
25	C₄×#88324	—	2	2	no	—
26	#13133224	—	2	2	no	—
27	#13133224	—	2	2	no	—
28	#13133224	—	4	1	no	—
29	#13133224	—	4	1	no	—
30	C₂×#11719917	—	2	1	no	—
31	C₂×#11719917	—	4	1	no	—
32	C₂²×#88322	—	4	1	no	—
33	C₂²×#88322	D₄	1	1	yes	characteristic
34	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
35	#13133153	—	2	1	no	—
36	#13133153	C₂×C₄	1	2	yes	—
37	#13133144	—	2	2	no	—
38	#13133144	—	2	2	no	—
39	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
40	C₈×Dic₅₃	D₄	1	1	yes	characteristic
41	C₈×Dic₅₃	D₄	1	1	yes	characteristic
42	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
43	C₂×#41×C₅₃	—	2	1	no	—
44	#135×C₅₃	—	2	1	no	—
45	#135×C₅₃	—	2	1	no	—
46	#135×C₅₃	C₈	1	2	yes	complemented
47	#135×C₅₃	C₂×C₄	1	2	yes	—
48	C₂×Q₁₆×C₅₃	C₈	1	1	yes	characteristic, complemented
49	C₂×SD₁₆×C₅₃	—	2	1	no	—
50	C₂×D₈×C₅₃	C₈	1	1	yes	characteristic, complemented
51	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
52	#41×C₅₃	—	2	2	no	—
53	#41×C₅₃	—	2	2	no	—
54	#41×C₅₃	—	4	1	no	—
55	#41×C₅₃	—	4	1	no	—
56	C₂×Q₈×C₅₃	—	2	1	no	—
57	C₂×D₄×C₅₃	—	2	1	no	—
58	C₂×D₄×C₅₃	—	4	1	no	—
59	C₂²×C₄×C₅₃	—	4	1	no	—
60	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×Dic₅₃	—	2	1	no	—
62	C₂×#88324	—	2	4	no	—
63	Q₁₆×C₅₃	—	2	1	no	—
64	Q₁₆×C₅₃	C₂×C₈	1	2	yes	—
65	SD₁₆×C₅₃	—	2	2	no	—
66	SD₁₆×C₅₃	—	2	2	no	—
67	D₈×C₅₃	—	2	1	no	—
68	D₈×C₅₃	C₂×C₈	1	2	yes	—
69	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
71	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
72	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
73	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
74	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₄×Dic₅₃	—	2	1	no	—
76	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
78	#11719917	—	2	2	no	—
79	#11719917	—	4	1	no	—
80	#11719917	—	4	2	no	—
81	#11719917	—	4	2	no	—
82	C₂×#88322	—	4	1	no	—
83	C₂×#88322	—	4	2	no	—
84	C₂×#88322	—	4	2	no	—
85	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×#88322	D₈	1	1	yes	characteristic
87	C₂×#88322	D₈	1	1	yes	characteristic
88	C₂×#88322	SD₁₆	1	1	yes	characteristic
89	C₂×#88322	SD₁₆	1	1	yes	characteristic
90	C₂×#88322	C₂×D₄	1	1	yes	characteristic
91	#11719910	—	4	2	no	—
92	C₂³×C₅₃	—	4	1	no	—
93	C₂³×C₅₃	#104	1	1	yes	characteristic
94	Q₈×C₅₃	—	2	2	no	—
95	Q₈×C₅₃	—	4	1	no	—
96	D₄×C₅₃	—	2	2	no	—
97	D₄×C₅₃	—	4	2	no	—
98	D₄×C₅₃	—	4	2	no	—
99	D₄×C₅₃	—	4	1	no	—
100	C₂×C₄×C₅₃	—	4	1	no	—
101	C₂×C₄×C₅₃	—	4	2	no	—
102	C₂×C₄×C₅₃	—	4	2	no	—
103	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
105	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
106	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
107	C₂×C₄×C₅₃	#104	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×Dic₅₃	—	2	1	no	—
110	C₂×Dic₅₃	—	2	1	no	—
111	C₂×Dic₅₃	—	4	2	no	—
112	#88324	—	4	4	no	—
113	C₈×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
114	C₈×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
115	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
116	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
117	#88322	—	4	2	no	—
118	#88322	#102	1	1	yes	characteristic
119	#88322	#102	1	1	yes	characteristic
120	#88322	C₂×D₈	1	1	yes	characteristic
121	#88322	C₂×SD₁₆	1	1	yes	characteristic
122	#11604	—	53	1	no	maximal, Sylow
123	C₂²×C₅₃	—	4	1	no	—
124	C₂²×C₅₃	—	4	2	no	—
125	C₂²×C₅₃	—	8	2	no	—
126	C₂²×C₅₃	D₄⋊C₈	1	1	yes	characteristic
127	C₂²×C₅₃	D₄⋊C₈	1	1	yes	characteristic
128	C₂²×C₅₃	C₂×#104	1	1	yes	characteristic
129	C₄×C₅₃	—	4	2	no	—
130	C₄×C₅₃	D₄⋊C₈	1	1	yes	characteristic
131	C₄×C₅₃	D₄⋊C₈	1	1	yes	characteristic
132	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
133	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
134	Dic₅₃	—	2	2	no	—
135	Dic₅₃	—	4	1	no	—
136	C₂×#443	—	53	1	no	—
137	C₂×#364	—	53	1	no	—
138	#1450	—	53	1	no	—
139	#1336	—	53	4	no	—
140	C₂×C₅₃	—	2	1	no	—
141	C₂×C₅₃	—	2	1	no	—
142	C₂×C₅₃	—	2	1	no	—
143	C₂×C₅₃	—	8	2	no	—
144	C₂×C₅₃	C₂×D₄⋊C₈	1	1	yes	characteristic
145	C₂×#135	#6157	1	1	yes	characteristic
146	C₂×#131	—	106	1	no	—
147	C₂×M₃₂	—	106	1	no	—
148	#443	—	53	2	no	—
149	#443	—	53	2	no	—
150	#443	—	106	1	no	—
151	#443	—	106	1	no	—
152	#431	—	106	1	no	—
153	C₂×#108	—	53	1	no	—
154	C₂×#104	—	106	1	no	—
155	C₂×C₄×C₈	—	53	1	no	—
156	#364	—	53	2	no	—
157	#364	—	106	1	no	—
158	#322	—	53	2	no	—
159	C₈²	—	106	1	no	—
160	C₅₃	#11604	1	1	yes	characteristic, complemented, Sylow
161	C₂×#41	—	2	1	no	—
162	#135	—	2	1	no	—
163	#135	—	2	1	no	—
164	#135	#88324	1	2	yes	complemented
165	#135	C₂×#6157	1	2	yes	—
166	C₂×Q₁₆	#88324	1	1	yes	characteristic, complemented
167	C₂×SD₁₆	—	2	1	no	—
168	C₂×D₈	#88324	1	1	yes	characteristic, complemented
169	#131	—	106	2	no	—
170	#131	—	106	2	no	—
171	#131	—	212	1	no	—
172	#131	—	212	1	no	—
173	C₂×M₁₆	—	106	1	no	—
174	C₂×M₁₆	—	212	1	no	—
175	C₂²×C₈	—	53	1	no	—
176	C₂²×C₈	—	212	1	no	—
177	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
178	C₂×C₄²	—	53	1	no	—
179	M₃₂	—	106	2	no	—
180	M₃₂	—	212	1	no	—
181	C₂×C₁₆	—	212	1	no	—
182	#108	—	53	2	no	—
183	#108	—	106	1	no	—
184	C₄⋊C₈	—	106	2	no	—
185	#104	—	106	2	no	—
186	#104	—	106	2	no	—
187	C₂²⋊C₈	—	106	2	no	—
188	C₄×C₈	—	53	1	no	—
189	C₄×C₈	—	53	1	no	—
190	C₄×C₈	—	53	1	no	—
191	C₄×C₈	—	53	1	no	—
192	C₄×C₈	—	106	2	no	—
193	#41	—	2	2	no	—
194	#41	—	2	2	no	—
195	#41	—	4	1	no	—
196	#41	—	4	1	no	—
197	C₂×Q₈	—	2	1	no	—
198	C₂×D₄	—	2	1	no	—
199	C₂×D₄	—	4	1	no	—
200	C₂²×C₄	—	4	1	no	—
201	C₂²×C₄	—	106	1	no	—
202	C₂²×C₄	C₂²⋊#6157	1	1	yes	characteristic
203	Q₁₆	—	2	1	no	—
204	Q₁₆	C₂×#88324	1	2	yes	—
205	SD₁₆	—	2	2	no	—
206	SD₁₆	—	2	2	no	—
207	D₈	—	2	1	no	—
208	D₈	C₂×#88324	1	2	yes	—
209	M₁₆	—	106	2	no	—
210	M₁₆	—	212	2	no	—
211	M₁₆	—	212	2	no	—
212	M₁₆	—	212	1	no	—
213	C₂×C₈	—	53	1	no	—
214	C₂×C₈	—	53	1	no	—
215	C₂×C₈	—	53	1	no	—
216	C₂×C₈	—	53	1	no	—
217	C₂×C₈	—	53	1	no	—
218	C₂×C₈	—	53	1	no	—
219	C₂×C₈	—	106	4	no	—
220	C₂×C₈	—	212	2	no	—
221	C₂×C₈	—	212	2	no	—
222	C₂×C₈	—	212	1	no	—
223	C₂×C₈	C₂×#88324	1	1	yes	characteristic
224	C₂×C₈	C₂×#88324	1	1	yes	characteristic
225	C₂×C₈	#11719940	1	1	yes	characteristic
226	C₂×C₈	#11719940	1	1	yes	characteristic
227	C₂×C₈	C₂²⋊#6157	1	1	yes	characteristic
228	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
229	C₄²	—	53	1	no	—
230	C₄²	—	53	1	no	—
231	C₄²	—	106	1	no	—
232	C₁₆	—	212	2	no	—
233	C₂³	—	4	1	no	—
234	C₂³	#13133281	1	1	yes	characteristic
235	Q₈	—	2	2	no	—
236	Q₈	—	4	1	no	—
237	D₄	—	2	2	no	—
238	D₄	—	4	2	no	—
239	D₄	—	4	2	no	—
240	D₄	—	4	1	no	—
241	C₂×C₄	—	4	1	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	106	1	no	—
245	C₂×C₄	—	106	1	no	—
246	C₂×C₄	—	212	2	no	—
247	C₂×C₄	C₂²⋊#88324	1	1	yes	characteristic
248	C₂×C₄	C₂²⋊#88324	1	1	yes	characteristic
249	C₂×C₄	#13133279	1	1	yes	characteristic
250	C₂×C₄	#13133279	1	1	yes	characteristic
251	C₂×C₄	#13133281	1	1	yes	characteristic
252	C₂×C₄	C₂×C₂²⋊#6157	1	1	yes	characteristic
253	C₈	—	53	1	no	—
254	C₈	—	53	1	no	—
255	C₈	—	53	1	no	—
256	C₈	—	53	1	no	—
257	C₈	—	212	4	no	—
258	C₈	—	212	2	no	—
259	C₈	C₂²⋊#88324	1	1	yes	characteristic
260	C₈	C₂²⋊#88324	1	1	yes	characteristic
261	C₈	C₂²×#88324	1	1	yes	characteristic
262	C₈	C₂×#11719940	1	1	yes	characteristic
263	C₂²	—	4	1	no	—
264	C₂²	—	4	2	no	—
265	C₂²	—	8	2	no	—
266	C₂²	D₄⋊#88324	1	1	yes	characteristic
267	C₂²	D₄⋊#88324	1	1	yes	characteristic
268	C₂²	C₂×#13133281	1	1	yes	characteristic
269	C₄	—	4	2	no	—
270	C₄	—	106	2	no	—
271	C₄	—	212	1	no	—
272	C₄	D₄⋊#88324	1	1	yes	characteristic
273	C₄	D₄⋊#88324	1	1	yes	characteristic
274	C₄	C₂×C₂²⋊#88324	1	1	yes	characteristic
275	C₄	C₂×#13133279	1	1	yes	characteristic
276	C₂	—	2	1	no	—
277	C₂	—	2	1	no	—
278	C₂	—	2	1	no	—
279	C₂	—	8	2	no	—
280	C₂	C₂×D₄⋊#88324	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#13133279
commutator	C₈×C₅₃	C₂²×C₈
Frattini	C₂×C₈	C₂²×#6157
Fitting, prime radical	C₂×#135×C₅₃	C₄
socle	C₂×C₅₃	C₂×D₄⋊C₈
cosocle	C₈×Dic₅₃	C₂³
hypercenter	C₂×#135	#6157
hypocenter	C₅₃	#11604
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#135

Complements

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

N	H ≅ G/N	type
1. C₂×#16811545	C₂	semidirect
2. #20289918	C₂	semidirect
3. #20289679	C₂	semidirect
8. #16811545	C₂²	semidirect
9. #16811464	C₂²	semidirect
46. #135×C₅₃	C₈	semidirect
48. C₂×Q₁₆×C₅₃	C₈	semidirect
50. C₂×D₈×C₅₃	C₈	semidirect
160. C₅₃	#11604	semidirect
164. #135	#88324	semidirect
166. C₂×Q₁₆	#88324	semidirect
168. C₂×D₈	#88324	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811545	2
#20289918	2
#20289679	2
C₂×#16810740	2
#11604	53