Subgroups — #28311160

Order 13568.

440 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811546	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16811546	C₂	1	1	yes	maximal, characteristic, complemented
3	#20290957	C₂	1	2	yes	maximal
4	#20289765	C₂	1	2	yes	maximal, complemented
5	C₄×#13133224	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133291	C₂²	1	2	yes	—
7	C₂×#13133291	C₂²	1	2	yes	—
8	#16811546	—	2	1	no	—
9	#16811546	—	2	1	no	—
10	#16811546	—	2	1	no	—
11	#16811546	—	2	1	no	—
12	#16811521	—	2	4	no	—
13	C₂×#13133224	C₄	1	1	yes	characteristic
14	C₂×#13133224	C₂²	1	1	yes	characteristic
15	#16810579	—	2	1	no	—
16	#16810579	—	2	1	no	—
17	#16810579	C₄	1	1	yes	characteristic
18	#16810579	C₂²	1	1	yes	characteristic
19	C₄×#11719917	—	2	1	no	—
20	C₄×#11719917	C₄	1	1	yes	characteristic
21	C₂×C₄×#88322	—	2	1	no	—
22	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
23	C₄×#41×C₅₃	C₄	1	1	yes	characteristic
24	#13133291	—	2	2	no	—
25	#13133291	—	2	2	no	—
26	#13133291	—	2	2	no	—
27	#13133291	—	2	2	no	—
28	C₂×#11719910	—	2	2	no	—
29	C₂×#11719910	—	2	2	no	—
30	#13133224	—	2	2	no	—
31	#13133224	—	2	1	no	—
32	#13133224	—	2	1	no	—
33	#13133224	C₈	1	4	yes	—
34	#13133224	D₄	1	2	yes	—
35	#13133224	Q₈	1	2	yes	—
36	C₂×#11719917	—	2	1	no	—
37	C₂×#11719917	—	2	1	no	—
38	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
39	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
40	C₂²×#88322	—	2	1	no	—
41	C₂²×#88322	—	2	1	no	—
42	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
43	C₂²×#88322	C₂³	1	1	yes	characteristic
44	#13133128	—	2	1	no	—
45	#13133128	—	2	1	no	—
46	#13133128	—	2	1	no	—
47	C₄×#88322	—	2	1	no	—
48	C₄×#88322	—	2	1	no	—
49	C₄×#88322	—	2	1	no	—
50	C₄×#88322	D₄	1	1	yes	characteristic
51	C₄×#88322	D₄	1	1	yes	characteristic
52	C₄×#88322	D₄	1	1	yes	characteristic
53	C₄×#88322	D₄	1	1	yes	characteristic
54	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₄×Q₈×C₅₃	—	2	1	no	—
56	C₄×D₄×C₅₃	—	2	2	no	—
57	C₄×D₄×C₅₃	—	2	1	no	—
58	#117×C₅₃	—	2	1	no	—
59	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²×C₅₃	—	2	1	no	—
61	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
62	#41×C₅₃	—	2	1	no	—
63	#41×C₅₃	—	2	1	no	—
64	#41×C₅₃	C₄⋊C₄	1	2	yes	—
65	#41×C₅₃	C₂×C₈	1	2	yes	—
66	C₂×Q₈×C₅₃	M₁₆	1	1	yes	characteristic
67	C₂×D₄×C₅₃	—	2	1	no	—
68	C₂×D₄×C₅₃	M₁₆	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	—	2	1	no	—
70	C₂²×C₄×C₅₃	—	2	2	no	—
71	C₂²×C₄×C₅₃	—	2	1	no	—
72	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₄⋊C₄×C₅₃	—	2	2	no	—
74	C₄⋊C₄×C₅₃	—	2	1	no	—
75	C₂²⋊C₄×C₅₃	—	2	2	no	—
76	C₂²⋊C₄×C₅₃	M₁₆	1	2	yes	—
77	C₄²×C₅₃	—	2	2	no	—
78	C₄²×C₅₃	—	2	1	no	—
79	C₄²×C₅₃	—	2	1	no	—
80	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
81	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
82	#11719917	—	2	1	no	—
83	#11719917	—	2	1	no	—
84	#11719917	—	2	4	no	—
85	#11719917	—	2	2	no	—
86	#11719917	—	2	2	no	—
87	#11719917	C₄⋊C₄	1	2	yes	—
88	#11719917	C₂×C₈	1	2	yes	—
89	C₂×#88322	—	2	1	no	—
90	C₂×#88322	—	2	1	no	—
91	C₂×#88322	—	2	4	no	—
92	C₂×#88322	—	2	2	no	—
93	C₂×#88322	—	2	2	no	—
94	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×#88322	C₂×C₈	1	2	yes	—
97	C₂×#88322	M₁₆	1	2	yes	—
98	C₂×#88322	C₂×D₄	1	1	yes	characteristic
99	C₂×#88322	C₂×D₄	1	1	yes	characteristic
100	C₂×#88322	C₂×D₄	1	1	yes	characteristic
101	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
102	C₂×#88322	#41	1	1	yes	characteristic
103	C₂×#88322	#41	1	1	yes	characteristic
104	#11719910	—	4	2	no	—
105	#11719910	—	4	2	no	—
106	C₂³×C₅₃	—	2	1	no	—
107	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
108	Q₈×C₅₃	—	2	1	no	—
109	Q₈×C₅₃	C₄⋊C₈	1	2	yes	—
110	D₄×C₅₃	—	2	2	no	—
111	D₄×C₅₃	—	2	2	no	—
112	D₄×C₅₃	—	2	1	no	—
113	D₄×C₅₃	C₄⋊C₈	1	2	yes	—
114	C₂×C₄×C₅₃	—	2	1	no	—
115	C₂×C₄×C₅₃	—	2	1	no	—
116	C₂×C₄×C₅₃	—	2	2	no	—
117	C₂×C₄×C₅₃	—	2	2	no	—
118	C₂×C₄×C₅₃	—	2	2	no	—
119	C₂×C₄×C₅₃	—	2	1	no	—
120	C₂×C₄×C₅₃	—	4	2	no	—
121	C₂×C₄×C₅₃	—	4	4	no	—
122	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
128	#88322	—	2	2	no	—
129	#88322	—	2	2	no	—
130	#88322	C₂²⋊C₈	1	2	yes	—
131	#88322	C₂²⋊C₈	1	2	yes	—
132	#88322	#121	1	1	yes	characteristic
133	#88322	#121	1	1	yes	characteristic
134	#88322	C₂²⋊Q₈	1	1	yes	characteristic
135	#88322	C₂²⋊Q₈	1	1	yes	characteristic
136	#12792	—	53	1	no	maximal, Sylow
137	C₂²×C₅₃	—	2	2	no	—
138	C₂²×C₅₃	—	4	1	no	—
139	C₂²×C₅₃	—	4	2	no	—
140	C₂²×C₅₃	#407	1	1	yes	characteristic
141	C₂²×C₅₃	#423	1	1	yes	characteristic
142	C₂²×C₅₃	#431	1	1	yes	characteristic
143	C₄×C₅₃	—	2	2	no	—
144	C₄×C₅₃	—	2	2	no	—
145	C₄×C₅₃	—	4	2	no	—
146	C₄×C₅₃	—	4	1	no	—
147	C₄×C₅₃	#380	1	1	yes	characteristic
148	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
149	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
150	C₄×C₅₃	#432	1	1	yes	characteristic
151	C₄×#131	—	53	1	no	—
152	C₂×#364	—	53	1	no	—
153	C₂×#364	—	53	1	no	—
154	#2117	—	53	2	no	—
155	#1378	—	53	2	no	—
156	C₂×C₅₃	—	2	1	no	—
157	C₂×C₅₃	—	2	1	no	—
158	C₂×C₅₃	—	2	1	no	—
159	C₂×C₅₃	—	4	2	no	—
160	C₂×C₅₃	#1797	1	1	yes	characteristic
161	C₂×#131	—	53	1	no	—
162	C₂×#131	—	53	1	no	—
163	C₄×#41	#6157	1	1	yes	characteristic
164	C₂×M₃₂	—	53	2	no	—
165	C₂×M₃₂	—	53	2	no	—
166	#405	—	53	1	no	—
167	#405	—	53	1	no	—
168	#405	—	106	1	no	—
169	#405	—	106	1	no	—
170	C₄×M₁₆	—	53	1	no	—
171	C₄×M₁₆	—	106	1	no	—
172	C₂×C₄×C₈	—	53	1	no	—
173	C₂×C₄×C₈	—	106	1	no	—
174	#364	—	106	1	no	—
175	#364	—	106	1	no	—
176	#364	—	106	1	no	—
177	#364	—	106	1	no	—
178	#350	—	106	4	no	—
179	C₅₃	#12792	1	1	yes	characteristic, complemented, Sylow
180	C₂×#41	C₂×#6157	1	1	yes	characteristic
181	#131	—	53	2	no	—
182	#131	—	53	2	no	—
183	#131	—	53	4	no	—
184	#131	—	106	1	no	—
185	#131	—	106	1	no	—
186	#131	—	106	2	no	—
187	C₂×M₁₆	—	53	1	no	—
188	C₂×M₁₆	—	53	1	no	—
189	C₂×M₁₆	—	106	1	no	—
190	C₂×M₁₆	—	106	1	no	—
191	C₂²×C₈	—	53	1	no	—
192	C₂²×C₈	—	53	1	no	—
193	C₂²×C₈	—	106	1	no	—
194	C₂²×C₈	—	106	1	no	—
195	C₄×Q₈	—	2	1	no	—
196	C₄×D₄	—	2	1	no	—
197	C₄×D₄	—	2	2	no	—
198	#117	—	2	1	no	—
199	#117	C₂×#6157	1	1	yes	characteristic
200	C₂×C₄²	—	2	1	no	—
201	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
202	M₃₂	—	106	2	no	—
203	M₃₂	—	106	2	no	—
204	M₃₂	—	106	2	no	—
205	M₃₂	—	106	2	no	—
206	C₂×C₁₆	—	106	2	no	—
207	C₂×C₁₆	—	106	2	no	—
208	#97	—	106	1	no	—
209	#97	—	106	1	no	—
210	#97	—	106	1	no	—
211	C₄×C₈	—	53	1	no	—
212	C₄×C₈	—	53	1	no	—
213	C₄×C₈	—	53	1	no	—
214	C₄×C₈	—	53	1	no	—
215	C₄×C₈	—	106	1	no	—
216	C₄×C₈	—	106	1	no	—
217	C₄×C₈	—	106	1	no	—
218	#41	—	2	1	no	—
219	#41	—	2	1	no	—
220	#41	D₅₃⋊C₈	1	2	yes	—
221	#41	C₄⋊#6157	1	2	yes	—
222	C₂×Q₈	#11719940	1	1	yes	characteristic
223	C₂×D₄	—	2	1	no	—
224	C₂×D₄	#11719940	1	1	yes	characteristic
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	1	no	—
228	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
229	M₁₆	—	53	2	no	—
230	M₁₆	—	53	2	no	—
231	M₁₆	—	106	2	no	—
232	M₁₆	—	106	2	no	—
233	M₁₆	—	106	4	no	—
234	M₁₆	—	106	1	no	—
235	M₁₆	—	106	1	no	—
236	C₂×C₈	—	53	1	no	—
237	C₂×C₈	—	53	1	no	—
238	C₂×C₈	—	53	1	no	—
239	C₂×C₈	—	53	1	no	—
240	C₂×C₈	—	53	1	no	—
241	C₂×C₈	—	53	1	no	—
242	C₂×C₈	—	53	1	no	—
243	C₂×C₈	—	53	1	no	—
244	C₂×C₈	—	53	2	no	—
245	C₂×C₈	—	53	2	no	—
246	C₂×C₈	—	106	2	no	—
247	C₂×C₈	—	106	2	no	—
248	C₂×C₈	—	106	4	no	—
249	C₂×C₈	—	106	1	no	—
250	C₂×C₈	—	106	1	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	1	no	—
253	C₂²⋊C₄	—	2	2	no	—
254	C₂²⋊C₄	#11719940	1	2	yes	—
255	C₄²	—	2	2	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	2	1	no	—
258	C₄²	C₂²⋊#6157	1	1	yes	characteristic
259	C₄²	C₂²⋊#6157	1	1	yes	characteristic
260	C₁₆	—	212	2	no	—
261	C₁₆	—	212	2	no	—
262	C₂³	—	2	1	no	—
263	C₂³	C₂×#11719940	1	1	yes	characteristic
264	Q₈	—	2	1	no	—
265	Q₈	#13133284	1	2	yes	—
266	D₄	—	2	2	no	—
267	D₄	—	2	2	no	—
268	D₄	—	2	1	no	—
269	D₄	#13133284	1	2	yes	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	C₂×C₂²⋊#6157	1	1	yes	characteristic
279	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
280	C₂×C₄	#13133296	1	1	yes	characteristic
281	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
282	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
283	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
284	C₈	—	53	1	no	—
285	C₈	—	53	1	no	—
286	C₈	—	53	1	no	—
287	C₈	—	53	1	no	—
288	C₈	—	53	2	no	—
289	C₈	—	53	2	no	—
290	C₈	—	106	2	no	—
291	C₈	—	106	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	4	1	no	—
294	C₂²	—	4	2	no	—
295	C₂²	#16811593	1	1	yes	characteristic
296	C₂²	#16811626	1	1	yes	characteristic
297	C₂²	#16811640	1	1	yes	characteristic
298	C₄	—	2	2	no	—
299	C₄	—	2	2	no	—
300	C₄	—	4	2	no	—
301	C₄	—	4	1	no	—
302	C₄	#16811565	1	1	yes	characteristic
303	C₄	D₅₃⋊(C₂²⋊C₈)	1	1	yes	characteristic
304	C₄	C₂×#13133284	1	1	yes	characteristic
305	C₄	#16811643	1	1	yes	characteristic
306	C₂	—	2	1	no	—
307	C₂	—	2	1	no	—
308	C₂	—	2	1	no	—
309	C₂	—	4	2	no	—
310	C₂	#20290571	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16811565
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₄×#41×C₅₃	C₄
socle	C₂×C₅₃	#1797
prime radical	C₂×#41×C₅₃	C₂×C₄
cosocle	C₂²×#88322	C₂³
hypercenter	C₄×#41	#6157
hypocenter	C₅₃	#12792
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. C₂×#16811546	C₂	semidirect
2. C₂×#16811546	C₂	semidirect
4. #20289765	C₂	semidirect
179. C₅₃	#12792	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811546	2
#20290957	2
#20289765	2
C₄×#13133224	2
#12792	53