Subgroups — C₄×#16811521 #28310482

Order 13568.

490 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811521	C₂	1	2	yes	maximal
2	C₄×#13133291	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×#11719910	C₂	1	1	yes	maximal, characteristic, complemented
4	#20289769	C₂	1	2	yes	maximal, complemented
5	C₄×#13133224	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133291	C₂²	1	2	yes	—
7	C₂²×#11719910	C₂²	1	2	yes	—
8	#16811521	C₄	1	8	yes	direct product factor
9	#16811511	—	2	1	no	—
10	C₄×#11719910	—	2	1	no	—
11	C₄×#11719910	—	2	1	no	—
12	C₄×#11719910	—	2	1	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	C₂×C₄	1	4	yes	complemented
26	C₂×#11719910	—	2	2	no	—
27	C₂×#11719910	—	2	2	no	—
28	C₂×#11719910	—	2	2	no	—
29	C₂×#11719910	C₂×C₄	1	4	yes	complemented
30	#13133224	—	2	2	no	—
31	#13133224	—	2	2	no	—
32	#13133224	C₈	1	4	yes	—
33	#13133224	C₂×C₄	1	4	yes	—
34	C₂×#11719917	—	2	1	no	—
35	C₂×#11719917	—	2	1	no	—
36	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
37	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
38	C₂²×#88322	—	2	1	no	—
39	C₂²×#88322	—	2	1	no	—
40	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
41	C₂²×#88322	C₂³	1	1	yes	characteristic
42	#13133128	—	2	1	no	—
43	#13133128	—	2	1	no	—
44	#13133128	—	2	1	no	—
45	C₄×#88322	—	2	1	no	—
46	C₄×#88322	—	2	1	no	—
47	C₄×#88322	—	2	1	no	—
48	C₄×#88322	D₄	1	1	yes	characteristic
49	C₄×#88322	D₄	1	1	yes	characteristic
50	C₄×#88322	D₄	1	1	yes	characteristic
51	C₄×#88322	D₄	1	1	yes	characteristic
52	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₄×Q₈×C₅₃	C₈	1	2	yes	—
54	C₄×D₄×C₅₃	—	2	2	no	—
55	C₄×D₄×C₅₃	C₈	1	2	yes	—
56	#117×C₅₃	—	2	1	no	—
57	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₅₃	—	2	1	no	—
59	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
60	#41×C₅₃	—	2	1	no	—
61	#41×C₅₃	—	2	1	no	—
62	#41×C₅₃	C₄²	1	2	yes	—
63	#41×C₅₃	C₂×C₈	1	2	yes	—
64	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
65	C₂×D₄×C₅₃	—	2	1	no	—
66	C₂×D₄×C₅₃	C₂×C₈	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	—	2	1	no	—
68	C₂²×C₄×C₅₃	—	2	2	no	—
69	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
70	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	C₄⋊C₄×C₅₃	—	2	2	no	—
72	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
73	C₂²⋊C₄×C₅₃	—	2	1	no	—
74	C₂²⋊C₄×C₅₃	—	2	2	no	—
75	C₄²×C₅₃	—	2	2	no	—
76	C₄²×C₅₃	—	2	1	no	—
77	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₄²×C₅₃	C₂×C₈	1	2	yes	—
80	#11719917	—	2	1	no	—
81	#11719917	—	2	1	no	—
82	#11719917	—	2	4	no	—
83	#11719917	—	2	4	no	—
84	#11719917	C₄²	1	2	yes	—
85	#11719917	C₂×C₈	1	2	yes	—
86	C₂×#88322	—	2	1	no	—
87	C₂×#88322	—	2	1	no	—
88	C₂×#88322	—	2	4	no	—
89	C₂×#88322	—	2	4	no	—
90	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×#88322	C₂×C₈	1	2	yes	—
93	C₂×#88322	M₁₆	1	2	yes	—
94	C₂×#88322	C₂²×C₄	1	2	yes	—
95	C₂×#88322	C₂×D₄	1	1	yes	characteristic
96	C₂×#88322	C₂×D₄	1	1	yes	characteristic
97	C₂×#88322	#41	1	2	yes	—
98	#11719910	—	2	4	no	—
99	#11719910	—	2	4	no	—
100	C₂³×C₅₃	—	2	1	no	—
101	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
102	Q₈×C₅₃	—	2	1	no	—
103	Q₈×C₅₃	C₄×C₈	1	2	yes	—
104	D₄×C₅₃	—	2	2	no	—
105	D₄×C₅₃	—	2	2	no	—
106	D₄×C₅₃	—	2	1	no	—
107	D₄×C₅₃	C₄×C₈	1	2	yes	—
108	C₂×C₄×C₅₃	—	2	1	no	—
109	C₂×C₄×C₅₃	—	2	4	no	—
110	C₂×C₄×C₅₃	—	2	2	no	—
111	C₂×C₄×C₅₃	—	2	2	no	—
112	C₂×C₄×C₅₃	—	2	2	no	—
113	C₂×C₄×C₅₃	—	4	4	no	—
114	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
115	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
116	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
122	#88322	—	2	2	no	—
123	#88322	—	2	2	no	—
124	#88322	C₂²⋊C₈	1	2	yes	—
125	#88322	C₂²⋊C₈	1	2	yes	—
126	#88322	C₄×D₄	1	2	yes	—
127	#88322	C₄×D₄	1	2	yes	—
128	C₄×#350	—	53	1	no	maximal, Sylow
129	C₂²×C₅₃	—	2	2	no	—
130	C₂²×C₅₃	—	2	2	no	—
131	C₂²×C₅₃	—	4	2	no	—
132	C₂²×C₅₃	C₄×M₁₆	1	1	yes	characteristic
133	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
134	C₂²×C₅₃	#431	1	1	yes	characteristic
135	C₄×C₅₃	—	2	2	no	—
136	C₄×C₅₃	—	2	2	no	—
137	C₄×C₅₃	—	4	2	no	—
138	C₄×C₅₃	#350	1	4	yes	complemented
139	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
140	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
141	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
142	C₄×C₅₃	#431	1	1	yes	characteristic
143	C₄×#131	—	53	1	no	—
144	C₂×#350	—	53	2	no	—
145	C₄×M₃₂	—	53	1	no	—
146	C₂×C₄×C₁₆	—	53	1	no	—
147	#1379	—	53	2	no	—
148	C₂×C₅₃	—	2	1	no	—
149	C₂×C₅₃	—	2	1	no	—
150	C₂×C₅₃	—	4	2	no	—
151	C₂×C₅₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
152	C₂×C₅₃	C₂×#350	1	2	yes	—
153	C₂×#131	—	53	1	no	—
154	C₂×#131	—	53	1	no	—
155	C₄×#41	#6157	1	1	yes	characteristic
156	C₂×M₃₂	—	53	2	no	—
157	C₂²×C₁₆	—	53	2	no	—
158	#405	—	53	1	no	—
159	#405	—	53	1	no	—
160	#405	—	106	1	no	—
161	#405	—	106	1	no	—
162	C₄×M₁₆	—	53	1	no	—
163	C₄×M₁₆	—	106	1	no	—
164	C₂×C₄×C₈	—	53	1	no	—
165	C₂×C₄×C₈	—	106	1	no	—
166	#350	—	53	8	no	—
167	#346	—	106	1	no	—
168	C₄×C₁₆	—	106	1	no	—
169	C₄×C₁₆	—	106	1	no	—
170	C₄×C₁₆	—	106	1	no	—
171	C₅₃	C₄×#350	1	1	yes	characteristic, complemented, Sylow
172	C₂×#41	C₂×#6157	1	1	yes	characteristic
173	#131	—	53	4	no	—
174	#131	—	53	4	no	—
175	#131	—	106	2	no	—
176	#131	—	106	2	no	—
177	C₂×M₁₆	—	53	1	no	—
178	C₂×M₁₆	—	53	1	no	—
179	C₂×M₁₆	—	106	1	no	—
180	C₂×M₁₆	—	106	1	no	—
181	C₂²×C₈	—	53	1	no	—
182	C₂²×C₈	—	53	1	no	—
183	C₂²×C₈	—	106	1	no	—
184	C₂²×C₈	—	106	1	no	—
185	C₄×Q₈	#88324	1	2	yes	—
186	C₄×D₄	—	2	2	no	—
187	C₄×D₄	#88324	1	2	yes	—
188	#117	—	2	1	no	—
189	#117	C₂×#6157	1	1	yes	characteristic
190	C₂×C₄²	—	2	1	no	—
191	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
192	M₃₂	—	53	4	no	—
193	M₃₂	—	106	2	no	—
194	C₂×C₁₆	—	53	4	no	—
195	C₂×C₁₆	—	106	2	no	—
196	C₂×C₁₆	—	106	2	no	—
197	C₂×C₁₆	—	106	2	no	—
198	#97	—	106	1	no	—
199	#97	—	106	1	no	—
200	#97	—	106	1	no	—
201	C₄×C₈	—	53	1	no	—
202	C₄×C₈	—	53	1	no	—
203	C₄×C₈	—	53	1	no	—
204	C₄×C₈	—	53	1	no	—
205	C₄×C₈	—	106	1	no	—
206	C₄×C₈	—	106	1	no	—
207	C₄×C₈	—	106	1	no	—
208	#41	—	2	1	no	—
209	#41	—	2	1	no	—
210	#41	D₅₃⋊C₈	1	2	yes	—
211	#41	C₄×#6157	1	2	yes	—
212	C₂×Q₈	C₂×#88324	1	1	yes	characteristic
213	C₂×D₄	—	2	1	no	—
214	C₂×D₄	C₂×#88324	1	1	yes	characteristic
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	#11719940	1	2	yes	—
218	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
219	M₁₆	—	53	2	no	—
220	M₁₆	—	53	2	no	—
221	M₁₆	—	106	4	no	—
222	M₁₆	—	106	4	no	—
223	M₁₆	—	106	1	no	—
224	M₁₆	—	106	1	no	—
225	C₂×C₈	—	53	2	no	—
226	C₂×C₈	—	53	1	no	—
227	C₂×C₈	—	53	1	no	—
228	C₂×C₈	—	53	1	no	—
229	C₂×C₈	—	53	1	no	—
230	C₂×C₈	—	53	2	no	—
231	C₂×C₈	—	53	2	no	—
232	C₂×C₈	—	53	2	no	—
233	C₂×C₈	—	106	4	no	—
234	C₂×C₈	—	106	4	no	—
235	C₂×C₈	—	106	1	no	—
236	C₂×C₈	—	106	1	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	C₂×#88324	1	2	yes	—
239	C₂²⋊C₄	—	2	1	no	—
240	C₂²⋊C₄	—	2	2	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	2	1	no	—
243	C₄²	C₂×#88324	1	2	yes	—
244	C₄²	C₂²⋊#6157	1	1	yes	characteristic
245	C₄²	C₂²⋊#6157	1	1	yes	characteristic
246	C₁₆	—	106	4	no	—
247	C₁₆	—	106	4	no	—
248	C₂³	—	2	1	no	—
249	C₂³	C₂×#11719940	1	1	yes	characteristic
250	Q₈	—	2	1	no	—
251	Q₈	C₄×#88324	1	2	yes	—
252	D₄	—	2	2	no	—
253	D₄	—	2	2	no	—
254	D₄	—	2	1	no	—
255	D₄	C₄×#88324	1	2	yes	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	C₂²⋊#88324	1	2	yes	—
263	C₂×C₄	C₂²⋊#88324	1	2	yes	—
264	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
265	C₂×C₄	C₂×C₂²⋊#6157	1	1	yes	characteristic
266	C₂×C₄	#13133296	1	1	yes	characteristic
267	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
268	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
269	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
270	C₈	—	53	2	no	—
271	C₈	—	53	2	no	—
272	C₈	—	53	2	no	—
273	C₈	—	53	2	no	—
274	C₈	—	106	2	no	—
275	C₈	—	106	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	4	2	no	—
279	C₂²	C₄×#11719940	1	1	yes	characteristic
280	C₂²	C₂×C₂²⋊#88324	1	1	yes	characteristic
281	C₂²	#16811640	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	4	2	no	—
285	C₄	#16811521	1	4	yes	direct product factor
286	C₄	C₄×C₂²⋊#6157	1	1	yes	characteristic
287	C₄	C₂×C₄×#88324	1	1	yes	characteristic
288	C₄	D₅₃⋊(C₂²⋊C₈)	1	1	yes	characteristic
289	C₄	#16811639	1	1	yes	characteristic
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	—	4	2	no	—
293	C₂	C₄×C₂²⋊#88324	1	1	yes	characteristic
294	C₂	C₂×#16811521	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂²⋊#6157
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₄×#41×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂×#41×C₅₃	C₂×C₄
cosocle	C₂²×#88322	C₂³
hypercenter	C₄×#41	#6157
hypocenter	C₅₃	C₄×#350
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
2. C₄×#13133291	C₂	semidirect
3. C₂×C₄×#11719910	C₂	semidirect
4. #20289769	C₂	semidirect
8. #16811521	C₄	direct (N × H)
25. #13133291	C₂×C₄	semidirect
29. C₂×#11719910	C₂×C₄	semidirect
138. C₄×C₅₃	#350	semidirect
171. C₅₃	C₄×#350	semidirect
285. C₄	#16811521	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16811521	2
C₄×#13133291	2
C₂×C₄×#11719910	2
#20289769	2
C₄×#13133224	2
C₄×#350	53

Subgroups — C4×#16811521 #28310482

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16811521 #28310482