Subgroups — C₄×#16810405 #27299132

Order 13568.

522 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	C₂×#16810405	C₂	1	2	yes	maximal
2	C₄×#13133157	C₂	1	2	yes	maximal, complemented
3	#20272799	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133157	C₂²	1	4	yes	—
6	#16810405	C₄	1	8	yes	direct product factor
7	#16810399	—	2	2	no	—
8	C₄×#11719908	—	2	2	no	—
9	C₂²×M₁₆×C₅₃	C₄	1	1	yes	characteristic
10	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
11	C₂²×C₄²×C₅₃	C₄	1	1	yes	characteristic
12	C₄×M₁₆×C₅₃	—	2	2	no	—
13	C₄×M₁₆×C₅₃	—	2	2	no	—
14	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
15	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	#13133157	—	2	4	no	—
19	#13133157	C₂×C₄	1	8	yes	complemented
20	C₂×#11719908	—	2	4	no	—
21	C₂³×C₄×C₅₃	C₈	1	2	yes	—
22	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×M₁₆×C₅₃	—	2	2	no	—
24	C₂×M₁₆×C₅₃	—	2	2	no	—
25	C₂×M₁₆×C₅₃	—	2	2	no	—
26	C₂×M₁₆×C₅₃	—	2	2	no	—
27	C₂×M₁₆×C₅₃	C₈	1	4	yes	—
28	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
29	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	—	2	1	no	—
34	C₂×C₄²×C₅₃	—	2	1	no	—
35	C₂×C₄²×C₅₃	—	2	2	no	—
36	C₂×C₄²×C₅₃	C₈	1	2	yes	—
37	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
39	#97×C₅₃	—	2	2	no	—
40	#97×C₅₃	—	2	2	no	—
41	C₄×C₈×C₅₃	—	2	1	no	—
42	C₄×C₈×C₅₃	—	2	1	no	—
43	C₄×C₈×C₅₃	D₄	1	2	yes	—
44	C₄×C₈×C₅₃	D₄	1	2	yes	—
45	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	—	2	1	no	—
47	C₂²×C₄×C₅₃	—	2	1	no	—
48	C₂²×C₄×C₅₃	—	2	1	no	—
49	C₂²×C₄×C₅₃	—	2	1	no	—
50	C₂²×C₄×C₅₃	—	2	2	no	—
51	C₂²×C₄×C₅₃	—	2	2	no	—
52	C₂²×C₄×C₅₃	—	4	4	no	—
53	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
54	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
56	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
57	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
58	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
59	M₁₆×C₅₃	—	2	8	no	—
60	M₁₆×C₅₃	—	2	8	no	—
61	C₂×C₈×C₅₃	—	2	1	no	—
62	C₂×C₈×C₅₃	—	2	1	no	—
63	C₂×C₈×C₅₃	—	2	1	no	—
64	C₂×C₈×C₅₃	—	2	1	no	—
65	C₂×C₈×C₅₃	C₄²	1	2	yes	—
66	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
67	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
68	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
69	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
70	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₅₃	#41	1	2	yes	—
73	C₄²×C₅₃	—	2	2	no	—
74	C₄²×C₅₃	—	2	2	no	—
75	C₄²×C₅₃	—	2	1	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	M₁₆	1	2	yes	—
80	#11719908	—	2	8	no	—
81	C₂³×C₅₃	—	2	1	no	—
82	C₂³×C₅₃	—	2	1	no	—
83	C₂³×C₅₃	—	4	2	no	—
84	C₂³×C₅₃	C₄×C₈	1	2	yes	—
85	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
86	C₂×C₄×C₅₃	—	2	1	no	—
87	C₂×C₄×C₅₃	—	2	1	no	—
88	C₂×C₄×C₅₃	—	2	4	no	—
89	C₂×C₄×C₅₃	—	2	2	no	—
90	C₂×C₄×C₅₃	—	2	2	no	—
91	C₂×C₄×C₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	—	2	2	no	—
93	C₂×C₄×C₅₃	—	2	2	no	—
94	C₂×C₄×C₅₃	—	4	4	no	—
95	C₂×C₄×C₅₃	—	4	4	no	—
96	C₂×C₄×C₅₃	—	4	2	no	—
97	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
98	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
99	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
100	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
106	C₈×C₅₃	—	2	2	no	—
107	C₈×C₅₃	—	2	2	no	—
108	C₈×C₅₃	C₂²⋊C₈	1	4	yes	—
109	C₈×C₅₃	C₄×D₄	1	4	yes	—
110	C₄×#349	—	53	1	no	maximal, Sylow
111	C₂²×C₅₃	—	2	2	no	—
112	C₂²×C₅₃	—	2	2	no	—
113	C₂²×C₅₃	—	4	2	no	—
114	C₂²×C₅₃	—	4	2	no	—
115	C₂²×C₅₃	—	4	2	no	—
116	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
117	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
118	C₂²×C₅₃	#431	1	1	yes	characteristic
119	C₄×C₅₃	—	2	2	no	—
120	C₄×C₅₃	—	2	2	no	—
121	C₄×C₅₃	—	4	2	no	—
122	C₄×C₅₃	#349	1	4	yes	complemented
123	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
124	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
125	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
126	C₄×C₅₃	#431	1	1	yes	characteristic
127	C₂×C₄×M₁₆	D₅₃	1	1	yes	characteristic
128	C₂×#349	—	53	2	no	—
129	C₄×M₃₂	—	53	2	no	—
130	#1377	—	53	2	no	—
131	C₂×C₅₃	—	2	1	no	—
132	C₂×C₅₃	—	2	1	no	—
133	C₂×C₅₃	—	4	2	no	—
134	C₂×C₅₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
135	C₂×C₅₃	C₂×#349	1	2	yes	—
136	C₂²×M₁₆	Dic₅₃	1	1	yes	characteristic
137	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
138	C₂²×C₄²	Dic₅₃	1	1	yes	characteristic
139	C₂×M₃₂	—	53	4	no	—
140	C₄×M₁₆	—	2	2	no	—
141	C₄×M₁₆	—	2	2	no	—
142	C₂×#97	Dic₅₃	1	1	yes	characteristic
143	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
144	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
145	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
146	#349	—	53	8	no	—
147	#346	—	106	2	no	—
148	C₄×C₁₆	—	106	2	no	—
149	C₅₃	C₄×#349	1	1	yes	characteristic, complemented, Sylow
150	C₂³×C₄	#88322	1	2	yes	—
151	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
152	C₂×M₁₆	—	2	2	no	—
153	C₂×M₁₆	—	2	2	no	—
154	C₂×M₁₆	—	2	2	no	—
155	C₂×M₁₆	—	2	2	no	—
156	C₂×M₁₆	#88322	1	4	yes	—
157	C₂×M₁₆	C₄×D₅₃	1	4	yes	—
158	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
159	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
160	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
161	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
162	C₂×C₄²	—	2	2	no	—
163	C₂×C₄²	—	2	1	no	—
164	C₂×C₄²	—	2	1	no	—
165	C₂×C₄²	#88322	1	2	yes	—
166	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
167	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
168	M₃₂	—	53	8	no	—
169	M₃₂	—	106	4	no	—
170	C₂×C₁₆	—	106	4	no	—
171	#97	—	2	2	no	—
172	#97	—	2	2	no	—
173	C₄×C₈	—	2	1	no	—
174	C₄×C₈	—	2	1	no	—
175	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
176	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
177	C₂⁴	C₂×#88322	1	1	yes	characteristic
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	—	2	1	no	—
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	—	4	4	no	—
185	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
186	C₂²×C₄	C₂×#88322	1	2	yes	—
187	C₂²×C₄	C₂×#88322	1	2	yes	—
188	C₂²×C₄	C₂×#88322	1	2	yes	—
189	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
190	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
191	M₁₆	—	2	8	no	—
192	M₁₆	—	2	8	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	C₂×#88322	1	2	yes	—
198	C₂×C₈	C₂×#88322	1	2	yes	—
199	C₂×C₈	#11719917	1	2	yes	—
200	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
201	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
202	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
203	C₂×C₈	#11719945	1	2	yes	—
204	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
205	C₄²	—	2	2	no	—
206	C₄²	—	2	2	no	—
207	C₄²	—	2	1	no	—
208	C₄²	—	2	1	no	—
209	C₄²	#11719917	1	2	yes	—
210	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
211	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
212	C₁₆	—	106	8	no	—
213	C₂³	—	2	1	no	—
214	C₂³	—	2	1	no	—
215	C₂³	—	4	2	no	—
216	C₂³	C₄×#88322	1	2	yes	—
217	C₂³	C₂²×#88322	1	1	yes	characteristic
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	—	4	4	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	C₄×#88322	1	2	yes	—
230	C₂×C₄	C₂²⋊#88322	1	2	yes	—
231	C₂×C₄	C₂²⋊#88322	1	2	yes	—
232	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
233	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
234	C₂×C₄	#13133176	1	1	yes	characteristic
235	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
236	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
237	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
238	C₈	—	2	2	no	—
239	C₈	—	2	2	no	—
240	C₈	C₂²⋊#88322	1	4	yes	—
241	C₈	C₄×C₂²⋊D₅₃	1	4	yes	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	4	2	no	—
245	C₂²	—	4	2	no	—
246	C₂²	—	4	2	no	—
247	C₂²	C₂×C₄×#88322	1	1	yes	characteristic
248	C₂²	C₂×C₂²⋊#88322	1	1	yes	characteristic
249	C₂²	#16810676	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	4	2	no	—
253	C₄	#16810405	1	4	yes	direct product factor
254	C₄	C₄×C₂²⋊Dic₅₃	1	1	yes	characteristic
255	C₄	C₄×#11719917	1	1	yes	characteristic
256	C₄	C₂×C₂²⋊#88322	1	1	yes	characteristic
257	C₄	#16810676	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	—	4	2	no	—
261	C₂	C₄×C₂²⋊#88322	1	1	yes	characteristic
262	C₂	C₂×#16810405	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂²⋊Dic₅₃
commutator	C₂²×C₅₃	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₄×M₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₅₃	C₂×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×C₄×M₁₆	D₅₃
hypocenter	C₅₃	C₄×#349
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₄×#13133157	C₂	semidirect
3. #20272799	C₂	semidirect
6. #16810405	C₄	direct (N × H)
19. #13133157	C₂×C₄	semidirect
122. C₄×C₅₃	#349	semidirect
149. C₅₃	C₄×#349	semidirect
253. C₄	#16810405	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×#16810405	2
C₄×#13133157	2
#20272799	2
C₂×C₄×M₁₆×C₅₃	2
C₄×#349	53

Subgroups — C4×#16810405 #27299132

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16810405 #27299132