Subgroups — C₄×#16810676 #27382217

Order 13568.

1318 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810676	C₂	1	2	yes	maximal
2	C₄×C₄⋊#88322	C₂	1	4	yes	maximal, complemented
3	C₄×C₂²⋊#88322	C₂	1	4	yes	maximal
4	#20274461	C₂	1	2	yes	maximal, complemented
5	C₄²×#88322	C₂	1	2	yes	maximal, complemented
6	C₂×C₄³×C₅₃	C₂	1	1	yes	maximal, characteristic
7	#16810676	C₄	1	16	yes	direct product factor
8	C₂×C₄⋊#88322	C₂²	1	4	yes	—
9	C₂×C₂²⋊#88322	C₂²	1	4	yes	—
10	C₂×C₄×#88322	C₂²	1	8	yes	—
11	C₂×C₄×#88322	C₂²	1	4	yes	—
12	#16810378	C₂²	1	8	yes	—
13	C₂²×C₄²×C₅₃	C₄	1	2	yes	—
14	C₂²×C₄²×C₅₃	C₄	1	2	yes	—
15	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
16	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
17	C₄³×C₅₃	C₄	1	2	yes	—
18	C₄³×C₅₃	C₄	1	2	yes	—
19	C₄³×C₅₃	C₂²	1	1	yes	characteristic
20	C₄³×C₅₃	C₂²	1	2	yes	—
21	C₄³×C₅₃	C₂²	1	1	yes	characteristic
22	C₂²×#88322	C₂³	1	8	yes	—
23	C₄⋊#88322	C₂×C₄	1	16	yes	complemented
24	C₂²⋊#88322	C₂×C₄	1	16	yes	—
25	C₄×#88322	—	2	16	no	—
26	C₄×#88322	C₂×C₄	1	16	yes	complemented
27	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₅₃	C₂×C₄	1	4	yes	—
29	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₅₃	—	2	8	no	—
32	C₂×C₄²×C₅₃	—	2	8	no	—
33	C₂×C₄²×C₅₃	C₈	1	16	yes	complemented
34	C₂×C₄²×C₅₃	C₂×C₄	1	8	yes	—
35	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
45	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
48	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	—	2	8	no	—
50	C₂²×C₄×C₅₃	—	2	4	no	—
51	C₂²×C₄×C₅₃	—	2	8	no	—
52	C₂²×C₄×C₅₃	—	2	4	no	—
53	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
54	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
55	C₂²×C₄×C₅₃	C₂×C₈	1	16	yes	—
56	C₂²×C₄×C₅₃	C₂×C₈	1	8	yes	—
57	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
60	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
61	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
66	C₄²×C₅₃	—	2	8	no	—
67	C₄²×C₅₃	—	2	4	no	—
68	C₄²×C₅₃	—	2	8	no	—
69	C₄²×C₅₃	—	2	4	no	—
70	C₄²×C₅₃	C₄²	1	4	yes	—
71	C₄²×C₅₃	C₄²	1	4	yes	—
72	C₄²×C₅₃	C₂×C₈	1	8	yes	—
73	C₄²×C₅₃	C₂×C₈	1	16	yes	—
74	C₄²×C₅₃	C₂×C₈	1	8	yes	—
75	C₄²×C₅₃	M₁₆	1	8	yes	—
76	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
77	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
78	C₄²×C₅₃	#41	1	8	yes	—
79	C₂×#88322	—	2	16	no	—
80	C₂×#88322	C₂²×C₄	1	16	yes	—
81	C₂³×C₅₃	—	2	2	no	—
82	C₂³×C₅₃	—	2	2	no	—
83	C₂³×C₅₃	C₂×C₄²	1	2	yes	—
84	C₂³×C₅₃	C₂²×C₈	1	4	yes	—
85	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
86	C₂×C₄×C₅₃	—	2	4	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	—	2	2	no	—
89	C₂×C₄×C₅₃	—	2	4	no	—
90	C₂×C₄×C₅₃	—	2	8	no	—
91	C₂×C₄×C₅₃	—	2	8	no	—
92	C₂×C₄×C₅₃	—	2	32	no	—
93	C₂×C₄×C₅₃	C₄×C₈	1	16	yes	complemented
94	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
95	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
96	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
97	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
98	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
99	C₂×C₄×C₅₃	#117	1	8	yes	—
100	C₂×C₄×C₅₃	C₂²×C₈	1	8	yes	—
101	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
102	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
103	C₂×C₄×C₅₃	C₂²×C₈	1	8	yes	—
104	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
105	C₂×C₄×C₅₃	C₂×M₁₆	1	8	yes	—
106	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
107	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
108	#88322	—	2	16	no	—
109	C₄×#431	—	53	1	no	maximal, Sylow
110	C₂²×C₅₃	—	2	2	no	—
111	C₂²×C₅₃	—	2	8	no	—
112	C₂²×C₅₃	—	2	2	no	—
113	C₂²×C₅₃	C₂×C₄×C₈	1	4	yes	—
114	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
115	C₂²×C₅₃	C₂×#117	1	2	yes	—
116	C₂²×C₅₃	C₂³×C₈	1	2	yes	—
117	C₂²×C₅₃	C₂²×M₁₆	1	2	yes	—
118	C₄×C₅₃	—	2	2	no	—
119	C₄×C₅₃	—	2	8	no	—
120	C₄×C₅₃	—	2	2	no	—
121	C₄×C₅₃	C₂×C₄×C₈	1	4	yes	—
122	C₄×C₅₃	C₂×C₄×C₈	1	4	yes	—
123	C₄×C₅₃	C₄×M₁₆	1	4	yes	—
124	C₄×C₅₃	#431	1	16	yes	complemented
125	C₄×C₅₃	C₄×#41	1	4	yes	—
126	C₂×#431	—	53	2	no	—
127	C₂×C₄³	D₅₃	1	1	yes	characteristic
128	C₄×C₄⋊C₈	—	53	4	no	—
129	C₄×C₂²⋊C₈	—	53	4	no	—
130	#1744	—	53	2	no	—
131	C₄²×C₈	—	53	2	no	—
132	C₂×C₅₃	—	2	4	no	—
133	C₂×C₅₃	C₄×#117	1	1	yes	characteristic
134	C₂×C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic
135	C₂×C₅₃	C₂×C₄×M₁₆	1	1	yes	characteristic
136	C₂×C₅₃	C₂×#431	1	4	yes	—
137	C₂²×C₄²	Dic₅₃	1	2	yes	—
138	C₂²×C₄²	Dic₅₃	1	2	yes	—
139	C₂²×C₄²	C₂×D₅₃	1	2	yes	—
140	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
141	#431	—	53	16	no	—
142	C₂×C₄⋊C₈	—	53	4	no	—
143	C₂×C₂²⋊C₈	—	53	4	no	—
144	C₂×C₄×C₈	—	53	4	no	—
145	C₂×C₄×C₈	—	53	8	no	—
146	C₄³	Dic₅₃	1	2	yes	—
147	C₄³	Dic₅₃	1	2	yes	—
148	C₄³	C₂×D₅₃	1	1	yes	characteristic
149	C₄³	C₂×D₅₃	1	2	yes	—
150	C₄³	C₂×D₅₃	1	1	yes	characteristic
151	#336	—	53	8	no	—
152	C₅₃	C₄×#431	1	1	yes	characteristic, complemented, Sylow
153	C₂³×C₄	C₂×Dic₅₃	1	4	yes	—
154	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
155	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
156	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
157	C₂²×C₈	—	53	8	no	—
158	C₂×C₄²	—	2	8	no	—
159	C₂×C₄²	—	2	8	no	—
160	C₂×C₄²	#88322	1	16	yes	complemented
161	C₂×C₄²	C₄×D₅₃	1	8	yes	—
162	C₂×C₄²	C₂×Dic₅₃	1	4	yes	—
163	C₂×C₄²	C₂×Dic₅₃	1	4	yes	—
164	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
165	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
166	C₂×C₄²	C₂×Dic₅₃	1	4	yes	—
167	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
168	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
169	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
170	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
171	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
172	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
173	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
174	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
175	C₄⋊C₈	—	53	16	no	—
176	C₂²⋊C₈	—	53	16	no	—
177	C₄×C₈	—	53	16	no	—
178	C₄×C₈	—	106	16	no	—
179	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
180	C₂²×C₄	—	2	8	no	—
181	C₂²×C₄	—	2	4	no	—
182	C₂²×C₄	—	2	8	no	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	C₂×#88322	1	16	yes	—
185	C₂²×C₄	C₂×#88322	1	8	yes	—
186	C₂²×C₄	C₄×Dic₅₃	1	4	yes	—
187	C₂²×C₄	C₄×Dic₅₃	1	4	yes	—
188	C₂²×C₄	C₂×C₄×D₅₃	1	4	yes	—
189	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
190	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
191	C₂²×C₄	C₂²×Dic₅₃	1	4	yes	—
192	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
193	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
194	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
195	C₂²×C₄	C₂²×Dic₅₃	1	4	yes	—
196	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
197	C₂×C₈	—	53	16	no	—
198	C₂×C₈	—	106	16	no	—
199	C₄²	—	2	8	no	—
200	C₄²	—	2	4	no	—
201	C₄²	—	2	8	no	—
202	C₄²	—	2	4	no	—
203	C₄²	C₂×#88322	1	8	yes	—
204	C₄²	C₂×#88322	1	16	yes	—
205	C₄²	C₂×#88322	1	8	yes	—
206	C₄²	#11719917	1	8	yes	—
207	C₄²	C₄×Dic₅₃	1	4	yes	—
208	C₄²	C₄×Dic₅₃	1	4	yes	—
209	C₄²	C₂×C₄×D₅₃	1	4	yes	—
210	C₄²	C₂×C₄×D₅₃	1	4	yes	—
211	C₄²	#11719945	1	8	yes	—
212	C₂³	—	2	2	no	—
213	C₂³	—	2	2	no	—
214	C₂³	C₂×C₄×Dic₅₃	1	2	yes	—
215	C₂³	C₂²×#88322	1	4	yes	—
216	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	8	no	—
222	C₂×C₄	—	2	8	no	—
223	C₂×C₄	—	2	32	no	—
224	C₂×C₄	C₄×#88322	1	16	yes	complemented
225	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
226	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
227	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
228	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
229	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
230	C₂×C₄	#13133176	1	8	yes	—
231	C₂×C₄	C₂²×#88322	1	8	yes	—
232	C₂×C₄	C₂²×#88322	1	4	yes	—
233	C₂×C₄	C₂²×#88322	1	4	yes	—
234	C₂×C₄	C₂²×#88322	1	8	yes	—
235	C₂×C₄	C₂×#11719917	1	4	yes	—
236	C₂×C₄	C₂×#11719917	1	8	yes	—
237	C₂×C₄	C₂²×C₄×D₅₃	1	2	yes	—
238	C₂×C₄	C₂×#11719945	1	4	yes	—
239	C₈	—	106	16	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	8	no	—
242	C₂²	—	2	2	no	—
243	C₂²	C₂×C₄×#88322	1	4	yes	—
244	C₂²	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
245	C₂²	C₂×#13133176	1	2	yes	—
246	C₂²	C₂³×#88322	1	2	yes	—
247	C₂²	C₂²×#11719917	1	2	yes	—
248	C₄	—	2	2	no	—
249	C₄	—	2	8	no	—
250	C₄	—	2	2	no	—
251	C₄	C₂×C₄×#88322	1	4	yes	—
252	C₄	C₂×C₄×#88322	1	4	yes	—
253	C₄	C₄×#11719917	1	4	yes	—
254	C₄	#16810676	1	16	yes	direct product factor
255	C₄	C₄×#11719945	1	4	yes	—
256	C₂	—	2	4	no	—
257	C₂	C₄×#13133176	1	1	yes	characteristic
258	C₂	C₂²×C₄×#88322	1	1	yes	characteristic
259	C₂	C₂×C₄×#11719917	1	1	yes	characteristic
260	C₂	C₂×#16810676	1	4	yes	—

	subgroup	G/H
center	C₄³	C₂×D₅₃
commutator	C₂×C₅₃	C₂²×C₄×C₈
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₄³×C₅₃	C₂
socle	C₂³×C₅₃	C₂³×C₄
prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×C₄³	D₅₃
hypocenter	C₅₃	C₄×#431
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄³ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
2. C₄×C₄⋊#88322	C₂	semidirect
4. #20274461	C₂	semidirect
5. C₄²×#88322	C₂	semidirect
7. #16810676	C₄	direct (N × H)
23. C₄⋊#88322	C₂×C₄	semidirect
26. C₄×#88322	C₂×C₄	semidirect
33. C₂×C₄²×C₅₃	C₈	semidirect
93. C₂×C₄×C₅₃	C₄×C₈	semidirect
124. C₄×C₅₃	#431	semidirect
152. C₅₃	C₄×#431	semidirect
160. C₂×C₄²	#88322	semidirect
224. C₂×C₄	C₄×#88322	semidirect
254. C₄	#16810676	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×#16810676	2
C₄×C₄⋊#88322	2
C₄×C₂²⋊#88322	2
#20274461	2
C₄²×#88322	2
C₂×C₄³×C₅₃	2
C₄×#431	53

Subgroups — C4×#16810676 #27382217

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16810676 #27382217