Subgroups — C₂²×C₄×#11719940 #28329173

Order 13568.

3950 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11719940	C₂	1	2	yes	maximal
2	C₂×C₄×#11719940	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#13133272	C₂	1	2	yes	maximal, complemented
4	C₂²×C₄×#88324	C₂	1	2	yes	maximal, complemented
5	C₂³×C₄×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719940	C₄	1	32	yes	direct product factor
7	C₂²×#11719940	C₂²	1	24	yes	—
8	C₂³×#88324	C₂²	1	4	yes	—
9	C₄×#11719940	C₂²	1	64	yes	direct product factor
10	C₂×#13133272	C₂²	1	24	yes	complemented
11	C₂×C₄×#88324	C₂²	1	24	yes	complemented
12	C₂⁴×Dic₅₃	C₄	1	1	yes	characteristic
13	C₂⁴×Dic₅₃	C₂²	1	1	yes	characteristic
14	C₂²×C₄×Dic₅₃	C₄	1	12	yes	—
15	C₂²×C₄×Dic₅₃	C₄	1	1	yes	characteristic
16	C₂²×C₄×Dic₅₃	C₄	1	1	yes	characteristic
17	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
18	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
19	C₂²×C₄×Dic₅₃	C₂²	1	12	yes	—
20	C₂⁴×C₄×C₅₃	C₄	1	1	yes	characteristic
21	C₂×#11719940	C₂×C₄	1	192	yes	direct product factor
22	C₂×#11719940	C₂³	1	32	yes	—
23	C₂²×#88324	C₂×C₄	1	32	yes	complemented
24	C₂²×#88324	C₂³	1	24	yes	—
25	#13133272	C₂³	1	32	yes	complemented
26	C₄×#88324	C₂³	1	32	yes	complemented
27	C₂³×Dic₅₃	—	2	8	no	—
28	C₂³×Dic₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂³×Dic₅₃	C₂×C₄	1	12	yes	—
30	C₂³×Dic₅₃	C₂×C₄	1	6	yes	—
31	C₂³×Dic₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂³×Dic₅₃	C₂×C₄	1	16	yes	—
33	C₂³×Dic₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂³×Dic₅₃	C₂³	1	1	yes	characteristic
35	C₂³×Dic₅₃	C₂³	1	6	yes	—
36	C₂×C₄×Dic₅₃	C₂×C₄	1	6	yes	—
37	C₂×C₄×Dic₅₃	C₂×C₄	1	48	yes	—
38	C₂×C₄×Dic₅₃	C₂×C₄	1	12	yes	—
39	C₂×C₄×Dic₅₃	C₂×C₄	1	12	yes	—
40	C₂×C₄×Dic₅₃	C₂×C₄	1	6	yes	—
41	C₂×C₄×Dic₅₃	C₂³	1	6	yes	—
42	C₂×C₄×Dic₅₃	C₂³	1	6	yes	—
43	C₂×C₄×Dic₅₃	C₂³	1	16	yes	—
44	C₂⁵×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂³×C₄×C₅₃	—	2	8	no	—
46	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂³×C₄×C₅₃	C₂×C₄	1	12	yes	—
48	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂⁴×C₅₃	—	2	4	no	—
50	C₂⁴×C₅₃	—	2	4	no	—
51	C₂⁴×C₅₃	C₄²	1	8	yes	—
52	C₂⁴×C₅₃	C₂²×C₄	1	6	yes	—
53	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	—	2	4	no	—
55	C₂²×C₄×C₅₃	—	2	48	no	—
56	C₂²×C₄×C₅₃	M₁₆	1	8	yes	complemented
57	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
58	C₂²×C₄×C₅₃	C₂²×C₄	1	16	yes	—
59	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
60	C₂²×Dic₅₃	—	2	4	no	—
61	C₂²×Dic₅₃	—	2	48	no	—
62	C₂²×Dic₅₃	C₄²	1	48	yes	—
63	C₂²×Dic₅₃	C₄²	1	8	yes	—
64	C₂²×Dic₅₃	M₁₆	1	8	yes	complemented
65	C₂²×Dic₅₃	C₂²×C₄	1	6	yes	—
66	C₂²×Dic₅₃	C₂²×C₄	1	16	yes	—
67	C₂²×Dic₅₃	C₂²×C₄	1	12	yes	—
68	C₂²×Dic₅₃	C₂²×C₄	1	3	yes	—
69	C₂²×Dic₅₃	C₂²×C₄	1	6	yes	—
70	C₂²×Dic₅₃	C₂²×C₄	1	8	yes	—
71	C₂²×Dic₅₃	C₂²×C₄	1	48	yes	—
72	C₂²×Dic₅₃	C₂²×C₄	1	6	yes	—
73	C₂²×Dic₅₃	C₂⁴	1	3	yes	—
74	C₂²×Dic₅₃	C₂⁴	1	4	yes	—
75	#11719940	C₂²×C₄	1	128	yes	direct product factor
76	C₂×#88324	C₂²×C₄	1	96	yes	complemented
77	C₂×#88324	C₂⁴	1	16	yes	—
78	C₄×Dic₅₃	C₂²×C₄	1	12	yes	—
79	C₄×Dic₅₃	C₂²×C₄	1	16	yes	—
80	C₄×Dic₅₃	C₂²×C₄	1	16	yes	—
81	C₄×Dic₅₃	C₂²×C₄	1	12	yes	—
82	C₄×Dic₅₃	C₂⁴	1	4	yes	—
83	C₄×Dic₅₃	C₂⁴	1	4	yes	—
84	C₂³×C₅₃	—	2	12	no	—
85	C₂³×C₅₃	—	2	12	no	—
86	C₂³×C₅₃	—	2	32	no	—
87	C₂³×C₅₃	C₂×C₄²	1	24	yes	—
88	C₂³×C₅₃	C₂×C₄²	1	4	yes	—
89	C₂³×C₅₃	C₂×M₁₆	1	4	yes	—
90	C₂³×C₅₃	C₂×M₁₆	1	4	yes	—
91	C₂³×C₅₃	C₂³×C₄	1	4	yes	—
92	C₂³×C₅₃	C₂³×C₄	1	3	yes	—
93	C₂×C₄×C₅₃	—	2	12	no	—
94	C₂×C₄×C₅₃	—	2	32	no	—
95	C₂×C₄×C₅₃	C₂×M₁₆	1	24	yes	complemented
96	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
97	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
98	C₂×Dic₅₃	—	2	12	no	—
99	C₂×Dic₅₃	—	2	32	no	—
100	C₂×Dic₅₃	C₂×C₄²	1	48	yes	—
101	C₂×Dic₅₃	C₂×C₄²	1	12	yes	—
102	C₂×Dic₅₃	C₂×C₄²	1	24	yes	—
103	C₂×Dic₅₃	C₂×M₁₆	1	24	yes	complemented
104	C₂×Dic₅₃	C₂³×C₄	1	4	yes	—
105	C₂×Dic₅₃	C₂³×C₄	1	4	yes	—
106	C₂×Dic₅₃	C₂³×C₄	1	12	yes	—
107	C₂×Dic₅₃	C₂³×C₄	1	16	yes	—
108	C₂×Dic₅₃	C₂³×C₄	1	3	yes	—
109	C₂×Dic₅₃	C₂³×C₄	1	4	yes	—
110	C₂×Dic₅₃	C₂⁵	1	1	yes	characteristic
111	#88324	C₂³×C₄	1	32	yes	complemented
112	C₂²×C₄×M₁₆	—	53	1	no	maximal, Sylow
113	C₂²×C₅₃	—	2	48	no	—
114	C₂²×C₅₃	—	2	4	no	—
115	C₂²×C₅₃	—	2	4	no	—
116	C₂²×C₅₃	C₄×M₁₆	1	16	yes	complemented
117	C₂²×C₅₃	C₂²×C₄²	1	8	yes	—
118	C₂²×C₅₃	C₂²×C₄²	1	6	yes	—
119	C₂²×C₅₃	C₂²×M₁₆	1	6	yes	—
120	C₂²×C₅₃	C₂²×M₁₆	1	6	yes	—
121	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
122	C₄×C₅₃	—	2	4	no	—
123	C₄×C₅₃	C₂²×M₁₆	1	8	yes	complemented
124	Dic₅₃	—	2	4	no	—
125	Dic₅₃	C₂²×C₄²	1	6	yes	—
126	Dic₅₃	C₂²×C₄²	1	8	yes	—
127	Dic₅₃	C₂²×M₁₆	1	8	yes	complemented
128	Dic₅₃	C₂⁴×C₄	1	2	yes	—
129	C₂³×M₁₆	—	53	2	no	—
130	C₂³×C₄²	—	53	1	no	—
131	C₂×C₄×M₁₆	—	53	24	no	—
132	C₂²×#97	—	53	2	no	—
133	C₂²×C₄×C₈	—	53	2	no	—
134	C₂×C₅₃	—	2	8	no	—
135	C₂×C₅₃	C₂×C₄×M₁₆	1	12	yes	complemented
136	C₂×C₅₃	C₂³×C₄²	1	1	yes	characteristic
137	C₂×C₅₃	C₂³×M₁₆	1	1	yes	characteristic
138	C₂×C₅₃	C₂³×M₁₆	1	1	yes	characteristic
139	C₂⁴×C₄	—	53	1	no	—
140	C₂⁴×C₄	—	53	1	no	—
141	C₂⁴×C₄	#6157	1	1	yes	characteristic
142	C₂²×M₁₆	—	53	24	no	—
143	C₂²×M₁₆	—	53	32	no	—
144	C₂³×C₈	—	53	4	no	—
145	C₂²×C₄²	—	53	1	no	—
146	C₂²×C₄²	—	53	12	no	—
147	C₂²×C₄²	—	53	1	no	—
148	C₂²×C₄²	—	53	12	no	—
149	C₂²×C₄²	—	53	1	no	—
150	C₂²×C₄²	—	53	1	no	—
151	C₄×M₁₆	—	53	64	no	—
152	C₂×#97	—	53	24	no	—
153	C₂×C₄×C₈	—	53	24	no	—
154	C₅₃	C₂²×C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
155	C₂⁵	C₂×#6157	1	1	yes	characteristic
156	C₂³×C₄	—	2	8	no	—
157	C₂³×C₄	—	53	16	no	—
158	C₂³×C₄	—	53	1	no	—
159	C₂³×C₄	—	53	6	no	—
160	C₂³×C₄	—	53	6	no	—
161	C₂³×C₄	—	53	1	no	—
162	C₂³×C₄	—	53	1	no	—
163	C₂³×C₄	—	53	12	no	—
164	C₂³×C₄	—	53	1	no	—
165	C₂³×C₄	—	106	8	no	—
166	C₂³×C₄	C₂×#6157	1	1	yes	characteristic
167	C₂³×C₄	C₂×#6157	1	12	yes	—
168	C₂³×C₄	C₂×#6157	1	1	yes	characteristic
169	C₂×M₁₆	—	53	32	no	—
170	C₂×M₁₆	—	53	192	no	—
171	C₂²×C₈	—	53	24	no	—
172	C₂²×C₈	—	53	32	no	—
173	C₂×C₄²	—	53	12	no	—
174	C₂×C₄²	—	53	6	no	—
175	C₂×C₄²	—	53	6	no	—
176	C₂×C₄²	—	53	48	no	—
177	C₂×C₄²	—	53	12	no	—
178	C₂×C₄²	—	53	6	no	—
179	C₂×C₄²	—	53	6	no	—
180	C₂×C₄²	—	53	16	no	—
181	#97	—	53	32	no	—
182	C₄×C₈	—	53	32	no	—
183	C₂⁴	—	2	4	no	—
184	C₂⁴	—	2	4	no	—
185	C₂⁴	C₄×#6157	1	8	yes	—
186	C₂⁴	C₂²×#6157	1	6	yes	—
187	C₂⁴	C₂²×#6157	1	1	yes	characteristic
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	2	48	no	—
190	C₂²×C₄	—	53	48	no	—
191	C₂²×C₄	—	53	8	no	—
192	C₂²×C₄	—	53	8	no	—
193	C₂²×C₄	—	53	48	no	—
194	C₂²×C₄	—	53	8	no	—
195	C₂²×C₄	—	53	12	no	—
196	C₂²×C₄	—	53	3	no	—
197	C₂²×C₄	—	53	6	no	—
198	C₂²×C₄	—	53	6	no	—
199	C₂²×C₄	—	53	3	no	—
200	C₂²×C₄	—	53	4	no	—
201	C₂²×C₄	—	53	16	no	—
202	C₂²×C₄	—	53	6	no	—
203	C₂²×C₄	—	106	48	no	—
204	C₂²×C₄	—	106	4	no	—
205	C₂²×C₄	#11719940	1	8	yes	direct product factor
206	C₂²×C₄	C₂²×#6157	1	6	yes	—
207	C₂²×C₄	C₂²×#6157	1	16	yes	—
208	C₂²×C₄	C₂²×#6157	1	6	yes	—
209	M₁₆	—	53	128	no	—
210	C₂×C₈	—	53	96	no	—
211	C₂×C₈	—	53	16	no	—
212	C₄²	—	53	4	no	—
213	C₄²	—	53	4	no	—
214	C₄²	—	53	16	no	—
215	C₄²	—	53	12	no	—
216	C₄²	—	53	12	no	—
217	C₄²	—	53	16	no	—
218	C₂³	—	2	32	no	—
219	C₂³	—	2	12	no	—
220	C₂³	—	2	12	no	—
221	C₂³	C₂×C₄×#6157	1	24	yes	—
222	C₂³	C₂×C₄×#6157	1	4	yes	—
223	C₂³	C₂×#11719940	1	4	yes	—
224	C₂³	D₅₃⋊M₁₆	1	4	yes	—
225	C₂³	C₂³×#6157	1	4	yes	—
226	C₂³	C₂³×#6157	1	3	yes	—
227	C₂×C₄	—	2	12	no	—
228	C₂×C₄	—	2	32	no	—
229	C₂×C₄	—	53	48	no	—
230	C₂×C₄	—	53	12	no	—
231	C₂×C₄	—	53	24	no	—
232	C₂×C₄	—	53	12	no	—
233	C₂×C₄	—	53	16	no	—
234	C₂×C₄	—	53	24	no	—
235	C₂×C₄	—	53	3	no	—
236	C₂×C₄	—	53	4	no	—
237	C₂×C₄	—	53	4	no	—
238	C₂×C₄	—	53	1	no	—
239	C₂×C₄	—	53	4	no	—
240	C₂×C₄	—	106	32	no	—
241	C₂×C₄	—	106	12	no	—
242	C₂×C₄	C₂×#11719940	1	24	yes	direct product factor
243	C₂×C₄	C₂³×#6157	1	4	yes	—
244	C₂×C₄	C₂³×#6157	1	4	yes	—
245	C₈	—	53	32	no	—
246	C₂²	—	2	4	no	—
247	C₂²	—	2	4	no	—
248	C₂²	—	2	48	no	—
249	C₂²	C₄×#11719940	1	16	yes	direct product factor
250	C₂²	C₂²×C₄×#6157	1	8	yes	—
251	C₂²	C₂²×C₄×#6157	1	6	yes	—
252	C₂²	C₂²×#11719940	1	6	yes	—
253	C₂²	C₂×D₅₃⋊M₁₆	1	6	yes	—
254	C₂²	C₂⁴×#6157	1	1	yes	characteristic
255	C₄	—	2	4	no	—
256	C₄	—	53	2	no	—
257	C₄	—	53	6	no	—
258	C₄	—	53	8	no	—
259	C₄	—	53	8	no	—
260	C₄	—	106	4	no	—
261	C₄	C₂²×#11719940	1	8	yes	direct product factor
262	C₂	—	2	8	no	—
263	C₂	C₂×C₄×#11719940	1	12	yes	direct product factor
264	C₂	C₂³×C₄×#6157	1	1	yes	characteristic
265	C₂	C₂³×#11719940	1	1	yes	characteristic
266	C₂	C₂²×D₅₃⋊M₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂×#6157
commutator	C₂×C₅₃	C₂³×C₄²
Frattini	C₂²	C₂⁴×#6157
Fitting	C₂⁴×C₄×C₅₃	C₄
socle	C₂⁴×C₅₃	C₂²×C₄
prime radical	C₂⁵×C₅₃	C₂×C₄
cosocle	C₂×Dic₅₃	C₂⁵
hypercenter	C₂⁴×C₄	#6157
hypocenter	C₅₃	C₂²×C₄×M₁₆
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×#11719940	C₂	direct (N × H)
3. C₂²×#13133272	C₂	semidirect
4. C₂²×C₄×#88324	C₂	semidirect
6. C₂²×#11719940	C₄	direct (N × H)
9. C₄×#11719940	C₂²	direct (N × H)
10. C₂×#13133272	C₂²	semidirect
11. C₂×C₄×#88324	C₂²	semidirect
21. C₂×#11719940	C₂×C₄	direct (N × H)
23. C₂²×#88324	C₂×C₄	semidirect
25. #13133272	C₂³	semidirect
26. C₄×#88324	C₂³	semidirect
56. C₂²×C₄×C₅₃	M₁₆	semidirect
64. C₂²×Dic₅₃	M₁₆	semidirect
75. #11719940	C₂²×C₄	direct (N × H)
76. C₂×#88324	C₂²×C₄	semidirect
95. C₂×C₄×C₅₃	C₂×M₁₆	semidirect
103. C₂×Dic₅₃	C₂×M₁₆	semidirect
111. #88324	C₂³×C₄	semidirect
116. C₂²×C₅₃	C₄×M₁₆	semidirect
123. C₄×C₅₃	C₂²×M₁₆	semidirect
127. Dic₅₃	C₂²×M₁₆	semidirect
135. C₂×C₅₃	C₂×C₄×M₁₆	semidirect
154. C₅₃	C₂²×C₄×M₁₆	semidirect
205. C₂²×C₄	#11719940	direct (N × H)
242. C₂×C₄	C₂×#11719940	direct (N × H)
249. C₂²	C₄×#11719940	direct (N × H)
261. C₄	C₂²×#11719940	direct (N × H)
263. C₂	C₂×C₄×#11719940	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#11719940	2
C₂×C₄×#11719940	2
C₂²×#13133272	2
C₂²×C₄×#88324	2
C₂³×C₄×Dic₅₃	2
C₂²×C₄×M₁₆	53

Subgroups — C22×C4×#11719940 #28329173

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×#11719940 #28329173