Subgroups — C₂×#20283134 #28247491

Order 13568.

3342 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11719917	C₂	1	2	yes	maximal
2	#20283134	C₂	1	16	yes	maximal, direct product factor
3	C₂×#16810579	C₂	1	8	yes	maximal, complemented
4	C₂×C₄×#11719917	C₂	1	4	yes	maximal, complemented
5	C₂²×#117×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719917	C₄	1	32	yes	complemented
7	C₂²×#11719917	C₂²	1	4	yes	—
8	C₂²×#11719917	C₂²	1	4	yes	—
9	C₂²×#11719917	C₂²	1	16	yes	—
10	C₂³×#88322	C₂²	1	4	yes	—
11	#16810579	C₂²	1	64	yes	complemented
12	C₄×#11719917	C₂²	1	32	yes	complemented
13	C₂×#13133128	C₂²	1	8	yes	complemented
14	C₂×C₄×#88322	C₂²	1	8	yes	complemented
15	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#117×C₅₃	C₄	1	4	yes	—
17	C₂×#117×C₅₃	C₄	1	8	yes	—
18	C₂×#117×C₅₃	C₂²	1	4	yes	—
19	C₂×#117×C₅₃	C₂²	1	8	yes	—
20	C₂²×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
21	C₂²×C₂²⋊C₄×C₅₃	C₄	1	2	yes	—
22	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
23	C₂×#11719917	C₂×C₄	1	32	yes	—
24	C₂×#11719917	C₂×C₄	1	32	yes	complemented
25	C₂×#11719917	C₂×C₄	1	128	yes	complemented
26	C₂×#11719917	C₂³	1	16	yes	—
27	C₂×#11719917	C₂³	1	16	yes	—
28	C₂²×#88322	C₂×C₄	1	32	yes	complemented
29	C₂²×#88322	C₂³	1	8	yes	—
30	C₂²×#88322	C₂³	1	16	yes	—
31	#13133128	C₂³	1	32	yes	complemented
32	C₄×#88322	C₂³	1	32	yes	complemented
33	C₂⁵×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄×C₅₃	C₂×C₄	1	4	yes	—
35	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
37	C₂³×C₄×C₅₃	C₂×C₄	1	4	yes	—
38	C₂³×C₄×C₅₃	C₂×C₄	1	16	yes	—
39	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
40	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
41	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
42	#117×C₅₃	C₂×C₄	1	16	yes	—
43	#117×C₅₃	C₂×C₄	1	32	yes	—
44	#117×C₅₃	C₂³	1	16	yes	—
45	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	8	yes	—
46	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	16	yes	—
47	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	8	yes	—
48	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	16	yes	—
49	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
50	C₂×C₄²×C₅₃	C₂×C₄	1	8	yes	—
51	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
52	C₂×C₄²×C₅₃	C₂³	1	8	yes	—
53	C₂⁴×C₅₃	—	4	4	no	—
54	C₂⁴×C₅₃	C₄²	1	8	yes	—
55	C₂⁴×C₅₃	C₂²×C₄	1	2	yes	—
56	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₂⁴×C₅₃	C₂²×C₄	1	4	yes	—
58	C₂²×C₄×C₅₃	—	4	8	no	—
59	C₂²×C₄×C₅₃	C₄²	1	8	yes	—
60	C₂²×C₄×C₅₃	C₄²	1	16	yes	—
61	C₂²×C₄×C₅₃	C₄²	1	32	yes	—
62	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
63	C₂²×C₄×C₅₃	C₂²×C₄	1	16	yes	—
64	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
67	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
69	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
70	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
71	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
72	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
73	C₂²×C₄×C₅₃	C₂²×C₄	1	32	yes	—
74	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
75	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
76	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
77	C₄⋊C₄×C₅₃	C₂²×C₄	1	32	yes	—
78	C₂²⋊C₄×C₅₃	C₂²×C₄	1	32	yes	—
79	C₄²×C₅₃	C₂²×C₄	1	8	yes	—
80	C₄²×C₅₃	C₂²×C₄	1	16	yes	—
81	C₄²×C₅₃	C₂⁴	1	8	yes	—
82	#11719917	C₂²×C₄	1	64	yes	—
83	#11719917	C₂²×C₄	1	64	yes	complemented
84	C₂×#88322	C₂²×C₄	1	32	yes	—
85	C₂×#88322	C₂²×C₄	1	64	yes	complemented
86	C₂×#88322	C₂⁴	1	16	yes	—
87	C₂³×C₅₃	—	2	4	no	—
88	C₂³×C₅₃	—	2	8	no	—
89	C₂³×C₅₃	—	4	16	no	—
90	C₂³×C₅₃	—	4	8	no	—
91	C₂³×C₅₃	C₂×C₄²	1	8	yes	—
92	C₂³×C₅₃	C₂×C₄²	1	4	yes	—
93	C₂³×C₅₃	C₂×C₄²	1	16	yes	—
94	C₂³×C₅₃	C₂³×C₄	1	2	yes	—
95	C₂³×C₅₃	C₂³×C₄	1	4	yes	—
96	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	—	2	16	no	—
98	C₂×C₄×C₅₃	—	4	16	no	—
99	C₂×C₄×C₅₃	C₂×C₄²	1	8	yes	—
100	C₂×C₄×C₅₃	C₂×C₄²	1	16	yes	—
101	C₂×C₄×C₅₃	C₂×C₄²	1	16	yes	—
102	C₂×C₄×C₅₃	C₂×C₄²	1	32	yes	—
103	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
104	C₂×C₄×C₅₃	C₂×C₄²	1	8	yes	—
105	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
106	C₂×C₄×C₅₃	C₂³×C₄	1	8	yes	—
107	C₂×C₄×C₅₃	C₂³×C₄	1	8	yes	—
108	C₂×C₄×C₅₃	C₂³×C₄	1	8	yes	—
109	C₂×C₄×C₅₃	C₂³×C₄	1	16	yes	—
110	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
111	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
113	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
114	#88322	C₂³×C₄	1	32	yes	—
115	C₂×#2873	—	53	1	no	maximal, Sylow
116	C₂²×C₅₃	—	2	8	no	—
117	C₂²×C₅₃	—	2	16	no	—
118	C₂²×C₅₃	—	2	4	no	—
119	C₂²×C₅₃	—	2	8	no	—
120	C₂²×C₅₃	—	4	16	no	—
121	C₂²×C₅₃	C₂²×C₄²	1	8	yes	—
122	C₂²×C₅₃	C₂²×C₄²	1	4	yes	—
123	C₂²×C₅₃	C₂²×C₄²	1	2	yes	—
124	C₂²×C₅₃	#568	1	4	yes	—
125	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
126	C₄×C₅₃	—	2	16	no	—
127	C₄×C₅₃	C₂²×C₄²	1	2	yes	—
128	C₄×C₅₃	C₂²×C₄²	1	8	yes	—
129	C₄×C₅₃	C₂²×C₄²	1	4	yes	—
130	C₄×C₅₃	C₂⁴×C₄	1	2	yes	—
131	C₂³×M₁₆	—	53	2	no	—
132	C₂²×#117	D₅₃	1	1	yes	characteristic
133	#2873	—	53	16	no	—
134	C₂×#405	—	53	8	no	—
135	C₂×C₄×M₁₆	—	53	4	no	—
136	C₂×C₅₃	—	2	4	no	—
137	C₂×C₅₃	—	2	8	no	—
138	C₂×C₅₃	#2873	1	4	yes	complemented
139	C₂×C₅₃	C₂³×C₄²	1	1	yes	characteristic
140	C₂×C₅₃	C₂×#568	1	2	yes	—
141	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
142	C₂²×M₁₆	—	53	32	no	—
143	C₂²×M₁₆	—	53	4	no	—
144	C₂²×M₁₆	—	53	4	no	—
145	C₂²×M₁₆	—	53	16	no	—
146	C₂³×C₈	—	53	4	no	—
147	C₂×#117	Dic₅₃	1	4	yes	—
148	C₂×#117	Dic₅₃	1	8	yes	—
149	C₂×#117	C₂×D₅₃	1	4	yes	—
150	C₂×#117	C₂×D₅₃	1	8	yes	—
151	C₂²×C₄⋊C₄	Dic₅₃	1	2	yes	—
152	C₂²×C₂²⋊C₄	Dic₅₃	1	2	yes	—
153	C₂²×C₄²	C₂×D₅₃	1	2	yes	—
154	#405	—	53	64	no	—
155	C₄×M₁₆	—	53	32	no	—
156	C₂×#97	—	53	8	no	—
157	C₂×C₄×C₈	—	53	8	no	—
158	C₅₃	C₂×#2873	1	1	yes	characteristic, complemented, Sylow
159	C₂⁵	C₂×Dic₅₃	1	1	yes	characteristic
160	C₂³×C₄	C₄×D₅₃	1	16	yes	—
161	C₂³×C₄	C₂×Dic₅₃	1	4	yes	—
162	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
163	C₂³×C₄	C₂×Dic₅₃	1	2	yes	—
164	C₂³×C₄	C₂×Dic₅₃	1	4	yes	—
165	C₂³×C₄	C₂²×D₅₃	1	4	yes	—
166	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
167	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
168	C₂×M₁₆	—	53	32	no	—
169	C₂×M₁₆	—	53	32	no	—
170	C₂×M₁₆	—	53	128	no	—
171	C₂×M₁₆	—	53	16	no	—
172	C₂×M₁₆	—	53	16	no	—
173	C₂²×C₈	—	53	32	no	—
174	C₂²×C₈	—	53	16	no	—
175	C₂²×C₈	—	53	8	no	—
176	#117	C₂×Dic₅₃	1	16	yes	—
177	#117	C₂×Dic₅₃	1	32	yes	—
178	#117	C₂²×D₅₃	1	16	yes	—
179	C₂×C₄⋊C₄	C₂×Dic₅₃	1	8	yes	—
180	C₂×C₄⋊C₄	C₂×Dic₅₃	1	16	yes	—
181	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	8	yes	—
182	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	16	yes	—
183	C₂×C₄²	C₂×Dic₅₃	1	4	yes	—
184	C₂×C₄²	C₂×Dic₅₃	1	8	yes	—
185	C₂×C₄²	C₂²×D₅₃	1	8	yes	—
186	C₂×C₄²	C₂²×D₅₃	1	4	yes	—
187	#97	—	53	32	no	—
188	C₄×C₈	—	53	32	no	—
189	C₂⁴	—	4	4	no	—
190	C₂⁴	C₄×Dic₅₃	1	8	yes	—
191	C₂⁴	C₂²×Dic₅₃	1	2	yes	—
192	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
193	C₂⁴	C₂²×Dic₅₃	1	4	yes	—
194	C₂²×C₄	—	4	8	no	—
195	C₂²×C₄	C₄×Dic₅₃	1	8	yes	—
196	C₂²×C₄	C₄×Dic₅₃	1	16	yes	—
197	C₂²×C₄	C₄×Dic₅₃	1	32	yes	—
198	C₂²×C₄	C₂×C₄×D₅₃	1	8	yes	—
199	C₂²×C₄	C₂×C₄×D₅₃	1	32	yes	—
200	C₂²×C₄	C₂×C₄×D₅₃	1	8	yes	—
201	C₂²×C₄	C₂×C₄×D₅₃	1	8	yes	—
202	C₂²×C₄	C₂²×Dic₅₃	1	8	yes	—
203	C₂²×C₄	C₂²×Dic₅₃	1	16	yes	—
204	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
205	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
206	C₂²×C₄	C₂²×Dic₅₃	1	4	yes	—
207	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
208	C₂²×C₄	C₂²×Dic₅₃	1	8	yes	—
209	C₂²×C₄	C₂²×Dic₅₃	1	4	yes	—
210	C₂²×C₄	C₂³×D₅₃	1	2	yes	—
211	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
212	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
213	M₁₆	—	53	64	no	—
214	M₁₆	—	53	64	no	—
215	C₂×C₈	—	53	32	no	—
216	C₂×C₈	—	53	64	no	—
217	C₂×C₈	—	53	16	no	—
218	C₄⋊C₄	C₂²×Dic₅₃	1	32	yes	—
219	C₂²⋊C₄	C₂²×Dic₅₃	1	32	yes	—
220	C₄²	C₂²×Dic₅₃	1	16	yes	—
221	C₄²	C₂²×Dic₅₃	1	8	yes	—
222	C₄²	C₂³×D₅₃	1	8	yes	—
223	C₂³	—	2	4	no	—
224	C₂³	—	2	8	no	—
225	C₂³	—	4	8	no	—
226	C₂³	—	4	16	no	—
227	C₂³	C₂×C₄×Dic₅₃	1	8	yes	—
228	C₂³	C₂×C₄×Dic₅₃	1	4	yes	—
229	C₂³	C₂×C₄×Dic₅₃	1	16	yes	—
230	C₂³	C₂³×Dic₅₃	1	4	yes	—
231	C₂³	C₂³×Dic₅₃	1	2	yes	—
232	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
233	C₂×C₄	—	2	16	no	—
234	C₂×C₄	—	4	16	no	—
235	C₂×C₄	C₂×C₄×Dic₅₃	1	16	yes	—
236	C₂×C₄	C₂×C₄×Dic₅₃	1	8	yes	—
237	C₂×C₄	C₂×C₄×Dic₅₃	1	16	yes	—
238	C₂×C₄	C₂×C₄×Dic₅₃	1	32	yes	—
239	C₂×C₄	C₂×C₄×Dic₅₃	1	8	yes	—
240	C₂×C₄	C₂×C₄×Dic₅₃	1	4	yes	—
241	C₂×C₄	C₂²×C₄×D₅₃	1	8	yes	—
242	C₂×C₄	C₂²×C₄×D₅₃	1	4	yes	—
243	C₂×C₄	C₂²×C₄×D₅₃	1	8	yes	—
244	C₂×C₄	C₂²×C₄×D₅₃	1	8	yes	—
245	C₂×C₄	C₂³×Dic₅₃	1	16	yes	—
246	C₂×C₄	C₂³×Dic₅₃	1	2	yes	—
247	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
248	C₂×C₄	C₂³×Dic₅₃	1	4	yes	—
249	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
250	C₈	—	53	32	no	—
251	C₂²	—	2	4	no	—
252	C₂²	—	2	8	no	—
253	C₂²	—	2	8	no	—
254	C₂²	—	2	16	no	—
255	C₂²	—	4	16	no	—
256	C₂²	C₂²×C₄×Dic₅₃	1	8	yes	—
257	C₂²	C₂²×C₄×Dic₅₃	1	4	yes	—
258	C₂²	C₂²×C₄×Dic₅₃	1	2	yes	—
259	C₂²	#16811390	1	4	yes	—
260	C₂²	C₂⁴×Dic₅₃	1	1	yes	characteristic
261	C₄	—	2	16	no	—
262	C₄	C₂²×C₄×Dic₅₃	1	8	yes	—
263	C₄	C₂²×C₄×Dic₅₃	1	2	yes	—
264	C₄	C₂²×C₄×Dic₅₃	1	4	yes	—
265	C₄	C₂³×C₄×D₅₃	1	2	yes	—
266	C₂	—	2	4	no	—
267	C₂	—	2	8	no	—
268	C₂	#20283134	1	4	yes	direct product factor
269	C₂	C₂³×C₄×Dic₅₃	1	1	yes	characteristic
270	C₂	C₂×#16811390	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂³×D₅₃
commutator	C₂×C₅₃	C₂³×C₄²
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂²×#117×C₅₃	C₂
socle	C₂³×C₅₃	C₂³×C₄
prime radical	C₂⁵×C₅₃	C₂×C₄
cosocle	C₂×C₄×C₅₃	C₂⁵
hypercenter	C₂²×#117	D₅₃
hypocenter	C₅₃	C₂×#2873
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117

Complements

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

N	H ≅ G/N	type
2. #20283134	C₂	direct (N × H)
3. C₂×#16810579	C₂	semidirect
4. C₂×C₄×#11719917	C₂	semidirect
6. C₂²×#11719917	C₄	semidirect
11. #16810579	C₂²	semidirect
12. C₄×#11719917	C₂²	semidirect
13. C₂×#13133128	C₂²	semidirect
14. C₂×C₄×#88322	C₂²	semidirect
24. C₂×#11719917	C₂×C₄	semidirect
25. C₂×#11719917	C₂×C₄	semidirect
28. C₂²×#88322	C₂×C₄	semidirect
31. #13133128	C₂³	semidirect
32. C₄×#88322	C₂³	semidirect
83. #11719917	C₂²×C₄	semidirect
85. C₂×#88322	C₂²×C₄	semidirect
138. C₂×C₅₃	#2873	semidirect
158. C₅₃	C₂×#2873	semidirect
268. C₂	#20283134	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×#11719917	2
#20283134	2
C₂×#16810579	2
C₂×C₄×#11719917	2
C₂²×#117×C₅₃	2
C₂×#2873	53

Subgroups — C2×#20283134 #28247491

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20283134 #28247491