Subgroups — C₂×#20283324 #28247788

Order 13568.

2586 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811390	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13133224	C₂	1	1	yes	maximal, characteristic
3	#20283324	C₂	1	16	yes	maximal, direct product factor
4	C₂×#16810671	C₂	1	6	yes	maximal, complemented
5	C₂²×#13133152	C₂	1	6	yes	maximal, complemented
6	C₂×C₅₃×#568	C₂	1	1	yes	maximal, characteristic
7	#16811390	—	2	4	no	—
8	#16811390	C₂²	1	8	yes	—
9	C₂×#13133224	—	2	4	no	—
10	C₂×#13133224	C₄	1	8	yes	—
11	C₂×#13133224	C₄	1	8	yes	—
12	C₂×#13133224	C₂²	1	8	yes	—
13	C₂²×#11719917	C₂²	1	6	yes	—
14	C₂²×#11719917	C₂²	1	3	yes	—
15	C₂³×#88322	C₂²	1	3	yes	—
16	#16810671	C₂²	1	48	yes	complemented
17	C₂×#13133152	C₂²	1	4	yes	—
18	C₂×#13133152	C₂²	1	12	yes	—
19	C₂×#13133152	C₂²	1	48	yes	complemented
20	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic
21	C₅₃×#568	—	2	4	no	—
22	C₅₃×#568	C₂²	1	8	yes	—
23	C₂×#131×C₅₃	—	2	4	no	—
24	C₂²×M₁₆×C₅₃	C₂²	1	6	yes	—
25	#13133224	—	2	32	no	—
26	#13133224	C₂×C₄	1	64	yes	—
27	C₂×#11719917	—	2	6	no	—
28	C₂×#11719917	—	2	12	no	—
29	C₂×#11719917	C₂×C₄	1	12	yes	—
30	C₂×#11719917	C₂×C₄	1	12	yes	—
31	C₂×#11719917	C₂³	1	4	yes	—
32	C₂×#11719917	C₂³	1	24	yes	—
33	C₂×#11719917	C₂³	1	12	yes	—
34	C₂²×#88322	—	2	6	no	—
35	C₂²×#88322	C₂×C₄	1	12	yes	—
36	C₂²×#88322	C₂×C₄	1	12	yes	—
37	C₂²×#88322	C₂³	1	1	yes	characteristic
38	C₂²×#88322	C₂³	1	3	yes	—
39	C₂²×#88322	C₂³	1	12	yes	—
40	#13133152	C₂³	1	16	yes	—
41	#13133152	C₂³	1	48	yes	—
42	C₂×#41×C₅₃	C₂×C₄	1	4	yes	—
43	C₂×#41×C₅₃	C₂×C₄	1	4	yes	—
44	C₂×#41×C₅₃	D₄	1	4	yes	—
45	C₂×#41×C₅₃	D₄	1	4	yes	—
46	C₂×#41×C₅₃	Q₈	1	4	yes	—
47	C₂×#41×C₅₃	C₂³	1	4	yes	—
48	C₂²×Q₈×C₅₃	Q₈	1	1	yes	characteristic
49	C₂²×D₄×C₅₃	Q₈	1	3	yes	—
50	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
51	#131×C₅₃	—	2	32	no	—
52	C₂×M₁₆×C₅₃	—	2	6	no	—
53	C₂×M₁₆×C₅₃	—	2	12	no	—
54	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
55	C₂×M₁₆×C₅₃	C₂³	1	24	yes	—
56	C₂²×C₈×C₅₃	—	2	6	no	—
57	C₂⁴×C₅₃	C₂×Q₈	1	3	yes	—
58	#41×C₅₃	C₄⋊C₄	1	16	yes	—
59	#41×C₅₃	C₄⋊C₄	1	16	yes	—
60	#41×C₅₃	C₂²×C₄	1	16	yes	—
61	#41×C₅₃	C₂×D₄	1	16	yes	—
62	C₂×Q₈×C₅₃	C₄⋊C₄	1	4	yes	—
63	C₂×Q₈×C₅₃	C₄⋊C₄	1	4	yes	—
64	C₂×Q₈×C₅₃	C₂×Q₈	1	4	yes	—
65	C₂×D₄×C₅₃	C₄⋊C₄	1	12	yes	—
66	C₂×D₄×C₅₃	C₄⋊C₄	1	12	yes	—
67	C₂×D₄×C₅₃	C₂×Q₈	1	12	yes	—
68	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
69	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
70	C₂²×C₄×C₅₃	C₂×D₄	1	6	yes	—
71	C₂²×C₄×C₅₃	C₂×D₄	1	6	yes	—
72	C₂²×C₄×C₅₃	C₂×Q₈	1	3	yes	—
73	C₂²×C₄×C₅₃	C₂×Q₈	1	6	yes	—
74	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
76	M₁₆×C₅₃	—	2	4	no	—
77	M₁₆×C₅₃	—	2	24	no	—
78	M₁₆×C₅₃	C₂⁴	1	8	yes	—
79	C₂×C₈×C₅₃	—	2	4	no	—
80	C₂×C₈×C₅₃	—	2	24	no	—
81	#11719917	—	2	4	no	—
82	#11719917	—	2	24	no	—
83	#11719917	C₂²×C₄	1	48	yes	—
84	#11719917	C₂⁴	1	8	yes	—
85	C₂×#88322	—	2	4	no	—
86	C₂×#88322	—	2	24	no	—
87	C₂×#88322	C₂²×C₄	1	2	yes	—
88	C₂×#88322	C₂²×C₄	1	2	yes	—
89	C₂×#88322	C₂²×C₄	1	6	yes	—
90	C₂×#88322	C₂²×C₄	1	6	yes	—
91	C₂×#88322	C₂²×C₄	1	48	yes	—
92	C₂×#88322	C₂⁴	1	2	yes	—
93	C₂×#88322	C₂⁴	1	6	yes	—
94	C₂³×C₅₃	—	4	6	no	—
95	C₂³×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
96	C₂³×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
97	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
98	C₂³×C₅₃	C₂²×Q₈	1	6	yes	—
99	Q₈×C₅₃	C₂×C₄⋊C₄	1	16	yes	—
100	D₄×C₅₃	C₂×C₄⋊C₄	1	48	yes	—
101	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
102	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
103	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	12	yes	—
104	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	12	yes	—
105	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂³×C₄	1	12	yes	—
108	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂²×D₄	1	12	yes	—
110	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂²×Q₈	1	6	yes	—
113	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
114	C₈×C₅₃	—	2	8	no	—
115	#88322	—	2	8	no	—
116	#88322	C₂³×C₄	1	4	yes	—
117	#88322	C₂³×C₄	1	12	yes	—
118	C₂×#2916	—	53	1	no	maximal, Sylow
119	C₂²×C₅₃	—	2	2	no	—
120	C₂²×C₅₃	—	2	12	no	—
121	C₂²×C₅₃	—	4	6	no	—
122	C₂²×C₅₃	—	4	6	no	—
123	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
124	C₂²×C₅₃	C₂²×C₄⋊C₄	1	12	yes	—
125	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
126	C₂²×C₅₃	C₂³×Q₈	1	1	yes	characteristic
127	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
128	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
129	C₄×C₅₃	C₂²×C₄⋊C₄	1	12	yes	—
130	C₄×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
131	C₄×C₅₃	C₂³×D₄	1	1	yes	characteristic
132	C₂×#568	—	53	1	no	—
133	C₂×#568	D₅₃	1	1	yes	characteristic
134	C₂²×#131	—	53	1	no	—
135	#2916	—	53	16	no	—
136	C₂×#430	—	53	6	no	—
137	C₂²×#108	—	53	6	no	—
138	C₂×C₅₃	—	2	1	no	—
139	C₂×C₅₃	—	2	1	no	—
140	C₂×C₅₃	—	2	12	no	—
141	C₂×C₅₃	#2916	1	2	yes	complemented
142	C₂×C₅₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
143	C₂²×#41	C₂×D₅₃	1	1	yes	characteristic
144	#568	—	2	4	no	—
145	#568	—	53	8	no	—
146	#568	—	106	4	no	—
147	#568	C₂×D₅₃	1	8	yes	—
148	C₂×#131	—	2	4	no	—
149	C₂×#131	—	53	8	no	—
150	C₂×#131	—	53	8	no	—
151	C₂×#131	—	53	8	no	—
152	C₂×#131	—	106	4	no	—
153	C₂²×M₁₆	—	53	3	no	—
154	C₂²×M₁₆	—	53	6	no	—
155	C₂²×M₁₆	C₂×D₅₃	1	6	yes	—
156	C₂³×C₈	—	53	3	no	—
157	#430	—	53	48	no	—
158	C₂×#108	—	53	4	no	—
159	C₂×#108	—	53	12	no	—
160	C₂×#108	—	53	48	no	—
161	C₅₃	C₂×#2916	1	1	yes	characteristic, complemented, Sylow
162	C₂×#41	Dic₁₀₆	1	4	yes	—
163	C₂×#41	C₄×D₅₃	1	4	yes	—
164	C₂×#41	C₄×D₅₃	1	4	yes	—
165	C₂×#41	C₂²⋊D₅₃	1	4	yes	—
166	C₂×#41	C₂²⋊D₅₃	1	4	yes	—
167	C₂×#41	C₂²×D₅₃	1	4	yes	—
168	C₂²×Q₈	Dic₁₀₆	1	1	yes	characteristic
169	C₂²×D₄	Dic₁₀₆	1	3	yes	—
170	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
171	#131	—	2	32	no	—
172	#131	—	53	64	no	—
173	#131	—	106	32	no	—
174	C₂×M₁₆	—	2	6	no	—
175	C₂×M₁₆	—	2	12	no	—
176	C₂×M₁₆	—	53	12	no	—
177	C₂×M₁₆	—	53	12	no	—
178	C₂×M₁₆	—	53	12	no	—
179	C₂×M₁₆	—	53	4	no	—
180	C₂×M₁₆	—	53	24	no	—
181	C₂×M₁₆	—	106	6	no	—
182	C₂×M₁₆	—	106	12	no	—
183	C₂×M₁₆	C₂²×D₅₃	1	4	yes	—
184	C₂×M₁₆	C₂²×D₅₃	1	24	yes	—
185	C₂²×C₈	—	2	6	no	—
186	C₂²×C₈	—	53	12	no	—
187	C₂²×C₈	—	53	12	no	—
188	C₂²×C₈	—	53	1	no	—
189	C₂²×C₈	—	53	3	no	—
190	C₂²×C₈	—	53	12	no	—
191	C₂²×C₈	—	106	6	no	—
192	#108	—	53	16	no	—
193	#108	—	53	48	no	—
194	C₂⁴	C₂×Dic₁₀₆	1	3	yes	—
195	#41	#11719919	1	16	yes	—
196	#41	#11719919	1	16	yes	—
197	#41	C₂×C₄×D₅₃	1	16	yes	—
198	#41	C₂×C₂²⋊D₅₃	1	16	yes	—
199	C₂×Q₈	#11719919	1	4	yes	—
200	C₂×Q₈	#11719919	1	4	yes	—
201	C₂×Q₈	C₂×Dic₁₀₆	1	4	yes	—
202	C₂×D₄	#11719919	1	12	yes	—
203	C₂×D₄	#11719919	1	12	yes	—
204	C₂×D₄	C₂×Dic₁₀₆	1	12	yes	—
205	C₂²×C₄	C₂×Dic₁₀₆	1	3	yes	—
206	C₂²×C₄	C₂×Dic₁₀₆	1	6	yes	—
207	C₂²×C₄	C₂×C₄×D₅₃	1	6	yes	—
208	C₂²×C₄	C₂×C₄×D₅₃	1	6	yes	—
209	C₂²×C₄	C₂×C₂²⋊D₅₃	1	6	yes	—
210	C₂²×C₄	C₂×C₂²⋊D₅₃	1	6	yes	—
211	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
212	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
213	M₁₆	—	2	4	no	—
214	M₁₆	—	2	24	no	—
215	M₁₆	—	53	48	no	—
216	M₁₆	—	53	8	no	—
217	M₁₆	—	106	4	no	—
218	M₁₆	—	106	24	no	—
219	M₁₆	C₂³×D₅₃	1	8	yes	—
220	C₂×C₈	—	2	24	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	53	2	no	—
223	C₂×C₈	—	53	2	no	—
224	C₂×C₈	—	53	6	no	—
225	C₂×C₈	—	53	6	no	—
226	C₂×C₈	—	53	48	no	—
227	C₂×C₈	—	53	2	no	—
228	C₂×C₈	—	53	6	no	—
229	C₂×C₈	—	106	24	no	—
230	C₂×C₈	—	106	4	no	—
231	C₂³	—	4	6	no	—
232	C₂³	C₂×#11719919	1	6	yes	—
233	C₂³	C₂×#11719919	1	6	yes	—
234	C₂³	C₂²×Dic₁₀₆	1	1	yes	characteristic
235	C₂³	C₂²×Dic₁₀₆	1	6	yes	—
236	Q₈	C₂×#11719919	1	16	yes	—
237	D₄	C₂×#11719919	1	48	yes	—
238	C₂×C₄	C₂×#11719919	1	6	yes	—
239	C₂×C₄	C₂×#11719919	1	6	yes	—
240	C₂×C₄	C₂×#11719919	1	12	yes	—
241	C₂×C₄	C₂×#11719919	1	12	yes	—
242	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
243	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
244	C₂×C₄	C₂²×C₄×D₅₃	1	12	yes	—
245	C₂×C₄	C₂²×C₂²⋊D₅₃	1	12	yes	—
246	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
247	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
248	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
249	C₂×C₄	C₂²×Dic₁₀₆	1	6	yes	—
250	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
251	C₈	—	2	8	no	—
252	C₈	—	53	4	no	—
253	C₈	—	53	12	no	—
254	C₈	—	106	8	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	2	12	no	—
257	C₂²	—	4	6	no	—
258	C₂²	—	4	6	no	—
259	C₂²	C₂²×#11719919	1	1	yes	characteristic
260	C₂²	C₂²×#11719919	1	12	yes	—
261	C₂²	C₂²×#11719919	1	1	yes	characteristic
262	C₂²	C₂³×Dic₁₀₆	1	1	yes	characteristic
263	C₄	C₂²×#11719919	1	1	yes	characteristic
264	C₄	C₂²×#11719919	1	1	yes	characteristic
265	C₄	C₂²×#11719919	1	12	yes	—
266	C₄	C₂³×C₄×D₅₃	1	1	yes	characteristic
267	C₄	C₂³×C₂²⋊D₅₃	1	1	yes	characteristic
268	C₂	—	2	1	no	—
269	C₂	—	2	1	no	—
270	C₂	—	2	12	no	—
271	C₂	#20283324	1	2	yes	direct product factor
272	C₂	C₂³×#11719919	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂×C₅₃×#568	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂²×#41×C₅₃	C₂²
cosocle	C₂×C₄×C₅₃	C₂⁵
hypercenter	C₂×#568	D₅₃
hypocenter	C₅₃	C₂×#2916
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#41 ◁ C₂×#568

Complements

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

N	H ≅ G/N	type
3. #20283324	C₂	direct (N × H)
4. C₂×#16810671	C₂	semidirect
5. C₂²×#13133152	C₂	semidirect
16. #16810671	C₂²	semidirect
19. C₂×#13133152	C₂²	semidirect
141. C₂×C₅₃	#2916	semidirect
161. C₅₃	C₂×#2916	semidirect
271. C₂	#20283324	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×#16811390	2
C₂²×#13133224	2
#20283324	2
C₂×#16810671	2
C₂²×#13133152	2
C₂×C₅₃×#568	2
C₂×#2916	53

Subgroups — C2×#20283324 #28247788

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20283324 #28247788