Subgroups — C₂×#20287690 #28264986

Order 13568.

2582 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811295	C₂	1	2	yes	maximal, complemented
2	#20287690	C₂	1	16	yes	maximal, direct product factor
3	C₂×#16811000	C₂	1	3	yes	maximal, complemented
4	C₂×#16810819	C₂	1	6	yes	maximal, complemented
5	C₂×C₄×D₄₂₄	C₂	1	3	yes	maximal, complemented
6	C₂×C₈×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#11719949	—	2	2	no	—
8	C₂²×D₄₂₄	C₂²	1	3	yes	—
9	C₂²×D₄₂₄	C₂²	1	6	yes	—
10	#16811295	—	2	8	no	—
11	#16811295	C₂²	1	16	yes	complemented
12	C₂×#13133214	C₂²	1	6	yes	—
13	C₂×#13133192	—	2	6	no	—
14	C₂×C₄×D₂₁₂	C₂²	1	6	yes	—
15	#16811000	C₂²	1	24	yes	complemented
16	#16810819	C₂²	1	48	yes	complemented
17	C₄×D₄₂₄	C₂²	1	24	yes	complemented
18	C₂×#13133151	C₂²	1	1	yes	characteristic
19	C₂×#13133139	C₂²	1	6	yes	—
20	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₈×Q₈×C₅₃	C₂²	1	8	yes	complemented
22	C₂×C₄⋊C₈×C₅₃	C₂²	1	3	yes	—
23	C₂×C₄×C₈×C₅₃	C₂²	1	3	yes	—
24	C₂×#11719949	—	2	16	no	—
25	C₂×#11719949	—	4	16	no	—
26	C₂²×D₂₁₂	—	2	6	no	—
27	C₂²×D₂₁₂	—	2	6	no	—
28	C₂²×D₂₁₂	C₂³	1	6	yes	—
29	C₂²×C₄×D₅₃	—	2	6	no	—
30	C₂×D₄₂₄	—	2	12	no	—
31	C₂×D₄₂₄	—	4	12	no	—
32	C₂×D₄₂₄	C₂³	1	12	yes	—
33	C₂×D₄₂₄	C₂³	1	24	yes	—
34	#13133214	—	2	12	no	—
35	#13133214	C₂³	1	24	yes	—
36	#13133192	—	2	48	no	—
37	C₄×D₂₁₂	—	2	12	no	—
38	C₄×D₂₁₂	C₂³	1	24	yes	—
39	C₂×C₄⋊Dic₅₃	C₂³	1	2	yes	—
40	C₂×#11719921	—	2	6	no	—
41	#13133151	C₂³	1	4	yes	—
42	#13133139	C₂³	1	24	yes	—
43	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₅₃	C₂³	1	3	yes	—
45	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
46	C₄×Q₈×C₅₃	D₄	1	4	yes	—
47	C₄×Q₈×C₅₃	C₂³	1	4	yes	—
48	C₂×C₄⋊C₄×C₅₃	D₄	1	3	yes	—
49	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
50	C₄⋊C₈×C₅₃	C₂³	1	12	yes	—
51	C₄×C₈×C₅₃	C₂³	1	12	yes	—
52	C₂³×D₅₃	—	2	6	no	—
53	C₂×Q₈×C₅₃	D₈	1	8	yes	complemented
54	C₂×Q₈×C₅₃	C₂×D₄	1	4	yes	—
55	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
56	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
57	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
58	C₂²×Dic₅₃	—	2	2	no	—
59	#11719949	—	4	64	no	—
60	C₂×D₂₁₂	—	2	12	no	—
61	C₂×D₂₁₂	—	2	12	no	—
62	C₂×D₂₁₂	—	2	6	no	—
63	C₂×D₂₁₂	—	2	24	no	—
64	C₂×D₂₁₂	—	2	24	no	—
65	C₂×D₂₁₂	—	4	24	no	—
66	C₂×D₂₁₂	—	8	12	no	—
67	C₂×D₂₁₂	C₂⁴	1	12	yes	—
68	C₂×C₄×D₅₃	—	2	24	no	—
69	C₂×C₄×D₅₃	—	4	24	no	—
70	C₂×C₈×C₅₃	—	2	6	no	—
71	C₂×C₈×C₅₃	#41	1	4	yes	complemented
72	C₂×C₈×C₅₃	C₂⁴	1	6	yes	—
73	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
74	C₄⋊C₄×C₅₃	C₂×D₄	1	12	yes	—
75	C₄²×C₅₃	C₂×D₄	1	6	yes	—
76	C₄²×C₅₃	C₂⁴	1	6	yes	—
77	#11719921	—	2	24	no	—
78	C₄⋊Dic₅₃	—	2	2	no	—
79	C₄⋊Dic₅₃	C₂⁴	1	4	yes	—
80	D₄₂₄	—	2	24	no	—
81	D₄₂₄	—	4	24	no	—
82	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
83	C₂²×D₅₃	—	2	12	no	—
84	C₂²×D₅₃	—	4	12	no	—
85	C₂²×D₅₃	—	4	12	no	—
86	C₂²×D₅₃	—	4	12	no	—
87	Q₈×C₅₃	C₂×D₈	1	16	yes	complemented
88	C₂×C₄×C₅₃	—	2	6	no	—
89	C₂×C₄×C₅₃	C₂×D₈	1	12	yes	—
90	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
91	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
93	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
94	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
95	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
96	C₂×Dic₅₃	—	2	4	no	—
97	C₂×Dic₅₃	—	4	4	no	—
98	D₂₁₂	—	2	12	no	—
99	D₂₁₂	—	2	12	no	—
100	D₂₁₂	—	4	12	no	—
101	D₂₁₂	—	4	48	no	—
102	D₂₁₂	—	8	24	no	—
103	C₄×D₅₃	—	4	48	no	—
104	C₈×C₅₃	—	2	6	no	—
105	C₈×C₅₃	C₂×#41	1	4	yes	complemented
106	C₂×#3391	—	53	1	no	maximal, Sylow
107	C₂²×C₅₃	#550	1	2	yes	—
108	C₂²×C₅₃	C₂²×D₈	1	2	yes	—
109	C₂²×C₅₃	#576	1	2	yes	—
110	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
111	C₂×D₅₃	—	4	12	no	—
112	C₂×D₅₃	—	4	12	no	—
113	C₂×D₅₃	—	4	12	no	—
114	C₂×D₅₃	—	8	24	no	—
115	C₄×C₅₃	—	2	6	no	—
116	C₄×C₅₃	#550	1	2	yes	—
117	C₄×C₅₃	C₂²×D₈	1	12	yes	—
118	C₄×C₅₃	C₂²×#41	1	1	yes	characteristic
119	C₄×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
120	Dic₅₃	—	4	4	no	—
121	C₂×#550	—	53	2	no	—
122	#3391	—	53	16	no	—
123	C₂×#493	—	53	3	no	—
124	C₂×#459	—	53	6	no	—
125	C₂×C₈×Q₈	D₅₃	1	1	yes	characteristic, complemented
126	C₂×C₄×D₈	—	53	3	no	—
127	C₂×C₅₃	#3391	1	4	yes	complemented
128	C₂×C₅₃	C₂×#550	1	1	yes	characteristic
129	C₂×C₅₃	C₂³×D₈	1	1	yes	characteristic
130	C₂×C₅₃	C₂×#576	1	1	yes	characteristic
131	D₅₃	—	8	12	no	—
132	C₂²×#41	—	106	2	no	—
133	C₂²×D₈	—	53	3	no	—
134	C₂²×D₈	—	53	6	no	—
135	#550	—	53	16	no	—
136	#550	—	106	8	no	—
137	C₂×#127	—	53	6	no	—
138	C₂×#121	—	106	6	no	—
139	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
140	C₂×C₄×D₄	—	53	6	no	—
141	#493	—	53	24	no	—
142	#459	—	53	48	no	—
143	C₈×Q₈	C₂×D₅₃	1	8	yes	complemented
144	C₄×D₈	—	53	24	no	—
145	C₂×#107	—	53	1	no	—
146	C₂×C₄⋊C₈	C₂×D₅₃	1	3	yes	—
147	C₂×#102	—	53	6	no	—
148	C₂×C₄×C₈	C₂×D₅₃	1	3	yes	—
149	C₅₃	C₂×#3391	1	1	yes	characteristic, complemented, Sylow
150	C₂×#41	—	106	16	no	—
151	C₂×#41	—	212	16	no	—
152	C₂²×Q₈	D₂₁₂	1	1	yes	characteristic
153	C₂²×D₄	—	53	6	no	—
154	C₂²×D₄	—	106	6	no	—
155	C₂²×D₄	—	106	6	no	—
156	C₂³×C₄	—	106	6	no	—
157	C₂×D₈	—	53	12	no	—
158	C₂×D₈	—	53	24	no	—
159	C₂×D₈	—	106	12	no	—
160	C₂×D₈	—	212	12	no	—
161	C₂²×C₈	C₂²×D₅₃	1	3	yes	—
162	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
163	#127	—	53	24	no	—
164	#127	—	106	12	no	—
165	#121	—	106	48	no	—
166	C₄×Q₈	D₂₁₂	1	4	yes	—
167	C₄×Q₈	C₂²×D₅₃	1	4	yes	—
168	C₄×D₄	—	53	24	no	—
169	C₄×D₄	—	106	12	no	—
170	C₂×C₄⋊C₄	—	53	2	no	—
171	C₂×C₄⋊C₄	D₂₁₂	1	3	yes	—
172	C₂×C₂²⋊C₄	—	106	6	no	—
173	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
174	#107	—	53	4	no	—
175	C₄⋊C₈	C₂²×D₅₃	1	12	yes	—
176	#102	—	53	24	no	—
177	C₄×C₈	C₂²×D₅₃	1	12	yes	—
178	C₂⁴	—	106	6	no	—
179	#41	—	212	64	no	—
180	C₂×Q₈	D₄₂₄	1	8	yes	complemented
181	C₂×Q₈	C₂×D₂₁₂	1	4	yes	—
182	C₂×D₄	—	53	12	no	—
183	C₂×D₄	—	106	24	no	—
184	C₂×D₄	—	106	6	no	—
185	C₂×D₄	—	106	24	no	—
186	C₂×D₄	—	106	12	no	—
187	C₂×D₄	—	106	12	no	—
188	C₂×D₄	—	424	12	no	—
189	C₂×D₄	—	212	24	no	—
190	C₂²×C₄	—	106	2	no	—
191	C₂²×C₄	—	106	24	no	—
192	C₂²×C₄	—	212	24	no	—
193	C₂²×C₄	C₂×D₂₁₂	1	3	yes	—
194	C₂²×C₄	C₂×D₂₁₂	1	3	yes	—
195	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
196	D₈	—	106	24	no	—
197	D₈	—	212	24	no	—
198	C₂×C₈	—	2	6	no	—
199	C₂×C₈	#11719949	1	4	yes	complemented
200	C₂×C₈	C₂³×D₅₃	1	6	yes	—
201	C₂×C₈	C₂³×D₅₃	1	2	yes	—
202	C₄⋊C₄	—	53	4	no	—
203	C₄⋊C₄	—	106	2	no	—
204	C₄⋊C₄	C₂×D₂₁₂	1	12	yes	—
205	C₂²⋊C₄	—	106	24	no	—
206	C₄²	C₂×D₂₁₂	1	6	yes	—
207	C₄²	C₂³×D₅₃	1	6	yes	—
208	C₂³	—	106	12	no	—
209	C₂³	—	212	12	no	—
210	C₂³	—	212	12	no	—
211	C₂³	—	212	12	no	—
212	C₂³	C₂²×D₂₁₂	1	1	yes	characteristic
213	Q₈	C₂×D₄₂₄	1	16	yes	complemented
214	D₄	—	106	12	no	—
215	D₄	—	106	12	no	—
216	D₄	—	424	24	no	—
217	D₄	—	212	12	no	—
218	D₄	—	212	48	no	—
219	C₂×C₄	—	2	6	no	—
220	C₂×C₄	—	106	4	no	—
221	C₂×C₄	—	212	4	no	—
222	C₂×C₄	—	212	48	no	—
223	C₂×C₄	C₂×D₄₂₄	1	12	yes	—
224	C₂×C₄	C₂²×D₂₁₂	1	6	yes	—
225	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
226	C₂×C₄	C₂²×D₂₁₂	1	6	yes	—
227	C₂×C₄	C₂×#11719949	1	2	yes	—
228	C₂×C₄	#13133263	1	2	yes	—
229	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
230	C₈	—	2	6	no	—
231	C₈	C₂×#11719949	1	4	yes	complemented
232	C₂²	—	424	24	no	—
233	C₂²	—	212	12	no	—
234	C₂²	—	212	12	no	—
235	C₂²	—	212	12	no	—
236	C₂²	#16811295	1	2	yes	—
237	C₂²	C₂²×D₄₂₄	1	2	yes	—
238	C₂²	#16811428	1	2	yes	—
239	C₂²	C₂³×D₂₁₂	1	1	yes	characteristic
240	C₄	—	2	6	no	—
241	C₄	—	212	4	no	—
242	C₄	#16811295	1	2	yes	—
243	C₄	C₂²×D₄₂₄	1	12	yes	—
244	C₄	C₂²×#11719949	1	1	yes	characteristic
245	C₄	C₂×#13133263	1	1	yes	characteristic
246	C₂	—	424	12	no	—
247	C₂	#20287690	1	4	yes	direct product factor
248	C₂	C₂×#16811295	1	1	yes	characteristic
249	C₂	C₂³×D₄₂₄	1	1	yes	characteristic
250	C₂	C₂×#16811428	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₂₁₂
commutator, cosocle	C₂×C₄×C₅₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂×C₈×Q₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×D₄
hypercenter	C₂×C₈×Q₈	D₅₃
hypocenter	C₅₃	C₂×#3391
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#16811295	C₂	semidirect
2. #20287690	C₂	direct (N × H)
3. C₂×#16811000	C₂	semidirect
4. C₂×#16810819	C₂	semidirect
5. C₂×C₄×D₄₂₄	C₂	semidirect
6. C₂×C₈×Q₈×C₅₃	C₂	semidirect
11. #16811295	C₂²	semidirect
15. #16811000	C₂²	semidirect
16. #16810819	C₂²	semidirect
17. C₄×D₄₂₄	C₂²	semidirect
21. C₈×Q₈×C₅₃	C₂²	semidirect
53. C₂×Q₈×C₅₃	D₈	semidirect
71. C₂×C₈×C₅₃	#41	semidirect
87. Q₈×C₅₃	C₂×D₈	semidirect
105. C₈×C₅₃	C₂×#41	semidirect
125. C₂×C₈×Q₈	D₅₃	semidirect
127. C₂×C₅₃	#3391	semidirect
143. C₈×Q₈	C₂×D₅₃	semidirect
149. C₅₃	C₂×#3391	semidirect
180. C₂×Q₈	D₄₂₄	semidirect
199. C₂×C₈	#11719949	semidirect
213. Q₈	C₂×D₄₂₄	semidirect
231. C₈	C₂×#11719949	semidirect
247. C₂	#20287690	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×#16811295	2
#20287690	2
C₂×#16811000	2
C₂×#16810819	2
C₂×C₄×D₄₂₄	2
C₂×C₈×Q₈×C₅₃	2
C₂×#3391	53

Subgroups — C2×#20287690 #28264986

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20287690 #28264986