Subgroups — C₂×#20283326 #28247790

Order 13568.

2586 conjugacy classes of subgroups.

The subgroup lattice has 240 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	2	yes	maximal
2	#20283326	C₂	1	16	yes	maximal, direct product factor
3	C₂×#16810672	C₂	1	6	yes	maximal, complemented
4	C₂²×#13133153	C₂	1	6	yes	maximal, complemented
5	C₂²×#131×C₅₃	C₂	1	1	yes	maximal, characteristic
6	#16811390	—	2	8	no	—
7	#16811390	C₂²	1	16	yes	—
8	C₂×#13133224	—	2	8	no	—
9	C₂²×#11719917	C₂²	1	12	yes	—
10	#16810672	C₂²	1	48	yes	complemented
11	C₂×#13133153	C₂²	1	4	yes	—
12	C₂×#13133153	C₂²	1	12	yes	—
13	C₂×#13133153	C₂²	1	48	yes	complemented
14	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#131×C₅₃	C₄	1	8	yes	—
16	C₂×#131×C₅₃	C₄	1	8	yes	—
17	C₂×#131×C₅₃	C₂²	1	8	yes	—
18	C₂²×M₁₆×C₅₃	C₂²	1	3	yes	—
19	C₂³×C₈×C₅₃	C₂²	1	3	yes	—
20	#13133224	—	2	64	no	—
21	C₂×#11719917	—	2	12	no	—
22	C₂×#11719917	—	2	24	no	—
23	C₂×#11719917	C₂³	1	8	yes	—
24	C₂×#11719917	C₂³	1	48	yes	—
25	C₂²×#88322	—	2	12	no	—
26	#13133153	C₂³	1	16	yes	—
27	#13133153	C₂³	1	48	yes	—
28	C₂×#41×C₅₃	C₂×C₄	1	4	yes	—
29	C₂×#41×C₅₃	C₂×C₄	1	4	yes	—
30	C₂×#41×C₅₃	D₄	1	4	yes	—
31	C₂×#41×C₅₃	D₄	1	4	yes	—
32	C₂×#41×C₅₃	Q₈	1	4	yes	—
33	C₂×#41×C₅₃	C₂³	1	4	yes	—
34	C₂²×Q₈×C₅₃	Q₈	1	1	yes	characteristic
35	C₂²×D₄×C₅₃	Q₈	1	3	yes	—
36	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
37	#131×C₅₃	C₂×C₄	1	64	yes	—
38	C₂×M₁₆×C₅₃	C₂×C₄	1	12	yes	—
39	C₂×M₁₆×C₅₃	C₂×C₄	1	12	yes	—
40	C₂×M₁₆×C₅₃	C₂³	1	12	yes	—
41	C₂²×C₈×C₅₃	C₂×C₄	1	12	yes	—
42	C₂²×C₈×C₅₃	C₂×C₄	1	12	yes	—
43	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₅₃	C₂³	1	3	yes	—
45	C₂²×C₈×C₅₃	C₂³	1	12	yes	—
46	C₂⁴×C₅₃	C₂×Q₈	1	3	yes	—
47	#41×C₅₃	C₄⋊C₄	1	16	yes	—
48	#41×C₅₃	C₄⋊C₄	1	16	yes	—
49	#41×C₅₃	C₂²×C₄	1	16	yes	—
50	#41×C₅₃	C₂×D₄	1	16	yes	—
51	C₂×Q₈×C₅₃	C₄⋊C₄	1	4	yes	—
52	C₂×Q₈×C₅₃	C₄⋊C₄	1	4	yes	—
53	C₂×Q₈×C₅₃	C₂×Q₈	1	4	yes	—
54	C₂×D₄×C₅₃	C₄⋊C₄	1	12	yes	—
55	C₂×D₄×C₅₃	C₄⋊C₄	1	12	yes	—
56	C₂×D₄×C₅₃	C₂×Q₈	1	12	yes	—
57	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
58	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
59	C₂²×C₄×C₅₃	C₂×D₄	1	6	yes	—
60	C₂²×C₄×C₅₃	C₂×D₄	1	6	yes	—
61	C₂²×C₄×C₅₃	C₂×Q₈	1	3	yes	—
62	C₂²×C₄×C₅₃	C₂×Q₈	1	6	yes	—
63	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
65	M₁₆×C₅₃	C₂²×C₄	1	48	yes	—
66	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₅₃	C₂²×C₄	1	6	yes	—
69	C₂×C₈×C₅₃	C₂²×C₄	1	6	yes	—
70	C₂×C₈×C₅₃	C₂²×C₄	1	48	yes	—
71	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
72	C₂×C₈×C₅₃	C₂⁴	1	6	yes	—
73	#11719917	—	2	8	no	—
74	#11719917	—	2	48	no	—
75	#11719917	C₂⁴	1	16	yes	—
76	C₂×#88322	—	2	8	no	—
77	C₂×#88322	—	2	48	no	—
78	C₂³×C₅₃	—	4	6	no	—
79	C₂³×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
80	C₂³×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
81	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
82	C₂³×C₅₃	C₂²×Q₈	1	6	yes	—
83	Q₈×C₅₃	C₂×C₄⋊C₄	1	16	yes	—
84	D₄×C₅₃	C₂×C₄⋊C₄	1	48	yes	—
85	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
86	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
87	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	12	yes	—
88	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	12	yes	—
89	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂³×C₄	1	12	yes	—
92	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂²×D₄	1	12	yes	—
94	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂²×Q₈	1	6	yes	—
97	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
98	C₈×C₅₃	C₂³×C₄	1	4	yes	—
99	C₈×C₅₃	C₂³×C₄	1	12	yes	—
100	#88322	—	2	16	no	—
101	C₂×#2916	—	53	1	no	maximal, Sylow
102	C₂²×C₅₃	—	2	2	no	—
103	C₂²×C₅₃	—	2	12	no	—
104	C₂²×C₅₃	—	4	6	no	—
105	C₂²×C₅₃	—	4	6	no	—
106	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
107	C₂²×C₅₃	C₂²×C₄⋊C₄	1	12	yes	—
108	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
109	C₂²×C₅₃	C₂³×Q₈	1	1	yes	characteristic
110	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
111	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
112	C₄×C₅₃	C₂²×C₄⋊C₄	1	12	yes	—
113	C₄×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
114	C₄×C₅₃	C₂³×D₄	1	1	yes	characteristic
115	C₂×#568	—	53	2	no	—
116	C₂²×#131	D₅₃	1	1	yes	characteristic
117	#2916	—	53	16	no	—
118	C₂×#430	—	53	6	no	—
119	C₂²×#108	—	53	6	no	—
120	C₂×C₅₃	—	2	1	no	—
121	C₂×C₅₃	—	2	1	no	—
122	C₂×C₅₃	—	2	12	no	—
123	C₂×C₅₃	#2916	1	2	yes	complemented
124	C₂×C₅₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
125	C₂²×#41	C₂×D₅₃	1	1	yes	characteristic
126	#568	—	53	16	no	—
127	#568	—	106	8	no	—
128	C₂×#131	—	106	8	no	—
129	C₂×#131	Dic₅₃	1	8	yes	—
130	C₂×#131	Dic₅₃	1	8	yes	—
131	C₂×#131	C₂×D₅₃	1	8	yes	—
132	C₂²×M₁₆	—	53	12	no	—
133	C₂²×M₁₆	C₂×D₅₃	1	3	yes	—
134	C₂³×C₈	C₂×D₅₃	1	3	yes	—
135	#430	—	53	48	no	—
136	C₂×#108	—	53	4	no	—
137	C₂×#108	—	53	12	no	—
138	C₂×#108	—	53	48	no	—
139	C₅₃	C₂×#2916	1	1	yes	characteristic, complemented, Sylow
140	C₂×#41	Dic₁₀₆	1	4	yes	—
141	C₂×#41	D₂₁₂	1	4	yes	—
142	C₂×#41	D₂₁₂	1	4	yes	—
143	C₂×#41	C₂×Dic₅₃	1	4	yes	—
144	C₂×#41	C₂×Dic₅₃	1	4	yes	—
145	C₂×#41	C₂²×D₅₃	1	4	yes	—
146	C₂²×Q₈	Dic₁₀₆	1	1	yes	characteristic
147	C₂²×D₄	Dic₁₀₆	1	3	yes	—
148	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
149	#131	—	106	64	no	—
150	#131	C₂×Dic₅₃	1	64	yes	—
151	C₂×M₁₆	—	53	8	no	—
152	C₂×M₁₆	—	53	48	no	—
153	C₂×M₁₆	—	106	12	no	—
154	C₂×M₁₆	—	106	24	no	—
155	C₂×M₁₆	C₂×Dic₅₃	1	12	yes	—
156	C₂×M₁₆	C₂×Dic₅₃	1	12	yes	—
157	C₂×M₁₆	C₂²×D₅₃	1	12	yes	—
158	C₂²×C₈	—	106	12	no	—
159	C₂²×C₈	C₂×Dic₅₃	1	12	yes	—
160	C₂²×C₈	C₂×Dic₅₃	1	12	yes	—
161	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
162	C₂²×C₈	C₂²×D₅₃	1	3	yes	—
163	C₂²×C₈	C₂²×D₅₃	1	12	yes	—
164	#108	—	53	16	no	—
165	#108	—	53	48	no	—
166	C₂⁴	C₂×Dic₁₀₆	1	3	yes	—
167	#41	C₄⋊Dic₅₃	1	16	yes	—
168	#41	C₄⋊Dic₅₃	1	16	yes	—
169	#41	C₂×D₂₁₂	1	16	yes	—
170	#41	C₂²×Dic₅₃	1	16	yes	—
171	C₂×Q₈	C₄⋊Dic₅₃	1	4	yes	—
172	C₂×Q₈	C₄⋊Dic₅₃	1	4	yes	—
173	C₂×Q₈	C₂×Dic₁₀₆	1	4	yes	—
174	C₂×D₄	C₄⋊Dic₅₃	1	12	yes	—
175	C₂×D₄	C₄⋊Dic₅₃	1	12	yes	—
176	C₂×D₄	C₂×Dic₁₀₆	1	12	yes	—
177	C₂²×C₄	C₂×Dic₁₀₆	1	3	yes	—
178	C₂²×C₄	C₂×Dic₁₀₆	1	6	yes	—
179	C₂²×C₄	C₂×D₂₁₂	1	6	yes	—
180	C₂²×C₄	C₂×D₂₁₂	1	6	yes	—
181	C₂²×C₄	C₂²×Dic₅₃	1	6	yes	—
182	C₂²×C₄	C₂²×Dic₅₃	1	6	yes	—
183	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
184	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
185	M₁₆	—	53	16	no	—
186	M₁₆	—	106	8	no	—
187	M₁₆	—	106	48	no	—
188	M₁₆	C₂²×Dic₅₃	1	48	yes	—
189	C₂×C₈	—	106	48	no	—
190	C₂×C₈	—	106	8	no	—
191	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
192	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
193	C₂×C₈	C₂²×Dic₅₃	1	6	yes	—
194	C₂×C₈	C₂²×Dic₅₃	1	6	yes	—
195	C₂×C₈	C₂²×Dic₅₃	1	48	yes	—
196	C₂×C₈	C₂³×D₅₃	1	2	yes	—
197	C₂×C₈	C₂³×D₅₃	1	6	yes	—
198	C₂³	—	4	6	no	—
199	C₂³	C₂×C₄⋊Dic₅₃	1	6	yes	—
200	C₂³	C₂×C₄⋊Dic₅₃	1	6	yes	—
201	C₂³	C₂²×Dic₁₀₆	1	1	yes	characteristic
202	C₂³	C₂²×Dic₁₀₆	1	6	yes	—
203	Q₈	C₂×C₄⋊Dic₅₃	1	16	yes	—
204	D₄	C₂×C₄⋊Dic₅₃	1	48	yes	—
205	C₂×C₄	C₂×C₄⋊Dic₅₃	1	6	yes	—
206	C₂×C₄	C₂×C₄⋊Dic₅₃	1	6	yes	—
207	C₂×C₄	C₂×C₄⋊Dic₅₃	1	12	yes	—
208	C₂×C₄	C₂×C₄⋊Dic₅₃	1	12	yes	—
209	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
210	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
211	C₂×C₄	C₂³×Dic₅₃	1	12	yes	—
212	C₂×C₄	C₂²×D₂₁₂	1	12	yes	—
213	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
214	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
215	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
216	C₂×C₄	C₂²×Dic₁₀₆	1	6	yes	—
217	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
218	C₈	—	106	16	no	—
219	C₈	C₂³×Dic₅₃	1	4	yes	—
220	C₈	C₂³×Dic₅₃	1	12	yes	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	12	no	—
223	C₂²	—	4	6	no	—
224	C₂²	—	4	6	no	—
225	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
226	C₂²	C₂²×C₄⋊Dic₅₃	1	12	yes	—
227	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
228	C₂²	C₂³×Dic₁₀₆	1	1	yes	characteristic
229	C₄	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
230	C₄	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
231	C₄	C₂²×C₄⋊Dic₅₃	1	12	yes	—
232	C₄	C₂⁴×Dic₅₃	1	1	yes	characteristic
233	C₄	C₂³×D₂₁₂	1	1	yes	characteristic
234	C₂	—	2	1	no	—
235	C₂	—	2	1	no	—
236	C₂	—	2	12	no	—
237	C₂	#20283326	1	2	yes	direct product factor
238	C₂	C₂³×C₄⋊Dic₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₂
commutator	C₄×C₅₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂²×#131×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂²×#41×C₅₃	C₂²
cosocle	C₂×C₄×C₅₃	C₂⁵
hypercenter	C₂²×#131	D₅₃
hypocenter	C₅₃	C₂×#2916
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#41 ◁ C₂²×#131

Complements

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

N	H ≅ G/N	type
2. #20283326	C₂	direct (N × H)
3. C₂×#16810672	C₂	semidirect
4. C₂²×#13133153	C₂	semidirect
10. #16810672	C₂²	semidirect
13. C₂×#13133153	C₂²	semidirect
123. C₂×C₅₃	#2916	semidirect
139. C₅₃	C₂×#2916	semidirect
237. C₂	#20283326	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
#20283326	2
C₂×#16810672	2
C₂²×#13133153	2
C₂²×#131×C₅₃	2
C₂×#2916	53

Subgroups — C2×#20283326 #28247790

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20283326 #28247790