Subgroups — C₂×#20274312 #27371741

Order 13568.

814 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811028	C₂	1	1	yes	maximal, characteristic
2	C₂×#16811021	C₂	1	2	yes	maximal
3	C₂×#16811003	C₂	1	1	yes	maximal, characteristic
4	C₂×#16810974	C₂	1	2	yes	maximal
5	#20274312	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₈²×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₁₂	—	2	2	no	—
8	C₂×#13133217	C₂²	1	2	yes	—
9	C₂×#13133217	C₂²	1	2	yes	—
10	#16811028	—	2	2	no	—
11	#16811028	C₂²	1	4	yes	—
12	#16811021	—	2	4	no	—
13	#16811021	C₂²	1	8	yes	—
14	#16811003	—	2	2	no	—
15	#16811003	C₂²	1	4	yes	—
16	#16810974	—	2	4	no	—
17	#16810974	C₂²	1	8	yes	—
18	C₂×#13133151	—	2	2	no	—
19	C₂×#13133149	—	2	4	no	—
20	C₂×#13133142	—	2	4	no	—
21	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₈²×C₅₃	C₂²	1	4	yes	—
25	C₂²×Dic₁₀₆	—	2	2	no	—
26	C₂²×Dic₁₀₆	—	2	2	no	—
27	C₂×Dic₂₁₂	—	2	8	no	—
28	C₂×Dic₂₁₂	—	4	8	no	—
29	#13133217	—	2	2	no	—
30	#13133217	—	2	2	no	—
31	#13133217	—	2	2	no	—
32	#13133217	—	2	2	no	—
33	#13133217	—	2	2	no	—
34	#13133217	—	2	2	no	—
35	#13133217	C₂³	1	4	yes	—
36	#13133217	C₂³	1	4	yes	—
37	C₂×C₄⋊Dic₅₃	—	2	2	no	—
38	C₂×C₄⋊Dic₅₃	—	2	2	no	—
39	C₂×C₄⋊Dic₅₃	—	2	2	no	—
40	C₂×C₄⋊Dic₅₃	—	2	2	no	—
41	#13133151	—	2	8	no	—
42	#13133149	—	2	16	no	—
43	#13133142	—	2	16	no	—
44	C₂²×C₈×C₅₃	D₄	1	2	yes	—
45	C₂²×C₈×C₅₃	D₄	1	2	yes	—
46	C₂²×C₈×C₅₃	D₄	1	2	yes	—
47	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
48	C₄×C₈×C₅₃	D₄	1	2	yes	—
49	C₄×C₈×C₅₃	D₄	1	2	yes	—
50	C₄×C₈×C₅₃	D₄	1	2	yes	—
51	C₄×C₈×C₅₃	C₂³	1	2	yes	—
52	C₄×C₈×C₅₃	C₂³	1	2	yes	—
53	C₄×C₈×C₅₃	C₂³	1	2	yes	—
54	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×Dic₅₃	—	4	2	no	—
58	C₂²×Dic₅₃	—	4	2	no	—
59	C₂×Dic₁₀₆	—	2	4	no	—
60	C₂×Dic₁₀₆	—	2	4	no	—
61	C₂×Dic₁₀₆	—	4	8	no	—
62	C₂×Dic₁₀₆	—	4	8	no	—
63	C₂×Dic₁₀₆	—	4	2	no	—
64	C₂×Dic₁₀₆	—	4	2	no	—
65	C₂×C₈×C₅₃	—	2	4	no	—
66	C₂×C₈×C₅₃	SD₁₆	1	8	yes	—
67	C₂×C₈×C₅₃	Q₁₆	1	8	yes	complemented
68	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
69	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
70	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
71	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
75	C₄⋊Dic₅₃	—	2	4	no	—
76	C₄⋊Dic₅₃	—	2	4	no	—
77	C₄⋊Dic₅₃	—	2	4	no	—
78	C₄⋊Dic₅₃	—	2	4	no	—
79	C₄⋊Dic₅₃	—	4	2	no	—
80	C₄⋊Dic₅₃	—	4	2	no	—
81	C₄⋊Dic₅₃	—	4	2	no	—
82	C₄⋊Dic₅₃	—	4	2	no	—
83	Dic₂₁₂	—	4	16	no	—
84	C₂³×C₅₃	#127	1	1	yes	characteristic
85	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
86	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
87	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂×SD₁₆	1	4	yes	—
89	C₂×C₄×C₅₃	C₂×Q₁₆	1	4	yes	—
90	C₂×C₄×C₅₃	#135	1	4	yes	—
91	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
94	C₂×Dic₅₃	—	4	4	no	—
95	C₂×Dic₅₃	—	4	4	no	—
96	C₂×Dic₅₃	—	8	4	no	—
97	C₂×Dic₅₃	—	8	4	no	—
98	Dic₁₀₆	—	4	8	no	—
99	Dic₁₀₆	—	4	8	no	—
100	Dic₁₀₆	—	8	4	no	—
101	Dic₁₀₆	—	8	4	no	—
102	C₈×C₅₃	—	2	4	no	—
103	C₈×C₅₃	C₂×SD₁₆	1	8	yes	—
104	C₈×C₅₃	C₂×Q₁₆	1	8	yes	complemented
105	C₂×#1707	—	53	1	no	maximal, Sylow
106	C₂²×C₅₃	#492	1	2	yes	—
107	C₂²×C₅₃	#494	1	2	yes	—
108	C₂²×C₅₃	#495	1	2	yes	—
109	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
110	C₄×C₅₃	#492	1	2	yes	—
111	C₄×C₅₃	#494	1	2	yes	—
112	C₄×C₅₃	#495	1	2	yes	—
113	C₄×C₅₃	C₂²×SD₁₆	1	2	yes	—
114	C₄×C₅₃	C₂²×Q₁₆	1	2	yes	—
115	C₄×C₅₃	C₂×#135	1	2	yes	—
116	Dic₅₃	—	8	4	no	—
117	Dic₅₃	—	8	4	no	—
118	C₂×#500	—	53	1	no	—
119	C₂×#498	—	53	2	no	—
120	C₂×#494	—	53	1	no	—
121	C₂×#487	—	53	2	no	—
122	#1707	—	53	8	no	—
123	C₂×C₈²	D₅₃	1	1	yes	characteristic
124	C₂×C₅₃	#1707	1	4	yes	complemented
125	C₂×C₅₃	C₂×#492	1	1	yes	characteristic
126	C₂×C₅₃	C₂×#494	1	1	yes	characteristic
127	C₂×C₅₃	C₂×#495	1	1	yes	characteristic
128	C₂²×Q₁₆	—	106	2	no	—
129	C₂×C₄⋊Q₈	—	53	2	no	—
130	C₂×C₄⋊Q₈	—	53	2	no	—
131	#500	—	53	4	no	—
132	#500	—	106	2	no	—
133	#498	—	53	8	no	—
134	#498	—	106	4	no	—
135	#494	—	53	4	no	—
136	#494	—	106	2	no	—
137	#487	—	53	8	no	—
138	#487	—	106	4	no	—
139	C₂×#107	—	106	2	no	—
140	C₂×#106	—	106	4	no	—
141	C₂×#103	—	106	4	no	—
142	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
143	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
144	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
145	C₈²	C₂×D₅₃	1	4	yes	—
146	C₅₃	C₂×#1707	1	1	yes	characteristic, complemented, Sylow
147	C₂²×Q₈	—	106	2	no	—
148	C₂²×Q₈	—	106	2	no	—
149	C₂×Q₁₆	—	106	8	no	—
150	C₂×Q₁₆	—	212	8	no	—
151	C₂²×C₈	D₂₁₂	1	2	yes	—
152	C₂²×C₈	D₂₁₂	1	2	yes	—
153	C₂²×C₈	D₂₁₂	1	2	yes	—
154	C₄⋊Q₈	—	53	4	no	—
155	C₄⋊Q₈	—	53	4	no	—
156	C₄⋊Q₈	—	106	2	no	—
157	C₄⋊Q₈	—	106	2	no	—
158	C₄⋊Q₈	—	106	2	no	—
159	C₄⋊Q₈	—	106	2	no	—
160	C₄⋊Q₈	—	106	2	no	—
161	C₄⋊Q₈	—	106	2	no	—
162	C₂×C₄⋊C₄	—	106	2	no	—
163	C₂×C₄⋊C₄	—	106	2	no	—
164	C₂×C₄⋊C₄	—	106	2	no	—
165	C₂×C₄⋊C₄	—	106	2	no	—
166	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
167	#107	—	106	8	no	—
168	#106	—	106	16	no	—
169	#103	—	106	16	no	—
170	C₄×C₈	D₂₁₂	1	2	yes	—
171	C₄×C₈	D₂₁₂	1	2	yes	—
172	C₄×C₈	D₂₁₂	1	2	yes	—
173	C₄×C₈	C₂²×D₅₃	1	2	yes	—
174	C₄×C₈	C₂²×D₅₃	1	2	yes	—
175	C₄×C₈	C₂²×D₅₃	1	2	yes	—
176	C₂×Q₈	—	106	4	no	—
177	C₂×Q₈	—	106	4	no	—
178	C₂×Q₈	—	212	8	no	—
179	C₂×Q₈	—	212	8	no	—
180	C₂×Q₈	—	212	2	no	—
181	C₂×Q₈	—	212	2	no	—
182	C₂²×C₄	—	212	2	no	—
183	C₂²×C₄	—	212	2	no	—
184	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
185	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
186	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
187	Q₁₆	—	212	16	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	#11719913	1	8	yes	—
190	C₂×C₈	Dic₂₁₂	1	8	yes	complemented
191	C₂×C₈	C₂×D₂₁₂	1	4	yes	—
192	C₂×C₈	C₂×D₂₁₂	1	4	yes	—
193	C₂×C₈	C₂×D₂₁₂	1	4	yes	—
194	C₄⋊C₄	—	106	4	no	—
195	C₄⋊C₄	—	106	4	no	—
196	C₄⋊C₄	—	106	4	no	—
197	C₄⋊C₄	—	106	4	no	—
198	C₄⋊C₄	—	212	2	no	—
199	C₄⋊C₄	—	212	2	no	—
200	C₄⋊C₄	—	212	2	no	—
201	C₄⋊C₄	—	212	2	no	—
202	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
203	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
204	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
205	C₄²	C₂³×D₅₃	1	1	yes	characteristic
206	C₂³	#13133214	1	1	yes	characteristic
207	Q₈	—	424	4	no	—
208	Q₈	—	424	4	no	—
209	Q₈	—	212	8	no	—
210	Q₈	—	212	8	no	—
211	C₂×C₄	—	424	4	no	—
212	C₂×C₄	—	424	4	no	—
213	C₂×C₄	—	212	4	no	—
214	C₂×C₄	—	212	4	no	—
215	C₂×C₄	#13133214	1	1	yes	characteristic
216	C₂×C₄	#13133214	1	1	yes	characteristic
217	C₂×C₄	#13133214	1	1	yes	characteristic
218	C₂×C₄	C₂×#11719913	1	4	yes	—
219	C₂×C₄	C₂×Dic₂₁₂	1	4	yes	—
220	C₂×C₄	#13133239	1	4	yes	—
221	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
222	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
223	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
224	C₈	—	2	4	no	—
225	C₈	C₂×#11719913	1	8	yes	—
226	C₈	C₂×Dic₂₁₂	1	8	yes	complemented
227	C₂²	#16810996	1	2	yes	—
228	C₂²	#16811003	1	2	yes	—
229	C₂²	#16811007	1	2	yes	—
230	C₂²	C₂×#13133214	1	1	yes	characteristic
231	C₄	—	424	4	no	—
232	C₄	—	424	4	no	—
233	C₄	#16810996	1	2	yes	—
234	C₄	#16811003	1	2	yes	—
235	C₄	#16811007	1	2	yes	—
236	C₄	C₂²×#11719913	1	2	yes	—
237	C₄	C₂²×Dic₂₁₂	1	2	yes	—
238	C₄	C₂×#13133239	1	2	yes	—
239	C₂	#20274312	1	4	yes	direct product factor
240	C₂	C₂×#16810996	1	1	yes	characteristic
241	C₂	C₂×#16811003	1	1	yes	characteristic
242	C₂	C₂×#16811007	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133214
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₈²×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	#127
hypercenter	C₂×C₈²	D₅₃
hypocenter	C₅₃	C₂×#1707
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
5. #20274312	C₂	direct (N × H)
67. C₂×C₈×C₅₃	Q₁₆	semidirect
104. C₈×C₅₃	C₂×Q₁₆	semidirect
124. C₂×C₅₃	#1707	semidirect
146. C₅₃	C₂×#1707	semidirect
190. C₂×C₈	Dic₂₁₂	semidirect
226. C₈	C₂×Dic₂₁₂	semidirect
239. C₂	#20274312	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×#16811028	2
C₂×#16811021	2
C₂×#16811003	2
C₂×#16810974	2
#20274312	2
C₂×C₈²×C₅₃	2
C₂×#1707	53

Subgroups — C2×#20274312 #27371741

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20274312 #27371741