Subgroups — C₂×#20273611 #27353833

Order 13568.

670 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810750	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
3	#20273611	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16810353	C₂	1	2	yes	maximal
5	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅₃×#335	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₁₂	C₄	1	1	yes	characteristic
8	C₂×C₄×Dic₁₀₆	C₂²	1	2	yes	—
9	#16810750	—	2	4	no	—
10	#16810750	C₂²	1	8	yes	—
11	C₄×Dic₂₁₂	C₄	1	4	yes	—
12	C₄×Dic₂₁₂	C₂²	1	4	yes	—
13	C₂×#13133151	C₄	1	1	yes	characteristic
14	C₂×#13133146	—	2	2	no	—
15	C₂×#13133142	C₄	1	2	yes	—
16	C₂×#13133127	—	2	2	no	—
17	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
18	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
19	#16810353	C₂²	1	8	yes	—
20	C₈×#88322	C₂²	1	4	yes	—
21	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₅₃×#335	C₂²	1	4	yes	—
24	C₂²×Dic₁₀₆	C₂×C₄	1	2	yes	—
25	C₂×Dic₂₁₂	—	2	4	no	—
26	C₂×Dic₂₁₂	C₂×C₄	1	4	yes	—
27	C₂²×#88322	—	2	1	no	—
28	C₂²×#88322	D₄	1	1	yes	characteristic
29	C₂²×#88322	D₄	1	1	yes	characteristic
30	C₄×Dic₁₀₆	—	2	2	no	—
31	C₄×Dic₁₀₆	—	2	2	no	—
32	C₄×Dic₁₀₆	C₂×C₄	1	4	yes	—
33	C₄×Dic₁₀₆	C₂³	1	4	yes	—
34	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
35	C₂×#11719919	—	2	2	no	—
36	C₂×C₄×Dic₅₃	—	2	2	no	—
37	#13133151	C₂×C₄	1	4	yes	—
38	#13133146	—	2	8	no	—
39	#13133142	C₂×C₄	1	8	yes	—
40	#13133127	—	2	8	no	—
41	C₄×#88322	—	2	1	no	—
42	C₄×#88322	D₄	1	2	yes	—
43	C₄×#88322	C₂³	1	2	yes	—
44	C₄×#88322	C₂³	1	2	yes	—
45	C₂²×C₈×C₅₃	—	2	2	no	—
46	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
49	C₄⋊C₈×C₅₃	—	2	2	no	—
50	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
51	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
52	C₄×C₈×C₅₃	C₂³	1	2	yes	—
53	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
56	C₂²×Dic₅₃	—	2	2	no	—
57	C₂²×Dic₅₃	—	2	2	no	—
58	C₂×Dic₁₀₆	—	2	4	no	—
59	C₂×Dic₁₀₆	—	2	4	no	—
60	C₂×Dic₁₀₆	—	2	2	no	—
61	C₂×Dic₁₀₆	C₂²×C₄	1	4	yes	—
62	C₂×C₈×C₅₃	—	2	2	no	—
63	C₂×C₈×C₅₃	—	2	4	no	—
64	C₂×C₈×C₅₃	—	4	4	no	—
65	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
66	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
68	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₅₃	#41	1	1	yes	characteristic
71	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
72	C₄⋊Dic₅₃	—	2	2	no	—
73	C₄⋊Dic₅₃	C₂²×C₄	1	4	yes	—
74	#11719919	—	2	4	no	—
75	#11719919	—	2	4	no	—
76	C₄×Dic₅₃	—	2	4	no	—
77	C₄×Dic₅₃	—	2	4	no	—
78	C₂×#88322	—	2	4	no	—
79	C₂×#88322	—	2	2	no	—
80	C₂×#88322	—	2	2	no	—
81	C₂×#88322	Q₁₆	1	4	yes	complemented
82	C₂×#88322	C₂×D₄	1	2	yes	—
83	C₂×#88322	C₂×D₄	1	2	yes	—
84	Dic₂₁₂	—	2	8	no	—
85	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
91	C₂×C₄×C₅₃	#131	1	2	yes	—
92	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
93	C₂×C₄×C₅₃	#135	1	2	yes	—
94	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
97	C₂×Dic₅₃	—	2	4	no	—
98	C₂×Dic₅₃	—	2	4	no	—
99	C₂×Dic₅₃	—	4	4	no	—
100	C₂×Dic₅₃	—	4	4	no	—
101	Dic₁₀₆	—	2	4	no	—
102	Dic₁₀₆	—	2	4	no	—
103	Dic₁₀₆	—	4	4	no	—
104	C₈×C₅₃	—	2	2	no	—
105	C₈×C₅₃	—	4	4	no	—
106	C₈×C₅₃	C₂×M₁₆	1	4	yes	—
107	#88322	—	2	4	no	—
108	#88322	—	2	2	no	—
109	#88322	C₂×Q₁₆	1	4	yes	complemented
110	C₂×#1591	—	53	1	no	maximal, Sylow
111	C₂²×C₅₃	#436	1	2	yes	—
112	C₂²×C₅₃	C₄×Q₁₆	1	2	yes	—
113	C₂²×C₅₃	#443	1	2	yes	—
114	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
115	C₄×C₅₃	—	2	1	no	—
116	C₄×C₅₃	—	2	1	no	—
117	C₄×C₅₃	#436	1	2	yes	—
118	C₄×C₅₃	C₄×Q₁₆	1	2	yes	—
119	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
120	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
121	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
122	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
123	Dic₅₃	—	4	4	no	—
124	Dic₅₃	—	4	4	no	—
125	C₂×#446	—	53	2	no	—
126	C₂×C₄×Q₁₆	—	53	1	no	—
127	#1591	—	53	8	no	—
128	C₂×#335	D₅₃	1	1	yes	characteristic
129	C₂×Q₈⋊C₈	—	53	2	no	—
130	C₂×C₈²	—	53	1	no	—
131	C₂×C₅₃	#1591	1	4	yes	complemented
132	C₂×C₅₃	C₂×#436	1	1	yes	characteristic
133	C₂×C₅₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
134	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
135	C₂²×Q₁₆	—	53	1	no	—
136	C₂×C₄×Q₈	—	53	2	no	—
137	#446	—	53	8	no	—
138	#446	—	106	4	no	—
139	C₄×Q₁₆	—	53	4	no	—
140	C₄×Q₁₆	—	53	4	no	—
141	C₂×#107	—	53	1	no	—
142	C₂×C₄⋊C₈	—	106	2	no	—
143	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
144	C₂×#103	—	53	2	no	—
145	C₂×#97	—	106	2	no	—
146	C₂×C₄×C₈	—	53	1	no	—
147	C₂×C₄×C₈	—	53	1	no	—
148	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
149	#335	C₂×D₅₃	1	4	yes	—
150	Q₈⋊C₈	—	53	8	no	—
151	C₈²	—	53	4	no	—
152	C₅₃	C₂×#1591	1	1	yes	characteristic, complemented, Sylow
153	C₂²×Q₈	—	53	2	no	—
154	C₂×Q₁₆	—	53	4	no	—
155	C₂×Q₁₆	—	106	4	no	—
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	—	53	1	no	—
158	C₂²×C₈	—	53	1	no	—
159	C₂²×C₈	—	106	1	no	—
160	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
161	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
162	C₄×Q₈	—	53	4	no	—
163	C₄×Q₈	—	53	4	no	—
164	C₄×Q₈	—	106	2	no	—
165	C₄×Q₈	—	106	2	no	—
166	C₂×C₄⋊C₄	—	53	2	no	—
167	C₂×C₄⋊C₄	—	106	2	no	—
168	C₂×C₄²	—	106	2	no	—
169	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
170	#107	—	53	4	no	—
171	C₄⋊C₈	—	2	2	no	—
172	C₄⋊C₈	—	106	8	no	—
173	C₄⋊C₈	C₂²×D₅₃	1	4	yes	—
174	#103	—	53	8	no	—
175	#97	—	106	8	no	—
176	C₄×C₈	—	53	2	no	—
177	C₄×C₈	—	53	2	no	—
178	C₄×C₈	—	53	2	no	—
179	C₄×C₈	—	106	1	no	—
180	C₄×C₈	C₄×D₅₃	1	2	yes	—
181	C₄×C₈	C₂²×D₅₃	1	2	yes	—
182	C₂×Q₈	—	53	4	no	—
183	C₂×Q₈	—	106	2	no	—
184	C₂×Q₈	—	106	4	no	—
185	C₂×Q₈	—	106	4	no	—
186	C₂²×C₄	—	106	2	no	—
187	C₂²×C₄	—	106	2	no	—
188	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
189	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
190	C₂²×C₄	#11719949	1	1	yes	characteristic
191	Q₁₆	—	106	8	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	4	no	—
194	C₂×C₈	—	4	4	no	—
195	C₂×C₈	—	53	4	no	—
196	C₂×C₈	—	53	2	no	—
197	C₂×C₈	—	53	2	no	—
198	C₂×C₈	—	106	4	no	—
199	C₂×C₈	—	106	2	no	—
200	C₂×C₈	—	106	2	no	—
201	C₂×C₈	#11719912	1	4	yes	—
202	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
203	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
204	C₄⋊C₄	—	53	4	no	—
205	C₄⋊C₄	—	106	2	no	—
206	C₄⋊C₄	—	106	4	no	—
207	C₄⋊C₄	—	106	4	no	—
208	C₄²	—	106	4	no	—
209	C₄²	—	106	4	no	—
210	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
211	C₄²	D₄×D₅₃	1	1	yes	characteristic
212	C₄²	#11719949	1	1	yes	characteristic
213	C₄²	C₂³×D₅₃	1	1	yes	characteristic
214	C₂³	#13133179	1	1	yes	characteristic
215	Q₈	—	106	4	no	—
216	Q₈	—	106	4	no	—
217	Q₈	—	212	4	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	106	4	no	—
220	C₂×C₄	—	106	4	no	—
221	C₂×C₄	—	212	4	no	—
222	C₂×C₄	—	212	4	no	—
223	C₂×C₄	#13133179	1	1	yes	characteristic
224	C₂×C₄	#13133179	1	1	yes	characteristic
225	C₂×C₄	#13133179	1	1	yes	characteristic
226	C₂×C₄	C₂×#11719912	1	2	yes	—
227	C₂×C₄	#13133225	1	2	yes	—
228	C₂×C₄	Q₁₆×D₅₃	1	2	yes	—
229	C₂×C₄	#13133236	1	2	yes	—
230	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
231	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
232	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
233	C₈	—	2	2	no	—
234	C₈	—	4	4	no	—
235	C₈	—	53	4	no	—
236	C₈	—	106	4	no	—
237	C₈	—	106	2	no	—
238	C₈	C₂×#11719912	1	4	yes	—
239	C₂²	#16810700	1	2	yes	—
240	C₂²	#16810717	1	2	yes	—
241	C₂²	#16810739	1	2	yes	—
242	C₂²	C₂×#13133179	1	1	yes	characteristic
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	212	4	no	—
246	C₄	—	212	4	no	—
247	C₄	#16810700	1	2	yes	—
248	C₄	#16810717	1	2	yes	—
249	C₄	C₂²×#11719912	1	1	yes	characteristic
250	C₄	C₂×#13133225	1	1	yes	characteristic
251	C₄	C₂×Q₁₆×D₅₃	1	1	yes	characteristic
252	C₄	C₂×#13133236	1	1	yes	characteristic
253	C₂	#20273611	1	4	yes	direct product factor
254	C₂	C₂×#16810700	1	1	yes	characteristic
255	C₂	C₂×#16810717	1	1	yes	characteristic
256	C₂	C₂×#16810739	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₅₃×#335	C₂
socle, prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂×#335	D₅₃
hypocenter	C₅₃	C₂×#1591
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
3. #20273611	C₂	direct (N × H)
81. C₂×#88322	Q₁₆	semidirect
109. #88322	C₂×Q₁₆	semidirect
131. C₂×C₅₃	#1591	semidirect
152. C₅₃	C₂×#1591	semidirect
253. C₂	#20273611	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×#16810750	2
C₂×C₄×Dic₂₁₂	2
#20273611	2
C₂×#16810353	2
C₂×C₈×#88322	2
C₂×C₅₃×#335	2
C₂×#1591	53

Subgroups — C2×#20273611 #27353833

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20273611 #27353833