Subgroups — C₂×#20272678 #27293826

Order 13568.

406 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic
3	#20272678	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₅₃×#335	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₁₂	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₁₀₆	—	2	1	no	—
7	C₂²×#11719908	—	2	1	no	—
8	C₄×Dic₂₁₂	—	2	1	no	—
9	C₄×Dic₂₁₂	—	2	1	no	—
10	C₄×Dic₂₁₂	C₄	1	2	yes	—
11	C₄×Dic₂₁₂	C₂²	1	2	yes	—
12	C₂×#13133151	C₄	1	1	yes	characteristic
13	C₂×#13133142	—	2	1	no	—
14	C₄×#11719908	—	2	1	no	—
15	C₄×#11719908	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#335	—	2	1	no	—
19	C₅₃×#335	C₂²	1	2	yes	—
20	C₂²×Dic₁₀₆	—	2	1	no	—
21	C₂×Dic₂₁₂	—	2	1	no	—
22	C₂×Dic₂₁₂	—	2	2	no	—
23	C₂×Dic₂₁₂	C₈	1	4	yes	complemented
24	C₂×Dic₂₁₂	C₂×C₄	1	2	yes	—
25	C₄×Dic₁₀₆	—	2	2	no	—
26	C₄×Dic₁₀₆	—	2	2	no	—
27	C₄×Dic₁₀₆	—	4	1	no	—
28	C₄×Dic₁₀₆	—	4	1	no	—
29	C₂×C₄⋊Dic₅₃	—	2	1	no	—
30	C₂×#11719919	—	4	1	no	—
31	C₂×C₄×Dic₅₃	—	4	1	no	—
32	C₂×#11719908	—	2	4	no	—
33	C₂×#11719908	—	2	2	no	—
34	#13133151	—	2	1	no	—
35	#13133151	C₂×C₄	1	2	yes	—
36	#13133142	—	2	2	no	—
37	#13133142	—	2	2	no	—
38	C₂²×C₈×C₅₃	—	4	1	no	—
39	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
42	C₄⋊C₈×C₅₃	—	2	2	no	—
43	C₄⋊C₈×C₅₃	—	4	1	no	—
44	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
46	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
47	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
51	C₂²×Dic₅₃	—	4	1	no	—
52	C₂²×Dic₅₃	—	4	1	no	—
53	C₂×Dic₁₀₆	—	2	4	no	—
54	C₂×Dic₁₀₆	—	2	2	no	—
55	C₂×Dic₁₀₆	—	4	1	no	—
56	C₂×Dic₁₀₆	—	4	2	no	—
57	C₂×C₈×C₅₃	—	2	2	no	—
58	C₂×C₈×C₅₃	—	4	2	no	—
59	C₂×C₈×C₅₃	—	8	2	no	—
60	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
63	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
64	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₄²×C₅₃	D₈	1	1	yes	characteristic
68	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
69	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₄⋊Dic₅₃	—	2	2	no	—
71	C₄⋊Dic₅₃	—	4	1	no	—
72	#11719919	—	4	2	no	—
73	#11719919	—	4	2	no	—
74	C₄×Dic₅₃	—	4	2	no	—
75	C₄×Dic₅₃	—	4	2	no	—
76	Dic₂₁₂	—	2	4	no	—
77	Dic₂₁₂	—	2	2	no	—
78	Dic₂₁₂	C₂×C₈	1	4	yes	complemented
79	#11719908	—	2	4	no	—
80	C₂³×C₅₃	#102	1	1	yes	characteristic
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
83	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
84	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
85	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
86	C₂×C₄×C₅₃	#104	1	2	yes	—
87	C₂×C₄×C₅₃	SD₃₂	1	2	yes	—
88	C₂×C₄×C₅₃	Q₃₂	1	2	yes	—
89	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
92	C₂×Dic₅₃	—	4	4	no	—
93	C₂×Dic₅₃	—	4	2	no	—
94	C₂×Dic₅₃	—	4	2	no	—
95	C₂×Dic₅₃	—	8	2	no	—
96	Dic₁₀₆	—	2	4	no	—
97	Dic₁₀₆	—	4	4	no	—
98	Dic₁₀₆	—	4	2	no	—
99	C₈×C₅₃	—	2	1	no	—
100	C₈×C₅₃	—	2	1	no	—
101	C₈×C₅₃	—	8	2	no	—
102	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
103	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
104	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
105	C₂×#1332	—	53	1	no	maximal, Sylow
106	C₂²×C₅₃	D₄⋊C₈	1	2	yes	—
107	C₂²×C₅₃	#358	1	2	yes	—
108	C₂²×C₅₃	#359	1	2	yes	—
109	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
110	C₄×C₅₃	—	2	1	no	—
111	C₄×C₅₃	—	2	1	no	—
112	C₄×C₅₃	D₄⋊C₈	1	2	yes	—
113	C₄×C₅₃	#358	1	2	yes	—
114	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
115	C₄×C₅₃	C₂×#104	1	1	yes	characteristic
116	C₄×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
117	C₄×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
118	Dic₅₃	—	4	4	no	—
119	Dic₅₃	—	8	2	no	—
120	C₂×C₄×Q₁₆	—	53	1	no	—
121	C₂×C₄×C₁₆	—	53	1	no	—
122	C₂×#335	D₅₃	1	1	yes	characteristic
123	#1332	—	53	4	no	—
124	C₂×C₅₃	#1332	1	4	yes	complemented
125	C₂×C₅₃	C₂×D₄⋊C₈	1	1	yes	characteristic
126	C₂×C₅₃	C₂×#358	1	1	yes	characteristic
127	C₂×C₅₃	C₂×#359	1	1	yes	characteristic
128	C₂²×Q₁₆	—	53	1	no	—
129	C₂×C₄×Q₈	—	106	1	no	—
130	C₂²×C₁₆	—	106	1	no	—
131	C₄×Q₁₆	—	53	2	no	—
132	C₄×Q₁₆	—	53	2	no	—
133	C₄×Q₁₆	—	106	1	no	—
134	C₄×Q₁₆	—	106	1	no	—
135	C₂×#107	—	53	1	no	—
136	C₂×C₄⋊C₈	—	2	1	no	—
137	C₂×#103	—	106	1	no	—
138	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
139	C₄×C₁₆	—	53	2	no	—
140	C₄×C₁₆	—	106	1	no	—
141	#335	—	2	1	no	—
142	#335	C₂×D₅₃	1	2	yes	—
143	C₅₃	C₂×#1332	1	1	yes	characteristic, complemented, Sylow
144	C₂²×Q₈	—	106	1	no	—
145	C₂×Q₁₆	—	53	2	no	—
146	C₂×Q₁₆	—	53	4	no	—
147	C₂×Q₁₆	—	106	2	no	—
148	C₂×Q₁₆	—	106	1	no	—
149	C₂²×C₈	—	4	1	no	—
150	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
151	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
152	C₄×Q₈	—	106	2	no	—
153	C₄×Q₈	—	106	2	no	—
154	C₄×Q₈	—	212	1	no	—
155	C₄×Q₈	—	212	1	no	—
156	C₂×C₄⋊C₄	—	106	1	no	—
157	C₂×C₄⋊C₄	—	212	1	no	—
158	C₂×C₄²	—	212	1	no	—
159	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
160	C₂×C₁₆	—	106	4	no	—
161	C₂×C₁₆	—	106	2	no	—
162	#107	—	53	2	no	—
163	#107	—	106	1	no	—
164	C₄⋊C₈	—	2	2	no	—
165	C₄⋊C₈	—	4	1	no	—
166	#103	—	106	2	no	—
167	#103	—	106	2	no	—
168	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
169	C₄×C₈	D₂₁₂	1	1	yes	characteristic
170	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
171	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
172	C₂×Q₈	—	106	4	no	—
173	C₂×Q₈	—	106	2	no	—
174	C₂×Q₈	—	212	1	no	—
175	C₂×Q₈	—	212	2	no	—
176	C₂²×C₄	—	212	1	no	—
177	C₂²×C₄	—	212	1	no	—
178	C₂²×C₄	#11719921	1	1	yes	characteristic
179	C₂²×C₄	#11719922	1	1	yes	characteristic
180	C₂²×C₄	#11719924	1	1	yes	characteristic
181	Q₁₆	—	53	4	no	—
182	Q₁₆	—	106	4	no	—
183	Q₁₆	—	106	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	4	2	no	—
186	C₂×C₈	—	8	2	no	—
187	C₂×C₈	C₈×D₅₃	1	2	yes	—
188	C₂×C₈	#11719912	1	2	yes	—
189	C₂×C₈	#11719921	1	1	yes	characteristic
190	C₂×C₈	#11719921	1	1	yes	characteristic
191	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
192	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
193	C₄⋊C₄	—	106	2	no	—
194	C₄⋊C₄	—	212	1	no	—
195	C₄⋊C₄	—	212	2	no	—
196	C₄⋊C₄	—	212	2	no	—
197	C₄²	—	212	2	no	—
198	C₄²	—	212	2	no	—
199	C₄²	#11719921	1	1	yes	characteristic
200	C₄²	#11719922	1	1	yes	characteristic
201	C₄²	#11719924	1	1	yes	characteristic
202	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
203	C₁₆	—	106	4	no	—
204	C₂³	#13133137	1	1	yes	characteristic
205	Q₈	—	106	4	no	—
206	Q₈	—	212	4	no	—
207	Q₈	—	212	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	424	2	no	—
210	C₂×C₄	—	212	2	no	—
211	C₂×C₄	—	212	4	no	—
212	C₂×C₄	—	212	2	no	—
213	C₂×C₄	#13133129	1	2	yes	—
214	C₂×C₄	#13133137	1	1	yes	characteristic
215	C₂×C₄	#13133137	1	1	yes	characteristic
216	C₂×C₄	#13133137	1	1	yes	characteristic
217	C₂×C₄	#13133143	1	2	yes	—
218	C₂×C₄	#13133161	1	2	yes	—
219	C₂×C₄	#13133163	1	2	yes	—
220	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
221	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
222	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
223	C₈	—	2	1	no	—
224	C₈	—	2	1	no	—
225	C₈	—	8	2	no	—
226	C₈	#13133129	1	2	yes	—
227	C₈	C₂×C₈×D₅₃	1	1	yes	characteristic
228	C₈	C₂×#11719912	1	1	yes	characteristic
229	C₂²	#16810350	1	2	yes	—
230	C₂²	#16810426	1	2	yes	—
231	C₂²	#16810429	1	2	yes	—
232	C₂²	C₂×#13133137	1	1	yes	characteristic
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	424	2	no	—
236	C₄	—	212	4	no	—
237	C₄	#16810350	1	2	yes	—
238	C₄	#16810426	1	2	yes	—
239	C₄	C₂×#13133129	1	1	yes	characteristic
240	C₄	C₂×#13133143	1	1	yes	characteristic
241	C₄	C₂×#13133161	1	1	yes	characteristic
242	C₄	C₂×#13133163	1	1	yes	characteristic
243	C₂	#20272678	1	4	yes	direct product factor
244	C₂	C₂×#16810350	1	1	yes	characteristic
245	C₂	C₂×#16810426	1	1	yes	characteristic
246	C₂	C₂×#16810429	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#11719922
commutator	C₈×C₅₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#335	C₂
socle, prime radical	C₂³×C₅₃	#102
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₂×#335	D₅₃
hypocenter	C₅₃	C₂×#1332
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
3. #20272678	C₂	direct (N × H)
23. C₂×Dic₂₁₂	C₈	semidirect
78. Dic₂₁₂	C₂×C₈	semidirect
124. C₂×C₅₃	#1332	semidirect
143. C₅₃	C₂×#1332	semidirect
243. C₂	#20272678	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×C₄×Dic₂₁₂	2
C₂×C₄×#11719908	2
#20272678	2
C₂×C₅₃×#335	2
C₂×#1332	53

Subgroups — C2×#20272678 #27293826

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20272678 #27293826