Subgroups — C₂×#20274324 #27371764

Order 13568.

750 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×#16811028	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810975	C₂	1	2	yes	maximal
3	C₂×#16810973	C₂	1	2	yes	maximal
4	#20274324	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅₃×#500	C₂	1	1	yes	maximal, characteristic
7	C₂×#13133217	C₂²	1	2	yes	—
8	#16811028	—	2	2	no	—
9	#16811028	C₂²	1	4	yes	—
10	#16810975	—	2	4	no	—
11	#16810975	C₂²	1	8	yes	—
12	#16810973	—	2	4	no	—
13	#16810973	C₂²	1	8	yes	—
14	C₂×#13133151	—	2	2	no	—
15	C₂×#13133141	—	2	4	no	—
16	C₂×#13133140	—	2	4	no	—
17	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
18	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
19	C₈×#88322	C₂²	1	4	yes	—
20	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
21	C₅₃×#500	—	2	2	no	—
22	C₅₃×#500	C₂²	1	4	yes	—
23	C₂×#107×C₅₃	—	2	2	no	—
24	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₂²×Dic₁₀₆	—	2	2	no	—
26	C₂²×#88322	D₄	1	2	yes	—
27	C₂²×#88322	D₄	1	2	yes	—
28	#13133217	—	2	2	no	—
29	#13133217	—	2	2	no	—
30	#13133217	—	2	2	no	—
31	#13133217	C₂³	1	4	yes	—
32	C₂×C₄⋊Dic₅₃	—	2	2	no	—
33	C₂×C₄⋊Dic₅₃	—	2	2	no	—
34	#13133151	—	2	8	no	—
35	#13133141	—	2	16	no	—
36	#13133140	—	2	16	no	—
37	C₄×#88322	D₄	1	2	yes	—
38	C₄×#88322	D₄	1	2	yes	—
39	C₄×#88322	C₂³	1	2	yes	—
40	C₄×#88322	C₂³	1	2	yes	—
41	C₂²×Q₈×C₅₃	—	2	2	no	—
42	C₂²×C₈×C₅₃	D₄	1	2	yes	—
43	C₄⋊Q₈×C₅₃	—	2	2	no	—
44	C₄⋊Q₈×C₅₃	—	2	2	no	—
45	C₄⋊Q₈×C₅₃	—	2	2	no	—
46	C₄⋊Q₈×C₅₃	C₂³	1	4	yes	—
47	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
48	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
49	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
50	#107×C₅₃	—	2	8	no	—
51	C₄×C₈×C₅₃	D₄	1	2	yes	—
52	C₄×C₈×C₅₃	C₂³	1	2	yes	—
53	C₂×Q₈×C₅₃	—	2	4	no	—
54	C₂×Q₈×C₅₃	—	4	2	no	—
55	C₂×Q₈×C₅₃	—	4	8	no	—
56	C₂²×C₄×C₅₃	—	4	2	no	—
57	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×Dic₅₃	—	4	2	no	—
61	C₂×Dic₁₀₆	—	2	4	no	—
62	C₂×Dic₁₀₆	—	4	8	no	—
63	C₂×Dic₁₀₆	—	4	2	no	—
64	C₂×C₈×C₅₃	Q₁₆	1	8	yes	—
65	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
66	C₄⋊C₄×C₅₃	—	2	4	no	—
67	C₄⋊C₄×C₅₃	—	2	4	no	—
68	C₄⋊C₄×C₅₃	—	4	2	no	—
69	C₄⋊C₄×C₅₃	—	4	2	no	—
70	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
74	C₄⋊Dic₅₃	—	2	4	no	—
75	C₄⋊Dic₅₃	—	2	4	no	—
76	C₄⋊Dic₅₃	—	4	2	no	—
77	C₄⋊Dic₅₃	—	4	2	no	—
78	C₂×#88322	—	2	4	no	—
79	C₂×#88322	—	2	4	no	—
80	C₂×#88322	C₂×D₄	1	4	yes	—
81	C₂×#88322	C₂×D₄	1	4	yes	—
82	C₂³×C₅₃	#127	1	1	yes	characteristic
83	Q₈×C₅₃	—	4	8	no	—
84	Q₈×C₅₃	—	8	4	no	—
85	C₂×C₄×C₅₃	—	4	4	no	—
86	C₂×C₄×C₅₃	—	8	4	no	—
87	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
88	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
89	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂×Q₁₆	1	4	yes	—
91	C₂×C₄×C₅₃	#135	1	4	yes	—
92	C₂×C₄×C₅₃	#135	1	4	yes	—
93	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
96	C₂×Dic₅₃	—	4	4	no	—
97	C₂×Dic₅₃	—	8	4	no	—
98	Dic₁₀₆	—	4	8	no	—
99	Dic₁₀₆	—	8	4	no	—
100	C₈×C₅₃	C₂×Q₁₆	1	8	yes	—
101	#88322	—	2	4	no	—
102	#88322	—	2	4	no	—
103	C₂×#1710	—	53	1	no	maximal, Sylow
104	C₂²×C₅₃	#494	1	2	yes	—
105	C₂²×C₅₃	#495	1	2	yes	—
106	C₂²×C₅₃	#495	1	2	yes	—
107	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
108	C₄×C₅₃	—	8	4	no	—
109	C₄×C₅₃	#494	1	2	yes	—
110	C₄×C₅₃	#495	1	2	yes	—
111	C₄×C₅₃	#495	1	2	yes	—
112	C₄×C₅₃	C₂²×Q₁₆	1	2	yes	—
113	C₄×C₅₃	C₂×#135	1	2	yes	—
114	C₄×C₅₃	C₂×#135	1	2	yes	—
115	Dic₅₃	—	8	4	no	—
116	C₂×#500	—	53	1	no	—
117	C₂×#500	D₅₃	1	1	yes	characteristic
118	C₂×#487	—	53	2	no	—
119	C₂×#487	—	53	2	no	—
120	#1710	—	53	8	no	—
121	C₂×C₈²	—	53	1	no	—
122	C₂×C₅₃	#1710	1	4	yes	complemented
123	C₂×C₅₃	C₂×#494	1	1	yes	characteristic
124	C₂×C₅₃	C₂×#495	1	1	yes	characteristic
125	C₂×C₅₃	C₂×#495	1	1	yes	characteristic
126	C₂×C₄⋊Q₈	—	53	2	no	—
127	C₂×C₄⋊Q₈	C₂×D₅₃	1	2	yes	—
128	#500	—	2	2	no	—
129	#500	—	53	4	no	—
130	#500	—	106	2	no	—
131	#500	C₂×D₅₃	1	4	yes	—
132	#487	—	53	8	no	—
133	#487	—	53	8	no	—
134	#487	—	106	4	no	—
135	#487	—	106	4	no	—
136	C₂×#107	—	2	2	no	—
137	C₂×#107	—	106	2	no	—
138	C₂×#103	—	106	4	no	—
139	C₂×#103	—	106	4	no	—
140	C₂×C₄×C₈	—	53	1	no	—
141	C₂×C₄×C₈	—	53	1	no	—
142	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
143	C₈²	—	53	4	no	—
144	C₅₃	C₂×#1710	1	1	yes	characteristic, complemented, Sylow
145	C₂²×Q₈	—	2	2	no	—
146	C₂²×Q₈	—	106	2	no	—
147	C₂²×C₈	—	53	2	no	—
148	C₂²×C₈	—	53	2	no	—
149	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
150	C₄⋊Q₈	—	2	2	no	—
151	C₄⋊Q₈	—	2	2	no	—
152	C₄⋊Q₈	—	2	2	no	—
153	C₄⋊Q₈	—	53	4	no	—
154	C₄⋊Q₈	—	106	2	no	—
155	C₄⋊Q₈	—	106	2	no	—
156	C₄⋊Q₈	—	106	2	no	—
157	C₄⋊Q₈	C₂²×D₅₃	1	4	yes	—
158	C₂×C₄⋊C₄	—	2	2	no	—
159	C₂×C₄⋊C₄	—	2	2	no	—
160	C₂×C₄⋊C₄	—	106	2	no	—
161	C₂×C₄⋊C₄	—	106	2	no	—
162	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
163	#107	—	2	8	no	—
164	#107	—	106	8	no	—
165	#103	—	106	16	no	—
166	#103	—	106	16	no	—
167	C₄×C₈	—	53	2	no	—
168	C₄×C₈	—	53	2	no	—
169	C₄×C₈	—	53	2	no	—
170	C₄×C₈	—	53	2	no	—
171	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
172	C₄×C₈	C₂²×D₅₃	1	2	yes	—
173	C₂×Q₈	—	2	4	no	—
174	C₂×Q₈	—	4	2	no	—
175	C₂×Q₈	—	4	8	no	—
176	C₂×Q₈	—	106	4	no	—
177	C₂×Q₈	—	212	8	no	—
178	C₂×Q₈	—	212	2	no	—
179	C₂²×C₄	—	4	2	no	—
180	C₂²×C₄	—	212	2	no	—
181	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
182	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
183	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
184	C₂×C₈	—	53	4	no	—
185	C₂×C₈	—	53	4	no	—
186	C₂×C₈	—	106	4	no	—
187	C₂×C₈	—	106	4	no	—
188	C₂×C₈	#11719925	1	8	yes	—
189	C₂×C₈	C₂×C₂²⋊D₅₃	1	4	yes	—
190	C₄⋊C₄	—	2	4	no	—
191	C₄⋊C₄	—	2	4	no	—
192	C₄⋊C₄	—	4	2	no	—
193	C₄⋊C₄	—	4	2	no	—
194	C₄⋊C₄	—	106	4	no	—
195	C₄⋊C₄	—	106	4	no	—
196	C₄⋊C₄	—	212	2	no	—
197	C₄⋊C₄	—	212	2	no	—
198	C₄²	D₄×D₅₃	1	1	yes	characteristic
199	C₄²	D₄×D₅₃	1	1	yes	characteristic
200	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
201	C₄²	C₂³×D₅₃	1	1	yes	characteristic
202	C₂³	#13133213	1	1	yes	characteristic
203	Q₈	—	4	8	no	—
204	Q₈	—	8	4	no	—
205	Q₈	—	424	4	no	—
206	Q₈	—	212	8	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	—	8	4	no	—
209	C₂×C₄	—	424	4	no	—
210	C₂×C₄	—	212	4	no	—
211	C₂×C₄	#13133213	1	1	yes	characteristic
212	C₂×C₄	#13133213	1	1	yes	characteristic
213	C₂×C₄	#13133213	1	1	yes	characteristic
214	C₂×C₄	C₂×#11719925	1	4	yes	—
215	C₂×C₄	#13133236	1	4	yes	—
216	C₂×C₄	#13133240	1	4	yes	—
217	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
218	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
219	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
220	C₈	—	106	4	no	—
221	C₈	—	106	4	no	—
222	C₈	C₂×#11719925	1	8	yes	—
223	C₂²	#16811002	1	2	yes	—
224	C₂²	#16811006	1	2	yes	—
225	C₂²	#16811008	1	2	yes	—
226	C₂²	C₂×#13133213	1	1	yes	characteristic
227	C₄	—	8	4	no	—
228	C₄	—	424	4	no	—
229	C₄	#16811002	1	2	yes	—
230	C₄	#16811006	1	2	yes	—
231	C₄	#16811008	1	2	yes	—
232	C₄	C₂²×#11719925	1	2	yes	—
233	C₄	C₂×#13133236	1	2	yes	—
234	C₄	C₂×#13133240	1	2	yes	—
235	C₂	#20274324	1	4	yes	direct product factor
236	C₂	C₂×#16811002	1	1	yes	characteristic
237	C₂	C₂×#16811006	1	1	yes	characteristic
238	C₂	C₂×#16811008	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133213
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₅₃×#500	C₂
socle, prime radical	C₂³×C₅₃	#127
hypercenter	C₂×#500	D₅₃
hypocenter	C₅₃	C₂×#1710
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
4. #20274324	C₂	direct (N × H)
122. C₂×C₅₃	#1710	semidirect
144. C₅₃	C₂×#1710	semidirect
235. C₂	#20274324	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×#16810975	2
C₂×#16810973	2
#20274324	2
C₂×C₈×#88322	2
C₂×C₅₃×#500	2
C₂×#1710	53

Subgroups — C2×#20274324 #27371764

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20274324 #27371764