Subgroups — C₂×#20274313 #27371742

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×#16811019	C₂	1	2	yes	maximal
2	C₂×#16811003	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810973	C₂	1	2	yes	maximal
4	#20274313	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	C₂²×Dic₂₁₂	—	2	2	no	—
8	C₂×#13133217	C₂²	1	2	yes	—
9	#16811019	—	2	4	no	—
10	#16811019	C₂²	1	8	yes	—
11	#16811003	—	2	2	no	—
12	#16811003	C₂²	1	4	yes	—
13	#16810973	—	2	4	no	—
14	#16810973	C₂²	1	8	yes	—
15	C₂×#13133148	—	2	4	no	—
16	C₂×#13133141	—	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	C₂×Dic₂₁₂	—	2	8	no	—
27	C₂×Dic₂₁₂	—	4	8	no	—
28	C₂²×#88322	D₄	1	2	yes	—
29	C₂²×#88322	D₄	1	2	yes	—
30	#13133217	—	2	2	no	—
31	#13133217	—	2	2	no	—
32	#13133217	—	2	2	no	—
33	#13133217	C₂³	1	4	yes	—
34	C₂×C₄⋊Dic₅₃	—	2	2	no	—
35	C₂×C₄⋊Dic₅₃	—	2	2	no	—
36	#13133148	—	2	16	no	—
37	#13133141	—	2	16	no	—
38	C₄×#88322	D₄	1	2	yes	—
39	C₄×#88322	D₄	1	2	yes	—
40	C₄×#88322	C₂³	1	2	yes	—
41	C₄×#88322	C₂³	1	2	yes	—
42	C₂²×Q₈×C₅₃	—	2	2	no	—
43	C₂²×C₈×C₅₃	D₄	1	2	yes	—
44	C₄⋊Q₈×C₅₃	—	2	2	no	—
45	C₄⋊Q₈×C₅₃	—	2	2	no	—
46	C₄⋊Q₈×C₅₃	—	2	2	no	—
47	C₄⋊Q₈×C₅₃	C₂³	1	4	yes	—
48	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
49	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
50	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
51	#107×C₅₃	—	2	8	no	—
52	C₄×C₈×C₅₃	D₄	1	2	yes	—
53	C₄×C₈×C₅₃	C₂³	1	2	yes	—
54	C₂×Q₈×C₅₃	—	2	4	no	—
55	C₂×Q₈×C₅₃	—	4	2	no	—
56	C₂×Q₈×C₅₃	—	4	8	no	—
57	C₂²×C₄×C₅₃	—	4	2	no	—
58	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×Dic₅₃	—	4	2	no	—
62	C₂×Dic₁₀₆	—	2	4	no	—
63	C₂×Dic₁₀₆	—	4	8	no	—
64	C₂×Dic₁₀₆	—	4	2	no	—
65	C₂×C₈×C₅₃	SD₁₆	1	8	yes	—
66	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
67	C₄⋊C₄×C₅₃	—	2	4	no	—
68	C₄⋊C₄×C₅₃	—	2	4	no	—
69	C₄⋊C₄×C₅₃	—	4	2	no	—
70	C₄⋊C₄×C₅₃	—	4	2	no	—
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	—	4	2	no	—
78	C₄⋊Dic₅₃	—	4	2	no	—
79	C₂×#88322	—	2	4	no	—
80	C₂×#88322	Q₁₆	1	8	yes	complemented
81	C₂×#88322	C₂×D₄	1	4	yes	—
82	C₂×#88322	C₂×D₄	1	4	yes	—
83	Dic₂₁₂	—	4	16	no	—
84	C₂³×C₅₃	#127	1	1	yes	characteristic
85	Q₈×C₅₃	—	4	8	no	—
86	Q₈×C₅₃	—	8	4	no	—
87	C₂×C₄×C₅₃	—	4	4	no	—
88	C₂×C₄×C₅₃	—	8	4	no	—
89	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
90	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
91	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂×SD₁₆	1	4	yes	—
93	C₂×C₄×C₅₃	C₂×Q₁₆	1	4	yes	—
94	C₂×C₄×C₅₃	#135	1	4	yes	—
95	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
98	C₂×Dic₅₃	—	4	4	no	—
99	C₂×Dic₅₃	—	8	4	no	—
100	Dic₁₀₆	—	4	8	no	—
101	Dic₁₀₆	—	8	4	no	—
102	C₈×C₅₃	C₂×SD₁₆	1	8	yes	—
103	#88322	—	2	4	no	—
104	#88322	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	—	8	4	no	—
111	C₄×C₅₃	#492	1	2	yes	—
112	C₄×C₅₃	#494	1	2	yes	—
113	C₄×C₅₃	#495	1	2	yes	—
114	C₄×C₅₃	C₂²×SD₁₆	1	2	yes	—
115	C₄×C₅₃	C₂²×Q₁₆	1	2	yes	—
116	C₄×C₅₃	C₂×#135	1	2	yes	—
117	Dic₅₃	—	8	4	no	—
118	C₂×#500	D₅₃	1	1	yes	characteristic
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	—	53	1	no	—
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	C₂×D₅₃	1	2	yes	—
131	#500	—	2	2	no	—
132	#500	C₂×D₅₃	1	4	yes	—
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	—	2	2	no	—
140	C₂×#106	—	106	4	no	—
141	C₂×#103	—	106	4	no	—
142	C₂×C₄×C₈	—	53	1	no	—
143	C₂×C₄×C₈	—	53	1	no	—
144	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
145	C₈²	—	53	4	no	—
146	C₅₃	C₂×#1707	1	1	yes	characteristic, complemented, Sylow
147	C₂²×Q₈	—	2	2	no	—
148	C₂²×Q₈	—	106	2	no	—
149	C₂×Q₁₆	—	106	8	no	—
150	C₂×Q₁₆	—	212	8	no	—
151	C₂²×C₈	—	53	2	no	—
152	C₂²×C₈	—	53	2	no	—
153	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
154	C₄⋊Q₈	—	2	2	no	—
155	C₄⋊Q₈	—	2	2	no	—
156	C₄⋊Q₈	—	2	2	no	—
157	C₄⋊Q₈	—	53	4	no	—
158	C₄⋊Q₈	—	106	2	no	—
159	C₄⋊Q₈	—	106	2	no	—
160	C₄⋊Q₈	—	106	2	no	—
161	C₄⋊Q₈	C₂²×D₅₃	1	4	yes	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄⋊C₄	—	2	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	—	2	8	no	—
168	#106	—	106	16	no	—
169	#103	—	106	16	no	—
170	C₄×C₈	—	53	2	no	—
171	C₄×C₈	—	53	2	no	—
172	C₄×C₈	—	53	2	no	—
173	C₄×C₈	—	53	2	no	—
174	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
175	C₄×C₈	C₂²×D₅₃	1	2	yes	—
176	C₂×Q₈	—	2	4	no	—
177	C₂×Q₈	—	4	2	no	—
178	C₂×Q₈	—	4	8	no	—
179	C₂×Q₈	—	106	4	no	—
180	C₂×Q₈	—	212	8	no	—
181	C₂×Q₈	—	212	2	no	—
182	C₂²×C₄	—	4	2	no	—
183	C₂²×C₄	—	212	2	no	—
184	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
185	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
186	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
187	Q₁₆	—	212	16	no	—
188	C₂×C₈	—	53	8	no	—
189	C₂×C₈	—	53	4	no	—
190	C₂×C₈	—	53	4	no	—
191	C₂×C₈	—	106	4	no	—
192	C₂×C₈	#11719924	1	8	yes	—
193	C₂×C₈	C₂×C₂²⋊D₅₃	1	4	yes	—
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	—	2	4	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄⋊C₄	—	106	4	no	—
199	C₄⋊C₄	—	106	4	no	—
200	C₄⋊C₄	—	212	2	no	—
201	C₄⋊C₄	—	212	2	no	—
202	C₄²	D₄×D₅₃	1	1	yes	characteristic
203	C₄²	D₄×D₅₃	1	1	yes	characteristic
204	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
205	C₄²	C₂³×D₅₃	1	1	yes	characteristic
206	C₂³	#13133213	1	1	yes	characteristic
207	Q₈	—	4	8	no	—
208	Q₈	—	8	4	no	—
209	Q₈	—	424	4	no	—
210	Q₈	—	212	8	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	—	8	4	no	—
213	C₂×C₄	—	424	4	no	—
214	C₂×C₄	—	212	4	no	—
215	C₂×C₄	#13133213	1	1	yes	characteristic
216	C₂×C₄	#13133213	1	1	yes	characteristic
217	C₂×C₄	#13133213	1	1	yes	characteristic
218	C₂×C₄	C₂×#11719924	1	4	yes	—
219	C₂×C₄	Q₁₆×D₅₃	1	4	yes	—
220	C₂×C₄	#13133236	1	4	yes	—
221	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
222	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
223	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
224	C₈	—	53	8	no	—
225	C₈	—	106	4	no	—
226	C₈	C₂×#11719924	1	8	yes	—
227	C₂²	#16810997	1	2	yes	—
228	C₂²	#16811001	1	2	yes	—
229	C₂²	#16811006	1	2	yes	—
230	C₂²	C₂×#13133213	1	1	yes	characteristic
231	C₄	—	8	4	no	—
232	C₄	—	424	4	no	—
233	C₄	#16810997	1	2	yes	—
234	C₄	#16811001	1	2	yes	—
235	C₄	#16811006	1	2	yes	—
236	C₄	C₂²×#11719924	1	2	yes	—
237	C₄	C₂×Q₁₆×D₅₃	1	2	yes	—
238	C₄	C₂×#13133236	1	2	yes	—
239	C₂	#20274313	1	4	yes	direct product factor
240	C₂	C₂×#16810997	1	1	yes	characteristic
241	C₂	C₂×#16811001	1	1	yes	characteristic
242	C₂	C₂×#16811006	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×#1707
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. #20274313	C₂	direct (N × H)
80. C₂×#88322	Q₁₆	semidirect
104. #88322	C₂×Q₁₆	semidirect
124. C₂×C₅₃	#1707	semidirect
146. C₅₃	C₂×#1707	semidirect
239. C₂	#20274313	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×#16811019	2
C₂×#16811003	2
C₂×#16810973	2
#20274313	2
C₂×C₈×#88322	2
C₂×C₅₃×#500	2
C₂×#1707	53

Subgroups — C2×#20274313 #27371742

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20274313 #27371742