Subgroups — C₂×#20273495 #27353159

Order 13568.

670 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810680	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810676	C₂	1	1	yes	maximal, characteristic
3	#20273495	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16810376	C₂	1	2	yes	maximal, complemented
5	C₂×#16810374	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₃×#431	C₂	1	1	yes	maximal, characteristic
7	#16810680	—	2	2	no	—
8	#16810680	C₂²	1	4	yes	—
9	#16810676	—	2	2	no	—
10	#16810676	C₂²	1	4	yes	—
11	C₂×C₄⋊#88322	C₂²	1	2	yes	—
12	C₂×C₄⋊#88322	C₂²	1	2	yes	—
13	C₂×C₂²⋊#88322	—	2	1	no	—
14	C₂×C₂²⋊#88322	—	2	1	no	—
15	C₂×#13133128	—	2	1	no	—
16	C₂×C₄×#88322	—	2	1	no	—
17	#16810376	C₂²	1	8	yes	complemented
18	#16810374	C₂²	1	8	yes	complemented
19	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
20	C₅₃×#431	C₄	1	4	yes	—
21	C₅₃×#431	C₂²	1	4	yes	—
22	C₂×C₄⋊C₈×C₅₃	C₄	1	2	yes	—
23	C₂×C₂²⋊C₈×C₅₃	C₄	1	2	yes	—
24	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₂²×#88322	—	2	2	no	—
27	C₂²×#88322	—	2	2	no	—
28	C₄⋊#88322	—	2	2	no	—
29	C₄⋊#88322	—	2	2	no	—
30	C₄⋊#88322	C₂³	1	4	yes	—
31	C₄⋊#88322	C₂³	1	4	yes	—
32	C₂²⋊#88322	—	2	4	no	—
33	C₂²⋊#88322	—	2	4	no	—
34	#13133128	—	2	4	no	—
35	C₄×#88322	—	2	4	no	—
36	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
39	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₅₃	D₄	1	2	yes	—
45	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
46	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
47	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
49	C₄⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
50	C₂²⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
51	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
52	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₅₃	C₂³	1	2	yes	—
54	C₄×C₈×C₅₃	C₂³	1	2	yes	—
55	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	—	2	4	no	—
57	C₂²×C₄×C₅₃	—	2	4	no	—
58	C₂²×C₄×C₅₃	—	4	2	no	—
59	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
71	C₂×C₈×C₅₃	—	2	4	no	—
72	C₂×C₈×C₅₃	—	2	4	no	—
73	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
74	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
75	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
76	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
77	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
78	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
80	C₄²×C₅₃	C₂×D₄	1	2	yes	—
81	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
83	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
84	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
85	C₂×#88322	—	2	4	no	—
86	C₂×#88322	—	2	4	no	—
87	C₂×#88322	—	4	4	no	—
88	C₂×#88322	—	4	4	no	—
89	C₂³×C₅₃	—	2	2	no	—
90	C₂³×C₅₃	—	2	2	no	—
91	C₂³×C₅₃	—	2	2	no	—
92	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
93	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₅₃	—	2	2	no	—
95	C₂×C₄×C₅₃	—	2	2	no	—
96	C₂×C₄×C₅₃	—	2	2	no	—
97	C₂×C₄×C₅₃	—	2	2	no	—
98	C₂×C₄×C₅₃	—	2	4	no	—
99	C₂×C₄×C₅₃	—	2	4	no	—
100	C₂×C₄×C₅₃	—	2	4	no	—
101	C₂×C₄×C₅₃	—	2	4	no	—
102	C₂×C₄×C₅₃	—	4	2	no	—
103	C₂×C₄×C₅₃	—	4	2	no	—
104	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
105	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
106	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
107	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₅₃	#131	1	4	yes	—
111	C₂×C₄×C₅₃	#135	1	4	yes	—
112	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
115	C₈×C₅₃	—	2	4	no	—
116	C₈×C₅₃	—	2	4	no	—
117	#88322	—	4	4	no	—
118	#88322	—	4	4	no	—
119	C₂×#1572	—	53	1	no	maximal, Sylow
120	C₂²×C₅₃	—	2	2	no	—
121	C₂²×C₅₃	—	2	2	no	—
122	C₂²×C₅₃	—	2	2	no	—
123	C₂²×C₅₃	—	4	4	no	—
124	C₂²×C₅₃	#424	1	2	yes	—
125	C₂²×C₅₃	#427	1	2	yes	—
126	C₂²×C₅₃	#430	1	2	yes	—
127	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
128	C₄×C₅₃	—	2	2	no	—
129	C₄×C₅₃	—	2	2	no	—
130	C₄×C₅₃	—	2	2	no	—
131	C₄×C₅₃	—	2	1	no	—
132	C₄×C₅₃	—	2	1	no	—
133	C₄×C₅₃	#424	1	2	yes	—
134	C₄×C₅₃	#427	1	2	yes	—
135	C₄×C₅₃	C₂×#131	1	2	yes	—
136	C₄×C₅₃	C₂×#135	1	2	yes	—
137	C₂×#432	—	53	1	no	—
138	C₂×#431	—	53	1	no	—
139	C₂×#431	D₅₃	1	1	yes	characteristic
140	#1572	—	53	8	no	—
141	C₂×#335	—	53	2	no	—
142	C₂×#334	—	53	2	no	—
143	C₂×C₅₃	—	4	2	no	—
144	C₂×C₅₃	#1572	1	4	yes	complemented
145	C₂×C₅₃	C₂×#424	1	1	yes	characteristic
146	C₂×C₅₃	C₂×#427	1	1	yes	characteristic
147	C₂×C₅₃	C₂×#430	1	1	yes	characteristic
148	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
149	#432	—	53	4	no	—
150	#432	—	106	2	no	—
151	#431	—	53	4	no	—
152	#431	—	106	2	no	—
153	#431	Dic₅₃	1	4	yes	—
154	#431	C₂×D₅₃	1	4	yes	—
155	C₂×C₄⋊C₈	—	53	2	no	—
156	C₂×C₄⋊C₈	—	53	2	no	—
157	C₂×C₄⋊C₈	Dic₅₃	1	2	yes	—
158	C₂×C₂²⋊C₈	—	106	1	no	—
159	C₂×C₂²⋊C₈	—	106	1	no	—
160	C₂×C₂²⋊C₈	Dic₅₃	1	2	yes	—
161	C₂×#97	—	106	1	no	—
162	C₂×C₄×C₈	—	106	1	no	—
163	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
164	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
165	#335	—	53	8	no	—
166	#334	—	53	8	no	—
167	C₅₃	C₂×#1572	1	1	yes	characteristic, complemented, Sylow
168	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
169	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
170	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
171	C₂²×C₈	—	106	2	no	—
172	C₂²×C₈	—	106	2	no	—
173	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
174	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
175	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
176	C₂×C₄²	Dic₁₀₆	1	2	yes	—
177	C₂×C₄²	D₂₁₂	1	2	yes	—
178	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
179	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
180	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
181	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
182	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
183	C₄⋊C₈	—	53	4	no	—
184	C₄⋊C₈	—	53	4	no	—
185	C₄⋊C₈	—	106	2	no	—
186	C₄⋊C₈	—	106	2	no	—
187	C₄⋊C₈	C₂×Dic₅₃	1	8	yes	—
188	C₂²⋊C₈	—	106	4	no	—
189	C₂²⋊C₈	—	106	4	no	—
190	C₂²⋊C₈	C₂×Dic₅₃	1	8	yes	—
191	#97	—	106	4	no	—
192	C₄×C₈	—	106	4	no	—
193	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
194	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
195	C₄×C₈	C₂²×D₅₃	1	2	yes	—
196	C₄×C₈	C₂²×D₅₃	1	2	yes	—
197	C₂⁴	C₄⋊Dic₅₃	1	1	yes	characteristic
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
202	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
203	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
204	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
205	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
206	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
207	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
208	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
209	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
210	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
211	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
212	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	106	4	no	—
216	C₂×C₈	—	106	4	no	—
217	C₂×C₈	—	212	4	no	—
218	C₂×C₈	—	212	4	no	—
219	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
220	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
221	C₄²	C₄⋊Dic₅₃	1	2	yes	—
222	C₄²	C₄⋊Dic₅₃	1	2	yes	—
223	C₄²	C₄⋊Dic₅₃	1	2	yes	—
224	C₄²	C₂×Dic₁₀₆	1	2	yes	—
225	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
226	C₄²	C₂×D₂₁₂	1	2	yes	—
227	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
228	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
229	C₄²	C₂²×Dic₅₃	1	2	yes	—
230	C₄²	C₂³×D₅₃	1	1	yes	characteristic
231	C₂³	—	2	2	no	—
232	C₂³	—	2	2	no	—
233	C₂³	—	2	2	no	—
234	C₂³	C₂×C₄⋊Dic₅₃	1	2	yes	—
235	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
247	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
248	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
249	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
250	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
251	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
252	C₂×C₄	#13133224	1	4	yes	—
253	C₂×C₄	#13133239	1	4	yes	—
254	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
255	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
256	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
257	C₈	—	2	4	no	—
258	C₈	—	2	4	no	—
259	C₈	—	212	4	no	—
260	C₈	—	212	4	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	4	4	no	—
265	C₂²	#16810650	1	2	yes	—
266	C₂²	#16810661	1	2	yes	—
267	C₂²	#16810672	1	2	yes	—
268	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
269	C₄	—	2	2	no	—
270	C₄	—	2	2	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	2	no	—
273	C₄	—	2	1	no	—
274	C₄	#16810650	1	2	yes	—
275	C₄	#16810661	1	2	yes	—
276	C₄	C₂×#13133224	1	2	yes	—
277	C₄	C₂×#13133239	1	2	yes	—
278	C₂	—	4	2	no	—
279	C₂	#20273495	1	4	yes	direct product factor
280	C₂	C₂×#16810650	1	1	yes	characteristic
281	C₂	C₂×#16810661	1	1	yes	characteristic
282	C₂	C₂×#16810672	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₅₃×#431	C₂
socle	C₂³×C₅₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₄⋊C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂×#431	D₅₃
hypocenter	C₅₃	C₂×#1572
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #20273495	C₂	direct (N × H)
4. C₂×#16810376	C₂	semidirect
5. C₂×#16810374	C₂	semidirect
17. #16810376	C₂²	semidirect
18. #16810374	C₂²	semidirect
144. C₂×C₅₃	#1572	semidirect
167. C₅₃	C₂×#1572	semidirect
279. C₂	#20273495	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×#16810680	2
C₂×#16810676	2
#20273495	2
C₂×#16810376	2
C₂×#16810374	2
C₂×C₅₃×#431	2
C₂×#1572	53

Subgroups — C2×#20273495 #27353159

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20273495 #27353159