Subgroups — C₂×#20276488 #27537104

Order 13568.

658 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133153	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810661	C₂	1	1	yes	maximal, characteristic
3	#20276488	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16810446	C₂	1	4	yes	maximal, complemented
5	C₂²×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719917	—	2	1	no	—
7	C₂×#13133176	—	2	1	no	—
8	C₂×#13133153	—	2	1	no	—
9	C₂×#13133153	—	2	2	no	—
10	C₂×#13133153	C₂²	1	2	yes	—
11	C₂×#13133153	C₂²	1	4	yes	—
12	#16810661	—	2	2	no	—
13	#16810661	C₂²	1	4	yes	—
14	C₂×#13133151	—	2	1	no	—
15	C₂×#13133149	C₂²	1	2	yes	—
16	#16810446	C₂²	1	16	yes	complemented
17	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×M₃₂×C₅₃	C₄	1	2	yes	complemented
19	C₂×M₃₂×C₅₃	C₄	1	4	yes	complemented
20	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
21	C₂×M₃₂×C₅₃	C₂²	1	4	yes	—
22	C₂²×C₁₆×C₅₃	C₄	1	2	yes	complemented
23	C₂×#11719917	—	2	2	no	—
24	C₂×#11719917	—	2	4	no	—
25	C₂×#11719917	—	4	1	no	—
26	C₂×#11719917	—	4	2	no	—
27	C₂²×#88322	—	4	1	no	—
28	#13133176	—	2	4	no	—
29	#13133176	—	4	2	no	—
30	C₂×C₄⋊Dic₅₃	—	2	2	no	—
31	C₂×C₂²⋊Dic₅₃	—	4	1	no	—
32	C₂×C₄×Dic₅₃	—	4	1	no	—
33	#13133153	—	2	2	no	—
34	#13133153	—	2	4	no	—
35	#13133153	C₂³	1	4	yes	—
36	#13133151	—	2	4	no	—
37	#13133149	—	2	2	no	—
38	#13133149	C₂³	1	4	yes	—
39	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₅₃	D₄	1	2	yes	—
45	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
46	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
47	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
48	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
50	M₃₂×C₅₃	C₂×C₄	1	4	yes	—
51	M₃₂×C₅₃	C₂×C₄	1	8	yes	complemented
52	M₃₂×C₅₃	C₂³	1	4	yes	—
53	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
54	C₂×C₁₆×C₅₃	C₂×C₄	1	8	yes	complemented
55	C₂⁴×C₅₃	SD₁₆	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
61	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
64	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
66	C₂²×Dic₅₃	—	4	2	no	—
67	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
69	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
70	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
72	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
73	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
74	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
81	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
82	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
83	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
84	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
85	C₂²⋊Dic₅₃	—	4	4	no	—
86	C₄⋊Dic₅₃	—	2	4	no	—
87	C₄⋊Dic₅₃	—	4	2	no	—
88	C₄×Dic₅₃	—	4	4	no	—
89	#11719917	—	2	4	no	—
90	#11719917	—	4	2	no	—
91	#11719917	—	4	4	no	—
92	C₂×#88322	—	4	4	no	—
93	C₂×#88322	—	4	2	no	—
94	C₁₆×C₅₃	C₂²×C₄	1	8	yes	—
95	C₂³×C₅₃	—	4	2	no	—
96	C₂³×C₅₃	#106	1	1	yes	characteristic
97	C₂³×C₅₃	#106	1	1	yes	characteristic
98	C₂³×C₅₃	#106	1	2	yes	—
99	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
100	C₂³×C₅₃	C₂×SD₁₆	1	2	yes	—
101	C₂×C₄×C₅₃	#106	1	2	yes	—
102	C₂×C₄×C₅₃	#106	1	2	yes	—
103	C₂×C₄×C₅₃	#106	1	2	yes	—
104	C₂×C₄×C₅₃	#106	1	2	yes	—
105	C₂×C₄×C₅₃	#106	1	2	yes	—
106	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
107	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
113	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
116	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
117	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
119	C₂×Dic₅₃	—	4	4	no	—
120	C₂×Dic₅₃	—	8	4	no	—
121	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
123	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
125	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
126	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
127	#88322	—	4	4	no	—
128	C₂×#2155	—	53	1	no	maximal, Sylow
129	C₂²×C₅₃	—	2	2	no	—
130	C₂²×C₅₃	—	2	2	no	—
131	C₂²×C₅₃	—	2	2	no	—
132	C₂²×C₅₃	—	4	2	no	—
133	C₂²×C₅₃	—	4	2	no	—
134	C₂²×C₅₃	C₂×#106	1	2	yes	—
135	C₂²×C₅₃	C₂×#106	1	2	yes	—
136	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
137	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
138	C₂²×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
139	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
140	C₄×C₅₃	C₂×#106	1	2	yes	—
141	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
142	C₄×C₅₃	C₂×#106	1	2	yes	—
143	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
144	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
145	Dic₅₃	—	8	4	no	—
146	C₂²×M₃₂	D₅₃	1	1	yes	characteristic
147	C₂²×#108	—	53	1	no	—
148	C₂×#427	—	53	1	no	—
149	#2155	—	53	8	no	—
150	C₂×#365	—	53	4	no	—
151	C₂×C₅₃	—	2	1	no	—
152	C₂×C₅₃	—	2	1	no	—
153	C₂×C₅₃	—	2	2	no	—
154	C₂×C₅₃	—	2	2	no	—
155	C₂×C₅₃	#2155	1	2	yes	complemented
156	C₂×C₅₃	C₂²×#106	1	1	yes	characteristic
157	C₂²×M₁₆	—	106	1	no	—
158	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
159	C₂×#117	—	106	1	no	—
160	C₂×M₃₂	Dic₅₃	1	2	yes	complemented
161	C₂×M₃₂	Dic₅₃	1	4	yes	complemented
162	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
163	C₂×M₃₂	C₂×D₅₃	1	4	yes	—
164	C₂²×C₁₆	Dic₅₃	1	2	yes	complemented
165	C₂×#108	—	53	2	no	—
166	C₂×#108	—	53	4	no	—
167	C₂×#108	—	106	1	no	—
168	C₂×#108	—	106	2	no	—
169	#427	—	53	4	no	—
170	#427	—	106	2	no	—
171	C₂×#107	—	106	1	no	—
172	C₂×#106	—	53	2	no	—
173	#365	—	53	16	no	—
174	C₅₃	C₂×#2155	1	1	yes	characteristic, complemented, Sylow
175	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
176	C₂×M₁₆	—	106	2	no	—
177	C₂×M₁₆	—	106	4	no	—
178	C₂×M₁₆	—	212	1	no	—
179	C₂×M₁₆	—	212	2	no	—
180	C₂²×C₈	—	212	1	no	—
181	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
182	C₂²×C₈	Dic₁₀₆	1	2	yes	—
183	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
184	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
185	C₂²×C₈	D₂₁₂	1	2	yes	—
186	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
187	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
188	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
189	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
190	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
191	#117	—	106	4	no	—
192	#117	—	212	2	no	—
193	C₂×C₄⋊C₄	—	106	2	no	—
194	C₂×C₂²⋊C₄	—	212	1	no	—
195	C₂×C₄²	—	212	1	no	—
196	M₃₂	C₂×Dic₅₃	1	4	yes	—
197	M₃₂	C₂×Dic₅₃	1	8	yes	complemented
198	M₃₂	C₂²×D₅₃	1	4	yes	—
199	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	—
200	C₂×C₁₆	C₂×Dic₅₃	1	8	yes	complemented
201	#108	—	53	4	no	—
202	#108	—	106	2	no	—
203	#108	—	106	4	no	—
204	#107	—	106	4	no	—
205	#106	—	53	4	no	—
206	#106	—	106	2	no	—
207	C₂⁴	#11719913	1	1	yes	characteristic
208	C₂²×C₄	—	212	2	no	—
209	C₂²×C₄	#11719913	1	2	yes	—
210	C₂²×C₄	#11719913	1	1	yes	characteristic
211	C₂²×C₄	#11719913	1	1	yes	characteristic
212	C₂²×C₄	#11719913	1	2	yes	—
213	C₂²×C₄	#11719913	1	1	yes	characteristic
214	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
215	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
216	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
219	M₁₆	—	106	4	no	—
220	M₁₆	—	212	2	no	—
221	M₁₆	—	212	4	no	—
222	C₂×C₈	—	212	2	no	—
223	C₂×C₈	—	212	4	no	—
224	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
225	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
226	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
227	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
228	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
229	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
230	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
231	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
232	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
233	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
234	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
235	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
236	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
237	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
238	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
239	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
240	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
241	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
242	C₄⋊C₄	—	106	4	no	—
243	C₄⋊C₄	—	212	2	no	—
244	C₂²⋊C₄	—	212	4	no	—
245	C₄²	—	212	4	no	—
246	C₁₆	C₂²×Dic₅₃	1	8	yes	—
247	C₂³	—	4	2	no	—
248	C₂³	#13133149	1	1	yes	characteristic
249	C₂³	#13133149	1	1	yes	characteristic
250	C₂³	#13133149	1	2	yes	—
251	C₂³	C₂×#11719913	1	1	yes	characteristic
252	C₂³	C₂×#11719913	1	2	yes	—
253	C₂×C₄	—	424	4	no	—
254	C₂×C₄	—	212	4	no	—
255	C₂×C₄	#13133149	1	2	yes	—
256	C₂×C₄	#13133149	1	2	yes	—
257	C₂×C₄	#13133149	1	1	yes	characteristic
258	C₂×C₄	#13133149	1	2	yes	—
259	C₂×C₄	#13133149	1	2	yes	—
260	C₂×C₄	#13133149	1	2	yes	—
261	C₂×C₄	#13133149	1	1	yes	characteristic
262	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
263	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
264	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
265	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
266	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
267	C₂×C₄	C₂×#11719913	1	2	yes	—
268	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
269	C₂×C₄	C₂×#11719913	1	2	yes	—
270	C₂×C₄	C₂×#11719913	1	2	yes	—
271	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
272	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
273	C₈	—	212	4	no	—
274	C₈	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
275	C₈	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
276	C₈	C₂×C₄⋊Dic₅₃	1	2	yes	—
277	C₈	C₂×C₄⋊Dic₅₃	1	2	yes	—
278	C₈	C₂³×Dic₅₃	1	1	yes	characteristic
279	C₈	C₂²×Dic₁₀₆	1	1	yes	characteristic
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	C₂×#13133149	1	2	yes	—
286	C₂²	C₂×#13133149	1	2	yes	—
287	C₂²	C₂×#13133149	1	1	yes	characteristic
288	C₂²	C₂×#13133149	1	1	yes	characteristic
289	C₂²	C₂²×#11719913	1	1	yes	characteristic
290	C₄	—	424	4	no	—
291	C₄	C₂×#13133149	1	1	yes	characteristic
292	C₄	C₂×#13133149	1	2	yes	—
293	C₄	C₂×#13133149	1	1	yes	characteristic
294	C₄	C₂×#13133149	1	2	yes	—
295	C₄	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
296	C₄	C₂²×#11719913	1	1	yes	characteristic
297	C₂	—	2	1	no	—
298	C₂	—	2	1	no	—
299	C₂	—	2	2	no	—
300	C₂	—	2	2	no	—
301	C₂	#20276488	1	2	yes	direct product factor
302	C₂	C₂²×#13133149	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133149
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂²×M₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#106
prime radical	C₂⁴×C₅₃	SD₁₆
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂²×M₃₂	D₅₃
hypocenter	C₅₃	C₂×#2155
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
3. #20276488	C₂	direct (N × H)
4. C₂×#16810446	C₂	semidirect
16. #16810446	C₂²	semidirect
18. C₂×M₃₂×C₅₃	C₄	semidirect
19. C₂×M₃₂×C₅₃	C₄	semidirect
22. C₂²×C₁₆×C₅₃	C₄	semidirect
51. M₃₂×C₅₃	C₂×C₄	semidirect
54. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
155. C₂×C₅₃	#2155	semidirect
160. C₂×M₃₂	Dic₅₃	semidirect
161. C₂×M₃₂	Dic₅₃	semidirect
164. C₂²×C₁₆	Dic₅₃	semidirect
174. C₅₃	C₂×#2155	semidirect
197. M₃₂	C₂×Dic₅₃	semidirect
200. C₂×C₁₆	C₂×Dic₅₃	semidirect
301. C₂	#20276488	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#13133153	2
C₂×#16810661	2
#20276488	2
C₂×#16810446	2
C₂²×M₃₂×C₅₃	2
C₂×#2155	53

Subgroups — C2×#20276488 #27537104

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276488 #27537104