Subgroups — C₂×#20272691 #27293883

Order 13568.

398 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×#16810444	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16810344	C₂	1	1	yes	maximal, characteristic, complemented
3	#20272691	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₅₃×#443	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133157	—	2	1	no	—
6	C₂×C₄×#88322	—	2	1	no	—
7	#16810444	—	2	1	no	—
8	#16810444	C₂²	1	2	yes	complemented
9	#16810344	—	2	1	no	—
10	#16810344	C₂²	1	2	yes	complemented
11	C₂×#135×C₅₃	C₄	1	1	yes	characteristic
12	C₂×#131×C₅₃	—	2	1	no	—
13	C₅₃×#443	—	2	1	no	—
14	C₅₃×#443	—	2	1	no	—
15	C₅₃×#443	C₄	1	2	yes	—
16	C₅₃×#443	C₂²	1	2	yes	—
17	C₂×#108×C₅₃	C₄	1	1	yes	characteristic
18	C₂×#104×C₅₃	—	2	1	no	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×#88322	—	2	2	no	—
21	#13133157	—	2	2	no	—
22	#13133157	—	4	1	no	—
23	C₂×#11719908	—	4	1	no	—
24	C₄×#88322	—	2	2	no	—
25	C₄×#88322	—	2	2	no	—
26	C₂×#41×C₅₃	—	2	1	no	—
27	#135×C₅₃	—	2	1	no	—
28	#135×C₅₃	—	2	1	no	—
29	#135×C₅₃	C₈	1	2	yes	complemented
30	#135×C₅₃	C₂×C₄	1	2	yes	—
31	C₂×Q₁₆×C₅₃	C₈	1	1	yes	characteristic, complemented
32	C₂×SD₁₆×C₅₃	—	2	1	no	—
33	C₂×D₈×C₅₃	C₈	1	1	yes	characteristic, complemented
34	#131×C₅₃	—	2	2	no	—
35	#131×C₅₃	—	2	2	no	—
36	#131×C₅₃	—	4	1	no	—
37	#131×C₅₃	—	4	1	no	—
38	C₂×M₁₆×C₅₃	—	2	1	no	—
39	C₂×M₁₆×C₅₃	—	4	1	no	—
40	C₂²×C₈×C₅₃	—	4	1	no	—
41	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
44	#108×C₅₃	—	2	1	no	—
45	#108×C₅₃	C₂×C₄	1	2	yes	—
46	#104×C₅₃	—	2	2	no	—
47	#104×C₅₃	—	2	2	no	—
48	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
50	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
51	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
52	#41×C₅₃	—	2	2	no	—
53	#41×C₅₃	—	2	2	no	—
54	#41×C₅₃	—	4	1	no	—
55	#41×C₅₃	—	4	1	no	—
56	C₂×Q₈×C₅₃	—	2	1	no	—
57	C₂×D₄×C₅₃	—	2	1	no	—
58	C₂×D₄×C₅₃	—	4	1	no	—
59	C₂²×C₄×C₅₃	—	2	1	no	—
60	C₂²×C₄×C₅₃	—	4	1	no	—
61	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
62	Q₁₆×C₅₃	—	2	1	no	—
63	Q₁₆×C₅₃	C₂×C₈	1	2	yes	complemented
64	SD₁₆×C₅₃	—	2	2	no	—
65	SD₁₆×C₅₃	—	2	2	no	—
66	D₈×C₅₃	—	2	1	no	—
67	D₈×C₅₃	C₂×C₈	1	2	yes	complemented
68	M₁₆×C₅₃	—	2	2	no	—
69	M₁₆×C₅₃	—	4	1	no	—
70	M₁₆×C₅₃	—	4	2	no	—
71	M₁₆×C₅₃	—	4	2	no	—
72	C₂×C₈×C₅₃	—	4	1	no	—
73	C₂×C₈×C₅₃	—	4	2	no	—
74	C₂×C₈×C₅₃	—	4	2	no	—
75	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
78	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
79	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
80	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
81	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
82	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
83	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
84	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
85	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₅₃	—	2	1	no	—
88	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
89	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
90	C₂×#88322	—	2	4	no	—
91	C₂×#88322	—	4	4	no	—
92	#11719908	—	4	2	no	—
93	C₂³×C₅₃	—	4	1	no	—
94	C₂³×C₅₃	#104	1	1	yes	characteristic
95	Q₈×C₅₃	—	2	2	no	—
96	Q₈×C₅₃	—	4	1	no	—
97	D₄×C₅₃	—	2	2	no	—
98	D₄×C₅₃	—	4	2	no	—
99	D₄×C₅₃	—	4	2	no	—
100	D₄×C₅₃	—	4	1	no	—
101	C₂×C₄×C₅₃	—	2	4	no	—
102	C₂×C₄×C₅₃	—	2	1	no	—
103	C₂×C₄×C₅₃	—	2	1	no	—
104	C₂×C₄×C₅₃	—	4	1	no	—
105	C₂×C₄×C₅₃	—	4	2	no	—
106	C₂×C₄×C₅₃	—	4	2	no	—
107	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
109	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
110	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
111	C₂×C₄×C₅₃	#104	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₈×C₅₃	—	4	2	no	—
114	C₈×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
115	C₈×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
116	C₈×C₅₃	#102	1	1	yes	characteristic
117	C₈×C₅₃	#102	1	1	yes	characteristic
118	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
119	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
120	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
121	C₈×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
122	#88322	—	4	4	no	—
123	C₂×#1336	—	53	1	no	maximal, Sylow
124	C₂²×C₅₃	—	2	2	no	—
125	C₂²×C₅₃	—	4	2	no	—
126	C₂²×C₅₃	—	8	2	no	—
127	C₂²×C₅₃	D₄⋊C₈	1	1	yes	characteristic
128	C₂²×C₅₃	D₄⋊C₈	1	1	yes	characteristic
129	C₂²×C₅₃	C₂×#104	1	1	yes	characteristic
130	C₄×C₅₃	—	2	2	no	—
131	C₄×C₅₃	—	2	2	no	—
132	C₄×C₅₃	—	4	2	no	—
133	C₄×C₅₃	D₄⋊C₈	1	1	yes	characteristic
134	C₄×C₅₃	D₄⋊C₈	1	1	yes	characteristic
135	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
136	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
137	C₂×#443	D₅₃	1	1	yes	characteristic
138	C₂×#364	—	53	1	no	—
139	C₂×#322	—	53	1	no	—
140	#1336	—	53	4	no	—
141	C₂×C₅₃	—	2	1	no	—
142	C₂×C₅₃	—	2	1	no	—
143	C₂×C₅₃	—	8	2	no	—
144	C₂×C₅₃	#1336	1	2	yes	complemented
145	C₂×C₅₃	C₂×D₄⋊C₈	1	1	yes	characteristic
146	C₂×#135	Dic₅₃	1	1	yes	characteristic
147	C₂×#131	—	2	1	no	—
148	C₂×M₃₂	—	106	1	no	—
149	#443	—	2	1	no	—
150	#443	—	2	1	no	—
151	#443	Dic₅₃	1	2	yes	—
152	#443	C₂×D₅₃	1	2	yes	—
153	C₂×#108	Dic₅₃	1	1	yes	characteristic
154	C₂×#104	—	2	1	no	—
155	C₂×C₄×C₈	—	106	1	no	—
156	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
157	#364	—	53	2	no	—
158	#364	—	106	1	no	—
159	#322	—	53	2	no	—
160	#322	—	106	1	no	—
161	C₅₃	C₂×#1336	1	1	yes	characteristic, complemented, Sylow
162	C₂×#41	—	2	1	no	—
163	#135	—	2	1	no	—
164	#135	—	2	1	no	—
165	#135	#88322	1	2	yes	complemented
166	#135	C₂×Dic₅₃	1	2	yes	—
167	C₂×Q₁₆	#88322	1	1	yes	characteristic, complemented
168	C₂×SD₁₆	—	2	1	no	—
169	C₂×D₈	#88322	1	1	yes	characteristic, complemented
170	#131	—	2	2	no	—
171	#131	—	2	2	no	—
172	#131	—	4	1	no	—
173	#131	—	4	1	no	—
174	C₂×M₁₆	—	2	1	no	—
175	C₂×M₁₆	—	4	1	no	—
176	C₂²×C₈	—	4	1	no	—
177	C₂²×C₈	—	106	2	no	—
178	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
179	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
180	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
181	M₃₂	—	106	2	no	—
182	M₃₂	—	212	1	no	—
183	C₂×C₁₆	—	212	1	no	—
184	#108	—	2	1	no	—
185	#108	C₂×Dic₅₃	1	2	yes	—
186	#104	—	2	2	no	—
187	#104	—	2	2	no	—
188	C₄×C₈	—	106	2	no	—
189	C₄×C₈	—	106	2	no	—
190	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
191	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
192	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
193	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
194	#41	—	2	2	no	—
195	#41	—	2	2	no	—
196	#41	—	4	1	no	—
197	#41	—	4	1	no	—
198	C₂×Q₈	—	2	1	no	—
199	C₂×D₄	—	2	1	no	—
200	C₂×D₄	—	4	1	no	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	4	1	no	—
203	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
204	Q₁₆	—	2	1	no	—
205	Q₁₆	C₂×#88322	1	2	yes	complemented
206	SD₁₆	—	2	2	no	—
207	SD₁₆	—	2	2	no	—
208	D₈	—	2	1	no	—
209	D₈	C₂×#88322	1	2	yes	complemented
210	M₁₆	—	2	2	no	—
211	M₁₆	—	4	1	no	—
212	M₁₆	—	4	2	no	—
213	M₁₆	—	4	2	no	—
214	C₂×C₈	—	4	1	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	—	4	2	no	—
217	C₂×C₈	—	106	4	no	—
218	C₂×C₈	—	212	4	no	—
219	C₂×C₈	C₂×#88322	1	1	yes	characteristic
220	C₂×C₈	C₂×#88322	1	1	yes	characteristic
221	C₂×C₈	#11719917	1	1	yes	characteristic
222	C₂×C₈	#11719917	1	1	yes	characteristic
223	C₂×C₈	#11719922	1	1	yes	characteristic
224	C₂×C₈	#11719922	1	1	yes	characteristic
225	C₂×C₈	#11719923	1	1	yes	characteristic
226	C₂×C₈	#11719923	1	1	yes	characteristic
227	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
228	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
229	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
230	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
231	C₄²	—	2	1	no	—
232	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
233	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
234	C₁₆	—	212	2	no	—
235	C₂³	—	4	1	no	—
236	C₂³	#13133144	1	1	yes	characteristic
237	Q₈	—	2	2	no	—
238	Q₈	—	4	1	no	—
239	D₄	—	2	2	no	—
240	D₄	—	4	2	no	—
241	D₄	—	4	2	no	—
242	D₄	—	4	1	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
250	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
251	C₂×C₄	#13133138	1	1	yes	characteristic
252	C₂×C₄	#13133138	1	1	yes	characteristic
253	C₂×C₄	#13133144	1	1	yes	characteristic
254	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
255	C₈	—	4	2	no	—
256	C₈	—	212	4	no	—
257	C₈	C₂²⋊#88322	1	1	yes	characteristic
258	C₈	C₂²⋊#88322	1	1	yes	characteristic
259	C₈	#13133138	1	1	yes	characteristic
260	C₈	#13133138	1	1	yes	characteristic
261	C₈	C₂²×#88322	1	1	yes	characteristic
262	C₈	C₂×#11719917	1	1	yes	characteristic
263	C₈	C₂×#11719922	1	1	yes	characteristic
264	C₈	C₂×#11719923	1	1	yes	characteristic
265	C₂²	—	2	2	no	—
266	C₂²	—	4	2	no	—
267	C₂²	—	8	2	no	—
268	C₂²	D₄⋊#88322	1	1	yes	characteristic
269	C₂²	D₄⋊#88322	1	1	yes	characteristic
270	C₂²	C₂×#13133144	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	4	2	no	—
274	C₄	D₄⋊#88322	1	1	yes	characteristic
275	C₄	D₄⋊#88322	1	1	yes	characteristic
276	C₄	C₂×C₂²⋊#88322	1	1	yes	characteristic
277	C₄	C₂×#13133138	1	1	yes	characteristic
278	C₂	—	2	1	no	—
279	C₂	—	2	1	no	—
280	C₂	—	8	2	no	—
281	C₂	#20272691	1	2	yes	direct product factor
282	C₂	C₂×D₄⋊#88322	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133138
commutator	C₈×C₅₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#443	C₂
socle	C₂²×C₅₃	D₄⋊C₈
prime radical	C₂×#135×C₅₃	C₄
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₂×#443	D₅₃
hypocenter	C₅₃	C₂×#1336
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×#443

Complements

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

N	H ≅ G/N	type
1. C₂×#16810444	C₂	semidirect
2. C₂×#16810344	C₂	semidirect
3. #20272691	C₂	direct (N × H)
8. #16810444	C₂²	semidirect
10. #16810344	C₂²	semidirect
29. #135×C₅₃	C₈	semidirect
31. C₂×Q₁₆×C₅₃	C₈	semidirect
33. C₂×D₈×C₅₃	C₈	semidirect
63. Q₁₆×C₅₃	C₂×C₈	semidirect
67. D₈×C₅₃	C₂×C₈	semidirect
144. C₂×C₅₃	#1336	semidirect
161. C₅₃	C₂×#1336	semidirect
165. #135	#88322	semidirect
167. C₂×Q₁₆	#88322	semidirect
169. C₂×D₈	#88322	semidirect
205. Q₁₆	C₂×#88322	semidirect
209. D₈	C₂×#88322	semidirect
281. C₂	#20272691	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×#16810444	2
C₂×#16810344	2
#20272691	2
C₂×C₅₃×#443	2
C₂×#1336	53

Subgroups — C2×#20272691 #27293883

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20272691 #27293883