Subgroups — C₂×#20272676 #27293823

Order 13568.

470 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16810376	C₂	1	1	yes	maximal, characteristic, complemented
3	#20272676	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719908	—	2	1	no	—
6	C₂×C₄⋊#88322	—	2	1	no	—
7	C₄×#11719908	—	2	1	no	—
8	C₄×#11719908	C₂²	1	2	yes	complemented
9	#16810376	—	2	1	no	—
10	#16810376	C₂²	1	2	yes	complemented
11	C₂²×D₈×C₅₃	C₄	1	1	yes	characteristic
12	C₂×C₄×D₄×C₅₃	—	2	1	no	—
13	C₄×D₈×C₅₃	—	2	1	no	—
14	C₄×D₈×C₅₃	—	2	1	no	—
15	C₄×D₈×C₅₃	C₄	1	2	yes	—
16	C₄×D₈×C₅₃	C₂²	1	2	yes	—
17	C₂×#107×C₅₃	C₄	1	1	yes	characteristic
18	C₂×#102×C₅₃	—	2	1	no	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×#88322	—	4	1	no	—
21	C₂×#11719908	—	2	4	no	—
22	C₂×#11719908	—	2	2	no	—
23	C₄⋊#88322	—	2	2	no	—
24	C₄⋊#88322	—	4	1	no	—
25	C₂²×D₄×C₅₃	—	2	1	no	—
26	C₂³×C₄×C₅₃	—	4	1	no	—
27	C₂×D₈×C₅₃	—	2	1	no	—
28	C₂×D₈×C₅₃	—	2	2	no	—
29	C₂×D₈×C₅₃	C₈	1	4	yes	complemented
30	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
33	C₄×D₄×C₅₃	—	2	2	no	—
34	C₄×D₄×C₅₃	—	2	2	no	—
35	C₄×D₄×C₅₃	—	4	1	no	—
36	C₄×D₄×C₅₃	—	4	1	no	—
37	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
38	C₂×C₂²⋊C₄×C₅₃	—	4	1	no	—
39	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
40	#107×C₅₃	—	2	1	no	—
41	#107×C₅₃	C₂×C₄	1	2	yes	—
42	#102×C₅₃	—	2	2	no	—
43	#102×C₅₃	—	2	2	no	—
44	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
46	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
47	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
48	C₂⁴×C₅₃	—	4	1	no	—
49	C₂×D₄×C₅₃	—	2	4	no	—
50	C₂×D₄×C₅₃	—	2	2	no	—
51	C₂×D₄×C₅₃	—	4	1	no	—
52	C₂×D₄×C₅₃	—	4	2	no	—
53	C₂²×C₄×C₅₃	—	4	1	no	—
54	C₂²×C₄×C₅₃	—	4	2	no	—
55	C₂²×C₄×C₅₃	—	4	2	no	—
56	C₂²×C₄×C₅₃	—	8	2	no	—
57	C₂²×C₄×C₅₃	—	8	2	no	—
58	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
61	D₈×C₅₃	—	2	4	no	—
62	D₈×C₅₃	—	2	2	no	—
63	D₈×C₅₃	C₂×C₈	1	4	yes	complemented
64	C₂×C₈×C₅₃	—	2	2	no	—
65	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
68	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
69	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₄⋊C₄×C₅₃	—	2	2	no	—
72	C₄⋊C₄×C₅₃	—	4	1	no	—
73	C₂²⋊C₄×C₅₃	—	4	2	no	—
74	C₂²⋊C₄×C₅₃	—	4	2	no	—
75	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₄²×C₅₃	D₈	1	1	yes	characteristic
77	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
78	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
79	C₂×#88322	—	4	2	no	—
80	C₂×#88322	—	8	2	no	—
81	#11719908	—	2	4	no	—
82	C₂³×C₅₃	—	4	4	no	—
83	C₂³×C₅₃	—	4	2	no	—
84	C₂³×C₅₃	—	8	2	no	—
85	C₂³×C₅₃	—	8	2	no	—
86	C₂³×C₅₃	#102	1	1	yes	characteristic
87	D₄×C₅₃	—	2	4	no	—
88	D₄×C₅₃	—	4	4	no	—
89	D₄×C₅₃	—	4	2	no	—
90	C₂×C₄×C₅₃	—	2	2	no	—
91	C₂×C₄×C₅₃	—	4	2	no	—
92	C₂×C₄×C₅₃	—	8	2	no	—
93	C₂×C₄×C₅₃	—	8	4	no	—
94	C₂×C₄×C₅₃	—	8	2	no	—
95	C₂×C₄×C₅₃	—	8	2	no	—
96	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
97	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
98	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
99	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
100	C₂×C₄×C₅₃	#104	1	2	yes	—
101	C₂×C₄×C₅₃	D₁₆	1	2	yes	—
102	C₂×C₄×C₅₃	SD₃₂	1	2	yes	—
103	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
106	C₈×C₅₃	—	2	1	no	—
107	C₈×C₅₃	—	2	1	no	—
108	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
109	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
110	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
111	#88322	—	8	2	no	—
112	C₂×#1331	—	53	1	no	maximal, Sylow
113	C₂²×C₅₃	—	4	4	no	—
114	C₂²×C₅₃	—	8	8	no	—
115	C₂²×C₅₃	—	8	2	no	—
116	C₂²×C₅₃	—	8	2	no	—
117	C₂²×C₅₃	D₄⋊C₈	1	2	yes	—
118	C₂²×C₅₃	#357	1	2	yes	—
119	C₂²×C₅₃	#359	1	2	yes	—
120	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
121	C₄×C₅₃	—	2	1	no	—
122	C₄×C₅₃	—	2	1	no	—
123	C₄×C₅₃	—	8	2	no	—
124	C₄×C₅₃	D₄⋊C₈	1	2	yes	—
125	C₄×C₅₃	#357	1	2	yes	—
126	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
127	C₄×C₅₃	C₂×#104	1	1	yes	characteristic
128	C₄×C₅₃	C₂×D₁₆	1	1	yes	characteristic
129	C₄×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
130	C₂×C₄×D₈	D₅₃	1	1	yes	characteristic
131	C₂×C₄×C₁₆	—	53	1	no	—
132	C₂×#335	—	53	1	no	—
133	#1331	—	53	4	no	—
134	C₂×C₅₃	—	8	4	no	—
135	C₂×C₅₃	#1331	1	4	yes	complemented
136	C₂×C₅₃	C₂×D₄⋊C₈	1	1	yes	characteristic
137	C₂×C₅₃	C₂×#357	1	1	yes	characteristic
138	C₂×C₅₃	C₂×#359	1	1	yes	characteristic
139	C₂²×D₈	Dic₅₃	1	1	yes	characteristic
140	C₂×C₄×D₄	—	2	1	no	—
141	C₂²×C₁₆	—	106	1	no	—
142	C₄×D₈	—	2	1	no	—
143	C₄×D₈	—	2	1	no	—
144	C₄×D₈	Dic₅₃	1	2	yes	—
145	C₄×D₈	C₂×D₅₃	1	2	yes	—
146	C₂×#107	Dic₅₃	1	1	yes	characteristic
147	C₂×C₄⋊C₈	—	106	1	no	—
148	C₂×#102	—	2	1	no	—
149	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
150	C₄×C₁₆	—	53	2	no	—
151	C₄×C₁₆	—	106	1	no	—
152	#335	—	53	2	no	—
153	#335	—	106	1	no	—
154	C₅₃	C₂×#1331	1	1	yes	characteristic, complemented, Sylow
155	C₂²×D₄	—	2	1	no	—
156	C₂³×C₄	—	4	1	no	—
157	C₂×D₈	—	2	2	no	—
158	C₂×D₈	—	2	1	no	—
159	C₂×D₈	#88322	1	4	yes	complemented
160	C₂×D₈	C₂×Dic₅₃	1	2	yes	—
161	C₂²×C₈	—	212	1	no	—
162	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
163	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
164	C₄×D₄	—	2	2	no	—
165	C₄×D₄	—	2	2	no	—
166	C₄×D₄	—	4	1	no	—
167	C₄×D₄	—	4	1	no	—
168	C₂×C₄⋊C₄	—	2	1	no	—
169	C₂×C₂²⋊C₄	—	4	1	no	—
170	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
171	C₂×C₁₆	—	106	4	no	—
172	C₂×C₁₆	—	106	2	no	—
173	#107	—	2	1	no	—
174	#107	C₂×Dic₅₃	1	2	yes	—
175	C₄⋊C₈	—	106	2	no	—
176	C₄⋊C₈	—	212	1	no	—
177	#102	—	2	2	no	—
178	#102	—	2	2	no	—
179	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
180	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
181	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
182	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
183	C₂⁴	—	4	1	no	—
184	C₂×D₄	—	2	4	no	—
185	C₂×D₄	—	2	2	no	—
186	C₂×D₄	—	4	1	no	—
187	C₂×D₄	—	4	2	no	—
188	C₂²×C₄	—	4	1	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	—	4	2	no	—
191	C₂²×C₄	—	8	2	no	—
192	C₂²×C₄	—	8	2	no	—
193	C₂²×C₄	#11719922	1	1	yes	characteristic
194	C₂²×C₄	#11719923	1	1	yes	characteristic
195	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
196	D₈	—	2	4	no	—
197	D₈	—	2	2	no	—
198	D₈	C₂×#88322	1	4	yes	complemented
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	424	2	no	—
201	C₂×C₈	—	212	2	no	—
202	C₂×C₈	C₂×#88322	1	2	yes	—
203	C₂×C₈	#11719917	1	2	yes	—
204	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
205	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
206	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
207	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	4	1	no	—
210	C₂²⋊C₄	—	4	2	no	—
211	C₂²⋊C₄	—	4	2	no	—
212	C₄²	#11719922	1	1	yes	characteristic
213	C₄²	#11719923	1	1	yes	characteristic
214	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
215	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
216	C₁₆	—	106	4	no	—
217	C₂³	—	4	4	no	—
218	C₂³	—	4	2	no	—
219	C₂³	—	8	2	no	—
220	C₂³	—	8	2	no	—
221	C₂³	#13133138	1	1	yes	characteristic
222	D₄	—	2	4	no	—
223	D₄	—	4	4	no	—
224	D₄	—	4	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	8	2	no	—
228	C₂×C₄	—	8	2	no	—
229	C₂×C₄	—	8	2	no	—
230	C₂×C₄	—	8	4	no	—
231	C₂×C₄	C₂²⋊#88322	1	2	yes	—
232	C₂×C₄	#13133138	1	1	yes	characteristic
233	C₂×C₄	#13133138	1	1	yes	characteristic
234	C₂×C₄	#13133138	1	1	yes	characteristic
235	C₂×C₄	#13133144	1	2	yes	—
236	C₂×C₄	#13133158	1	2	yes	—
237	C₂×C₄	#13133161	1	2	yes	—
238	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
239	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
240	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
241	C₈	—	2	1	no	—
242	C₈	—	2	1	no	—
243	C₈	—	424	2	no	—
244	C₈	C₂²⋊#88322	1	2	yes	—
245	C₈	C₂²×#88322	1	1	yes	characteristic
246	C₈	C₂×#11719917	1	1	yes	characteristic
247	C₂²	—	4	4	no	—
248	C₂²	—	8	8	no	—
249	C₂²	—	8	2	no	—
250	C₂²	—	8	2	no	—
251	C₂²	D₄⋊#88322	1	2	yes	—
252	C₂²	#16810424	1	2	yes	—
253	C₂²	#16810430	1	2	yes	—
254	C₂²	C₂×#13133138	1	1	yes	characteristic
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	8	2	no	—
258	C₄	D₄⋊#88322	1	2	yes	—
259	C₄	#16810424	1	2	yes	—
260	C₄	C₂×C₂²⋊#88322	1	1	yes	characteristic
261	C₄	C₂×#13133144	1	1	yes	characteristic
262	C₄	C₂×#13133158	1	1	yes	characteristic
263	C₄	C₂×#13133161	1	1	yes	characteristic
264	C₂	—	8	4	no	—
265	C₂	#20272676	1	4	yes	direct product factor
266	C₂	C₂×D₄⋊#88322	1	1	yes	characteristic
267	C₂	C₂×#16810424	1	1	yes	characteristic
268	C₂	C₂×#16810430	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#11719922
commutator	C₈×C₅₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₂×C₄×D₈×C₅₃	C₂
socle	C₂³×C₅₃	#102
prime radical	C₂²×D₈×C₅₃	C₄
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₂×C₄×D₈	D₅₃
hypocenter	C₅₃	C₂×#1331
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₄×#11719908	C₂	semidirect
2. C₂×#16810376	C₂	semidirect
3. #20272676	C₂	direct (N × H)
8. C₄×#11719908	C₂²	semidirect
10. #16810376	C₂²	semidirect
29. C₂×D₈×C₅₃	C₈	semidirect
63. D₈×C₅₃	C₂×C₈	semidirect
135. C₂×C₅₃	#1331	semidirect
154. C₅₃	C₂×#1331	semidirect
159. C₂×D₈	#88322	semidirect
198. D₈	C₂×#88322	semidirect
265. C₂	#20272676	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×C₄×#11719908	2
C₂×#16810376	2
#20272676	2
C₂×C₄×D₈×C₅₃	2
C₂×#1331	53

Subgroups — C2×#20272676 #27293823

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20272676 #27293823