Subgroups — C₂×D₈×#88322 #27353449

Order 13568.

910 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×D₄×#88322	C₂	1	2	yes	maximal, complemented
2	D₈×#88322	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16810376	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₄⋊#88322	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×#88322	—	2	2	no	—
8	D₄×#88322	—	2	4	no	—
9	D₄×#88322	C₂²	1	8	yes	complemented
10	C₂×C₄⋊#88322	C₂²	1	2	yes	—
11	C₂×C₂²⋊#88322	—	2	2	no	—
12	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
13	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
14	#16810376	C₂²	1	4	yes	complemented
15	D₄⋊#88322	C₂²	1	8	yes	complemented
16	C₈×#88322	C₂²	1	4	yes	complemented
17	C₂²×D₈×C₅₃	C₄	1	1	yes	characteristic
18	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
19	C₄×D₈×C₅₃	C₄	1	4	yes	—
20	C₄×D₈×C₅₃	C₂²	1	4	yes	—
21	C₂×#107×C₅₃	C₄	1	1	yes	characteristic
22	C₂×#102×C₅₃	C₄	1	2	yes	—
23	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₂²×#88322	—	2	8	no	—
25	C₂²×#88322	—	2	2	no	—
26	C₂²×#88322	—	2	1	no	—
27	C₂²×#88322	—	4	8	no	—
28	C₂²×#88322	D₄	1	1	yes	characteristic
29	C₂²×#88322	D₄	1	1	yes	characteristic
30	C₄⋊#88322	—	2	2	no	—
31	C₄⋊#88322	C₂³	1	4	yes	—
32	C₂²⋊#88322	—	2	8	no	—
33	C₄×#88322	—	2	1	no	—
34	C₄×#88322	D₄	1	2	yes	—
35	C₄×#88322	C₂³	1	2	yes	—
36	C₄×#88322	C₂³	1	2	yes	—
37	C₂²×D₄×C₅₃	C₂×C₄	1	2	yes	—
38	C₂³×C₄×C₅₃	—	2	2	no	—
39	C₂×D₈×C₅₃	C₈	1	8	yes	complemented
40	C₂×D₈×C₅₃	C₂×C₄	1	4	yes	—
41	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₄×D₄×C₅₃	—	2	2	no	—
44	C₄×D₄×C₅₃	—	2	2	no	—
45	C₄×D₄×C₅₃	C₂×C₄	1	4	yes	—
46	C₄×D₄×C₅₃	C₂³	1	4	yes	—
47	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
48	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
49	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
50	#107×C₅₃	C₂×C₄	1	4	yes	—
51	#102×C₅₃	C₂×C₄	1	8	yes	—
52	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₅₃	C₂³	1	2	yes	—
54	C₂⁴×C₅₃	—	2	2	no	—
55	C₂×D₄×C₅₃	—	2	2	no	—
56	C₂×D₄×C₅₃	—	2	4	no	—
57	C₂×D₄×C₅₃	C₂×C₈	1	8	yes	complemented
58	C₂×D₄×C₅₃	C₂²×C₄	1	4	yes	—
59	C₂²×C₄×C₅₃	—	2	2	no	—
60	C₂²×C₄×C₅₃	—	2	4	no	—
61	C₂²×C₄×C₅₃	—	2	4	no	—
62	C₂²×C₄×C₅₃	—	4	4	no	—
63	C₂²×C₄×C₅₃	—	4	4	no	—
64	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
67	D₈×C₅₃	C₂×C₈	1	16	yes	complemented
68	C₂×C₈×C₅₃	—	2	2	no	—
69	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	complemented
70	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
72	C₄⋊C₄×C₅₃	—	2	2	no	—
73	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
74	C₂²⋊C₄×C₅₃	—	2	4	no	—
75	C₂²⋊C₄×C₅₃	—	2	4	no	—
76	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₅₃	#41	1	1	yes	characteristic
79	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
80	C₂×#88322	—	2	4	no	—
81	C₂×#88322	—	2	2	no	—
82	C₂×#88322	—	2	2	no	—
83	C₂×#88322	—	2	4	no	—
84	C₂×#88322	—	4	4	no	—
85	C₂×#88322	—	4	16	no	—
86	C₂×#88322	D₈	1	4	yes	direct product factor
87	C₂×#88322	C₂×D₄	1	2	yes	—
88	C₂×#88322	C₂×D₄	1	2	yes	—
89	C₂³×C₅₃	—	2	4	no	—
90	C₂³×C₅₃	—	2	8	no	—
91	C₂³×C₅₃	—	4	4	no	—
92	C₂³×C₅₃	—	4	4	no	—
93	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
94	D₄×C₅₃	—	2	4	no	—
95	D₄×C₅₃	—	2	8	no	—
96	D₄×C₅₃	C₂²×C₈	1	8	yes	complemented
97	C₂×C₄×C₅₃	—	2	2	no	—
98	C₂×C₄×C₅₃	—	2	4	no	—
99	C₂×C₄×C₅₃	—	4	4	no	—
100	C₂×C₄×C₅₃	—	4	4	no	—
101	C₂×C₄×C₅₃	—	4	4	no	—
102	C₂×C₄×C₅₃	—	4	8	no	—
103	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
107	C₂×C₄×C₅₃	#131	1	2	yes	—
108	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
109	C₂×C₄×C₅₃	#135	1	2	yes	—
110	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
113	C₈×C₅₃	—	2	2	no	—
114	C₈×C₅₃	C₂²×C₈	1	4	yes	complemented
115	#88322	—	2	4	no	—
116	#88322	—	2	2	no	—
117	#88322	—	4	4	no	—
118	#88322	C₂×D₈	1	4	yes	direct product factor
119	C₂×C₈×D₈	—	53	1	no	maximal, Sylow
120	C₂²×C₅₃	—	2	8	no	—
121	C₂²×C₅₃	—	4	16	no	—
122	C₂²×C₅₃	—	4	4	no	—
123	C₂²×C₅₃	—	4	4	no	—
124	C₂²×C₅₃	C₈×D₄	1	2	yes	—
125	C₂²×C₅₃	C₄×D₈	1	2	yes	—
126	C₂²×C₅₃	#443	1	2	yes	—
127	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
128	C₄×C₅₃	—	2	1	no	—
129	C₄×C₅₃	—	2	1	no	—
130	C₄×C₅₃	—	4	4	no	—
131	C₄×C₅₃	C₈×D₄	1	2	yes	—
132	C₄×C₅₃	C₄×D₈	1	2	yes	—
133	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
134	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
135	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
136	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
137	C₂×C₄×D₈	D₅₃	1	1	yes	characteristic
138	C₂×C₈×D₄	—	53	2	no	—
139	C₈×D₈	—	53	8	no	—
140	C₂×#335	—	53	1	no	—
141	C₂×D₄⋊C₈	—	53	2	no	—
142	C₂×C₈²	—	53	1	no	—
143	C₂×C₅₃	—	4	8	no	—
144	C₂×C₅₃	C₈×D₈	1	4	yes	complemented
145	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
146	C₂×C₅₃	C₂×C₄×D₈	1	1	yes	characteristic
147	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
148	C₂²×D₈	Dic₅₃	1	1	yes	characteristic
149	C₂³×C₈	—	106	2	no	—
150	C₂×C₄×D₄	C₂×D₅₃	1	2	yes	—
151	C₄×D₈	Dic₅₃	1	4	yes	—
152	C₄×D₈	C₂×D₅₃	1	4	yes	—
153	C₈×D₄	—	53	8	no	—
154	C₈×D₄	—	106	4	no	—
155	C₂×#107	Dic₅₃	1	1	yes	characteristic
156	C₂×C₄⋊C₈	—	53	2	no	—
157	C₂×#102	Dic₅₃	1	2	yes	—
158	C₂×C₂²⋊C₈	—	106	2	no	—
159	C₂×C₄×C₈	—	53	1	no	—
160	C₂×C₄×C₈	—	53	1	no	—
161	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
162	#335	—	53	4	no	—
163	D₄⋊C₈	—	53	8	no	—
164	C₈²	—	53	4	no	—
165	C₅₃	C₂×C₈×D₈	1	1	yes	characteristic, complemented, Sylow
166	C₂²×D₄	C₂×Dic₅₃	1	2	yes	—
167	C₂³×C₄	—	2	2	no	—
168	C₂×D₈	#88322	1	8	yes	direct product factor
169	C₂×D₈	C₂×Dic₅₃	1	4	yes	—
170	C₂²×C₈	—	53	1	no	—
171	C₂²×C₈	—	53	1	no	—
172	C₂²×C₈	—	106	8	no	—
173	C₂²×C₈	—	106	1	no	—
174	C₂²×C₈	—	106	2	no	—
175	C₂²×C₈	—	212	8	no	—
176	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
177	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
178	C₄×D₄	—	2	2	no	—
179	C₄×D₄	—	2	2	no	—
180	C₄×D₄	C₂×Dic₅₃	1	4	yes	—
181	C₄×D₄	C₂²×D₅₃	1	4	yes	—
182	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
183	C₂×C₂²⋊C₄	—	2	2	no	—
184	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
185	#107	C₂×Dic₅₃	1	4	yes	—
186	C₄⋊C₈	—	53	4	no	—
187	C₄⋊C₈	—	106	2	no	—
188	#102	C₂×Dic₅₃	1	8	yes	—
189	C₂²⋊C₈	—	106	8	no	—
190	C₄×C₈	—	53	2	no	—
191	C₄×C₈	—	53	2	no	—
192	C₄×C₈	—	53	2	no	—
193	C₄×C₈	—	106	1	no	—
194	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
195	C₄×C₈	C₂²×D₅₃	1	2	yes	—
196	C₂⁴	—	2	2	no	—
197	C₂×D₄	—	2	2	no	—
198	C₂×D₄	—	2	4	no	—
199	C₂×D₄	C₂×#88322	1	8	yes	complemented
200	C₂×D₄	C₂²×Dic₅₃	1	4	yes	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	—	4	4	no	—
205	C₂²×C₄	—	4	4	no	—
206	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
207	C₂²×C₄	#11719947	1	1	yes	characteristic
208	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
209	D₈	C₂×#88322	1	16	yes	direct product factor
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	53	4	no	—
212	C₂×C₈	—	53	2	no	—
213	C₂×C₈	—	53	2	no	—
214	C₂×C₈	—	106	4	no	—
215	C₂×C₈	—	106	2	no	—
216	C₂×C₈	—	106	2	no	—
217	C₂×C₈	—	106	4	no	—
218	C₂×C₈	—	212	16	no	—
219	C₂×C₈	—	212	4	no	—
220	C₂×C₈	C₂×#88322	1	4	yes	complemented
221	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
222	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	C₂²×Dic₅₃	1	4	yes	—
225	C₂²⋊C₄	—	2	4	no	—
226	C₂²⋊C₄	—	2	4	no	—
227	C₄²	D₄×D₅₃	1	1	yes	characteristic
228	C₄²	#11719947	1	1	yes	characteristic
229	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
230	C₄²	C₂³×D₅₃	1	1	yes	characteristic
231	C₂³	—	2	4	no	—
232	C₂³	—	2	8	no	—
233	C₂³	—	4	4	no	—
234	C₂³	—	4	4	no	—
235	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
236	D₄	—	2	4	no	—
237	D₄	—	2	8	no	—
238	D₄	C₂²×#88322	1	8	yes	complemented
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	4	8	no	—
245	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
246	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
247	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
248	C₂×C₄	C₂²×#88322	1	2	yes	—
249	C₂×C₄	#13133224	1	2	yes	—
250	C₂×C₄	D₈×D₅₃	1	2	yes	—
251	C₂×C₄	#13133236	1	2	yes	—
252	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
253	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
254	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
255	C₈	—	2	2	no	—
256	C₈	—	53	4	no	—
257	C₈	—	106	4	no	—
258	C₈	—	106	2	no	—
259	C₈	—	212	4	no	—
260	C₈	C₂²×#88322	1	4	yes	complemented
261	C₂²	—	2	8	no	—
262	C₂²	—	4	4	no	—
263	C₂²	—	4	16	no	—
264	C₂²	—	4	4	no	—
265	C₂²	D₄×#88322	1	2	yes	—
266	C₂²	D₈×Dic₅₃	1	2	yes	—
267	C₂²	#16810740	1	2	yes	—
268	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	4	4	no	—
272	C₄	D₄×#88322	1	2	yes	—
273	C₄	D₈×Dic₅₃	1	2	yes	—
274	C₄	C₂³×#88322	1	1	yes	characteristic
275	C₄	C₂×#13133224	1	1	yes	characteristic
276	C₄	C₂×D₈×D₅₃	1	1	yes	characteristic
277	C₄	C₂×#13133236	1	1	yes	characteristic
278	C₂	—	4	8	no	—
279	C₂	D₈×#88322	1	4	yes	direct product factor
280	C₂	C₂×D₄×#88322	1	1	yes	characteristic
281	C₂	C₂×D₈×Dic₅₃	1	1	yes	characteristic
282	C₂	C₂×#16810740	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅₃
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₄×D₈×C₅₃	C₂
socle	C₂³×C₅₃	C₄×D₄
prime radical	C₂²×D₈×C₅₃	C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂×C₄×D₈	D₅₃
hypocenter	C₅₃	C₂×C₈×D₈
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
1. C₂×D₄×#88322	C₂	semidirect
2. D₈×#88322	C₂	direct (N × H)
3. C₂×#16810376	C₂	semidirect
4. C₂×D₄⋊#88322	C₂	semidirect
5. C₂×C₈×#88322	C₂	semidirect
9. D₄×#88322	C₂²	semidirect
14. #16810376	C₂²	semidirect
15. D₄⋊#88322	C₂²	semidirect
16. C₈×#88322	C₂²	semidirect
39. C₂×D₈×C₅₃	C₈	semidirect
57. C₂×D₄×C₅₃	C₂×C₈	semidirect
67. D₈×C₅₃	C₂×C₈	semidirect
69. C₂×C₈×C₅₃	C₂×C₈	semidirect
86. C₂×#88322	D₈	direct (N × H)
96. D₄×C₅₃	C₂²×C₈	semidirect
114. C₈×C₅₃	C₂²×C₈	semidirect
118. #88322	C₂×D₈	direct (N × H)
144. C₂×C₅₃	C₈×D₈	semidirect
165. C₅₃	C₂×C₈×D₈	semidirect
168. C₂×D₈	#88322	direct (N × H)
199. C₂×D₄	C₂×#88322	semidirect
209. D₈	C₂×#88322	direct (N × H)
220. C₂×C₈	C₂×#88322	semidirect
238. D₄	C₂²×#88322	semidirect
260. C₈	C₂²×#88322	semidirect
279. C₂	D₈×#88322	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×D₄×#88322	2
D₈×#88322	2
C₂×#16810376	2
C₂×D₄⋊#88322	2
C₂×C₈×#88322	2
C₂×C₄×D₈×C₅₃	2
C₂×C₈×D₈	53

Subgroups — C2×D8×#88322 #27353449

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₈×#88322 #27353449