Subgroups — C₈×#88063 #15753478

Order 3328.

542 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₅₂	C₂	1	1	yes	maximal, characteristic
2	C₄×#88063	C₂	1	1	yes	maximal, characteristic
3	#13113156	C₂	1	1	yes	maximal, characteristic
4	C₈×#6099	C₂	1	1	yes	maximal, characteristic, complemented
5	#13112687	C₂	1	1	yes	maximal, characteristic, complemented
6	#13112685	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₁₃	C₄	1	2	yes	—
9	C₂×C₄×D₅₂	C₂²	1	1	yes	characteristic
10	C₁₃×#431	C₂²	1	1	yes	characteristic
11	C₄×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×#88063	C₄	1	2	yes	—
14	C₂×#88060	C₄	1	2	yes	—
15	#11718586	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
17	C₄×#6099	C₂²	1	1	yes	characteristic
18	C₈×D₅₂	—	2	1	no	—
19	C₈×D₅₂	—	2	1	no	—
20	C₈×D₅₂	—	2	1	no	—
21	C₈×D₅₂	—	2	1	no	—
22	C₁₃×#322	—	2	1	no	—
23	C₈²×C₁₃	—	2	1	no	—
24	#11718146	C₄	1	2	yes	—
25	#11718060	C₄	1	4	yes	—
26	#11718046	—	2	1	no	—
27	C₈×#1055	—	2	1	no	—
28	C₂²×D₅₂	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×D₁₃	C₂×C₄	1	2	yes	—
30	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
34	C₂×#6103	C₂×C₄	1	2	yes	—
35	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂×#6099	C₂×C₄	1	2	yes	—
37	C₂²×#1055	C₂×C₄	1	2	yes	—
38	C₂×C₈×D₁₃	—	2	2	no	—
39	C₂×C₈×D₁₃	—	2	2	no	—
40	C₂×C₈×D₁₃	—	4	4	no	—
41	C₄×D₅₂	—	2	1	no	—
42	C₄×D₅₂	—	2	1	no	—
43	C₄×D₅₂	—	2	1	no	—
44	C₄×D₅₂	—	2	1	no	—
45	C₁₃×C₄⋊C₈	—	2	1	no	—
46	C₁₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
47	C₁₃×#97	—	2	1	no	—
48	C₄×C₈×C₁₃	—	2	1	no	—
49	C₄×C₈×C₁₃	—	2	1	no	—
50	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
51	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
52	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
53	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
54	C₂²⋊#1055	C₂×C₄	1	2	yes	—
55	#88063	C₈	1	8	yes	direct product factor
56	#88060	—	2	2	no	—
57	#88060	—	2	2	no	—
58	C₄⋊#1055	—	2	1	no	—
59	C₄⋊#1055	—	2	1	no	—
60	C₄⋊#1055	—	2	1	no	—
61	#88043	—	2	1	no	—
62	C₄×#1055	—	2	1	no	—
63	C₄×#1055	—	2	1	no	—
64	C₈×#100	—	13	1	no	maximal, Sylow
65	C₂³×D₁₃	C₄²	1	2	yes	—
66	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×Dic₁₃	C₄²	1	2	yes	—
70	C₂×D₅₂	—	2	1	no	—
71	C₂×D₅₂	—	2	1	no	—
72	C₂×D₅₂	—	2	2	no	—
73	C₂×D₅₂	C₂×C₈	1	4	yes	—
74	C₂×C₄×D₁₃	—	2	2	no	—
75	C₂×C₄×D₁₃	—	2	2	no	—
76	C₂×C₄×D₁₃	—	4	2	no	—
77	C₂×C₄×D₁₃	—	4	2	no	—
78	C₁₃×M₁₆	—	2	2	no	—
79	C₁₃×M₁₆	C₂×C₈	1	4	yes	complemented
80	C₂×C₈×C₁₃	—	2	2	no	—
81	C₂×C₈×C₁₃	—	2	2	no	—
82	C₂×C₈×C₁₃	—	2	2	no	—
83	C₂×C₈×C₁₃	—	4	1	no	—
84	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
88	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₁₃	#41	1	1	yes	characteristic
91	C₄²×C₁₃	#41	1	1	yes	characteristic
92	#6103	—	2	2	no	—
93	#6103	—	2	2	no	—
94	C₄⋊Dic₁₃	—	2	1	no	—
95	C₄⋊Dic₁₃	—	2	1	no	—
96	#6099	—	2	2	no	—
97	#6099	C₂×C₈	1	4	yes	complemented
98	C₂×#1055	—	2	2	no	—
99	C₂×#1055	—	2	2	no	—
100	C₂×#1055	—	2	2	no	—
101	C₂×#1055	—	4	2	no	—
102	C₈×D₁₃	—	4	4	no	—
103	C₈×D₁₃	—	4	4	no	—
104	C₂×C₈×D₄	—	13	1	no	—
105	C₄×#100	—	13	1	no	—
106	#1460	—	13	1	no	—
107	C₈×M₁₆	—	13	1	no	—
108	C₈×M₁₆	D₁₃	1	1	yes	characteristic, complemented
109	#1283	—	13	1	no	—
110	#1283	—	13	1	no	—
111	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
112	C₂²×D₁₃	—	2	2	no	—
113	C₂²×D₁₃	—	2	2	no	—
114	C₂²×D₁₃	—	4	2	no	—
115	C₂²×D₁₃	—	4	2	no	—
116	C₂²×D₁₃	C₄×C₈	1	4	yes	—
117	C₂×C₄×C₁₃	—	2	1	no	—
118	C₂×C₄×C₁₃	—	2	1	no	—
119	C₂×C₄×C₁₃	—	4	1	no	—
120	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
127	C₂×C₄×C₁₃	#131	1	2	yes	—
128	C₂×Dic₁₃	—	2	2	no	—
129	C₂×Dic₁₃	—	4	2	no	—
130	D₅₂	—	2	4	no	—
131	D₅₂	—	2	4	no	—
132	C₄×D₁₃	—	4	4	no	—
133	C₄×D₁₃	—	4	2	no	—
134	C₄×D₁₃	—	4	2	no	—
135	C₈×C₁₃	—	2	1	no	—
136	C₈×C₁₃	—	2	1	no	—
137	C₈×C₁₃	—	2	1	no	—
138	C₈×C₁₃	—	2	4	no	—
139	C₈×C₁₃	—	4	2	no	—
140	C₈×C₁₃	#100	1	2	yes	complemented
141	#1055	—	2	4	no	—
142	#1055	—	4	2	no	—
143	#1055	—	4	2	no	—
144	C₂³×C₈	—	13	2	no	—
145	C₂×C₄×D₄	—	13	1	no	—
146	C₈×D₄	—	26	1	no	—
147	C₈×D₄	—	26	1	no	—
148	C₈×D₄	—	26	1	no	—
149	C₈×D₄	—	26	1	no	—
150	#431	—	13	1	no	—
151	#431	C₂×D₁₃	1	1	yes	characteristic
152	C₂×C₄⋊C₈	—	13	1	no	—
153	C₂×#100	—	13	2	no	—
154	C₂×C₂²⋊C₈	—	13	2	no	—
155	C₄×M₁₆	—	13	1	no	—
156	C₄×M₁₆	C₂×D₁₃	1	1	yes	characteristic
157	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
158	#343	—	13	2	no	—
159	#323	—	13	4	no	—
160	#322	—	2	1	no	—
161	#322	—	26	1	no	—
162	C₈²	—	2	1	no	—
163	C₈²	—	26	1	no	—
164	C₂²×C₁₃	—	2	1	no	—
165	C₂²×C₁₃	—	2	1	no	—
166	C₂²×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
167	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
168	C₂²×C₁₃	#405	1	1	yes	characteristic
169	C₂×D₁₃	—	2	4	no	—
170	C₂×D₁₃	—	4	4	no	—
171	C₂×D₁₃	—	4	2	no	—
172	C₂×D₁₃	—	4	2	no	—
173	C₂×D₁₃	—	4	4	no	—
174	C₄×C₁₃	—	2	1	no	—
175	C₄×C₁₃	—	2	1	no	—
176	C₄×C₁₃	—	2	1	no	—
177	C₄×C₁₃	C₂×#100	1	1	yes	characteristic
178	C₄×C₁₃	#413	1	1	yes	characteristic
179	C₄×C₁₃	C₈×D₄	1	2	yes	—
180	C₄×C₁₃	C₈×D₄	1	2	yes	—
181	Dic₁₃	—	4	2	no	—
182	C₂²×D₄	—	13	1	no	—
183	C₂³×C₄	—	13	2	no	—
184	C₂×M₁₆	—	13	2	no	—
185	C₂×M₁₆	C₄×D₁₃	1	2	yes	—
186	C₂²×C₈	—	13	2	no	—
187	C₂²×C₈	—	26	2	no	—
188	C₂²×C₈	—	26	2	no	—
189	C₂²×C₈	—	52	4	no	—
190	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
191	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
192	C₄×D₄	—	26	1	no	—
193	C₄×D₄	—	26	1	no	—
194	C₄×D₄	—	26	1	no	—
195	C₄×D₄	—	26	1	no	—
196	C₂×C₄⋊C₄	—	13	1	no	—
197	C₂×C₂²⋊C₄	—	13	2	no	—
198	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
199	C₄⋊C₈	—	2	1	no	—
200	C₄⋊C₈	—	26	1	no	—
201	C₄⋊C₈	—	26	1	no	—
202	C₄⋊C₈	—	26	1	no	—
203	#100	—	13	8	no	—
204	C₂²⋊C₈	—	13	2	no	—
205	C₂²⋊C₈	—	26	2	no	—
206	C₂²⋊C₈	—	26	2	no	—
207	C₂²⋊C₈	C₄×D₁₃	1	2	yes	—
208	#97	—	2	1	no	—
209	#97	—	26	1	no	—
210	C₄×C₈	—	2	1	no	—
211	C₄×C₈	—	2	1	no	—
212	C₄×C₈	—	26	1	no	—
213	C₄×C₈	—	26	1	no	—
214	C₄×C₈	D₅₂	1	1	yes	characteristic
215	C₄×C₈	D₅₂	1	1	yes	characteristic
216	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
217	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
218	C₂×C₁₃	—	2	1	no	—
219	C₂×C₁₃	—	2	1	no	—
220	C₂×C₁₃	C₈×C₂²⋊C₄	1	1	yes	characteristic
221	C₂×C₁₃	C₄×#100	1	1	yes	characteristic
222	C₂×C₁₃	#1757	1	1	yes	characteristic
223	D₁₃	—	4	4	no	—
224	C₂⁴	—	13	2	no	—
225	C₂×D₄	—	13	4	no	—
226	C₂×D₄	—	26	1	no	—
227	C₂×D₄	—	26	2	no	—
228	C₂×D₄	—	26	1	no	—
229	C₂²×C₄	—	13	2	no	—
230	C₂²×C₄	—	26	2	no	—
231	C₂²×C₄	—	26	2	no	—
232	C₂²×C₄	—	52	2	no	—
233	C₂²×C₄	—	52	2	no	—
234	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
235	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
237	M₁₆	—	2	2	no	—
238	M₁₆	—	13	4	no	—
239	M₁₆	—	26	2	no	—
240	M₁₆	C₈×D₁₃	1	4	yes	complemented
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	4	1	no	—
245	C₂×C₈	—	26	2	no	—
246	C₂×C₈	—	26	2	no	—
247	C₂×C₈	—	26	2	no	—
248	C₂×C₈	—	52	4	no	—
249	C₂×C₈	—	52	4	no	—
250	C₂×C₈	—	52	2	no	—
251	C₂×C₈	#6103	1	1	yes	characteristic
252	C₂×C₈	#6103	1	1	yes	characteristic
253	C₂×C₈	#6103	1	1	yes	characteristic
254	C₂×C₈	#6103	1	1	yes	characteristic
255	C₄⋊C₄	—	26	1	no	—
256	C₄⋊C₄	—	26	1	no	—
257	C₂²⋊C₄	—	26	2	no	—
258	C₂²⋊C₄	—	26	2	no	—
259	C₄²	C₂×D₅₂	1	1	yes	characteristic
260	C₄²	#6127	1	1	yes	characteristic
261	C₄²	#6127	1	1	yes	characteristic
262	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
263	C₁₃	C₈×#100	1	1	yes	characteristic, complemented, Sylow
264	C₂³	—	13	4	no	—
265	C₂³	—	26	2	no	—
266	C₂³	—	26	2	no	—
267	C₂³	—	52	2	no	—
268	C₂³	—	52	2	no	—
269	C₂³	C₄²×D₁₃	1	1	yes	characteristic
270	D₄	—	26	4	no	—
271	D₄	—	26	4	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	26	2	no	—
276	C₂×C₄	—	52	4	no	—
277	C₂×C₄	—	52	2	no	—
278	C₂×C₄	—	52	2	no	—
279	C₂×C₄	—	52	2	no	—
280	C₂×C₄	#88126	1	1	yes	characteristic
281	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
282	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
283	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
284	C₂×C₄	#88155	1	2	yes	—
285	C₂×C₄	C₂×#6103	1	1	yes	characteristic
286	C₂×C₄	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
287	C₂×C₄	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
288	C₈	—	2	4	no	—
289	C₈	—	2	1	no	—
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	—	4	2	no	—
293	C₈	—	26	4	no	—
294	C₈	—	52	2	no	—
295	C₈	—	52	2	no	—
296	C₈	#88063	1	2	yes	direct product factor
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	26	4	no	—
300	C₂²	—	52	4	no	—
301	C₂²	—	52	4	no	—
302	C₂²	—	52	2	no	—
303	C₂²	—	52	2	no	—
304	C₂²	C₄×C₈×D₁₃	1	1	yes	characteristic
305	C₂²	#11718350	1	1	yes	characteristic
306	C₂²	C₄×#6103	1	1	yes	characteristic
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	52	2	no	—
311	C₄	C₈×D₅₂	1	2	yes	—
312	C₄	C₈×C₂²⋊D₁₃	1	2	yes	—
313	C₄	C₂×#88063	1	1	yes	characteristic
314	C₄	#11718793	1	1	yes	characteristic
315	C₂	—	2	1	no	—
316	C₂	—	2	1	no	—
317	C₂	—	52	4	no	—
318	C₂	C₈×#6103	1	1	yes	characteristic
319	C₂	C₄×#88063	1	1	yes	characteristic
320	C₂	#13114637	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	#6103
commutator	C₂²×C₁₃	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₈×C₁₃×M₁₆	C₂
socle	C₂²×C₁₃	C₄×C₂²⋊C₄
prime radical	C₂²×D₅₂	C₂×C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₈×M₁₆	D₁₃
hypocenter	C₁₃	C₈×#100
LCS	G ▷ C₂²×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
4. C₈×#6099	C₂	semidirect
5. #13112687	C₂	semidirect
6. #13112685	C₂	semidirect
7. C₈×C₁₃×M₁₆	C₂	semidirect
55. #88063	C₈	direct (N × H)
79. C₁₃×M₁₆	C₂×C₈	semidirect
97. #6099	C₂×C₈	semidirect
108. C₈×M₁₆	D₁₃	semidirect
140. C₈×C₁₃	#100	semidirect
240. M₁₆	C₈×D₁₃	semidirect
263. C₁₃	C₈×#100	semidirect
296. C₈	#88063	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×C₈×D₅₂	2
C₄×#88063	2
#13113156	2
C₈×#6099	2
#13112687	2
#13112685	2
C₈×C₁₃×M₁₆	2
C₈×#100	13

Subgroups — C8×#88063 #15753478

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#88063 #15753478