Subgroups — C₈×#14581713 #35295166

Order 21248.

606 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×C₄⋊Dic₈₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#14581713	C₂	1	1	yes	maximal, characteristic
3	C₂×D₄⋊(C₈₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×C₈₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#24905302	C₂	1	1	yes	maximal, characteristic, complemented
6	#24905297	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₈×D₄×C₈₃	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
9	C₂×#14581713	C₄	1	2	yes	—
10	C₂×C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₈×Dic₈₃	—	2	1	no	—
13	C₄×C₄⋊Dic₈₃	C₂²	1	1	yes	characteristic
14	#19127463	C₄	1	2	yes	—
15	#19127455	—	2	1	no	—
16	D₄⋊(C₈₃⋊C₈)	C₄	1	4	yes	complemented
17	C₈×C₈₃⋊C₈	—	2	2	no	—
18	C₂³×C₈×C₈₃	—	2	1	no	—
19	C₂×C₄×D₄×C₈₃	C₂²	1	1	yes	characteristic
20	C₈×D₄×C₈₃	—	2	2	no	—
21	C₈×D₄×C₈₃	C₄	1	4	yes	complemented
22	C₂×C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊C₈×C₈₃	—	2	1	no	—
24	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
25	C₂²×C₈₃⋊C₈	—	2	1	no	—
26	C₂²×C₈₃⋊C₈	—	2	1	no	—
27	C₂²×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
28	C₂×C₄⋊Dic₈₃	C₂×C₄	1	2	yes	—
29	C₂×C₄×Dic₈₃	—	2	1	no	—
30	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
31	C₄⋊(C₈₃⋊C₈)	C₂×C₄	1	2	yes	—
32	#14581713	C₈	1	8	yes	direct product factor
33	C₄×C₈₃⋊C₈	—	2	1	no	—
34	C₄×C₈₃⋊C₈	—	2	1	no	—
35	C₄×C₈₃⋊C₈	—	2	1	no	—
36	C₄×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
37	C₈×Dic₈₃	—	4	2	no	—
38	C₈₃⋊#95	—	2	1	no	—
39	C₂²×D₄×C₈₃	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₈₃	—	2	1	no	—
41	C₂²×C₈×C₈₃	—	2	1	no	—
42	C₂²×C₈×C₈₃	—	2	4	no	—
43	C₂²×C₈×C₈₃	—	4	2	no	—
44	C₂²×C₈×C₈₃	—	4	2	no	—
45	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
47	C₄×D₄×C₈₃	—	2	1	no	—
48	C₄×D₄×C₈₃	—	2	1	no	—
49	C₄×D₄×C₈₃	C₂×C₄	1	2	yes	—
50	C₄×D₄×C₈₃	C₂×C₄	1	2	yes	—
51	C₂×C₄⋊C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₂²⋊C₄×C₈₃	—	2	1	no	—
53	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
54	C₄⋊C₈×C₈₃	—	2	1	no	—
55	C₄⋊C₈×C₈₃	C₂×C₄	1	2	yes	—
56	C₂²⋊C₈×C₈₃	—	2	4	no	—
57	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₈₃	D₄	1	2	yes	—
59	C₂²×Dic₈₃	—	2	2	no	—
60	C₂×C₈₃⋊C₈	—	2	4	no	—
61	C₂×C₈₃⋊C₈	—	2	2	no	—
62	C₂×C₈₃⋊C₈	—	2	2	no	—
63	C₂×C₈₃⋊C₈	—	2	2	no	—
64	C₂×C₈₃⋊C₈	—	2	2	no	—
65	C₂×C₈₃⋊C₈	—	4	1	no	—
66	C₂×C₈₃⋊C₈	—	4	1	no	—
67	C₂×C₈₃⋊C₈	C₂×C₈	1	4	yes	complemented
68	C₄⋊Dic₈₃	—	2	2	no	—
69	C₄⋊Dic₈₃	C₂×C₈	1	4	yes	complemented
70	C₄×Dic₈₃	—	4	1	no	—
71	C₄×Dic₈₃	—	4	1	no	—
72	C₂⁴×C₈₃	—	2	1	no	—
73	C₂×D₄×C₈₃	—	2	1	no	—
74	C₂×D₄×C₈₃	—	2	2	no	—
75	C₂×D₄×C₈₃	C₄²	1	2	yes	—
76	C₂×D₄×C₈₃	C₂×C₈	1	4	yes	—
77	C₂²×C₄×C₈₃	—	2	1	no	—
78	C₂²×C₄×C₈₃	—	2	2	no	—
79	C₂²×C₄×C₈₃	—	2	2	no	—
80	C₂²×C₄×C₈₃	—	4	1	no	—
81	C₂²×C₄×C₈₃	—	4	1	no	—
82	C₂²×C₄×C₈₃	—	4	1	no	—
83	C₂²×C₄×C₈₃	—	4	1	no	—
84	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
87	C₂×C₈×C₈₃	—	2	1	no	—
88	C₂×C₈×C₈₃	—	2	1	no	—
89	C₂×C₈×C₈₃	—	2	2	no	—
90	C₂×C₈×C₈₃	—	4	1	no	—
91	C₂×C₈×C₈₃	—	4	1	no	—
92	C₂×C₈×C₈₃	—	4	8	no	—
93	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
94	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
95	C₂×C₈×C₈₃	D₈	1	2	yes	—
96	C₂×C₈×C₈₃	SD₁₆	1	2	yes	—
97	C₄⋊C₄×C₈₃	—	2	1	no	—
98	C₄⋊C₄×C₈₃	C₄²	1	2	yes	—
99	C₂²⋊C₄×C₈₃	—	2	2	no	—
100	C₂²⋊C₄×C₈₃	—	2	2	no	—
101	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₈₃	#41	1	1	yes	characteristic
105	C₂³×C₈₃	—	2	2	no	—
106	C₂³×C₈₃	—	2	4	no	—
107	C₂³×C₈₃	—	4	1	no	—
108	C₂³×C₈₃	—	4	1	no	—
109	C₂³×C₈₃	—	4	2	no	—
110	C₂³×C₈₃	C₄×D₄	1	1	yes	characteristic
111	D₄×C₈₃	—	2	2	no	—
112	D₄×C₈₃	—	2	4	no	—
113	D₄×C₈₃	C₄×C₈	1	4	yes	complemented
114	C₂×C₄×C₈₃	—	2	2	no	—
115	C₂×C₄×C₈₃	—	2	2	no	—
116	C₂×C₄×C₈₃	—	4	1	no	—
117	C₂×C₄×C₈₃	—	4	1	no	—
118	C₂×C₄×C₈₃	—	4	2	no	—
119	C₂×C₄×C₈₃	—	4	2	no	—
120	C₂×C₄×C₈₃	—	4	2	no	—
121	C₂×C₄×C₈₃	—	4	2	no	—
122	C₂×C₄×C₈₃	C₂×C₄²	1	1	yes	characteristic
123	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₈₃	#117	1	1	yes	characteristic
125	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
126	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
127	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₈₃	C₂²×C₈	1	2	yes	—
129	C₂×C₄×C₈₃	#131	1	2	yes	—
130	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
132	C₂×C₄×C₈₃	#135	1	1	yes	characteristic
133	C₂×C₄×C₈₃	#135	1	1	yes	characteristic
134	C₂×Dic₈₃	—	2	4	no	—
135	C₂×Dic₈₃	—	4	2	no	—
136	C₂×Dic₈₃	—	4	2	no	—
137	C₈×C₈₃	—	2	2	no	—
138	C₈×C₈₃	—	4	2	no	—
139	C₈×C₈₃	#102	1	4	yes	complemented
140	C₈₃⋊C₈	—	2	4	no	—
141	C₈₃⋊C₈	—	2	4	no	—
142	C₈₃⋊C₈	—	4	2	no	—
143	C₂²×C₈₃	—	2	4	no	—
144	C₂²×C₈₃	—	4	4	no	—
145	C₂²×C₈₃	—	4	4	no	—
146	C₂²×C₈₃	—	4	2	no	—
147	C₂²×C₈₃	—	4	2	no	—
148	C₂²×C₈₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
149	C₂²×C₈₃	C₈×D₄	1	2	yes	—
150	C₂²×C₈₃	C₄×D₈	1	1	yes	characteristic
151	C₂²×C₈₃	C₄×SD₁₆	1	1	yes	characteristic
152	C₂²×C₈₃	#443	1	2	yes	—
153	C₄×C₈₃	—	2	1	no	—
154	C₄×C₈₃	—	2	1	no	—
155	C₄×C₈₃	—	4	2	no	—
156	C₄×C₈₃	C₂×C₄×C₈	1	1	yes	characteristic
157	C₄×C₈₃	#405	1	1	yes	characteristic
158	C₄×C₈₃	C₂×#102	1	1	yes	characteristic
159	C₄×C₈₃	#416	1	1	yes	characteristic
160	C₄×C₈₃	C₈×D₄	1	2	yes	—
161	C₄×C₈₃	C₄×D₈	1	1	yes	characteristic
162	C₄×C₈₃	C₄×SD₁₆	1	1	yes	characteristic
163	Dic₈₃	—	4	4	no	—
164	C₈×#102	—	83	1	no	maximal, Sylow
165	C₂×C₈₃	—	4	4	no	—
166	C₂×C₈₃	C₈×D₈	1	2	yes	—
167	C₂×C₈₃	C₈×SD₁₆	1	2	yes	—
168	C₂×C₈₃	C₈×C₂²⋊C₄	1	1	yes	characteristic
169	C₂×C₈₃	C₄×#102	1	1	yes	characteristic
170	C₂×C₈₃	#1764	1	1	yes	characteristic
171	C₂×C₈×D₄	D₈₃	1	1	yes	characteristic
172	C₈×C₄⋊C₄	—	83	1	no	—
173	C₄×#102	—	83	1	no	—
174	C₂×D₄⋊C₈	—	83	1	no	—
175	C₂×C₈²	—	83	1	no	—
176	#1279	—	83	1	no	—
177	#1277	—	83	1	no	—
178	C₈₃	C₈×#102	1	1	yes	characteristic, complemented, Sylow
179	C₂³×C₈	—	2	1	no	—
180	C₂×C₄×D₄	C₂×D₈₃	1	1	yes	characteristic
181	C₈×D₄	—	2	2	no	—
182	C₈×D₄	Dic₈₃	1	4	yes	complemented
183	C₂×C₄⋊C₈	—	83	1	no	—
184	C₂×C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
185	C₂×#102	—	83	2	no	—
186	C₂×C₂²⋊C₈	—	2	1	no	—
187	C₂×C₄×C₈	—	83	1	no	—
188	C₂×C₄×C₈	—	83	1	no	—
189	C₂×C₄×C₈	—	166	1	no	—
190	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
191	C₄×C₄⋊C₄	—	83	1	no	—
192	#340	—	83	2	no	—
193	#336	—	166	1	no	—
194	D₄⋊C₈	—	83	4	no	—
195	C₈²	—	166	2	no	—
196	C₂²×D₄	C₄×D₈₃	1	1	yes	characteristic
197	C₂³×C₄	—	2	1	no	—
198	C₂²×C₈	—	2	1	no	—
199	C₂²×C₈	—	2	4	no	—
200	C₂²×C₈	—	4	2	no	—
201	C₂²×C₈	—	4	2	no	—
202	C₂²×C₈	—	83	2	no	—
203	C₂²×C₈	—	166	1	no	—
204	C₂²×C₈	—	166	1	no	—
205	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
206	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
207	C₄×D₄	—	2	1	no	—
208	C₄×D₄	—	2	1	no	—
209	C₄×D₄	C₄×D₈₃	1	2	yes	—
210	C₄×D₄	C₂×Dic₈₃	1	2	yes	—
211	C₂×C₄⋊C₄	—	83	2	no	—
212	C₂×C₄⋊C₄	C₄×D₈₃	1	1	yes	characteristic
213	C₂×C₂²⋊C₄	—	2	1	no	—
214	C₂×C₄²	—	166	1	no	—
215	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
216	C₄⋊C₈	—	2	1	no	—
217	C₄⋊C₈	—	83	2	no	—
218	C₄⋊C₈	—	166	1	no	—
219	C₄⋊C₈	C₂×Dic₈₃	1	2	yes	—
220	#102	—	83	8	no	—
221	C₂²⋊C₈	—	2	4	no	—
222	C₄×C₈	—	332	2	no	—
223	C₄×C₈	—	83	2	no	—
224	C₄×C₈	—	166	1	no	—
225	C₄×C₈	—	166	1	no	—
226	C₄×C₈	—	166	1	no	—
227	C₄×C₈	C₂×Dic₈₃	1	2	yes	—
228	C₄×C₈	C₂²⋊D₈₃	1	2	yes	—
229	#95	—	166	1	no	—
230	C₂⁴	—	2	1	no	—
231	C₂×D₄	—	2	2	no	—
232	C₂×D₄	—	2	1	no	—
233	C₂×D₄	C₄×Dic₈₃	1	2	yes	—
234	C₂×D₄	C₈×D₈₃	1	4	yes	—
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	4	1	no	—
239	C₂²×C₄	—	4	1	no	—
240	C₂²×C₄	—	4	1	no	—
241	C₂²×C₄	—	4	1	no	—
242	C₂²×C₄	—	166	2	no	—
243	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
244	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
245	C₂²×C₄	#13050590	1	1	yes	characteristic
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	4	1	no	—
250	C₂×C₈	—	4	1	no	—
251	C₂×C₈	—	4	8	no	—
252	C₂×C₈	—	332	1	no	—
253	C₂×C₈	—	332	1	no	—
254	C₂×C₈	—	83	4	no	—
255	C₂×C₈	—	166	2	no	—
256	C₂×C₈	—	166	4	no	—
257	C₂×C₈	—	166	2	no	—
258	C₂×C₈	—	166	2	no	—
259	C₂×C₈	—	166	2	no	—
260	C₂×C₈	C₂²⋊Dic₈₃	1	2	yes	—
261	C₂×C₈	C₂²⋊Dic₈₃	1	2	yes	—
262	C₂×C₈	#13050575	1	2	yes	—
263	C₂×C₈	#13050578	1	2	yes	—
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	83	4	no	—
266	C₄⋊C₄	—	166	2	no	—
267	C₄⋊C₄	C₄×Dic₈₃	1	2	yes	—
268	C₂²⋊C₄	—	2	2	no	—
269	C₂²⋊C₄	—	2	2	no	—
270	C₄²	—	332	1	no	—
271	C₄²	—	332	1	no	—
272	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
273	C₄²	C₂²×Dic₈₃	1	1	yes	characteristic
274	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
275	C₄²	#13050590	1	1	yes	characteristic
276	C₂³	—	2	4	no	—
277	C₂³	—	2	2	no	—
278	C₂³	—	4	2	no	—
279	C₂³	—	4	1	no	—
280	C₂³	—	4	1	no	—
281	C₂³	C₄×C₂²⋊D₈₃	1	1	yes	characteristic
282	D₄	—	2	2	no	—
283	D₄	—	2	4	no	—
284	D₄	C₈×Dic₈₃	1	4	yes	complemented
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	332	2	no	—
294	C₂×C₄	—	332	2	no	—
295	C₂×C₄	—	166	4	no	—
296	C₂×C₄	C₂×C₄×Dic₈₃	1	1	yes	characteristic
297	C₂×C₄	C₂×C₂²⋊Dic₈₃	1	1	yes	characteristic
298	C₂×C₄	#14581751	1	1	yes	characteristic
299	C₂×C₄	C₄×C₂²⋊D₈₃	1	1	yes	characteristic
300	C₂×C₄	C₄×C₂²⋊D₈₃	1	1	yes	characteristic
301	C₂×C₄	C₄×C₂²⋊D₈₃	1	1	yes	characteristic
302	C₂×C₄	C₂×C₈×D₈₃	1	2	yes	—
303	C₂×C₄	#14581798	1	2	yes	—
304	C₂×C₄	C₂×#13050575	1	1	yes	characteristic
305	C₂×C₄	C₂×#13050578	1	1	yes	characteristic
306	C₂×C₄	#14581812	1	1	yes	characteristic
307	C₂×C₄	#14581812	1	1	yes	characteristic
308	C₈	—	2	2	no	—
309	C₈	—	4	2	no	—
310	C₈	—	332	2	no	—
311	C₈	—	166	4	no	—
312	C₈	—	166	4	no	—
313	C₈	#14581713	1	4	yes	direct product factor
314	C₂²	—	2	4	no	—
315	C₂²	—	4	2	no	—
316	C₂²	—	4	4	no	—
317	C₂²	—	4	4	no	—
318	C₂²	—	4	2	no	—
319	C₂²	C₄×C₂²⋊Dic₈₃	1	1	yes	characteristic
320	C₂²	C₈×C₂²⋊D₈₃	1	2	yes	—
321	C₂²	C₄×#13050575	1	1	yes	characteristic
322	C₂²	C₄×#13050578	1	1	yes	characteristic
323	C₂²	#19127816	1	2	yes	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	4	2	no	—
327	C₄	—	332	4	no	—
328	C₄	C₂×C₈×Dic₈₃	1	1	yes	characteristic
329	C₄	#19127656	1	1	yes	characteristic
330	C₄	C₂×#14581713	1	1	yes	characteristic
331	C₄	#19127697	1	1	yes	characteristic
332	C₄	C₈×C₂²⋊D₈₃	1	2	yes	—
333	C₄	C₄×#13050575	1	1	yes	characteristic
334	C₄	C₄×#13050578	1	1	yes	characteristic
335	C₂	—	4	4	no	—
336	C₂	C₈×#13050575	1	2	yes	—
337	C₂	C₈×#13050578	1	2	yes	—
338	C₂	C₈×C₂²⋊Dic₈₃	1	1	yes	characteristic
339	C₂	C₄×#14581713	1	1	yes	characteristic
340	C₂	#24907298	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²⋊D₈₃
commutator	C₄×C₈₃	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₂×C₈×D₄×C₈₃	C₂
socle	C₂³×C₈₃	C₄×D₄
prime radical	C₂²×D₄×C₈₃	C₂×C₄
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	C₂×C₈×D₄	D₈₃
hypocenter	C₈₃	C₈×#102
LCS	G ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. C₈×C₄⋊Dic₈₃	C₂	semidirect
4. C₂×C₈×C₈₃⋊C₈	C₂	semidirect
5. #24905302	C₂	semidirect
6. #24905297	C₂	semidirect
16. D₄⋊(C₈₃⋊C₈)	C₄	semidirect
21. C₈×D₄×C₈₃	C₄	semidirect
32. #14581713	C₈	direct (N × H)
67. C₂×C₈₃⋊C₈	C₂×C₈	semidirect
69. C₄⋊Dic₈₃	C₂×C₈	semidirect
113. D₄×C₈₃	C₄×C₈	semidirect
139. C₈×C₈₃	#102	semidirect
178. C₈₃	C₈×#102	semidirect
182. C₈×D₄	Dic₈₃	semidirect
284. D₄	C₈×Dic₈₃	semidirect
313. C₈	#14581713	direct (N × H)

Maximal subgroups

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

subgroup	index
C₈×C₄⋊Dic₈₃	2
C₄×#14581713	2
C₂×D₄⋊(C₈₃⋊C₈)	2
C₂×C₈×C₈₃⋊C₈	2
#24905302	2
#24905297	2
C₂×C₈×D₄×C₈₃	2
C₈×#102	83

Subgroups — C8×#14581713 #35295166

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#14581713 #35295166