Subgroups — C₈×#15667934 #36445192

Order 22784.

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	C₂×C₈×D₃₅₆	C₂	1	1	yes	maximal, characteristic
2	C₈×C₄⋊Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#15667934	C₂	1	1	yes	maximal, characteristic
4	C₂×#19151327	C₂	1	1	yes	maximal, characteristic
5	#26018092	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018087	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₈²×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₈₉	—	2	1	no	—
9	C₂×C₄×D₃₅₆	C₂²	1	1	yes	characteristic
10	C₈×D₃₅₆	—	2	2	no	—
11	C₈×D₃₅₆	C₄	1	4	yes	complemented
12	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
14	C₂×#15667934	C₄	1	2	yes	—
15	C₂×#15667924	—	2	1	no	—
16	C₂×C₈×Dic₈₉	—	2	1	no	—
17	C₄×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
18	#19151361	C₄	1	2	yes	—
19	#19151352	—	2	1	no	—
20	#19151327	C₄	1	4	yes	complemented
21	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
24	C₈²×C₈₉	—	2	2	no	—
25	C₂²×D₃₅₆	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×D₈₉	—	2	1	no	—
27	C₂²×#10623494	—	2	1	no	—
28	C₂²×#10623494	—	2	1	no	—
29	C₂×C₈×D₈₉	—	2	4	no	—
30	C₂×C₈×D₈₉	—	4	2	no	—
31	C₂×C₈×D₈₉	—	4	2	no	—
32	C₄×D₃₅₆	—	2	1	no	—
33	C₄×D₃₅₆	—	2	1	no	—
34	C₄×D₃₅₆	C₂×C₄	1	2	yes	—
35	C₄×D₃₅₆	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
38	C₂×#13083551	—	2	1	no	—
39	C₂×C₄×Dic₈₉	—	2	1	no	—
40	C₄⋊#10623494	—	2	1	no	—
41	C₄⋊#10623494	—	2	1	no	—
42	C₄⋊#10623494	C₂×C₄	1	2	yes	—
43	C₄⋊#10623494	C₂×C₄	1	2	yes	—
44	#15667934	C₈	1	8	yes	direct product factor
45	#15667924	—	2	4	no	—
46	C₈×Dic₈₉	—	4	2	no	—
47	#15667919	—	2	1	no	—
48	C₂²×C₈×C₈₉	—	2	1	no	—
49	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
50	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
53	C₄×C₈×C₈₉	—	2	1	no	—
54	C₄×C₈×C₈₉	—	2	1	no	—
55	C₄×C₈×C₈₉	—	2	1	no	—
56	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
57	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₈₉	D₄	1	2	yes	—
59	C₂³×D₈₉	—	2	1	no	—
60	C₂×D₃₅₆	—	2	1	no	—
61	C₂×D₃₅₆	—	2	2	no	—
62	C₂×D₃₅₆	C₄²	1	2	yes	—
63	C₂×D₃₅₆	C₂×C₈	1	4	yes	—
64	C₂²×Dic₈₉	—	2	1	no	—
65	C₂²×Dic₈₉	—	2	2	no	—
66	C₂×C₄×D₈₉	—	2	2	no	—
67	C₂×C₄×D₈₉	—	2	2	no	—
68	C₂×C₄×D₈₉	—	4	1	no	—
69	C₂×C₄×D₈₉	—	4	1	no	—
70	C₂×C₄×D₈₉	—	4	1	no	—
71	C₂×C₄×D₈₉	—	4	1	no	—
72	C₂×#10623494	—	2	2	no	—
73	C₂×#10623494	—	2	2	no	—
74	C₂×#10623494	—	4	1	no	—
75	C₂×#10623494	—	4	1	no	—
76	C₂×#10623494	—	4	1	no	—
77	C₂×#10623494	—	4	1	no	—
78	C₈×D₈₉	—	4	8	no	—
79	C₄⋊Dic₈₉	—	2	1	no	—
80	C₄⋊Dic₈₉	—	2	2	no	—
81	C₄⋊Dic₈₉	C₄²	1	2	yes	—
82	C₄⋊Dic₈₉	C₂×C₈	1	4	yes	complemented
83	#13083551	—	2	2	no	—
84	#13083551	—	2	2	no	—
85	C₄×Dic₈₉	—	4	1	no	—
86	C₄×Dic₈₉	—	4	1	no	—
87	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
90	C₂×C₈×C₈₉	—	2	4	no	—
91	C₂×C₈×C₈₉	—	2	2	no	—
92	C₂×C₈×C₈₉	—	2	2	no	—
93	C₂×C₈×C₈₉	—	2	2	no	—
94	C₂×C₈×C₈₉	—	2	1	no	—
95	C₂×C₈×C₈₉	—	2	1	no	—
96	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
97	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
98	C₂×C₈×C₈₉	C₂×C₈	1	4	yes	complemented
99	C₂×C₈×C₈₉	D₈	1	2	yes	—
100	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
101	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₈₉	#41	1	1	yes	characteristic
105	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
106	C₂²×D₈₉	—	2	2	no	—
107	C₂²×D₈₉	—	2	4	no	—
108	C₂²×D₈₉	—	4	1	no	—
109	C₂²×D₈₉	—	4	1	no	—
110	C₂²×D₈₉	—	4	2	no	—
111	C₂×C₄×C₈₉	—	2	2	no	—
112	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
113	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
115	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
119	C₂×C₄×C₈₉	#131	1	2	yes	—
120	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
122	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
123	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
124	C₂×Dic₈₉	—	2	2	no	—
125	C₂×Dic₈₉	—	2	4	no	—
126	C₂×Dic₈₉	—	4	2	no	—
127	C₂×Dic₈₉	—	4	1	no	—
128	C₂×Dic₈₉	—	4	1	no	—
129	C₂×Dic₈₉	—	4	2	no	—
130	D₃₅₆	—	2	2	no	—
131	D₃₅₆	—	2	4	no	—
132	D₃₅₆	C₄×C₈	1	4	yes	complemented
133	C₄×D₈₉	—	4	2	no	—
134	C₄×D₈₉	—	4	2	no	—
135	C₄×D₈₉	—	4	2	no	—
136	C₄×D₈₉	—	4	2	no	—
137	C₈×C₈₉	—	2	4	no	—
138	C₈×C₈₉	—	2	2	no	—
139	C₈×C₈₉	—	2	4	no	—
140	C₈×C₈₉	#102	1	4	yes	complemented
141	#10623494	—	4	2	no	—
142	#10623494	—	4	2	no	—
143	C₂²×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
144	C₂²×C₈₉	C₈×D₄	1	2	yes	—
145	C₂²×C₈₉	C₄×D₈	1	1	yes	characteristic
146	C₂²×C₈₉	C₄×SD₁₆	1	1	yes	characteristic
147	C₂²×C₈₉	#443	1	2	yes	—
148	C₂×D₈₉	—	2	4	no	—
149	C₂×D₈₉	—	4	4	no	—
150	C₂×D₈₉	—	4	4	no	—
151	C₂×D₈₉	—	4	2	no	—
152	C₂×D₈₉	—	4	2	no	—
153	C₄×C₈₉	—	2	1	no	—
154	C₄×C₈₉	—	2	1	no	—
155	C₄×C₈₉	C₂×C₄×C₈	1	1	yes	characteristic
156	C₄×C₈₉	#405	1	1	yes	characteristic
157	C₄×C₈₉	C₂×#102	1	1	yes	characteristic
158	C₄×C₈₉	#416	1	1	yes	characteristic
159	C₄×C₈₉	C₈×D₄	1	2	yes	—
160	C₄×C₈₉	C₄×D₈	1	1	yes	characteristic
161	C₄×C₈₉	C₄×SD₁₆	1	1	yes	characteristic
162	Dic₈₉	—	4	4	no	—
163	Dic₈₉	—	4	2	no	—
164	C₈×#102	—	89	1	no	maximal, Sylow
165	C₂×C₈₉	C₈×D₈	1	2	yes	—
166	C₂×C₈₉	C₈×SD₁₆	1	2	yes	—
167	C₂×C₈₉	C₈×C₂²⋊C₄	1	1	yes	characteristic
168	C₂×C₈₉	C₄×#102	1	1	yes	characteristic
169	C₂×C₈₉	#1764	1	1	yes	characteristic
170	D₈₉	—	4	4	no	—
171	C₂×C₈×D₄	—	89	1	no	—
172	C₈×C₄⋊C₄	—	89	1	no	—
173	C₄×#102	—	89	1	no	—
174	C₂×D₄⋊C₈	—	89	1	no	—
175	C₂×C₈²	D₈₉	1	1	yes	characteristic, complemented
176	#1279	—	89	1	no	—
177	#1277	—	89	1	no	—
178	C₈₉	C₈×#102	1	1	yes	characteristic, complemented, Sylow
179	C₂³×C₈	—	178	1	no	—
180	C₂×C₄×D₄	—	89	1	no	—
181	C₈×D₄	—	89	4	no	—
182	C₈×D₄	—	178	2	no	—
183	C₂×C₄⋊C₈	—	89	1	no	—
184	C₂×C₄⋊C₈	—	89	1	no	—
185	C₂×#102	—	89	2	no	—
186	C₂×C₂²⋊C₈	—	178	1	no	—
187	C₂×C₄×C₈	—	178	1	no	—
188	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
189	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
190	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
191	C₄×C₄⋊C₄	—	89	1	no	—
192	#340	—	89	2	no	—
193	#336	—	178	1	no	—
194	D₄⋊C₈	—	89	4	no	—
195	C₈²	—	2	2	no	—
196	C₂²×D₄	—	89	1	no	—
197	C₂³×C₄	—	178	1	no	—
198	C₂²×C₈	—	2	1	no	—
199	C₂²×C₈	—	356	2	no	—
200	C₂²×C₈	—	356	2	no	—
201	C₂²×C₈	—	178	1	no	—
202	C₂²×C₈	—	178	1	no	—
203	C₂²×C₈	—	178	4	no	—
204	C₂²×C₈	C₄×D₈₉	1	2	yes	—
205	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
206	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
207	C₄×D₄	—	89	2	no	—
208	C₄×D₄	—	89	2	no	—
209	C₄×D₄	—	178	1	no	—
210	C₄×D₄	—	178	1	no	—
211	C₂×C₄⋊C₄	—	89	1	no	—
212	C₂×C₄⋊C₄	—	89	2	no	—
213	C₂×C₂²⋊C₄	—	178	1	no	—
214	C₂×C₄²	—	178	1	no	—
215	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
216	C₄⋊C₈	—	89	2	no	—
217	C₄⋊C₈	—	89	2	no	—
218	C₄⋊C₈	—	178	1	no	—
219	C₄⋊C₈	—	178	1	no	—
220	#102	—	89	8	no	—
221	C₂²⋊C₈	—	178	4	no	—
222	C₄×C₈	—	2	1	no	—
223	C₄×C₈	—	2	1	no	—
224	C₄×C₈	—	2	1	no	—
225	C₄×C₈	—	356	2	no	—
226	C₄×C₈	C₄×D₈₉	1	2	yes	—
227	C₄×C₈	C₄×D₈₉	1	2	yes	—
228	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
229	#95	—	178	1	no	—
230	C₂⁴	—	178	1	no	—
231	C₂×D₄	—	89	4	no	—
232	C₂×D₄	—	89	2	no	—
233	C₂×D₄	—	178	1	no	—
234	C₂×D₄	—	178	2	no	—
235	C₂²×C₄	—	356	1	no	—
236	C₂²×C₄	—	356	1	no	—
237	C₂²×C₄	—	356	1	no	—
238	C₂²×C₄	—	356	1	no	—
239	C₂²×C₄	—	178	2	no	—
240	C₂²×C₄	—	178	1	no	—
241	C₂²×C₄	—	178	2	no	—
242	C₂²×C₄	—	178	2	no	—
243	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
245	C₂²×C₄	#13083574	1	1	yes	characteristic
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	356	1	no	—
253	C₂×C₈	—	356	1	no	—
254	C₂×C₈	—	356	1	no	—
255	C₂×C₈	—	356	1	no	—
256	C₂×C₈	—	356	8	no	—
257	C₂×C₈	—	178	2	no	—
258	C₂×C₈	—	178	2	no	—
259	C₂×C₈	#13083551	1	2	yes	—
260	C₂×C₈	#13083551	1	2	yes	—
261	C₂×C₈	C₈×D₈₉	1	4	yes	complemented
262	C₂×C₈	D₇₁₂	1	2	yes	—
263	C₂×C₈	#13083563	1	2	yes	—
264	C₄⋊C₄	—	89	4	no	—
265	C₄⋊C₄	—	89	2	no	—
266	C₄⋊C₄	—	178	1	no	—
267	C₄⋊C₄	—	178	2	no	—
268	C₂²⋊C₄	—	178	2	no	—
269	C₂²⋊C₄	—	178	2	no	—
270	C₄²	—	356	1	no	—
271	C₄²	—	356	1	no	—
272	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
273	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
274	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
275	C₄²	#13083574	1	1	yes	characteristic
276	C₂³	—	356	1	no	—
277	C₂³	—	356	1	no	—
278	C₂³	—	356	2	no	—
279	C₂³	—	178	4	no	—
280	C₂³	—	178	2	no	—
281	C₂³	C₄×D₃₅₆	1	1	yes	characteristic
282	D₄	—	89	4	no	—
283	D₄	—	178	4	no	—
284	D₄	—	178	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	356	2	no	—
287	C₂×C₄	—	356	1	no	—
288	C₂×C₄	—	356	1	no	—
289	C₂×C₄	—	356	2	no	—
290	C₂×C₄	—	356	2	no	—
291	C₂×C₄	—	356	2	no	—
292	C₂×C₄	—	356	2	no	—
293	C₂×C₄	—	356	2	no	—
294	C₂×C₄	—	178	4	no	—
295	C₂×C₄	—	178	2	no	—
296	C₂×C₄	C₄²×D₈₉	1	1	yes	characteristic
297	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
298	C₂×C₄	#15667969	1	1	yes	characteristic
299	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
300	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
301	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
302	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
303	C₂×C₄	#15668018	1	2	yes	—
304	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
305	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
306	C₂×C₄	#15668034	1	1	yes	characteristic
307	C₂×C₄	#15668034	1	1	yes	characteristic
308	C₈	—	2	4	no	—
309	C₈	—	2	4	no	—
310	C₈	—	2	2	no	—
311	C₈	—	356	2	no	—
312	C₈	—	356	2	no	—
313	C₈	#15667934	1	4	yes	direct product factor
314	C₂²	—	356	2	no	—
315	C₂²	—	356	4	no	—
316	C₂²	—	356	4	no	—
317	C₂²	—	356	2	no	—
318	C₂²	—	178	4	no	—
319	C₂²	C₄×#13083551	1	1	yes	characteristic
320	C₂²	C₈×D₃₅₆	1	2	yes	—
321	C₂²	C₄×D₇₁₂	1	1	yes	characteristic
322	C₂²	C₄×#13083563	1	1	yes	characteristic
323	C₂²	#19151716	1	2	yes	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	356	2	no	—
327	C₄	—	356	4	no	—
328	C₄	C₄×C₈×D₈₉	1	1	yes	characteristic
329	C₄	#19151552	1	1	yes	characteristic
330	C₄	C₂×#15667934	1	1	yes	characteristic
331	C₄	#19151595	1	1	yes	characteristic
332	C₄	C₈×C₂²⋊D₈₉	1	2	yes	—
333	C₄	C₄×D₇₁₂	1	1	yes	characteristic
334	C₄	C₄×#13083563	1	1	yes	characteristic
335	C₂	—	356	4	no	—
336	C₂	C₈×D₇₁₂	1	2	yes	—
337	C₂	C₈×#13083563	1	2	yes	—
338	C₂	C₈×#13083551	1	1	yes	characteristic
339	C₂	C₄×#15667934	1	1	yes	characteristic
340	C₂	#26020089	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	D₃₅₆
commutator	C₄×C₈₉	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₂×C₈²×C₈₉	C₂
socle	C₂³×C₈₉	C₄×D₄
prime radical	C₂²×D₃₅₆	C₂×C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂×C₈²	D₈₉
hypocenter	C₈₉	C₈×#102
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
2. C₈×C₄⋊Dic₈₉	C₂	semidirect
5. #26018092	C₂	semidirect
6. #26018087	C₂	semidirect
7. C₂×C₈²×C₈₉	C₂	semidirect
11. C₈×D₃₅₆	C₄	semidirect
20. #19151327	C₄	semidirect
44. #15667934	C₈	direct (N × H)
82. C₄⋊Dic₈₉	C₂×C₈	semidirect
98. C₂×C₈×C₈₉	C₂×C₈	semidirect
132. D₃₅₆	C₄×C₈	semidirect
140. C₈×C₈₉	#102	semidirect
175. C₂×C₈²	D₈₉	semidirect
178. C₈₉	C₈×#102	semidirect
261. C₂×C₈	C₈×D₈₉	semidirect
313. C₈	#15667934	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×D₃₅₆	2
C₈×C₄⋊Dic₈₉	2
C₄×#15667934	2
C₂×#19151327	2
#26018092	2
#26018087	2
C₂×C₈²×C₈₉	2
C₈×#102	89

Subgroups — C8×#15667934 #36445192

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#15667934 #36445192