Subgroups — #36523668

Order 22784.

610 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×#13083574	C₂	1	1	yes	maximal, characteristic
2	#26028917	C₂	1	2	yes	maximal
3	#26019031	C₂	1	4	yes	maximal, complemented
4	#26019022	C₂	1	2	yes	maximal, complemented
5	#26019014	C₂	1	2	yes	maximal, complemented
6	#26019011	C₂	1	1	yes	maximal, characteristic, complemented
7	#26018999	C₂	1	2	yes	maximal, complemented
8	C₂×#335×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#13083574	C₂²	1	1	yes	characteristic
10	C₈×Dic₁₇₈	C₄	1	1	yes	characteristic
11	C₈×Dic₁₇₈	C₄	1	1	yes	characteristic
12	C₈×D₃₅₆	C₄	1	1	yes	characteristic
13	C₈×D₃₅₆	C₄	1	1	yes	characteristic
14	C₈×C₂²⋊D₈₉	C₄	1	2	yes	—
15	C₈×C₂²⋊D₈₉	C₄	1	2	yes	—
16	#19151658	—	2	4	no	—
17	#19151654	C₂²	1	4	yes	—
18	#19151651	C₂²	1	1	yes	characteristic
19	#19151650	C₂²	1	4	yes	—
20	#19151649	C₂²	1	2	yes	—
21	#19151622	C₂²	1	2	yes	—
22	#19151564	—	2	2	no	—
23	#19151553	—	2	2	no	—
24	C₄×C₈×D₈₉	C₂²	1	2	yes	—
25	#19151351	C₂²	1	2	yes	—
26	#19151351	C₂²	1	2	yes	—
27	#19151348	C₂²	1	4	yes	—
28	#19151348	C₂²	1	4	yes	—
29	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
31	#335×C₈₉	C₂²	1	1	yes	characteristic
32	#335×C₈₉	C₂²	1	1	yes	characteristic
33	#335×C₈₉	C₂²	1	2	yes	—
34	C₂×#13083574	D₄	1	1	yes	characteristic
35	C₂×#13083558	—	2	2	no	—
36	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
37	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
38	C₄×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
39	C₄×Dic₁₇₈	Q₈	1	1	yes	characteristic
40	C₄×D₃₅₆	C₂×C₄	1	1	yes	characteristic
41	C₄×D₃₅₆	Q₈	1	1	yes	characteristic
42	C₄×C₂²⋊D₈₉	C₂×C₄	1	2	yes	—
43	C₄×C₂²⋊D₈₉	Q₈	1	2	yes	—
44	#15667971	D₄	1	1	yes	characteristic
45	#15667969	D₄	1	2	yes	—
46	C₄²×D₈₉	C₂³	1	2	yes	—
47	C₄⋊#10623494	—	2	2	no	—
48	C₄⋊#10623494	C₂×C₄	1	1	yes	characteristic
49	C₄⋊#10623494	C₂×C₄	1	1	yes	characteristic
50	C₄⋊#10623494	C₂³	1	4	yes	—
51	#15667941	C₂×C₄	1	2	yes	—
52	#15667941	C₂×C₄	1	2	yes	—
53	C₂²⋊#10623494	C₂×C₄	1	2	yes	—
54	#15667924	—	2	4	no	—
55	#15667924	—	2	4	no	—
56	#15667924	C₂×C₄	1	2	yes	—
57	#15667924	C₂×C₄	1	2	yes	—
58	#15667922	—	2	4	no	—
59	C₄×#10623494	C₂³	1	2	yes	—
60	C₈×Dic₈₉	—	2	4	no	—
61	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
62	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
63	C₂²×C₈×C₈₉	—	2	2	no	—
64	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
65	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
67	C₄⋊C₈×C₈₉	—	2	2	no	—
68	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
69	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
70	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
74	#13083574	—	2	2	no	—
75	#13083574	—	2	2	no	—
76	C₂×Dic₁₇₈	C₄⋊C₄	1	1	yes	characteristic
77	C₂×D₃₅₆	C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₂²⋊D₈₉	C₄⋊C₄	1	2	yes	—
79	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
80	C₂×C₄×D₈₉	C₂×D₄	1	2	yes	—
81	C₂×C₄×D₈₉	C₂×Q₈	1	2	yes	—
82	#13083558	—	4	4	no	—
83	C₂×#10623494	—	2	4	no	—
84	C₂×#10623494	C₂²×C₄	1	2	yes	—
85	C₂×#10623494	C₂²×C₄	1	2	yes	—
86	C₈×D₈₉	—	2	4	no	—
87	C₈×D₈₉	—	4	2	no	—
88	C₄⋊Dic₈₉	C₄⋊C₄	1	1	yes	characteristic
89	C₄⋊Dic₈₉	C₄⋊C₄	1	1	yes	characteristic
90	#13083553	C₄⋊C₄	1	2	yes	—
91	#13083553	C₄⋊C₄	1	2	yes	—
92	C₂²⋊Dic₈₉	C₄⋊C₄	1	2	yes	—
93	#13083551	C₄⋊C₄	1	2	yes	—
94	#13083551	C₄⋊C₄	1	2	yes	—
95	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
96	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
97	C₄×Dic₈₉	C₂×Q₈	1	2	yes	—
98	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
101	C₂×C₈×C₈₉	—	2	2	no	—
102	C₂×C₈×C₈₉	—	2	2	no	—
103	C₂×C₈×C₈₉	—	2	2	no	—
104	C₂×C₈×C₈₉	—	4	1	no	—
105	C₂×C₈×C₈₉	—	4	4	no	—
106	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
113	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
114	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂²×D₈₉	C₂×C₄⋊C₄	1	2	yes	—
116	C₂×C₄×C₈₉	—	2	1	no	—
117	C₂×C₄×C₈₉	—	2	1	no	—
118	C₂×C₄×C₈₉	—	4	1	no	—
119	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₈₉	#135	1	2	yes	—
123	C₂×C₄×C₈₉	#135	1	2	yes	—
124	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
127	C₂²⋊D₈₉	—	2	4	no	—
128	C₂×Dic₈₉	C₂×C₄⋊C₄	1	2	yes	—
129	C₂×Dic₈₉	C₂×C₄⋊C₄	1	2	yes	—
130	C₂×Dic₈₉	C₂×C₄⋊C₄	1	2	yes	—
131	D₃₅₆	—	2	2	no	—
132	C₄×D₈₉	—	2	2	no	—
133	C₄×D₈₉	—	2	2	no	—
134	C₄×D₈₉	—	2	2	no	—
135	C₄×D₈₉	—	2	2	no	—
136	C₄×D₈₉	—	4	2	no	—
137	Dic₁₇₈	—	2	2	no	—
138	C₈×C₈₉	—	2	1	no	—
139	C₈×C₈₉	—	2	1	no	—
140	C₈×C₈₉	—	2	1	no	—
141	C₈×C₈₉	—	4	2	no	—
142	C₈×C₈₉	—	4	2	no	—
143	C₈×C₈₉	#131	1	2	yes	—
144	#10623494	—	2	2	no	—
145	#10623494	—	2	2	no	—
146	#10623494	—	4	4	no	—
147	C₂²×C₈₉	—	2	1	no	—
148	C₂²×C₈₉	—	2	1	no	—
149	C₂²×C₈₉	#427	1	2	yes	—
150	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
151	C₂×D₈₉	—	2	2	no	—
152	C₂×D₈₉	—	2	2	no	—
153	C₂×D₈₉	—	2	2	no	—
154	C₄×C₈₉	—	2	1	no	—
155	C₄×C₈₉	—	2	1	no	—
156	C₄×C₈₉	—	2	1	no	—
157	C₄×C₈₉	#427	1	2	yes	—
158	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
159	C₄×C₈₉	#568	1	1	yes	characteristic
160	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
161	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
162	Dic₈₉	—	2	2	no	—
163	Dic₈₉	—	2	2	no	—
164	Dic₈₉	—	2	2	no	—
165	#17829	—	89	1	no	maximal, Sylow
166	C₂×C₈₉	—	2	2	no	—
167	C₂×C₈₉	#2906	1	1	yes	characteristic
168	C₂×C₈₉	C₂×#427	1	1	yes	characteristic
169	C₂×C₈₉	#2916	1	1	yes	characteristic
170	D₈₉	—	4	2	no	—
171	C₈×#41	—	89	1	no	—
172	#2925	—	89	2	no	—
173	#1572	—	89	4	no	—
174	#1570	—	89	2	no	—
175	#1568	—	89	2	no	—
176	#1567	—	89	1	no	—
177	#1564	—	89	2	no	—
178	C₂×#335	D₈₉	1	1	yes	characteristic, complemented
179	C₈₉	#17829	1	1	yes	characteristic, complemented, Sylow
180	C₄×#41	—	89	1	no	—
181	C₈×Q₈	—	89	1	no	—
182	C₈×Q₈	—	89	1	no	—
183	C₈×D₄	—	89	2	no	—
184	C₈×D₄	—	89	2	no	—
185	C₈×D₄	—	89	1	no	—
186	C₈×D₄	—	89	1	no	—
187	#433	—	178	4	no	—
188	#432	—	89	4	no	—
189	#431	—	89	1	no	—
190	#431	—	89	2	no	—
191	#431	—	89	4	no	—
192	#423	—	89	2	no	—
193	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
194	#408	—	178	2	no	—
195	#405	—	178	2	no	—
196	C₂×C₄×C₈	—	89	2	no	—
197	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
198	#335	—	89	2	no	—
199	#335	—	89	2	no	—
200	#335	C₂×D₈₉	1	1	yes	characteristic
201	#335	C₂×D₈₉	1	1	yes	characteristic
202	#335	C₂×D₈₉	1	2	yes	—
203	#334	—	89	4	no	—
204	#334	—	89	4	no	—
205	C₂×#41	—	89	1	no	—
206	C₂×M₁₆	—	178	2	no	—
207	C₂²×C₈	—	2	2	no	—
208	C₂²×C₈	—	89	2	no	—
209	C₂²×C₈	—	89	2	no	—
210	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
211	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
212	C₄×Q₈	—	89	1	no	—
213	C₄×Q₈	—	89	1	no	—
214	C₄×D₄	—	89	2	no	—
215	C₄×D₄	—	89	2	no	—
216	C₄×D₄	—	89	1	no	—
217	C₄×D₄	—	89	1	no	—
218	#117	—	89	1	no	—
219	#117	—	89	2	no	—
220	C₂×C₄²	—	89	2	no	—
221	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
222	C₄⋊C₈	—	2	2	no	—
223	C₄⋊C₈	—	89	2	no	—
224	C₄⋊C₈	—	89	2	no	—
225	C₄⋊C₈	—	89	1	no	—
226	C₄⋊C₈	—	89	1	no	—
227	C₄⋊C₈	—	89	4	no	—
228	C₄⋊C₈	—	178	2	no	—
229	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
230	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
231	C₂²⋊C₈	—	89	2	no	—
232	C₂²⋊C₈	—	89	2	no	—
233	C₂²⋊C₈	—	89	2	no	—
234	C₂²⋊C₈	—	178	4	no	—
235	C₂²⋊C₈	—	178	4	no	—
236	#97	—	178	4	no	—
237	C₄×C₈	—	89	2	no	—
238	C₄×C₈	—	89	2	no	—
239	C₄×C₈	—	89	2	no	—
240	C₄×C₈	—	178	4	no	—
241	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
242	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
243	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
244	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
245	#41	—	178	2	no	—
246	#41	—	178	2	no	—
247	C₂×Q₈	—	89	1	no	—
248	C₂×D₄	—	89	2	no	—
249	C₂×D₄	—	89	1	no	—
250	C₂²×C₄	—	89	2	no	—
251	C₂²×C₄	—	89	2	no	—
252	C₂²×C₄	—	89	2	no	—
253	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
254	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
255	C₂²×C₄	Q₈×D₈₉	1	1	yes	characteristic
256	M₁₆	—	356	4	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	4	1	no	—
261	C₂×C₈	—	4	4	no	—
262	C₂×C₈	—	89	2	no	—
263	C₂×C₈	—	89	2	no	—
264	C₂×C₈	—	356	2	no	—
265	C₂×C₈	—	178	4	no	—
266	C₂×C₈	—	178	4	no	—
267	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
268	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
269	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
270	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
271	C₄⋊C₄	—	89	2	no	—
272	C₄⋊C₄	—	89	2	no	—
273	C₄⋊C₄	—	89	1	no	—
274	C₄⋊C₄	—	89	1	no	—
275	C₂²⋊C₄	—	89	2	no	—
276	C₂²⋊C₄	—	89	2	no	—
277	C₂²⋊C₄	—	89	2	no	—
278	C₄²	—	89	2	no	—
279	C₄²	—	89	2	no	—
280	C₄²	—	89	2	no	—
281	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
282	C₄²	D₄×D₈₉	1	1	yes	characteristic
283	C₄²	Q₈×D₈₉	1	1	yes	characteristic
284	C₄²	C₂³×D₈₉	1	1	yes	characteristic
285	C₂³	—	89	2	no	—
286	C₂³	C₄⋊C₄×D₈₉	1	1	yes	characteristic
287	Q₈	—	178	2	no	—
288	D₄	—	178	4	no	—
289	D₄	—	178	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	89	2	no	—
294	C₂×C₄	—	89	2	no	—
295	C₂×C₄	—	89	2	no	—
296	C₂×C₄	—	356	2	no	—
297	C₂×C₄	—	178	2	no	—
298	C₂×C₄	—	178	2	no	—
299	C₂×C₄	—	178	2	no	—
300	C₂×C₄	—	178	2	no	—
301	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
302	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
303	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
304	C₂×C₄	#15668031	1	2	yes	—
305	C₂×C₄	#15668035	1	2	yes	—
306	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
307	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
308	C₂×C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	4	2	no	—
313	C₈	—	4	2	no	—
314	C₈	—	356	4	no	—
315	C₈	—	178	2	no	—
316	C₈	—	178	2	no	—
317	C₈	#15668018	1	2	yes	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	178	2	no	—
321	C₂²	—	178	2	no	—
322	C₂²	—	178	2	no	—
323	C₂²	#19151635	1	2	yes	—
324	C₂²	C₂×C₄⋊C₄×D₈₉	1	1	yes	characteristic
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	178	2	no	—
329	C₄	—	178	2	no	—
330	C₄	—	178	2	no	—
331	C₄	#19151635	1	2	yes	—
332	C₄	C₂×#15668018	1	1	yes	characteristic
333	C₄	#19152364	1	1	yes	characteristic
334	C₄	C₂×#15668031	1	1	yes	characteristic
335	C₄	C₂×#15668035	1	1	yes	characteristic
336	C₂	—	2	2	no	—
337	C₂	—	356	2	no	—
338	C₂	#26028817	1	1	yes	characteristic
339	C₂	C₂×#19151635	1	1	yes	characteristic
340	C₂	#26028860	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂×#335×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×C₄⋊C₄
prime radical	C₂×#13083574	D₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	C₂×#335	D₈₉
hypocenter	C₈₉	#17829
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
3. #26019031	C₂	semidirect
4. #26019022	C₂	semidirect
5. #26019014	C₂	semidirect
6. #26019011	C₂	semidirect
7. #26018999	C₂	semidirect
8. C₂×#335×C₈₉	C₂	semidirect
178. C₂×#335	D₈₉	semidirect
179. C₈₉	#17829	semidirect

Maximal subgroups

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

subgroup	index
C₈×#13083574	2
#26028917	2
#26019031	2
#26019022	2
#26019014	2
#26019011	2
#26018999	2
C₂×#335×C₈₉	2
#17829	89