Subgroups — #36447302

Order 22784.

462 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26019000	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19151349	C₂	1	1	yes	maximal, characteristic, complemented
3	#26018639	C₂	1	2	yes	maximal
4	#26018103	C₂	1	2	yes	maximal, complemented
5	C₄×#131×C₈₉	C₂	1	1	yes	maximal, characteristic
6	#19151651	C₂²	1	2	yes	—
7	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
8	#19151351	—	2	1	no	—
9	#19151349	—	2	1	no	—
10	#19151349	—	2	1	no	—
11	#19151349	—	2	1	no	—
12	Q₈⋊#10623494	—	2	2	no	—
13	D₄⋊#10623494	—	2	2	no	—
14	C₂×#131×C₈₉	C₄	1	1	yes	characteristic
15	C₂×#131×C₈₉	C₄	1	1	yes	characteristic
16	C₄×#41×C₈₉	C₂²	1	1	yes	characteristic
17	#405×C₈₉	—	2	1	no	—
18	#405×C₈₉	—	2	1	no	—
19	#405×C₈₉	C₄	1	1	yes	characteristic
20	#405×C₈₉	C₄	1	1	yes	characteristic
21	C₄×M₁₆×C₈₉	—	2	1	no	—
22	C₄×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₈₉	—	2	1	no	—
24	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
25	C₂²×#10623494	—	2	2	no	—
26	C₄⋊#10623494	—	2	2	no	—
27	C₄⋊#10623494	—	2	2	no	—
28	C₄⋊#10623494	—	2	2	no	—
29	C₂²⋊#10623494	—	2	2	no	—
30	C₄×#10623494	—	2	2	no	—
31	C₂×#41×C₈₉	C₂×C₄	1	1	yes	characteristic
32	#131×C₈₉	—	2	2	no	—
33	#131×C₈₉	—	2	2	no	—
34	#131×C₈₉	—	2	2	no	—
35	#131×C₈₉	C₈	1	4	yes	complemented
36	C₂×M₁₆×C₈₉	—	2	1	no	—
37	C₂×M₁₆×C₈₉	—	2	1	no	—
38	C₂×M₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
39	C₂×M₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₈₉	—	2	1	no	—
41	C₂²×C₈×C₈₉	—	2	1	no	—
42	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
44	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
45	C₄×Q₈×C₈₉	Q₈	1	1	yes	characteristic
46	C₄×D₄×C₈₉	—	2	1	no	—
47	C₄×D₄×C₈₉	—	2	1	no	—
48	C₄×D₄×C₈₉	D₄	1	1	yes	characteristic
49	C₄×D₄×C₈₉	Q₈	1	1	yes	characteristic
50	#117×C₈₉	—	2	1	no	—
51	#117×C₈₉	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₈₉	—	2	1	no	—
53	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
54	#97×C₈₉	—	2	1	no	—
55	#97×C₈₉	—	2	1	no	—
56	#97×C₈₉	—	2	1	no	—
57	C₄×C₈×C₈₉	—	2	1	no	—
58	C₄×C₈×C₈₉	—	2	1	no	—
59	C₄×C₈×C₈₉	—	2	1	no	—
60	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
61	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
62	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
63	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
64	C₂×#10623494	—	4	2	no	—
65	C₂×#10623494	—	4	2	no	—
66	C₂×#10623494	—	8	2	no	—
67	#41×C₈₉	—	2	1	no	—
68	#41×C₈₉	—	2	1	no	—
69	#41×C₈₉	C₂×C₈	1	2	yes	—
70	#41×C₈₉	M₁₆	1	2	yes	—
71	C₂×Q₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
72	C₂×D₄×C₈₉	—	2	1	no	—
73	C₂×D₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₈₉	—	2	1	no	—
75	C₂²×C₄×C₈₉	—	2	1	no	—
76	C₂²×C₄×C₈₉	—	2	1	no	—
77	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
79	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
80	M₁₆×C₈₉	—	2	4	no	—
81	M₁₆×C₈₉	—	2	1	no	—
82	M₁₆×C₈₉	—	2	1	no	—
83	M₁₆×C₈₉	—	2	4	no	—
84	M₁₆×C₈₉	C₂×C₈	1	2	yes	—
85	M₁₆×C₈₉	M₁₆	1	2	yes	—
86	C₂×C₈×C₈₉	—	2	4	no	—
87	C₂×C₈×C₈₉	—	2	1	no	—
88	C₂×C₈×C₈₉	—	2	1	no	—
89	C₂×C₈×C₈₉	—	2	1	no	—
90	C₂×C₈×C₈₉	—	2	1	no	—
91	C₂×C₈×C₈₉	—	2	4	no	—
92	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
97	C₂×C₈×C₈₉	M₁₆	1	2	yes	—
98	C₄⋊C₄×C₈₉	—	2	1	no	—
99	C₄⋊C₄×C₈₉	—	2	1	no	—
100	C₄⋊C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
101	C₄⋊C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
102	C₂²⋊C₄×C₈₉	—	2	1	no	—
103	C₂²⋊C₄×C₈₉	—	2	1	no	—
104	C₂²⋊C₄×C₈₉	—	2	1	no	—
105	C₄²×C₈₉	—	2	1	no	—
106	C₄²×C₈₉	—	2	1	no	—
107	C₄²×C₈₉	—	2	1	no	—
108	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
112	C₂³×C₈₉	—	2	1	no	—
113	C₂³×C₈₉	#117	1	1	yes	characteristic
114	Q₈×C₈₉	—	2	1	no	—
115	Q₈×C₈₉	C₄⋊C₈	1	2	yes	—
116	D₄×C₈₉	—	2	2	no	—
117	D₄×C₈₉	—	2	2	no	—
118	D₄×C₈₉	—	2	1	no	—
119	D₄×C₈₉	C₄⋊C₈	1	2	yes	—
120	C₂×C₄×C₈₉	—	2	1	no	—
121	C₂×C₄×C₈₉	—	2	2	no	—
122	C₂×C₄×C₈₉	—	2	2	no	—
123	C₂×C₄×C₈₉	—	2	2	no	—
124	C₂×C₄×C₈₉	—	2	1	no	—
125	C₂×C₄×C₈₉	—	2	1	no	—
126	C₂×C₄×C₈₉	—	4	1	no	—
127	C₂×C₄×C₈₉	—	4	2	no	—
128	C₂×C₄×C₈₉	—	4	2	no	—
129	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
132	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
133	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
134	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
135	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
136	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
137	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
138	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
139	C₈×C₈₉	—	2	2	no	—
140	C₈×C₈₉	—	2	1	no	—
141	C₈×C₈₉	—	2	2	no	—
142	C₈×C₈₉	—	2	1	no	—
143	C₈×C₈₉	C₂²⋊C₈	1	2	yes	—
144	C₈×C₈₉	C₂²⋊C₈	1	2	yes	—
145	#10623494	—	8	2	no	—
146	#10623494	—	8	2	no	—
147	C₂²×C₈₉	—	2	1	no	—
148	C₂²×C₈₉	—	2	1	no	—
149	C₂²×C₈₉	—	2	2	no	—
150	C₂²×C₈₉	—	4	1	no	—
151	C₂²×C₈₉	—	4	1	no	—
152	C₂²×C₈₉	#380	1	1	yes	characteristic
153	C₂²×C₈₉	#431	1	1	yes	characteristic
154	C₂²×C₈₉	#432	1	1	yes	characteristic
155	C₄×C₈₉	—	2	2	no	—
156	C₄×C₈₉	—	2	1	no	—
157	C₄×C₈₉	—	2	1	no	—
158	C₄×C₈₉	—	2	1	no	—
159	C₄×C₈₉	—	4	1	no	—
160	C₄×C₈₉	—	4	1	no	—
161	C₄×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
162	C₄×C₈₉	#407	1	1	yes	characteristic
163	C₄×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
164	C₄×C₈₉	#423	1	1	yes	characteristic
165	C₄×C₈₉	#471	1	2	yes	—
166	#8604	—	89	1	no	maximal, Sylow
167	C₂×C₈₉	—	2	1	no	—
168	C₂×C₈₉	—	2	1	no	—
169	C₂×C₈₉	—	4	2	no	—
170	C₂×C₈₉	#1797	1	1	yes	characteristic
171	C₂×C₈₉	#1814	1	1	yes	characteristic
172	C₂×C₈₉	#1815	1	1	yes	characteristic
173	C₄×#131	D₈₉	1	1	yes	characteristic
174	#1564	—	89	1	no	—
175	C₂×#334	—	89	1	no	—
176	#1477	—	89	2	no	—
177	#1283	—	89	2	no	—
178	C₈₉	#8604	1	1	yes	characteristic, complemented, Sylow
179	C₂×#131	Dic₈₉	1	1	yes	characteristic
180	C₂×#131	Dic₈₉	1	1	yes	characteristic
181	C₄×#41	C₂×D₈₉	1	1	yes	characteristic
182	#431	—	89	2	no	—
183	C₂×C₄⋊C₈	—	89	2	no	—
184	#405	—	2	1	no	—
185	#405	—	2	1	no	—
186	#405	Dic₈₉	1	1	yes	characteristic
187	#405	Dic₈₉	1	1	yes	characteristic
188	C₄×M₁₆	—	2	1	no	—
189	C₄×M₁₆	C₂×D₈₉	1	1	yes	characteristic
190	C₂×C₄×C₈	—	2	1	no	—
191	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
192	#335	—	178	1	no	—
193	#334	—	178	1	no	—
194	#334	—	178	1	no	—
195	#334	—	178	1	no	—
196	Q₈⋊C₈	—	178	2	no	—
197	D₄⋊C₈	—	178	2	no	—
198	C₂×#41	C₂×Dic₈₉	1	1	yes	characteristic
199	#131	—	2	2	no	—
200	#131	—	2	2	no	—
201	#131	—	2	2	no	—
202	#131	#10623494	1	4	yes	complemented
203	C₂×M₁₆	—	2	1	no	—
204	C₂×M₁₆	—	2	1	no	—
205	C₂×M₁₆	C₂×Dic₈₉	1	1	yes	characteristic
206	C₂×M₁₆	C₂×Dic₈₉	1	1	yes	characteristic
207	C₂²×C₈	—	2	1	no	—
208	C₂²×C₈	—	2	1	no	—
209	C₂²×C₈	—	178	2	no	—
210	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
211	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
212	C₄×Q₈	Dic₁₇₈	1	1	yes	characteristic
213	C₄×Q₈	D₃₅₆	1	1	yes	characteristic
214	C₄×D₄	—	2	1	no	—
215	C₄×D₄	—	2	1	no	—
216	C₄×D₄	Dic₁₇₈	1	1	yes	characteristic
217	C₄×D₄	D₃₅₆	1	1	yes	characteristic
218	#117	—	2	1	no	—
219	#117	C₂×Dic₈₉	1	1	yes	characteristic
220	C₂×C₄²	—	2	1	no	—
221	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
222	C₄⋊C₈	—	178	2	no	—
223	C₄⋊C₈	—	178	2	no	—
224	C₄⋊C₈	—	178	2	no	—
225	C₂²⋊C₈	—	178	2	no	—
226	#97	—	2	1	no	—
227	#97	—	2	1	no	—
228	#97	—	2	1	no	—
229	C₄×C₈	—	2	1	no	—
230	C₄×C₈	—	2	1	no	—
231	C₄×C₈	—	2	1	no	—
232	C₄×C₈	—	178	2	no	—
233	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
234	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
235	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
236	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
237	#41	—	2	1	no	—
238	#41	—	2	1	no	—
239	#41	C₂×#10623494	1	2	yes	—
240	#41	#13083558	1	2	yes	—
241	C₂×Q₈	C₄⋊Dic₈₉	1	1	yes	characteristic
242	C₂×D₄	—	2	1	no	—
243	C₂×D₄	C₄⋊Dic₈₉	1	1	yes	characteristic
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
248	C₂²×C₄	#13083574	1	1	yes	characteristic
249	C₂²×C₄	#13083574	1	1	yes	characteristic
250	M₁₆	—	2	4	no	—
251	M₁₆	—	2	1	no	—
252	M₁₆	—	2	1	no	—
253	M₁₆	—	2	4	no	—
254	M₁₆	C₂×#10623494	1	2	yes	—
255	M₁₆	#13083558	1	2	yes	—
256	C₂×C₈	—	2	4	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	4	no	—
262	C₂×C₈	—	356	2	no	—
263	C₂×C₈	—	356	2	no	—
264	C₂×C₈	—	712	2	no	—
265	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
266	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
267	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
268	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
269	C₂×C₈	C₂×#10623494	1	2	yes	—
270	C₂×C₈	#13083558	1	2	yes	—
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
274	C₄⋊C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
275	C₂²⋊C₄	—	2	1	no	—
276	C₂²⋊C₄	—	2	1	no	—
277	C₂²⋊C₄	—	2	1	no	—
278	C₄²	—	2	1	no	—
279	C₄²	—	2	1	no	—
280	C₄²	—	2	1	no	—
281	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
282	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
283	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
284	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
285	C₂³	—	2	1	no	—
286	C₂³	#15667971	1	1	yes	characteristic
287	Q₈	—	2	1	no	—
288	Q₈	C₄⋊#10623494	1	2	yes	—
289	D₄	—	2	2	no	—
290	D₄	—	2	2	no	—
291	D₄	—	2	1	no	—
292	D₄	C₄⋊#10623494	1	2	yes	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
303	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
304	C₂×C₄	#15667986	1	1	yes	characteristic
305	C₂×C₄	#15667986	1	1	yes	characteristic
306	C₂×C₄	#15667991	1	1	yes	characteristic
307	C₂×C₄	#15667991	1	1	yes	characteristic
308	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
309	C₂×C₄	C₂×#13083558	1	1	yes	characteristic
310	C₂×C₄	C₂×#13083558	1	1	yes	characteristic
311	C₂×C₄	C₂×#13083558	1	1	yes	characteristic
312	C₈	—	2	2	no	—
313	C₈	—	2	1	no	—
314	C₈	—	2	2	no	—
315	C₈	—	2	1	no	—
316	C₈	—	712	2	no	—
317	C₈	—	712	2	no	—
318	C₈	C₂²⋊#10623494	1	2	yes	—
319	C₈	C₂²⋊#10623494	1	2	yes	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	#19151458	1	1	yes	characteristic
326	C₂²	#19151651	1	1	yes	characteristic
327	C₂²	#19151655	1	1	yes	characteristic
328	C₄	—	2	2	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	4	1	no	—
334	C₄	C₂×C₂²⋊#10623494	1	1	yes	characteristic
335	C₄	#19151562	1	1	yes	characteristic
336	C₄	C₂×C₄⋊#10623494	1	1	yes	characteristic
337	C₄	#19151622	1	1	yes	characteristic
338	C₄	#19151861	1	2	yes	—
339	C₂	—	2	1	no	—
340	C₂	—	2	1	no	—
341	C₂	—	4	2	no	—
342	C₂	#26020232	1	1	yes	characteristic
343	C₂	#26020306	1	1	yes	characteristic
344	C₂	#26020311	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#15667986
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₄×#131×C₈₉	C₂
socle	C₂²×C₈₉	#380
prime radical	C₂×#41×C₈₉	C₂×C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₄×#131	D₈₉
hypocenter	C₈₉	#8604
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
1. #26019000	C₂	semidirect
2. C₂×#19151349	C₂	semidirect
4. #26018103	C₂	semidirect
35. #131×C₈₉	C₈	semidirect
178. C₈₉	#8604	semidirect
202. #131	#10623494	semidirect

Maximal subgroups

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

subgroup	index
#26019000	2
C₂×#19151349	2
#26018639	2
#26018103	2
C₄×#131×C₈₉	2
#8604	89