Subgroups — #36523568

Order 22784.

602 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029363	C₂	1	1	yes	maximal, characteristic
2	#26028896	C₂	1	1	yes	maximal, characteristic
3	#26019043	C₂	1	2	yes	maximal, complemented
4	#26019038	C₂	1	2	yes	maximal, complemented
5	#26019033	C₂	1	2	yes	maximal, complemented
6	#26019028	C₂	1	2	yes	maximal, complemented
7	#26019005	C₂	1	1	yes	maximal, characteristic, complemented
8	#26019002	C₂	1	1	yes	maximal, characteristic, complemented
9	#26018998	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#19151350	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#2989	C₂	1	1	yes	maximal, characteristic
12	Q₈×#10623494	—	2	1	no	—
13	D₄×#10623494	—	2	2	no	—
14	D₄×#10623494	—	2	1	no	—
15	#19151659	C₄	1	4	yes	—
16	#19151651	C₂²	1	1	yes	characteristic
17	#19151651	C₂²	1	2	yes	—
18	#19151651	C₂²	1	2	yes	—
19	#19151651	C₂²	1	2	yes	—
20	#19151625	C₄	1	2	yes	—
21	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
23	#19151565	C₄	1	2	yes	—
24	C₄×#13083558	C₂²	1	1	yes	characteristic
25	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
26	#19151350	C₂²	1	2	yes	—
27	#19151350	C₂²	1	2	yes	—
28	#19151350	C₂²	1	1	yes	characteristic
29	#19151350	C₂²	1	1	yes	characteristic
30	#19151350	C₂²	1	2	yes	—
31	#19151350	C₂²	1	2	yes	—
32	#19151348	C₂²	1	1	yes	characteristic
33	#19151348	C₂²	1	1	yes	characteristic
34	#19151348	C₂²	1	2	yes	—
35	#19151348	C₂²	1	2	yes	—
36	C₄×#41×C₈₉	C₂²	1	1	yes	characteristic
37	#446×C₈₉	—	2	1	no	—
38	#436×C₈₉	—	2	1	no	—
39	#435×C₈₉	—	2	2	no	—
40	#432×C₈₉	C₂²	1	2	yes	—
41	#431×C₈₉	C₂²	1	1	yes	characteristic
42	#423×C₈₉	C₂²	1	2	yes	—
43	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
44	C₂×#13083558	C₂×C₄	1	2	yes	—
45	C₂²×#10623494	—	2	1	no	—
46	C₂²×#10623494	—	2	2	no	—
47	C₂²×#10623494	C₂×C₄	1	2	yes	—
48	C₄⋊#10623494	—	2	1	no	—
49	C₄⋊#10623494	—	2	2	no	—
50	C₄⋊#10623494	C₂×C₄	1	4	yes	—
51	C₄⋊#10623494	C₂×C₄	1	4	yes	—
52	C₄⋊#10623494	C₂³	1	2	yes	—
53	C₄⋊#10623494	C₂³	1	1	yes	characteristic
54	C₄⋊#10623494	C₂³	1	1	yes	characteristic
55	C₂²⋊#10623494	—	2	1	no	—
56	C₂²⋊#10623494	—	2	2	no	—
57	C₂²⋊#10623494	C₂×C₄	1	4	yes	—
58	C₂²⋊#10623494	C₂×C₄	1	4	yes	—
59	#15667923	C₂×C₄	1	2	yes	—
60	C₄×#10623494	—	2	1	no	—
61	C₄×#10623494	—	2	2	no	—
62	C₄×#10623494	C₂×C₄	1	2	yes	—
63	C₄×#10623494	C₂³	1	1	yes	characteristic
64	C₄×#10623494	C₂³	1	1	yes	characteristic
65	C₄×#10623494	C₂³	1	2	yes	—
66	C₂×#41×C₈₉	C₂×C₄	1	1	yes	characteristic
67	C₂×M₁₆×C₈₉	—	2	2	no	—
68	C₂²×C₈×C₈₉	—	2	1	no	—
69	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
70	C₄×Q₈×C₈₉	Q₈	1	1	yes	characteristic
71	C₄×D₄×C₈₉	D₄	1	2	yes	—
72	C₄×D₄×C₈₉	D₄	1	1	yes	characteristic
73	C₄×D₄×C₈₉	Q₈	1	2	yes	—
74	C₄×D₄×C₈₉	Q₈	1	1	yes	characteristic
75	#117×C₈₉	C₂×C₄	1	1	yes	characteristic
76	#117×C₈₉	C₂×C₄	1	2	yes	—
77	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
78	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
79	C₄⋊C₈×C₈₉	—	2	1	no	—
80	C₄⋊C₈×C₈₉	—	2	2	no	—
81	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
82	C₄⋊C₈×C₈₉	C₂³	1	1	yes	characteristic
83	C₄⋊C₈×C₈₉	C₂³	1	1	yes	characteristic
84	C₂²⋊C₈×C₈₉	—	2	1	no	—
85	C₂²⋊C₈×C₈₉	—	2	2	no	—
86	#97×C₈₉	—	2	2	no	—
87	C₄×C₈×C₈₉	—	2	1	no	—
88	#13083558	—	2	4	no	—
89	C₂×#10623494	—	2	4	no	—
90	C₂×#10623494	—	2	2	no	—
91	C₂×#10623494	—	2	1	no	—
92	C₂×#10623494	—	2	1	no	—
93	C₂×#10623494	—	8	1	no	—
94	C₂×#10623494	—	8	2	no	—
95	C₂×#10623494	C₂²×C₄	1	2	yes	—
96	C₂×#10623494	C₂²×C₄	1	2	yes	—
97	C₂×#10623494	C₂²×C₄	1	4	yes	—
98	#41×C₈₉	—	2	4	no	—
99	C₂×Q₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
100	C₂×D₄×C₈₉	C₄⋊C₄	1	2	yes	—
101	C₂×D₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
102	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
103	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
106	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
107	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
108	M₁₆×C₈₉	—	8	2	no	—
109	C₂×C₈×C₈₉	—	2	1	no	—
110	C₂×C₈×C₈₉	—	2	1	no	—
111	C₂×C₈×C₈₉	—	2	2	no	—
112	C₂×C₈×C₈₉	—	8	1	no	—
113	C₄⋊C₄×C₈₉	C₄⋊C₄	1	2	yes	—
114	C₄⋊C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
115	C₄⋊C₄×C₈₉	C₄⋊C₄	1	2	yes	—
116	C₄⋊C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
117	C₂²⋊C₄×C₈₉	C₄⋊C₄	1	2	yes	—
118	C₂²⋊C₄×C₈₉	C₄⋊C₄	1	2	yes	—
119	C₂²⋊C₄×C₈₉	C₄⋊C₄	1	2	yes	—
120	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
121	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
122	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₈₉	C₂×D₄	1	2	yes	—
124	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
125	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
126	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
127	C₂³×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
128	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
129	Q₈×C₈₉	—	2	2	no	—
130	D₄×C₈₉	—	2	4	no	—
131	D₄×C₈₉	—	2	2	no	—
132	C₂×C₄×C₈₉	—	2	4	no	—
133	C₂×C₄×C₈₉	—	2	1	no	—
134	C₂×C₄×C₈₉	—	2	2	no	—
135	C₂×C₄×C₈₉	—	2	1	no	—
136	C₂×C₄×C₈₉	—	2	2	no	—
137	C₂×C₄×C₈₉	—	4	1	no	—
138	C₂×C₄×C₈₉	—	4	2	no	—
139	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
141	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
143	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
144	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₈₉	#131	1	4	yes	—
146	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
147	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
149	C₈×C₈₉	—	4	2	no	—
150	C₈×C₈₉	—	4	1	no	—
151	C₈×C₈₉	—	4	1	no	—
152	#10623494	—	2	2	no	—
153	#10623494	—	2	2	no	—
154	#10623494	—	2	4	no	—
155	#10623494	—	4	2	no	—
156	#10623494	—	4	1	no	—
157	#10623494	—	4	1	no	—
158	C₂²×C₈₉	—	2	1	no	—
159	C₂²×C₈₉	—	2	1	no	—
160	C₂²×C₈₉	—	2	2	no	—
161	C₂²×C₈₉	—	2	2	no	—
162	C₂²×C₈₉	—	2	2	no	—
163	C₂²×C₈₉	#424	1	2	yes	—
164	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
165	C₄×C₈₉	—	2	2	no	—
166	C₄×C₈₉	—	2	1	no	—
167	C₄×C₈₉	—	2	2	no	—
168	C₄×C₈₉	—	2	1	no	—
169	C₄×C₈₉	—	2	1	no	—
170	C₄×C₈₉	—	2	2	no	—
171	C₄×C₈₉	#424	1	2	yes	—
172	C₄×C₈₉	C₂×#131	1	2	yes	—
173	C₄×C₈₉	#576	1	1	yes	characteristic
174	C₄×C₈₉	#578	1	1	yes	characteristic
175	#17812	—	89	1	no	maximal, Sylow
176	C₂×C₈₉	—	2	2	no	—
177	C₂×C₈₉	—	4	2	no	—
178	C₂×C₈₉	C₂×#424	1	1	yes	characteristic
179	C₂×C₈₉	#2913	1	1	yes	characteristic
180	C₂×C₈₉	#2916	1	1	yes	characteristic
181	#2989	D₈₉	1	1	yes	characteristic
182	#2988	—	89	1	no	—
183	#2922	—	89	1	no	—
184	#1574	—	89	2	no	—
185	#1573	—	89	2	no	—
186	#1572	—	89	2	no	—
187	#1571	—	89	2	no	—
188	#1565	—	89	1	no	—
189	#1565	—	89	1	no	—
190	#1564	—	89	1	no	—
191	C₂×#335	—	89	1	no	—
192	C₈₉	#17812	1	1	yes	characteristic, complemented, Sylow
193	C₄×#41	C₂×D₈₉	1	1	yes	characteristic
194	#446	—	2	1	no	—
195	C₈×Q₈	—	178	1	no	—
196	#436	—	2	1	no	—
197	#435	—	2	2	no	—
198	C₈×D₄	—	178	2	no	—
199	C₈×D₄	—	178	1	no	—
200	#433	—	89	4	no	—
201	#432	C₂×D₈₉	1	2	yes	—
202	#431	—	89	2	no	—
203	#431	—	89	2	no	—
204	#431	—	89	1	no	—
205	#431	—	89	2	no	—
206	#431	C₂×D₈₉	1	1	yes	characteristic
207	#424	—	89	2	no	—
208	#423	C₂×D₈₉	1	2	yes	—
209	C₂×C₄⋊C₈	—	89	1	no	—
210	C₂×C₄⋊C₈	—	89	2	no	—
211	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
212	#408	—	89	2	no	—
213	C₄×M₁₆	—	89	1	no	—
214	C₂×C₄×C₈	—	89	1	no	—
215	#335	—	89	2	no	—
216	#335	—	89	2	no	—
217	#335	—	89	1	no	—
218	#335	—	89	1	no	—
219	#335	—	89	2	no	—
220	#335	—	89	2	no	—
221	#334	—	89	1	no	—
222	#334	—	89	1	no	—
223	#334	—	89	2	no	—
224	#334	—	89	2	no	—
225	C₂×#41	C₄×D₈₉	1	1	yes	characteristic
226	C₂×M₁₆	—	2	2	no	—
227	C₂×M₁₆	—	89	2	no	—
228	C₂²×C₈	—	2	1	no	—
229	C₂²×C₈	—	89	2	no	—
230	C₂²×C₈	—	178	1	no	—
231	C₂²×C₈	—	178	2	no	—
232	C₄×Q₈	Dic₁₇₈	1	1	yes	characteristic
233	C₄×Q₈	C₂²⋊D₈₉	1	1	yes	characteristic
234	C₄×D₄	Dic₁₇₈	1	2	yes	—
235	C₄×D₄	Dic₁₇₈	1	1	yes	characteristic
236	C₄×D₄	C₂²⋊D₈₉	1	2	yes	—
237	C₄×D₄	C₂²⋊D₈₉	1	1	yes	characteristic
238	#117	C₄×D₈₉	1	1	yes	characteristic
239	#117	C₄×D₈₉	1	2	yes	—
240	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
241	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
242	C₄⋊C₈	—	2	1	no	—
243	C₄⋊C₈	—	2	2	no	—
244	C₄⋊C₈	—	89	4	no	—
245	C₄⋊C₈	—	89	4	no	—
246	C₄⋊C₈	—	89	1	no	—
247	C₄⋊C₈	—	89	1	no	—
248	C₄⋊C₈	—	89	2	no	—
249	C₄⋊C₈	—	178	1	no	—
250	C₄⋊C₈	—	178	2	no	—
251	C₄⋊C₈	C₂²×D₈₉	1	1	yes	characteristic
252	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
253	C₄⋊C₈	C₂²×D₈₉	1	1	yes	characteristic
254	C₂²⋊C₈	—	2	1	no	—
255	C₂²⋊C₈	—	2	2	no	—
256	C₂²⋊C₈	—	89	4	no	—
257	C₂²⋊C₈	—	89	4	no	—
258	C₂²⋊C₈	—	178	1	no	—
259	C₂²⋊C₈	—	178	2	no	—
260	#97	—	2	2	no	—
261	#97	—	89	2	no	—
262	C₄×C₈	—	2	1	no	—
263	C₄×C₈	—	89	2	no	—
264	C₄×C₈	—	89	1	no	—
265	C₄×C₈	—	89	2	no	—
266	C₄×C₈	—	89	1	no	—
267	C₄×C₈	—	178	1	no	—
268	C₄×C₈	—	178	2	no	—
269	#41	—	2	4	no	—
270	C₂×Q₈	#13083553	1	1	yes	characteristic
271	C₂×D₄	#13083553	1	2	yes	—
272	C₂×D₄	#13083553	1	1	yes	characteristic
273	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
274	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
275	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
276	C₂²×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
277	C₂²×C₄	C₂×Dic₁₇₈	1	2	yes	—
278	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
279	M₁₆	—	8	2	no	—
280	M₁₆	—	178	4	no	—
281	C₂×C₈	—	2	1	no	—
282	C₂×C₈	—	2	1	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	8	1	no	—
285	C₂×C₈	—	89	2	no	—
286	C₂×C₈	—	89	2	no	—
287	C₂×C₈	—	89	4	no	—
288	C₂×C₈	—	178	1	no	—
289	C₂×C₈	—	178	1	no	—
290	C₂×C₈	—	178	2	no	—
291	C₂×C₈	—	178	4	no	—
292	C₂×C₈	—	712	1	no	—
293	C₂×C₈	—	712	2	no	—
294	C₄⋊C₄	#13083553	1	2	yes	—
295	C₄⋊C₄	#13083553	1	1	yes	characteristic
296	C₄⋊C₄	#13083553	1	2	yes	—
297	C₄⋊C₄	#13083553	1	1	yes	characteristic
298	C₂²⋊C₄	#13083553	1	2	yes	—
299	C₂²⋊C₄	#13083553	1	2	yes	—
300	C₂²⋊C₄	#13083553	1	2	yes	—
301	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
302	C₄²	C₂×C₄×D₈₉	1	2	yes	—
303	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
304	C₄²	C₂×C₂²⋊D₈₉	1	2	yes	—
305	C₄²	C₂×Dic₁₇₈	1	2	yes	—
306	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
307	C₄²	C₂³×D₈₉	1	1	yes	characteristic
308	C₂³	C₂×#13083553	1	2	yes	—
309	C₂³	C₂×#13083553	1	1	yes	characteristic
310	Q₈	—	2	2	no	—
311	D₄	—	2	4	no	—
312	D₄	—	2	2	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	4	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
321	C₂×C₄	C₂×#13083553	1	2	yes	—
322	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
323	C₂×C₄	C₂×#13083553	1	2	yes	—
324	C₂×C₄	C₂×#13083553	1	2	yes	—
325	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
326	C₂×C₄	#15668020	1	4	yes	—
327	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
328	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
329	C₂×C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
330	C₈	—	4	2	no	—
331	C₈	—	4	1	no	—
332	C₈	—	4	1	no	—
333	C₈	—	356	2	no	—
334	C₈	—	356	1	no	—
335	C₈	—	356	1	no	—
336	C₈	—	178	2	no	—
337	C₈	—	178	2	no	—
338	C₈	—	178	4	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	2	no	—
342	C₂²	—	2	2	no	—
343	C₂²	—	2	2	no	—
344	C₂²	#19151626	1	2	yes	—
345	C₂²	C₂²×#13083553	1	1	yes	characteristic
346	C₄	—	2	2	no	—
347	C₄	—	2	1	no	—
348	C₄	—	2	2	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	2	no	—
352	C₄	#19151626	1	2	yes	—
353	C₄	C₂×#15668020	1	2	yes	—
354	C₄	#19152401	1	1	yes	characteristic
355	C₄	#19152413	1	1	yes	characteristic
356	C₂	—	2	2	no	—
357	C₂	—	4	2	no	—
358	C₂	C₂×#19151626	1	1	yes	characteristic
359	C₂	#26028850	1	1	yes	characteristic
360	C₂	#26028861	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#2989	C₂
socle	C₂²×C₈₉	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₈₉	C₂×C₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	#2989	D₈₉
hypocenter	C₈₉	#17812
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ #2989

Complements

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

N	H ≅ G/N	type
3. #26019043	C₂	semidirect
4. #26019038	C₂	semidirect
5. #26019033	C₂	semidirect
6. #26019028	C₂	semidirect
7. #26019005	C₂	semidirect
8. #26019002	C₂	semidirect
9. #26018998	C₂	semidirect
10. C₂×#19151350	C₂	semidirect
192. C₈₉	#17812	semidirect

Maximal subgroups

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

subgroup	index
#26029363	2
#26028896	2
#26019043	2
#26019038	2
#26019033	2
#26019028	2
#26019005	2
#26019002	2
#26018998	2
C₂×#19151350	2
C₈₉×#2989	2
#17812	89