Subgroups — #36523532

Order 22784.

646 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26028885	C₂	1	1	yes	maximal, characteristic
2	#26028885	C₂	1	1	yes	maximal, characteristic
3	#26019040	C₂	1	4	yes	maximal, complemented
4	#26019030	C₂	1	4	yes	maximal, complemented
5	#26019002	C₂	1	4	yes	maximal, complemented
6	C₂×#431×C₈₉	C₂	1	1	yes	maximal, characteristic
7	#19151655	—	2	1	no	—
8	#19151655	C₄	1	2	yes	—
9	#19151655	C₄	1	2	yes	—
10	#19151655	C₂²	1	2	yes	—
11	#19151651	—	2	1	no	—
12	#19151651	C₂²	1	2	yes	—
13	#19151651	C₂²	1	2	yes	—
14	#19151651	C₂²	1	2	yes	—
15	#19151622	C₄	1	2	yes	—
16	#19151622	C₂²	1	2	yes	—
17	#19151562	—	2	1	no	—
18	#19151562	C₄	1	2	yes	—
19	C₄×#13083558	—	2	1	no	—
20	C₄×#13083558	C₂²	1	2	yes	—
21	#19151350	C₂²	1	8	yes	complemented
22	#19151348	C₂²	1	8	yes	complemented
23	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
24	#431×C₈₉	—	2	1	no	—
25	#431×C₈₉	—	2	1	no	—
26	#431×C₈₉	C₂²	1	2	yes	—
27	#431×C₈₉	C₂²	1	2	yes	—
28	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
29	C₂×C₂²⋊C₈×C₈₉	—	2	1	no	—
30	C₂×C₄×C₈×C₈₉	—	2	1	no	—
31	C₂×#13083558	—	2	2	no	—
32	C₂×#13083558	C₂×C₄	1	4	yes	—
33	C₄⋊#10623494	—	2	2	no	—
34	C₄⋊#10623494	C₂×C₄	1	4	yes	—
35	C₄⋊#10623494	C₂×C₄	1	4	yes	—
36	C₄⋊#10623494	C₂³	1	4	yes	—
37	C₂²⋊#10623494	—	2	2	no	—
38	C₂²⋊#10623494	—	2	2	no	—
39	C₂²⋊#10623494	C₂×C₄	1	4	yes	—
40	C₂²⋊#10623494	C₂×C₄	1	4	yes	—
41	#15667923	—	2	2	no	—
42	#15667923	C₂×C₄	1	4	yes	—
43	C₄×#10623494	—	2	2	no	—
44	C₄×#10623494	C₂³	1	4	yes	—
45	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
46	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
47	C₂³×C₄×C₈₉	Q₈	1	1	yes	characteristic
48	C₂²×C₈×C₈₉	—	2	2	no	—
49	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₈₉	Q₈	1	1	yes	characteristic
56	C₂×C₄²×C₈₉	Q₈	1	1	yes	characteristic
57	C₂×C₄²×C₈₉	Q₈	1	1	yes	characteristic
58	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
61	C₄⋊C₈×C₈₉	—	2	2	no	—
62	C₄⋊C₈×C₈₉	C₂³	1	4	yes	—
63	C₂²⋊C₈×C₈₉	—	2	2	no	—
64	C₂²⋊C₈×C₈₉	—	2	2	no	—
65	C₄×C₈×C₈₉	—	2	2	no	—
66	C₄×C₈×C₈₉	—	2	2	no	—
67	#13083558	—	2	8	no	—
68	#13083558	—	4	4	no	—
69	C₂×#10623494	—	2	4	no	—
70	C₂×#10623494	C₂²×C₄	1	8	yes	—
71	C₂⁴×C₈₉	C₄⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₈₉	—	2	4	no	—
73	C₂²×C₄×C₈₉	—	4	2	no	—
74	C₂²×C₄×C₈₉	—	4	2	no	—
75	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
91	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
92	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
93	C₂×C₈×C₈₉	—	2	4	no	—
94	C₂×C₈×C₈₉	—	4	4	no	—
95	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
96	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
97	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
98	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
99	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
100	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
101	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
108	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
109	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
110	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
111	C₂³×C₈₉	—	2	2	no	—
112	C₂³×C₈₉	—	4	1	no	—
113	C₂³×C₈₉	—	4	1	no	—
114	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₈₉	—	2	2	no	—
118	C₂×C₄×C₈₉	—	2	2	no	—
119	C₂×C₄×C₈₉	—	2	2	no	—
120	C₂×C₄×C₈₉	—	2	2	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	2	no	—
125	C₂×C₄×C₈₉	—	2	2	no	—
126	C₂×C₄×C₈₉	—	2	2	no	—
127	C₂×C₄×C₈₉	—	2	2	no	—
128	C₂×C₄×C₈₉	—	4	1	no	—
129	C₂×C₄×C₈₉	—	4	1	no	—
130	C₂×C₄×C₈₉	—	4	1	no	—
131	C₂×C₄×C₈₉	—	4	1	no	—
132	C₂×C₄×C₈₉	—	4	1	no	—
133	C₂×C₄×C₈₉	—	4	1	no	—
134	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
138	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₈₉	#131	1	4	yes	—
144	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
145	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
147	C₈×C₈₉	—	4	4	no	—
148	#10623494	—	2	8	no	—
149	#10623494	—	4	4	no	—
150	C₂²×C₈₉	—	2	1	no	—
151	C₂²×C₈₉	—	2	1	no	—
152	C₂²×C₈₉	—	2	1	no	—
153	C₂²×C₈₉	—	2	1	no	—
154	C₂²×C₈₉	—	2	1	no	—
155	C₂²×C₈₉	—	2	1	no	—
156	C₂²×C₈₉	—	2	1	no	—
157	C₂²×C₈₉	—	2	1	no	—
158	C₂²×C₈₉	—	4	4	no	—
159	C₂²×C₈₉	#424	1	2	yes	—
160	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
161	C₄×C₈₉	—	2	1	no	—
162	C₄×C₈₉	—	2	1	no	—
163	C₄×C₈₉	—	2	1	no	—
164	C₄×C₈₉	—	2	1	no	—
165	C₄×C₈₉	—	2	1	no	—
166	C₄×C₈₉	—	2	1	no	—
167	C₄×C₈₉	—	2	1	no	—
168	C₄×C₈₉	—	2	1	no	—
169	C₄×C₈₉	#424	1	2	yes	—
170	C₄×C₈₉	#430	1	2	yes	—
171	C₄×C₈₉	C₂×#131	1	2	yes	—
172	C₄×C₈₉	#575	1	2	yes	—
173	#17808	—	89	1	no	maximal, Sylow
174	C₂×C₈₉	—	2	2	no	—
175	C₂×C₈₉	—	4	1	no	—
176	C₂×C₈₉	—	4	1	no	—
177	C₂×C₈₉	C₂×#424	1	1	yes	characteristic
178	C₂×C₈₉	#2911	1	1	yes	characteristic
179	C₂×C₈₉	C₂×#430	1	1	yes	characteristic
180	#2920	—	89	1	no	—
181	#2920	—	89	1	no	—
182	C₂×#431	D₈₉	1	1	yes	characteristic
183	#1574	—	89	4	no	—
184	#1572	—	89	4	no	—
185	#1565	—	89	4	no	—
186	C₈₉	#17808	1	1	yes	characteristic, complemented, Sylow
187	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
188	#432	—	89	2	no	—
189	#432	—	89	2	no	—
190	#432	—	89	2	no	—
191	#432	—	178	1	no	—
192	#431	—	2	1	no	—
193	#431	—	2	1	no	—
194	#431	—	89	2	no	—
195	#431	—	89	2	no	—
196	#431	—	89	2	no	—
197	#431	—	178	1	no	—
198	#431	C₂×D₈₉	1	2	yes	—
199	#431	C₂×D₈₉	1	2	yes	—
200	#423	—	89	2	no	—
201	#423	—	89	2	no	—
202	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
203	#407	—	89	2	no	—
204	#407	—	178	1	no	—
205	C₂×C₂²⋊C₈	—	2	1	no	—
206	C₄×M₁₆	—	89	2	no	—
207	C₄×M₁₆	—	178	1	no	—
208	C₂×C₄×C₈	—	2	1	no	—
209	#335	—	89	8	no	—
210	#334	—	89	8	no	—
211	C₂³×C₄	Dic₁₇₈	1	1	yes	characteristic
212	C₂³×C₄	C₄×D₈₉	1	1	yes	characteristic
213	C₂³×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
214	C₂×M₁₆	—	89	4	no	—
215	C₂×M₁₆	—	178	2	no	—
216	C₂²×C₈	—	2	2	no	—
217	C₂×C₄²	Dic₁₇₈	1	1	yes	characteristic
218	C₂×C₄²	Dic₁₇₈	1	1	yes	characteristic
219	C₂×C₄²	Dic₁₇₈	1	1	yes	characteristic
220	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
221	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
222	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
223	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
224	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
225	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
226	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
227	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
228	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
229	C₄⋊C₈	—	2	2	no	—
230	C₄⋊C₈	—	89	4	no	—
231	C₄⋊C₈	—	89	4	no	—
232	C₄⋊C₈	—	89	4	no	—
233	C₄⋊C₈	—	178	2	no	—
234	C₄⋊C₈	C₂²×D₈₉	1	4	yes	—
235	C₂²⋊C₈	—	2	2	no	—
236	C₂²⋊C₈	—	2	2	no	—
237	C₂²⋊C₈	—	89	4	no	—
238	C₂²⋊C₈	—	89	4	no	—
239	C₂²⋊C₈	—	178	2	no	—
240	C₂²⋊C₈	—	178	2	no	—
241	#97	—	89	4	no	—
242	#97	—	178	2	no	—
243	C₄×C₈	—	2	2	no	—
244	C₄×C₈	—	2	2	no	—
245	C₄×C₈	—	89	4	no	—
246	C₄×C₈	—	178	2	no	—
247	C₂⁴	#13083553	1	1	yes	characteristic
248	C₂²×C₄	—	2	4	no	—
249	C₂²×C₄	—	4	2	no	—
250	C₂²×C₄	—	4	2	no	—
251	C₂²×C₄	#13083553	1	1	yes	characteristic
252	C₂²×C₄	#13083553	1	1	yes	characteristic
253	C₂²×C₄	#13083553	1	1	yes	characteristic
254	C₂²×C₄	#13083553	1	1	yes	characteristic
255	C₂²×C₄	#13083553	1	1	yes	characteristic
256	C₂²×C₄	#13083553	1	1	yes	characteristic
257	C₂²×C₄	#13083553	1	1	yes	characteristic
258	C₂²×C₄	#13083553	1	1	yes	characteristic
259	C₂²×C₄	#13083553	1	1	yes	characteristic
260	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
261	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
262	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
263	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
264	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
265	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
266	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
267	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
268	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
269	M₁₆	—	356	4	no	—
270	M₁₆	—	178	8	no	—
271	C₂×C₈	—	2	4	no	—
272	C₂×C₈	—	4	4	no	—
273	C₂×C₈	—	89	8	no	—
274	C₂×C₈	—	178	4	no	—
275	C₄²	#13083553	1	1	yes	characteristic
276	C₄²	#13083553	1	1	yes	characteristic
277	C₄²	#13083553	1	1	yes	characteristic
278	C₄²	#13083553	1	1	yes	characteristic
279	C₄²	#13083553	1	1	yes	characteristic
280	C₄²	#13083553	1	1	yes	characteristic
281	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
282	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
283	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
284	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
285	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
286	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
287	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
288	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
289	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
290	C₄²	C₂³×D₈₉	1	1	yes	characteristic
291	C₂³	—	2	2	no	—
292	C₂³	—	4	1	no	—
293	C₂³	—	4	1	no	—
294	C₂³	C₂×#13083553	1	1	yes	characteristic
295	C₂³	C₂×#13083553	1	1	yes	characteristic
296	C₂³	C₂×#13083553	1	1	yes	characteristic
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
315	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
316	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
317	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
318	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
319	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
320	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
321	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
322	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
323	C₂×C₄	#15668018	1	4	yes	—
324	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
325	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
326	C₂×C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
327	C₈	—	4	4	no	—
328	C₈	—	356	4	no	—
329	C₈	—	178	8	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	2	1	no	—
338	C₂²	—	4	4	no	—
339	C₂²	#19151624	1	2	yes	—
340	C₂²	C₂²×#13083553	1	1	yes	characteristic
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	2	1	no	—
345	C₄	—	2	1	no	—
346	C₄	—	2	1	no	—
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	#19151624	1	2	yes	—
350	C₄	#19151646	1	2	yes	—
351	C₄	C₂×#15668018	1	2	yes	—
352	C₄	#19152396	1	2	yes	—
353	C₂	—	2	2	no	—
354	C₂	—	4	1	no	—
355	C₂	—	4	1	no	—
356	C₂	C₂×#19151624	1	1	yes	characteristic
357	C₂	#26028841	1	1	yes	characteristic
358	C₂	C₂×#19151646	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂×#431×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₈₉	C₄⋊C₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	C₂×#431	D₈₉
hypocenter	C₈₉	#17808
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #26019040	C₂	semidirect
4. #26019030	C₂	semidirect
5. #26019002	C₂	semidirect
21. #19151350	C₂²	semidirect
22. #19151348	C₂²	semidirect
186. C₈₉	#17808	semidirect

Maximal subgroups

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

subgroup	index
#26028885	2
#26019040	2
#26019030	2
#26019002	2
C₂×#431×C₈₉	2
#17808	89