Subgroups — #36454180

Order 22784.

578 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151945	C₂	1	1	yes	maximal, characteristic
2	#26021213	C₂	1	1	yes	maximal, characteristic, complemented
3	#26021058	C₂	1	1	yes	maximal, characteristic
4	#26021058	C₂	1	1	yes	maximal, characteristic
5	#26021054	C₂	1	1	yes	maximal, characteristic, complemented
6	#26020380	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₈²×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15668012	C₂²	1	1	yes	characteristic
9	C₂×#15668009	C₂²	1	1	yes	characteristic
10	#19151945	—	2	2	no	—
11	#19151945	C₄	1	4	yes	complemented
12	C₂×#15667937	—	2	2	no	—
13	C₂×#15667934	—	2	2	no	—
14	C₂×#15667934	—	2	2	no	—
15	C₂×#15667934	—	2	2	no	—
16	#19151468	C₂²	1	1	yes	characteristic
17	#19151468	C₂²	1	1	yes	characteristic
18	#19151361	—	2	2	no	—
19	#19151361	—	2	2	no	—
20	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
23	C₈²×C₈₉	—	2	2	no	—
24	C₂²×Dic₁₇₈	—	2	1	no	—
25	C₂²×D₃₅₆	—	2	1	no	—
26	C₂²×D₃₅₆	—	2	1	no	—
27	C₂²×D₃₅₆	—	2	1	no	—
28	#15668012	—	2	1	no	—
29	#15668012	—	2	1	no	—
30	#15668012	—	2	1	no	—
31	#15668012	C₂×C₄	1	2	yes	—
32	#15668009	—	2	1	no	—
33	#15668009	—	2	1	no	—
34	#15668009	—	2	1	no	—
35	#15668009	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊Dic₈₉	—	2	1	no	—
37	C₂×C₄⋊Dic₈₉	—	2	1	no	—
38	C₂×C₄⋊Dic₈₉	—	2	1	no	—
39	C₂×C₄⋊Dic₈₉	—	2	1	no	—
40	C₂×C₄⋊Dic₈₉	—	2	1	no	—
41	C₂×C₄⋊Dic₈₉	—	2	1	no	—
42	C₂×C₄⋊Dic₈₉	—	2	1	no	—
43	C₂×C₄⋊Dic₈₉	—	2	1	no	—
44	#15667937	—	4	4	no	—
45	#15667934	—	2	8	no	—
46	#15667934	—	4	4	no	—
47	#15667934	—	4	4	no	—
48	C₂²×C₈×C₈₉	D₄	1	2	yes	—
49	C₂²×C₈×C₈₉	D₄	1	2	yes	—
50	C₂²×C₈×C₈₉	D₄	1	2	yes	—
51	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
52	C₄×C₈×C₈₉	—	2	1	no	—
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	D₄	1	2	yes	—
58	C₂³×D₈₉	—	4	1	no	—
59	C₂×Dic₁₇₈	—	2	2	no	—
60	C₂×Dic₁₇₈	—	4	2	no	—
61	C₂×Dic₁₇₈	—	4	2	no	—
62	C₂×Dic₁₇₈	—	4	1	no	—
63	C₂×D₃₅₆	—	2	2	no	—
64	C₂×D₃₅₆	—	2	2	no	—
65	C₂×D₃₅₆	—	2	2	no	—
66	C₂×D₃₅₆	—	4	2	no	—
67	C₂×D₃₅₆	—	4	2	no	—
68	C₂×D₃₅₆	—	4	2	no	—
69	C₂×D₃₅₆	—	4	2	no	—
70	C₂×D₃₅₆	—	4	2	no	—
71	C₂×D₃₅₆	—	4	2	no	—
72	C₂×D₃₅₆	—	4	1	no	—
73	C₂×D₃₅₆	—	4	1	no	—
74	C₂×D₃₅₆	—	4	1	no	—
75	C₂²×Dic₈₉	—	4	1	no	—
76	C₂²×Dic₈₉	—	4	1	no	—
77	C₂²×Dic₈₉	—	4	1	no	—
78	C₄⋊Dic₈₉	—	2	2	no	—
79	C₄⋊Dic₈₉	—	2	2	no	—
80	C₄⋊Dic₈₉	—	4	1	no	—
81	C₄⋊Dic₈₉	—	4	1	no	—
82	C₄⋊Dic₈₉	—	4	2	no	—
83	C₄⋊Dic₈₉	—	4	2	no	—
84	C₄⋊Dic₈₉	—	4	2	no	—
85	C₄⋊Dic₈₉	—	4	2	no	—
86	C₄⋊Dic₈₉	—	4	2	no	—
87	C₄⋊Dic₈₉	—	4	2	no	—
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×D₄	1	1	yes	characteristic
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	2	no	—
95	C₂×C₈×C₈₉	—	2	2	no	—
96	C₂×C₈×C₈₉	—	2	2	no	—
97	C₂×C₈×C₈₉	C₂²⋊C₄	1	4	yes	—
98	C₂×C₈×C₈₉	D₈	1	4	yes	—
99	C₂×C₈×C₈₉	SD₁₆	1	4	yes	—
100	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₈₉	#41	1	1	yes	characteristic
103	C₄²×C₈₉	#41	1	1	yes	characteristic
104	C₂³×C₈₉	#127	1	1	yes	characteristic
105	C₂²×D₈₉	—	4	2	no	—
106	C₂²×D₈₉	—	8	1	no	—
107	C₂²×D₈₉	—	8	1	no	—
108	C₂²×D₈₉	—	8	1	no	—
109	C₂²×D₈₉	—	8	1	no	—
110	C₂²×D₈₉	—	8	1	no	—
111	C₂²×D₈₉	—	8	1	no	—
112	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
116	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
117	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
118	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
119	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
120	C₂×C₄×C₈₉	#135	1	2	yes	—
121	C₂×C₄×C₈₉	#135	1	2	yes	—
122	C₂×C₄×C₈₉	#135	1	2	yes	—
123	C₂×C₄×C₈₉	#135	1	2	yes	—
124	C₂×Dic₈₉	—	4	2	no	—
125	C₂×Dic₈₉	—	8	1	no	—
126	C₂×Dic₈₉	—	8	1	no	—
127	C₂×Dic₈₉	—	8	1	no	—
128	C₂×Dic₈₉	—	8	1	no	—
129	C₂×Dic₈₉	—	8	1	no	—
130	C₂×Dic₈₉	—	8	1	no	—
131	C₂×Dic₈₉	—	8	1	no	—
132	C₂×Dic₈₉	—	8	1	no	—
133	D₃₅₆	—	4	4	no	—
134	D₃₅₆	—	4	4	no	—
135	D₃₅₆	—	4	4	no	—
136	D₃₅₆	—	8	2	no	—
137	D₃₅₆	—	8	2	no	—
138	D₃₅₆	—	8	2	no	—
139	Dic₁₇₈	—	4	4	no	—
140	Dic₁₇₈	—	8	2	no	—
141	C₈×C₈₉	—	2	4	no	—
142	C₈×C₈₉	—	2	4	no	—
143	C₈×C₈₉	—	2	4	no	—
144	C₂²×C₈₉	#390	1	1	yes	characteristic
145	C₂²×C₈₉	#492	1	1	yes	characteristic
146	C₂²×C₈₉	#493	1	1	yes	characteristic
147	C₂²×C₈₉	#495	1	1	yes	characteristic
148	C₂²×C₈₉	#495	1	1	yes	characteristic
149	C₂²×C₈₉	#495	1	1	yes	characteristic
150	C₂²×C₈₉	#495	1	1	yes	characteristic
151	C₂×D₈₉	—	8	2	no	—
152	C₂×D₈₉	—	8	2	no	—
153	C₂×D₈₉	—	8	2	no	—
154	C₂×D₈₉	—	16	1	no	—
155	C₂×D₈₉	—	16	1	no	—
156	C₂×D₈₉	—	16	1	no	—
157	C₂×D₈₉	—	16	1	no	—
158	C₄×C₈₉	#416	1	2	yes	—
159	C₄×C₈₉	C₄×D₈	1	2	yes	—
160	C₄×C₈₉	C₄×SD₁₆	1	2	yes	—
161	C₄×C₈₉	#465	1	2	yes	—
162	C₄×C₈₉	#466	1	2	yes	—
163	C₄×C₈₉	#488	1	2	yes	—
164	Dic₈₉	—	8	2	no	—
165	Dic₈₉	—	16	1	no	—
166	Dic₈₉	—	16	1	no	—
167	#9822	—	89	1	no	maximal, Sylow
168	C₂×C₈₉	#1706	1	1	yes	characteristic
169	C₂×C₈₉	#1708	1	1	yes	characteristic
170	C₂×C₈₉	#1709	1	1	yes	characteristic
171	C₂×C₈₉	#1711	1	1	yes	characteristic
172	C₂×C₈₉	#1958	1	1	yes	characteristic
173	C₂×C₈₉	#1968	1	1	yes	characteristic
174	C₂×C₈₉	#1970	1	1	yes	characteristic
175	D₈₉	—	16	2	no	—
176	C₂×#486	—	89	1	no	—
177	#1990	—	89	1	no	—
178	#1957	—	89	1	no	—
179	#1957	—	89	1	no	—
180	#1956	—	89	1	no	—
181	#1834	—	89	1	no	—
182	C₂×C₈²	D₈₉	1	1	yes	characteristic, complemented
183	C₈₉	#9822	1	1	yes	characteristic, complemented, Sylow
184	C₂×C₄⋊Q₈	—	89	1	no	—
185	C₂×#127	—	89	1	no	—
186	#486	—	89	4	no	—
187	#486	—	178	2	no	—
188	C₂×#103	—	178	2	no	—
189	C₂×#102	—	178	2	no	—
190	C₂×#102	—	178	2	no	—
191	C₂×#102	—	178	2	no	—
192	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
193	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
194	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
195	#384	—	89	1	no	—
196	#384	—	89	1	no	—
197	#340	—	178	2	no	—
198	#340	—	178	2	no	—
199	C₈²	—	2	2	no	—
200	C₂²×Q₈	—	178	1	no	—
201	C₂²×D₄	—	178	1	no	—
202	C₂²×D₄	—	178	1	no	—
203	C₂²×D₄	—	178	1	no	—
204	C₂²×C₈	D₃₅₆	1	2	yes	—
205	C₂²×C₈	D₃₅₆	1	2	yes	—
206	C₂²×C₈	D₃₅₆	1	2	yes	—
207	C₄⋊Q₈	—	89	2	no	—
208	C₄⋊Q₈	—	178	1	no	—
209	C₄⋊Q₈	—	178	1	no	—
210	C₄⋊Q₈	—	178	1	no	—
211	#127	—	89	2	no	—
212	#127	—	178	1	no	—
213	#127	—	178	1	no	—
214	#127	—	178	1	no	—
215	C₂×C₄⋊C₄	—	178	1	no	—
216	C₂×C₄⋊C₄	—	178	1	no	—
217	C₂×C₄⋊C₄	—	178	1	no	—
218	C₂×C₄⋊C₄	—	178	1	no	—
219	C₂×C₄⋊C₄	—	178	1	no	—
220	C₂×C₄⋊C₄	—	178	1	no	—
221	C₂×C₄⋊C₄	—	178	1	no	—
222	C₂×C₄⋊C₄	—	178	1	no	—
223	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
224	#103	—	356	4	no	—
225	#102	—	356	4	no	—
226	#102	—	356	4	no	—
227	#102	—	178	8	no	—
228	C₄×C₈	—	2	1	no	—
229	C₄×C₈	—	2	1	no	—
230	C₄×C₈	—	2	1	no	—
231	C₄×C₈	—	2	1	no	—
232	C₄×C₈	C₄×D₈₉	1	2	yes	—
233	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
234	C₂⁴	—	356	1	no	—
235	C₂×Q₈	—	356	1	no	—
236	C₂×Q₈	—	356	2	no	—
237	C₂×Q₈	—	356	2	no	—
238	C₂×Q₈	—	178	2	no	—
239	C₂×D₄	—	356	1	no	—
240	C₂×D₄	—	356	1	no	—
241	C₂×D₄	—	356	1	no	—
242	C₂×D₄	—	356	2	no	—
243	C₂×D₄	—	356	2	no	—
244	C₂×D₄	—	356	2	no	—
245	C₂×D₄	—	356	2	no	—
246	C₂×D₄	—	356	2	no	—
247	C₂×D₄	—	356	2	no	—
248	C₂×D₄	—	178	2	no	—
249	C₂×D₄	—	178	2	no	—
250	C₂×D₄	—	178	2	no	—
251	C₂²×C₄	—	356	1	no	—
252	C₂²×C₄	—	356	1	no	—
253	C₂²×C₄	—	356	1	no	—
254	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
255	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	#13083551	1	4	yes	—
264	C₂×C₈	D₇₁₂	1	4	yes	—
265	C₂×C₈	#13083563	1	4	yes	—
266	C₄⋊C₄	—	356	1	no	—
267	C₄⋊C₄	—	356	1	no	—
268	C₄⋊C₄	—	356	2	no	—
269	C₄⋊C₄	—	356	2	no	—
270	C₄⋊C₄	—	356	2	no	—
271	C₄⋊C₄	—	356	2	no	—
272	C₄⋊C₄	—	356	2	no	—
273	C₄⋊C₄	—	356	2	no	—
274	C₄⋊C₄	—	178	2	no	—
275	C₄⋊C₄	—	178	2	no	—
276	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
277	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
278	C₄²	#13083574	1	1	yes	characteristic
279	C₄²	#13083574	1	1	yes	characteristic
280	C₂³	—	356	2	no	—
281	C₂³	—	712	1	no	—
282	C₂³	—	712	1	no	—
283	C₂³	—	712	1	no	—
284	C₂³	—	712	1	no	—
285	C₂³	—	712	1	no	—
286	C₂³	—	712	1	no	—
287	C₂³	#15668009	1	1	yes	characteristic
288	Q₈	—	356	4	no	—
289	Q₈	—	712	2	no	—
290	D₄	—	356	4	no	—
291	D₄	—	356	4	no	—
292	D₄	—	356	4	no	—
293	D₄	—	712	2	no	—
294	D₄	—	712	2	no	—
295	D₄	—	712	2	no	—
296	C₂×C₄	—	356	2	no	—
297	C₂×C₄	—	712	1	no	—
298	C₂×C₄	—	712	1	no	—
299	C₂×C₄	—	712	1	no	—
300	C₂×C₄	—	712	1	no	—
301	C₂×C₄	—	712	1	no	—
302	C₂×C₄	—	712	1	no	—
303	C₂×C₄	—	712	1	no	—
304	C₂×C₄	—	712	1	no	—
305	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
306	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
307	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
308	C₂×C₄	#15667986	1	1	yes	characteristic
309	C₂×C₄	#15667986	1	1	yes	characteristic
310	C₂×C₄	#15668000	1	1	yes	characteristic
311	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
312	C₂×C₄	C₂×#13083563	1	2	yes	—
313	C₂×C₄	#15668034	1	2	yes	—
314	C₂×C₄	#15668034	1	2	yes	—
315	C₂×C₄	#15668034	1	2	yes	—
316	C₂×C₄	#15668034	1	2	yes	—
317	C₈	—	2	4	no	—
318	C₈	—	2	4	no	—
319	C₈	—	2	4	no	—
320	C₂²	—	1424	1	no	—
321	C₂²	—	1424	1	no	—
322	C₂²	—	1424	1	no	—
323	C₂²	—	1424	1	no	—
324	C₂²	—	712	2	no	—
325	C₂²	—	712	2	no	—
326	C₂²	—	712	2	no	—
327	C₂²	#19151500	1	1	yes	characteristic
328	C₂²	#19151971	1	1	yes	characteristic
329	C₂²	#19151975	1	1	yes	characteristic
330	C₂²	#19151982	1	1	yes	characteristic
331	C₂²	#19151982	1	1	yes	characteristic
332	C₂²	#19151982	1	1	yes	characteristic
333	C₂²	#19151982	1	1	yes	characteristic
334	C₄	—	1424	1	no	—
335	C₄	—	1424	1	no	—
336	C₄	—	712	2	no	—
337	C₄	#19151595	1	2	yes	—
338	C₄	C₄×D₇₁₂	1	2	yes	—
339	C₄	C₄×#13083563	1	2	yes	—
340	C₄	#19151827	1	2	yes	—
341	C₄	#19151834	1	2	yes	—
342	C₄	#19151953	1	2	yes	—
343	C₂	—	1424	2	no	—
344	C₂	#26019844	1	1	yes	characteristic
345	C₂	#26019853	1	1	yes	characteristic
346	C₂	#26019858	1	1	yes	characteristic
347	C₂	#26019864	1	1	yes	characteristic
348	C₂	#26021063	1	1	yes	characteristic
349	C₂	#26021106	1	1	yes	characteristic
350	C₂	#26021118	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15668009
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₂×C₈²×C₈₉	C₂
socle	C₂³×C₈₉	#127
prime radical	C₂×#15668009	C₂²
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂×C₈²	D₈₉
hypocenter	C₈₉	#9822
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
2. #26021213	C₂	semidirect
5. #26021054	C₂	semidirect
6. #26020380	C₂	semidirect
7. C₂×C₈²×C₈₉	C₂	semidirect
11. #19151945	C₄	semidirect
182. C₂×C₈²	D₈₉	semidirect
183. C₈₉	#9822	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151945	2
#26021213	2
#26021058	2
#26021054	2
#26020380	2
C₂×C₈²×C₈₉	2
#9822	89