Subgroups — #36448919

Order 22784.

426 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26028998	C₂	1	1	yes	maximal, characteristic
2	#26021027	C₂	1	1	yes	maximal, characteristic, complemented
3	#26019023	C₂	1	1	yes	maximal, characteristic, complemented
4	#26018899	C₂	1	1	yes	maximal, characteristic
5	#26018704	C₂	1	1	yes	maximal, characteristic
6	#26018087	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15668020	—	2	1	no	—
9	#19152148	C₂²	1	1	yes	characteristic
10	#19151677	—	2	1	no	—
11	#19151671	—	2	2	no	—
12	C₈×D₃₅₆	—	2	1	no	—
13	C₂×#15667947	—	2	1	no	—
14	#19151622	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
17	#19151566	—	2	1	no	—
18	#19151362	—	2	1	no	—
19	#19151349	—	2	1	no	—
20	#19151349	—	2	1	no	—
21	#19151343	C₄	1	1	yes	characteristic
22	#19151343	C₄	1	1	yes	characteristic
23	#19151341	C₄	1	1	yes	characteristic
24	#19151341	C₄	1	1	yes	characteristic
25	#19151327	—	2	2	no	—
26	#431×C₈₉	C₂²	1	1	yes	characteristic
27	C₄×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
29	C₂×#13083574	—	2	1	no	—
30	#15668020	—	4	2	no	—
31	#15668020	—	4	2	no	—
32	C₂×#13083558	—	2	1	no	—
33	C₂×#13083557	—	2	2	no	—
34	C₂²×#10623494	—	2	1	no	—
35	C₂²×#10623494	—	2	1	no	—
36	C₂×C₈×D₈₉	—	2	2	no	—
37	#15668012	C₂×C₄	1	1	yes	characteristic
38	#15668009	C₂×C₄	1	1	yes	characteristic
39	#15668000	—	2	1	no	—
40	#15667986	—	2	1	no	—
41	#15667986	Q₈	1	2	yes	—
42	C₄×D₃₅₆	—	2	1	no	—
43	C₄×D₃₅₆	D₄	1	2	yes	—
44	#15667947	—	4	2	no	—
45	C₄⋊#10623494	—	2	1	no	—
46	C₄⋊#10623494	—	2	1	no	—
47	C₄⋊#10623494	—	2	1	no	—
48	C₄⋊#10623494	—	2	1	no	—
49	C₄⋊#10623494	C₂×C₄	1	1	yes	characteristic
50	C₄⋊#10623494	C₂×C₄	1	1	yes	characteristic
51	C₄⋊#10623494	D₄	1	1	yes	characteristic
52	C₄⋊#10623494	D₄	1	1	yes	characteristic
53	#15667924	—	2	2	no	—
54	#15667924	—	2	2	no	—
55	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
56	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
57	C₂²×C₈×C₈₉	—	2	1	no	—
58	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
59	C₄⋊C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
60	C₄⋊C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
61	C₂²⋊C₈×C₈₉	—	2	1	no	—
62	#97×C₈₉	—	2	1	no	—
63	C₄×C₈×C₈₉	—	2	1	no	—
64	C₄×C₈×C₈₉	—	2	1	no	—
65	C₄×C₈×C₈₉	—	2	1	no	—
66	C₄×C₈×C₈₉	—	2	1	no	—
67	#13083574	—	4	1	no	—
68	#13083574	—	4	1	no	—
69	#13083574	—	4	1	no	—
70	#13083574	—	4	1	no	—
71	C₂×Dic₁₇₈	—	2	1	no	—
72	C₂×D₃₅₆	—	2	1	no	—
73	C₂×D₃₅₆	—	2	1	no	—
74	C₂×D₃₅₆	C₄⋊C₄	1	2	yes	—
75	C₂×C₂²⋊D₈₉	—	2	2	no	—
76	C₂×C₄×D₈₉	—	2	2	no	—
77	#13083558	—	4	1	no	—
78	#13083558	—	4	1	no	—
79	#13083557	—	4	4	no	—
80	C₂×#10623494	—	2	1	no	—
81	C₂×#10623494	—	2	1	no	—
82	C₂×#10623494	—	2	1	no	—
83	C₂×#10623494	—	2	1	no	—
84	C₂×#10623494	—	4	1	no	—
85	C₂×#10623494	—	4	1	no	—
86	C₂×#10623494	—	4	1	no	—
87	C₂×#10623494	—	4	1	no	—
88	C₂×#10623494	—	4	1	no	—
89	C₈×D₈₉	—	4	4	no	—
90	C₄⋊Dic₈₉	—	2	1	no	—
91	C₄⋊Dic₈₉	C₄⋊C₄	1	2	yes	—
92	#13083551	—	2	2	no	—
93	#13083551	—	2	2	no	—
94	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
96	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
97	M₁₆×C₈₉	—	2	1	no	—
98	M₁₆×C₈₉	—	2	1	no	—
99	M₁₆×C₈₉	—	2	1	no	—
100	M₁₆×C₈₉	SD₁₆	1	2	yes	—
101	C₂×C₈×C₈₉	—	2	1	no	—
102	C₂×C₈×C₈₉	—	2	2	no	—
103	C₂×C₈×C₈₉	—	2	1	no	—
104	C₂×C₈×C₈₉	—	2	1	no	—
105	C₂×C₈×C₈₉	—	2	1	no	—
106	C₂×C₈×C₈₉	—	4	1	no	—
107	C₂×C₈×C₈₉	—	4	1	no	—
108	C₄²×C₈₉	C₂²×C₄	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	#41	1	1	yes	characteristic
112	C₂³×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂²×D₈₉	—	2	2	no	—
114	C₂×C₄×C₈₉	—	2	1	no	—
115	C₂×C₄×C₈₉	—	2	1	no	—
116	C₂×C₄×C₈₉	—	4	1	no	—
117	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
123	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
125	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
126	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂²⋊D₈₉	—	4	2	no	—
128	C₂²⋊D₈₉	—	4	2	no	—
129	C₂×Dic₈₉	—	2	2	no	—
130	D₃₅₆	—	4	2	no	—
131	D₃₅₆	—	4	1	no	—
132	D₃₅₆	—	4	1	no	—
133	D₃₅₆	—	4	1	no	—
134	D₃₅₆	—	4	1	no	—
135	C₄×D₈₉	—	4	2	no	—
136	C₄×D₈₉	—	4	2	no	—
137	Dic₁₇₈	—	4	1	no	—
138	Dic₁₇₈	—	4	1	no	—
139	C₈×C₈₉	—	2	2	no	—
140	C₈×C₈₉	—	4	2	no	—
141	C₈×C₈₉	—	4	1	no	—
142	C₈×C₈₉	—	4	1	no	—
143	C₈×C₈₉	—	4	1	no	—
144	#10623494	—	4	1	no	—
145	#10623494	—	4	1	no	—
146	#10623494	—	4	1	no	—
147	#10623494	—	4	1	no	—
148	#10623494	—	8	1	no	—
149	C₂²×C₈₉	—	2	1	no	—
150	C₂²×C₈₉	—	2	1	no	—
151	C₂²×C₈₉	#385	1	1	yes	characteristic
152	C₂²×C₈₉	#476	1	1	yes	characteristic
153	C₂²×C₈₉	#478	1	1	yes	characteristic
154	C₂×D₈₉	—	4	2	no	—
155	C₂×D₈₉	—	4	2	no	—
156	C₂×D₈₉	—	4	2	no	—
157	C₄×C₈₉	—	2	1	no	—
158	C₄×C₈₉	—	2	1	no	—
159	C₄×C₈₉	—	2	1	no	—
160	C₄×C₈₉	C₄×SD₁₆	1	1	yes	characteristic
161	C₄×C₈₉	#442	1	1	yes	characteristic
162	C₄×C₈₉	#443	1	1	yes	characteristic
163	C₄×C₈₉	#444	1	1	yes	characteristic
164	C₄×C₈₉	#449	1	1	yes	characteristic
165	C₄×C₈₉	#450	1	1	yes	characteristic
166	Dic₈₉	—	4	2	no	—
167	#8932	—	89	1	no	maximal, Sylow
168	C₂×C₈₉	—	2	1	no	—
169	C₂×C₈₉	—	2	1	no	—
170	C₂×C₈₉	#1858	1	1	yes	characteristic
171	C₂×C₈₉	#1864	1	1	yes	characteristic
172	C₂×C₈₉	#1866	1	1	yes	characteristic
173	D₈₉	—	8	2	no	—
174	#2935	—	89	1	no	—
175	#1950	—	89	1	no	—
176	#1570	—	89	1	no	—
177	#1540	—	89	1	no	—
178	#1492	—	89	1	no	—
179	#1278	D₈₉	1	1	yes	characteristic, complemented
180	#1277	—	89	1	no	—
181	C₈₉	#8932	1	1	yes	characteristic, complemented, Sylow
182	C₂×#131	—	178	1	no	—
183	#532	—	89	1	no	—
184	#436	—	178	1	no	—
185	#435	—	178	2	no	—
186	C₈×D₄	—	178	1	no	—
187	#431	C₂×D₈₉	1	1	yes	characteristic
188	C₂×#108	—	178	1	no	—
189	#423	—	89	1	no	—
190	C₂×C₄⋊C₈	—	89	1	no	—
191	C₂×C₄⋊C₈	—	89	1	no	—
192	#408	—	178	1	no	—
193	C₄×M₁₆	C₂×D₈₉	1	1	yes	characteristic
194	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
195	#341	—	178	1	no	—
196	#334	—	178	1	no	—
197	#334	—	178	1	no	—
198	#332	—	89	1	no	—
199	#332	—	89	1	no	—
200	#331	—	89	1	no	—
201	#331	—	89	1	no	—
202	D₄⋊C₈	—	178	2	no	—
203	C₂×#41	—	178	1	no	—
204	#131	—	356	2	no	—
205	#131	—	356	2	no	—
206	C₂×M₁₆	—	178	2	no	—
207	C₂×M₁₆	—	178	1	no	—
208	C₂×M₁₆	D₃₅₆	1	1	yes	characteristic
209	C₂×M₁₆	D₃₅₆	1	1	yes	characteristic
210	C₂²×C₈	—	2	1	no	—
211	C₂²×C₈	—	178	2	no	—
212	C₂²×C₈	—	178	1	no	—
213	C₂²×C₈	—	178	1	no	—
214	C₄⋊Q₈	—	89	1	no	—
215	#127	—	89	1	no	—
216	#124	—	178	1	no	—
217	#121	—	89	2	no	—
218	#121	—	178	1	no	—
219	C₄×D₄	—	89	2	no	—
220	C₄×D₄	—	178	1	no	—
221	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
222	#108	—	356	2	no	—
223	C₄⋊C₈	—	89	1	no	—
224	C₄⋊C₈	—	89	1	no	—
225	C₄⋊C₈	—	89	1	no	—
226	C₄⋊C₈	—	89	1	no	—
227	C₄⋊C₈	—	178	1	no	—
228	C₄⋊C₈	—	178	1	no	—
229	C₄⋊C₈	—	178	1	no	—
230	C₄⋊C₈	—	178	1	no	—
231	C₄⋊C₈	C₄×D₈₉	1	1	yes	characteristic
232	C₄⋊C₈	C₄×D₈₉	1	1	yes	characteristic
233	C₂²⋊C₈	—	2	1	no	—
234	C₂²⋊C₈	—	178	2	no	—
235	C₂²⋊C₈	—	178	2	no	—
236	#97	—	2	1	no	—
237	C₄×C₈	—	2	1	no	—
238	C₄×C₈	—	2	1	no	—
239	C₄×C₈	—	2	1	no	—
240	C₄×C₈	—	2	1	no	—
241	#41	—	356	1	no	—
242	#41	—	356	1	no	—
243	#41	—	356	1	no	—
244	#41	—	356	1	no	—
245	C₂×Q₈	—	178	1	no	—
246	C₂×D₄	—	89	2	no	—
247	C₂×D₄	—	178	2	no	—
248	C₂×D₄	—	178	1	no	—
249	C₂×D₄	—	178	1	no	—
250	C₂²×C₄	—	178	2	no	—
251	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
252	C₂²×C₄	Q₈×D₈₉	1	1	yes	characteristic
253	C₂²×C₄	#13083573	1	1	yes	characteristic
254	M₁₆	—	2	1	no	—
255	M₁₆	—	2	1	no	—
256	M₁₆	—	2	1	no	—
257	M₁₆	—	356	4	no	—
258	M₁₆	—	356	1	no	—
259	M₁₆	—	356	1	no	—
260	M₁₆	#13083563	1	2	yes	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	4	1	no	—
267	C₂×C₈	—	4	1	no	—
268	C₂×C₈	—	356	4	no	—
269	C₂×C₈	—	356	1	no	—
270	C₂×C₈	—	356	1	no	—
271	C₂×C₈	—	356	1	no	—
272	C₂×C₈	—	356	1	no	—
273	C₂×C₈	—	356	1	no	—
274	C₂×C₈	—	178	1	no	—
275	C₂×C₈	—	178	1	no	—
276	C₂×C₈	—	178	1	no	—
277	C₂×C₈	—	178	1	no	—
278	C₄⋊C₄	—	89	2	no	—
279	C₄⋊C₄	—	178	1	no	—
280	C₂²⋊C₄	—	178	2	no	—
281	C₂²⋊C₄	—	178	2	no	—
282	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
283	C₄²	D₄×D₈₉	1	1	yes	characteristic
284	C₄²	D₄×D₈₉	1	1	yes	characteristic
285	C₄²	#13083574	1	1	yes	characteristic
286	C₂³	—	178	2	no	—
287	C₂³	#15667992	1	1	yes	characteristic
288	Q₈	—	356	1	no	—
289	Q₈	—	356	1	no	—
290	D₄	—	356	2	no	—
291	D₄	—	356	2	no	—
292	D₄	—	356	2	no	—
293	D₄	—	356	1	no	—
294	D₄	—	356	1	no	—
295	D₄	—	356	1	no	—
296	D₄	—	356	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	356	2	no	—
301	C₂×C₄	—	356	2	no	—
302	C₂×C₄	—	178	2	no	—
303	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
304	C₂×C₄	#15667972	1	1	yes	characteristic
305	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
306	C₂×C₄	#15667982	1	1	yes	characteristic
307	C₂×C₄	#15667990	1	1	yes	characteristic
308	C₂×C₄	#15667994	1	1	yes	characteristic
309	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
310	C₂×C₄	#15668034	1	1	yes	characteristic
311	C₂×C₄	#15668040	1	1	yes	characteristic
312	C₂×C₄	#15668040	1	1	yes	characteristic
313	C₈	—	2	2	no	—
314	C₈	—	4	2	no	—
315	C₈	—	4	1	no	—
316	C₈	—	4	1	no	—
317	C₈	—	4	1	no	—
318	C₈	—	356	1	no	—
319	C₈	—	356	1	no	—
320	C₈	—	356	1	no	—
321	C₈	—	356	1	no	—
322	C₈	—	712	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	356	2	no	—
326	C₂²	—	356	2	no	—
327	C₂²	—	356	2	no	—
328	C₂²	#19151472	1	1	yes	characteristic
329	C₂²	#19151889	1	1	yes	characteristic
330	C₂²	#19151899	1	1	yes	characteristic
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	356	2	no	—
335	C₄	C₄×#13083563	1	1	yes	characteristic
336	C₄	#19151711	1	1	yes	characteristic
337	C₄	#19151712	1	1	yes	characteristic
338	C₄	#19151717	1	1	yes	characteristic
339	C₄	#19151745	1	1	yes	characteristic
340	C₄	#19151751	1	1	yes	characteristic
341	C₂	—	2	1	no	—
342	C₂	—	2	1	no	—
343	C₂	—	712	2	no	—
344	C₂	#26020473	1	1	yes	characteristic
345	C₂	#26020516	1	1	yes	characteristic
346	C₂	#26020528	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#15667982
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1278	C₂
socle	C₂²×C₈₉	#385
prime radical	#19152148	C₂²
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1278	D₈₉
hypocenter	C₈₉	#8932
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1278

Complements

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

N	H ≅ G/N	type
2. #26021027	C₂	semidirect
3. #26019023	C₂	semidirect
6. #26018087	C₂	semidirect
7. C₈₉×#1278	C₂	semidirect
179. #1278	D₈₉	semidirect
181. C₈₉	#8932	semidirect

Maximal subgroups

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

subgroup	index
#26028998	2
#26021027	2
#26019023	2
#26018899	2
#26018704	2
#26018087	2
C₈₉×#1278	2
#8932	89