Subgroups — #36453774

Order 22784.

514 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030680	C₂	1	1	yes	maximal, characteristic
2	#26019000	C₂	1	1	yes	maximal, characteristic, complemented
3	#26018746	C₂	1	2	yes	maximal, complemented
4	#26018650	C₂	1	2	yes	maximal, complemented
5	C₈×C₄⋊C₄×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15668034	—	2	1	no	—
7	C₂×#15668034	—	2	1	no	—
8	#19152148	C₂²	1	2	yes	—
9	#19151982	—	2	1	no	—
10	#19151982	—	2	1	no	—
11	#19151978	C₄	1	1	yes	characteristic, complemented
12	#19151975	C₄	1	1	yes	characteristic, complemented
13	#19151971	C₄	1	2	yes	complemented
14	#19151651	C₂²	1	2	yes	—
15	#19151351	—	2	1	no	—
16	#19151349	—	2	1	no	—
17	#19151331	—	2	4	no	—
18	#19151325	—	2	4	no	—
19	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₈₉	—	2	1	no	—
21	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
23	#336×C₈₉	—	2	1	no	—
24	C₂×#13083574	—	2	2	no	—
25	#15668034	—	4	2	no	—
26	#15668034	—	4	2	no	—
27	#15668034	—	4	2	no	—
28	#15668034	—	4	2	no	—
29	C₂×Dic₃₅₆	—	2	1	no	—
30	C₂×Dic₃₅₆	—	2	1	no	—
31	C₂×#13083563	—	2	2	no	—
32	C₂×#13083563	—	2	2	no	—
33	C₂×D₇₁₂	—	2	1	no	—
34	C₂×D₇₁₂	—	2	1	no	—
35	#15668012	C₂×C₄	1	2	yes	—
36	#15668009	C₂×C₄	1	2	yes	—
37	#15668000	—	2	2	no	—
38	#15667986	—	2	2	no	—
39	#15667986	—	2	2	no	—
40	C₄×D₃₅₆	—	2	2	no	—
41	C₄×D₃₅₆	—	2	2	no	—
42	C₄⋊#10623494	—	2	2	no	—
43	C₂²⋊#10623494	—	2	2	no	—
44	C₄×#10623494	—	2	2	no	—
45	C₂²×C₈×C₈₉	—	2	1	no	—
46	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
49	C₂×C₄²×C₈₉	—	2	1	no	—
50	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
51	C₄⋊C₈×C₈₉	—	2	1	no	—
52	C₄⋊C₈×C₈₉	D₄	1	2	yes	—
53	C₄×C₈×C₈₉	—	4	2	no	—
54	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
57	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
58	#95×C₈₉	D₄	1	2	yes	—
59	#13083574	—	4	2	no	—
60	#13083574	—	4	2	no	—
61	#13083574	—	4	2	no	—
62	#13083574	—	4	2	no	—
63	C₂×Dic₁₇₈	—	2	2	no	—
64	C₂×D₃₅₆	—	2	2	no	—
65	C₂×D₃₅₆	—	2	2	no	—
66	C₂×D₃₅₆	—	2	2	no	—
67	C₂×C₂²⋊D₈₉	—	4	2	no	—
68	C₂×C₄×D₈₉	—	4	2	no	—
69	Dic₃₅₆	—	4	2	no	—
70	Dic₃₅₆	—	4	2	no	—
71	#13083563	—	4	4	no	—
72	#13083563	—	4	4	no	—
73	D₇₁₂	—	4	2	no	—
74	D₇₁₂	—	4	2	no	—
75	C₂×#10623494	—	4	2	no	—
76	C₄⋊Dic₈₉	—	2	2	no	—
77	C₄⋊Dic₈₉	—	2	2	no	—
78	#13083551	—	4	2	no	—
79	#13083551	—	4	2	no	—
80	C₂²×C₄×C₈₉	—	2	2	no	—
81	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
84	C₂×C₈×C₈₉	—	2	1	no	—
85	C₂×C₈×C₈₉	—	2	1	no	—
86	C₂×C₈×C₈₉	—	2	2	no	—
87	C₂×C₈×C₈₉	—	4	1	no	—
88	C₂×C₈×C₈₉	—	4	1	no	—
89	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₈×C₈₉	D₈	1	2	yes	—
94	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
95	C₄⋊C₄×C₈₉	—	2	4	no	—
96	C₄²×C₈₉	—	4	1	no	—
97	C₄²×C₈₉	—	4	1	no	—
98	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
99	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₈₉	#41	1	1	yes	characteristic
102	C₂³×C₈₉	#121	1	1	yes	characteristic
103	C₂²×D₈₉	—	4	2	no	—
104	C₂×C₄×C₈₉	—	2	2	no	—
105	C₂×C₄×C₈₉	—	2	4	no	—
106	C₂×C₄×C₈₉	—	4	1	no	—
107	C₂×C₄×C₈₉	—	4	2	no	—
108	C₂×C₄×C₈₉	—	4	2	no	—
109	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
113	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
114	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
115	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
117	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
118	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₂²⋊D₈₉	—	8	2	no	—
120	C₂²⋊D₈₉	—	8	2	no	—
121	C₂×Dic₈₉	—	4	2	no	—
122	D₃₅₆	—	4	2	no	—
123	D₃₅₆	—	4	2	no	—
124	D₃₅₆	—	4	4	no	—
125	D₃₅₆	—	8	2	no	—
126	C₄×D₈₉	—	8	2	no	—
127	C₄×D₈₉	—	8	2	no	—
128	Dic₁₇₈	—	4	2	no	—
129	Dic₁₇₈	—	4	2	no	—
130	C₈×C₈₉	—	2	1	no	—
131	C₈×C₈₉	—	2	1	no	—
132	C₈×C₈₉	—	4	2	no	—
133	C₈×C₈₉	#102	1	2	yes	—
134	C₈×C₈₉	#102	1	2	yes	—
135	#10623494	—	8	2	no	—
136	C₂²×C₈₉	—	2	1	no	—
137	C₂²×C₈₉	—	2	1	no	—
138	C₂²×C₈₉	#390	1	1	yes	characteristic
139	C₂²×C₈₉	#459	1	1	yes	characteristic
140	C₂²×C₈₉	#460	1	1	yes	characteristic
141	C₂×D₈₉	—	8	2	no	—
142	C₂×D₈₉	—	8	2	no	—
143	C₂×D₈₉	—	8	2	no	—
144	C₄×C₈₉	—	2	1	no	—
145	C₄×C₈₉	—	2	1	no	—
146	C₄×C₈₉	—	2	1	no	—
147	C₄×C₈₉	—	4	4	no	—
148	C₄×C₈₉	C₂×#102	1	1	yes	characteristic
149	C₄×C₈₉	#418	1	1	yes	characteristic
150	C₄×C₈₉	#443	1	2	yes	—
151	C₄×C₈₉	#459	1	1	yes	characteristic
152	C₄×C₈₉	#460	1	1	yes	characteristic
153	Dic₈₉	—	8	2	no	—
154	#9750	—	89	1	no	maximal, Sylow
155	C₂×C₈₉	—	2	1	no	—
156	C₂×C₈₉	—	2	1	no	—
157	C₂×C₈₉	#1974	1	1	yes	characteristic
158	C₂×C₈₉	#1976	1	1	yes	characteristic
159	C₂×C₈₉	#1982	1	1	yes	characteristic
160	D₈₉	—	16	2	no	—
161	#3147	—	89	1	no	—
162	C₈×C₄⋊C₄	D₈₉	1	1	yes	characteristic, complemented
163	#1564	—	89	1	no	—
164	#1502	—	89	2	no	—
165	#1480	—	89	2	no	—
166	C₈₉	#9750	1	1	yes	characteristic, complemented, Sylow
167	C₂×#135	—	178	1	no	—
168	C₂×#135	—	178	1	no	—
169	#532	—	89	2	no	—
170	#495	—	178	1	no	—
171	#495	—	178	1	no	—
172	#494	—	89	1	no	—
173	#493	—	89	1	no	—
174	#492	—	89	2	no	—
175	#431	—	89	2	no	—
176	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
177	C₂×C₄×C₈	—	2	1	no	—
178	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
179	C₄×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
180	#336	—	2	1	no	—
181	#335	—	178	1	no	—
182	#334	—	178	1	no	—
183	#327	—	178	4	no	—
184	D₄⋊C₈	—	178	4	no	—
185	C₂×#41	—	178	2	no	—
186	#135	—	356	2	no	—
187	#135	—	356	2	no	—
188	#135	—	356	2	no	—
189	#135	—	356	2	no	—
190	C₂×Q₁₆	—	178	1	no	—
191	C₂×Q₁₆	—	178	1	no	—
192	C₂×SD₁₆	—	178	2	no	—
193	C₂×SD₁₆	—	178	2	no	—
194	C₂×D₈	—	178	1	no	—
195	C₂×D₈	—	178	1	no	—
196	C₂²×C₈	—	2	1	no	—
197	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
198	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
199	C₄⋊Q₈	—	89	2	no	—
200	#127	—	89	2	no	—
201	#124	—	178	2	no	—
202	#121	—	178	2	no	—
203	#121	—	178	2	no	—
204	C₄×D₄	—	178	2	no	—
205	C₄×D₄	—	178	2	no	—
206	C₂×C₄⋊C₄	—	2	1	no	—
207	C₂×C₄²	—	2	1	no	—
208	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
209	C₄⋊C₈	—	2	1	no	—
210	C₄⋊C₈	—	178	2	no	—
211	C₄⋊C₈	D₃₅₆	1	2	yes	—
212	C₂²⋊C₈	—	178	2	no	—
213	C₄×C₈	—	4	2	no	—
214	C₄×C₈	—	178	2	no	—
215	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
216	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
217	C₄×C₈	D₃₅₆	1	1	yes	characteristic
218	C₄×C₈	D₃₅₆	1	1	yes	characteristic
219	#95	D₃₅₆	1	2	yes	—
220	#41	—	356	2	no	—
221	#41	—	356	2	no	—
222	#41	—	356	2	no	—
223	#41	—	356	2	no	—
224	C₂×Q₈	—	178	2	no	—
225	C₂×D₄	—	356	2	no	—
226	C₂×D₄	—	178	2	no	—
227	C₂×D₄	—	178	2	no	—
228	C₂×D₄	—	178	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	356	2	no	—
231	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
233	C₂²×C₄	#13083574	1	1	yes	characteristic
234	Q₁₆	—	356	2	no	—
235	Q₁₆	—	356	2	no	—
236	SD₁₆	—	356	4	no	—
237	SD₁₆	—	356	4	no	—
238	D₈	—	356	2	no	—
239	D₈	—	356	2	no	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	4	1	no	—
244	C₂×C₈	—	4	1	no	—
245	C₂×C₈	—	356	2	no	—
246	C₂×C₈	#13083551	1	1	yes	characteristic
247	C₂×C₈	#13083551	1	1	yes	characteristic
248	C₂×C₈	#13083551	1	1	yes	characteristic
249	C₂×C₈	#13083551	1	1	yes	characteristic
250	C₂×C₈	#13083559	1	2	yes	—
251	C₂×C₈	#13083561	1	2	yes	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	178	2	no	—
254	C₄⋊C₄	—	178	2	no	—
255	C₂²⋊C₄	—	356	2	no	—
256	C₂²⋊C₄	—	356	2	no	—
257	C₄²	—	4	1	no	—
258	C₄²	—	4	1	no	—
259	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
260	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
261	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
262	C₄²	#13083574	1	1	yes	characteristic
263	C₂³	—	356	2	no	—
264	C₂³	#15667986	1	1	yes	characteristic
265	Q₈	—	356	2	no	—
266	Q₈	—	356	2	no	—
267	D₄	—	356	4	no	—
268	D₄	—	356	2	no	—
269	D₄	—	356	2	no	—
270	D₄	—	712	2	no	—
271	D₄	—	712	2	no	—
272	D₄	—	712	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	356	2	no	—
279	C₂×C₄	—	712	2	no	—
280	C₂×C₄	—	712	2	no	—
281	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
282	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
283	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
284	C₂×C₄	#15667986	1	1	yes	characteristic
285	C₂×C₄	#15668000	1	1	yes	characteristic
286	C₂×C₄	#15668009	1	1	yes	characteristic
287	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
288	C₂×C₄	C₂×#13083561	1	1	yes	characteristic
289	C₂×C₄	#15668037	1	1	yes	characteristic
290	C₂×C₄	#15668037	1	1	yes	characteristic
291	C₈	—	2	1	no	—
292	C₈	—	2	1	no	—
293	C₈	—	4	2	no	—
294	C₈	—	712	2	no	—
295	C₈	#15667932	1	2	yes	—
296	C₈	#15667932	1	2	yes	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	712	2	no	—
300	C₂²	—	712	2	no	—
301	C₂²	—	712	2	no	—
302	C₂²	#19151500	1	1	yes	characteristic
303	C₂²	#19151793	1	1	yes	characteristic
304	C₂²	#19151798	1	1	yes	characteristic
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	4	4	no	—
309	C₄	—	712	2	no	—
310	C₄	C₂×#15667932	1	1	yes	characteristic
311	C₄	#19151602	1	1	yes	characteristic
312	C₄	#19151716	1	2	yes	—
313	C₄	#19151793	1	1	yes	characteristic
314	C₄	#19151798	1	1	yes	characteristic
315	C₂	—	2	1	no	—
316	C₂	—	2	1	no	—
317	C₂	—	1424	2	no	—
318	C₂	#26021140	1	1	yes	characteristic
319	C₂	#26021150	1	1	yes	characteristic
320	C₂	#26021179	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#15667986
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈×C₄⋊C₄×C₈₉	C₂
socle	C₂²×C₈₉	#390
prime radical	#26030680	C₂
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₈×C₄⋊C₄	D₈₉
hypocenter	C₈₉	#9750
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. #26019000	C₂	semidirect
3. #26018746	C₂	semidirect
4. #26018650	C₂	semidirect
5. C₈×C₄⋊C₄×C₈₉	C₂	semidirect
11. #19151978	C₄	semidirect
12. #19151975	C₄	semidirect
13. #19151971	C₄	semidirect
162. C₈×C₄⋊C₄	D₈₉	semidirect
166. C₈₉	#9750	semidirect

Maximal subgroups

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

subgroup	index
#26030680	2
#26019000	2
#26018746	2
#26018650	2
C₈×C₄⋊C₄×C₈₉	2
#9750	89