Subgroups — #36708118

Order 22784.

532 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#13083574	C₂	1	1	yes	maximal, characteristic
2	#26022154	C₂	1	2	yes	maximal, complemented
3	#26022083	C₂	1	2	yes	maximal, complemented
4	#26022082	C₂	1	4	yes	maximal, complemented
5	#26022051	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022050	C₂	1	2	yes	maximal, complemented
7	#26022006	C₂	1	2	yes	maximal, complemented
8	C₈₉×#2150	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#13083574	C₄	1	1	yes	characteristic
10	C₂×#15667951	C₂²	1	4	yes	—
11	C₈×Dic₁₇₈	C₄	1	1	yes	characteristic
12	C₈×Dic₁₇₈	C₂²	1	1	yes	characteristic
13	C₈×D₃₅₆	C₄	1	1	yes	characteristic
14	C₈×D₃₅₆	C₂²	1	1	yes	characteristic
15	C₈×C₂²⋊D₈₉	C₄	1	2	yes	—
16	C₈×C₂²⋊D₈₉	C₂²	1	2	yes	—
17	#19151651	C₄	1	1	yes	characteristic
18	#19151649	C₄	1	2	yes	—
19	C₄×C₈×D₈₉	C₂²	1	2	yes	—
20	C₄⋊#13083549	C₂²	1	1	yes	characteristic
21	C₄⋊#13083549	C₂²	1	1	yes	characteristic
22	#19151416	C₂²	1	4	yes	—
23	C₂²⋊#13083549	C₂²	1	2	yes	—
24	#19151377	C₂²	1	4	yes	—
25	C₄×#13083549	C₂²	1	2	yes	—
26	C₁₆×Dic₈₉	C₂²	1	4	yes	—
27	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
29	C₄⋊C₁₆×C₈₉	C₂²	1	2	yes	—
30	C₄⋊C₁₆×C₈₉	C₂²	1	2	yes	—
31	C₂×#13083574	C₈	1	1	yes	characteristic
32	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
33	C₂×C₈×D₈₉	C₂³	1	2	yes	—
34	C₄×Dic₁₇₈	C₈	1	1	yes	characteristic
35	C₄×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
36	C₄×D₃₅₆	C₈	1	1	yes	characteristic
37	C₄×D₃₅₆	C₂×C₄	1	1	yes	characteristic
38	C₄×C₂²⋊D₈₉	C₈	1	2	yes	—
39	C₄×C₂²⋊D₈₉	C₂×C₄	1	2	yes	—
40	#15667971	C₈	1	1	yes	characteristic
41	#15667969	C₈	1	2	yes	—
42	C₄²×D₈₉	C₂×C₄	1	2	yes	—
43	#15667951	—	4	4	no	—
44	C₂×#13083549	C₂³	1	2	yes	—
45	C₂×#13083549	C₂³	1	2	yes	—
46	C₄⋊#10623494	C₂×C₄	1	1	yes	characteristic
47	C₄⋊#10623494	C₂×C₄	1	1	yes	characteristic
48	#15667941	C₂×C₄	1	2	yes	—
49	#15667941	C₂×C₄	1	2	yes	—
50	C₂²⋊#10623494	C₂×C₄	1	2	yes	—
51	#15667924	C₂×C₄	1	2	yes	—
52	#15667924	C₂×C₄	1	2	yes	—
53	C₄×#10623494	C₂×C₄	1	2	yes	—
54	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
55	C₈×Dic₈₉	C₂³	1	2	yes	—
56	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
59	M₃₂×C₈₉	—	4	2	no	—
60	C₂×C₁₆×C₈₉	C₂³	1	4	yes	—
61	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
64	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
65	#13083574	—	2	4	no	—
66	C₂×Dic₁₇₈	C₂×C₈	1	1	yes	characteristic
67	C₂×D₃₅₆	C₂×C₈	1	1	yes	characteristic
68	C₂×C₂²⋊D₈₉	C₂×C₈	1	2	yes	—
69	C₂×C₄×D₈₉	C₂×C₈	1	2	yes	—
70	C₂×C₄×D₈₉	C₂×C₈	1	2	yes	—
71	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
72	C₂×#10623494	C₂²×C₄	1	2	yes	—
73	C₂×#10623494	C₂²×C₄	1	2	yes	—
74	C₈×D₈₉	—	2	2	no	—
75	C₈×D₈₉	—	2	2	no	—
76	C₈×D₈₉	—	4	2	no	—
77	C₄⋊Dic₈₉	C₂×C₈	1	1	yes	characteristic
78	C₄⋊Dic₈₉	C₂×C₈	1	1	yes	characteristic
79	#13083553	C₂×C₈	1	2	yes	—
80	#13083553	C₂×C₈	1	2	yes	—
81	C₂²⋊Dic₈₉	C₂×C₈	1	2	yes	—
82	#13083551	C₂×C₈	1	2	yes	—
83	#13083551	C₂×C₈	1	2	yes	—
84	C₄×Dic₈₉	C₂×C₈	1	2	yes	—
85	C₄×Dic₈₉	C₂×C₈	1	2	yes	—
86	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
87	#13083549	—	2	2	no	—
88	#13083549	—	2	2	no	—
89	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
90	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
91	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₈₉	—	2	1	no	—
93	C₂×C₈×C₈₉	—	2	1	no	—
94	C₂×C₈×C₈₉	—	4	1	no	—
95	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
99	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
100	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
101	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
103	C₁₆×C₈₉	—	2	4	no	—
104	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
105	C₂²×D₈₉	C₂²×C₈	1	2	yes	—
106	C₂×C₄×C₈₉	—	2	1	no	—
107	C₂×C₄×C₈₉	—	2	1	no	—
108	C₂×C₄×C₈₉	—	2	2	no	—
109	C₂×C₄×C₈₉	—	2	2	no	—
110	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
113	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
115	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
116	C₂²⋊D₈₉	—	2	4	no	—
117	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
118	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
119	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
120	D₃₅₆	—	2	2	no	—
121	C₄×D₈₉	—	2	2	no	—
122	C₄×D₈₉	—	2	2	no	—
123	C₄×D₈₉	—	2	4	no	—
124	C₄×D₈₉	—	2	4	no	—
125	Dic₁₇₈	M₃₂	1	4	yes	complemented
126	C₈×C₈₉	—	2	1	no	—
127	C₈×C₈₉	—	2	1	no	—
128	C₈×C₈₉	—	2	1	no	—
129	C₈×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
130	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
131	#10623494	—	2	2	no	—
132	#10623494	—	2	2	no	—
133	C₂²×C₈₉	—	2	1	no	—
134	C₂²×C₈₉	—	2	1	no	—
135	C₂²×C₈₉	—	2	1	no	—
136	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
137	C₂×D₈₉	—	2	2	no	—
138	C₂×D₈₉	—	2	2	no	—
139	C₂×D₈₉	—	2	2	no	—
140	C₄×C₈₉	—	2	1	no	—
141	C₄×C₈₉	—	2	1	no	—
142	C₄×C₈₉	—	2	1	no	—
143	C₄×C₈₉	C₂×M₃₂	1	2	yes	—
144	C₄×C₈₉	#504	1	2	yes	—
145	C₄×C₈₉	#520	1	1	yes	characteristic
146	C₄×C₈₉	#568	1	1	yes	characteristic
147	Dic₈₉	—	2	2	no	—
148	Dic₈₉	—	2	2	no	—
149	Dic₈₉	C₂×M₃₂	1	4	yes	—
150	#33505	—	89	1	no	maximal, Sylow
151	C₂×C₈₉	—	4	1	no	—
152	C₂×C₈₉	#2988	1	1	yes	characteristic
153	C₂×C₈₉	C₂²×M₃₂	1	1	yes	characteristic
154	C₂×C₈₉	C₂×#504	1	1	yes	characteristic
155	D₈₉	—	4	2	no	—
156	C₈×#41	—	89	1	no	—
157	Q₈×C₁₆	—	89	2	no	—
158	#2169	—	89	4	no	—
159	#2169	—	89	2	no	—
160	#2162	—	89	1	no	—
161	#2162	—	89	2	no	—
162	#2150	—	89	2	no	—
163	#2150	D₈₉	1	1	yes	characteristic, complemented
164	C₈₉	#33505	1	1	yes	characteristic, complemented, Sylow
165	C₄×#41	—	89	1	no	—
166	C₂×M₃₂	—	89	4	no	—
167	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
168	C₈×Q₈	—	89	1	no	—
169	C₈×Q₈	—	89	1	no	—
170	C₈×D₄	—	89	2	no	—
171	C₈×D₄	—	89	2	no	—
172	C₈×D₄	—	89	1	no	—
173	C₈×D₄	—	89	1	no	—
174	#431	—	89	1	no	—
175	#431	—	89	2	no	—
176	C₂×C₄×C₈	—	89	2	no	—
177	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
178	C₄⋊C₁₆	—	89	4	no	—
179	C₄⋊C₁₆	—	89	1	no	—
180	C₄⋊C₁₆	—	89	1	no	—
181	C₄⋊C₁₆	C₂×D₈₉	1	2	yes	—
182	C₄⋊C₁₆	C₂×D₈₉	1	2	yes	—
183	C₂²⋊C₁₆	—	89	2	no	—
184	C₂²⋊C₁₆	—	89	4	no	—
185	C₄×C₁₆	—	89	4	no	—
186	C₄×C₁₆	—	89	2	no	—
187	C₂×#41	—	89	1	no	—
188	C₂²×C₈	—	89	2	no	—
189	C₂²×C₈	—	89	2	no	—
190	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
191	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
192	C₄×Q₈	—	89	1	no	—
193	C₄×Q₈	—	89	1	no	—
194	C₄×D₄	—	89	2	no	—
195	C₄×D₄	—	89	2	no	—
196	C₄×D₄	—	89	1	no	—
197	C₄×D₄	—	89	1	no	—
198	#117	—	89	1	no	—
199	#117	—	89	2	no	—
200	C₂×C₄²	—	89	2	no	—
201	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
202	M₃₂	—	4	2	no	—
203	M₃₂	—	356	4	no	—
204	C₂×C₁₆	—	89	2	no	—
205	C₂×C₁₆	—	89	2	no	—
206	C₂×C₁₆	C₂²×D₈₉	1	4	yes	—
207	C₄⋊C₈	—	89	2	no	—
208	C₄⋊C₈	—	89	2	no	—
209	C₄⋊C₈	—	89	1	no	—
210	C₄⋊C₈	—	89	1	no	—
211	C₂²⋊C₈	—	89	2	no	—
212	C₂²⋊C₈	—	89	2	no	—
213	C₂²⋊C₈	—	89	2	no	—
214	C₄×C₈	—	89	2	no	—
215	C₄×C₈	—	89	2	no	—
216	C₄×C₈	—	89	2	no	—
217	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
218	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
219	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
220	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
221	#41	—	178	4	no	—
222	C₂×Q₈	—	89	1	no	—
223	C₂×D₄	—	89	2	no	—
224	C₂×D₄	—	89	1	no	—
225	C₂²×C₄	—	89	2	no	—
226	C₂²×C₄	—	89	2	no	—
227	C₂²×C₄	—	89	2	no	—
228	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
229	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	2	1	no	—
233	C₂×C₈	—	4	1	no	—
234	C₂×C₈	—	89	2	no	—
235	C₂×C₈	—	89	2	no	—
236	C₂×C₈	—	356	2	no	—
237	C₂×C₈	—	178	2	no	—
238	C₂×C₈	—	178	2	no	—
239	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
240	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
241	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
242	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
243	C₄⋊C₄	—	89	2	no	—
244	C₄⋊C₄	—	89	2	no	—
245	C₄⋊C₄	—	89	1	no	—
246	C₄⋊C₄	—	89	1	no	—
247	C₂²⋊C₄	—	89	2	no	—
248	C₂²⋊C₄	—	89	2	no	—
249	C₂²⋊C₄	—	89	2	no	—
250	C₄²	—	89	2	no	—
251	C₄²	—	89	2	no	—
252	C₄²	—	89	2	no	—
253	C₄²	C₈×D₈₉	1	1	yes	characteristic
254	C₄²	C₈×D₈₉	1	1	yes	characteristic
255	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
256	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
257	C₁₆	—	2	4	no	—
258	C₁₆	—	178	2	no	—
259	C₁₆	—	178	2	no	—
260	C₂³	—	89	2	no	—
261	C₂³	C₂×C₈×D₈₉	1	1	yes	characteristic
262	Q₈	—	89	4	no	—
263	D₄	—	178	4	no	—
264	D₄	—	178	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	89	2	no	—
270	C₂×C₄	—	89	2	no	—
271	C₂×C₄	—	89	2	no	—
272	C₂×C₄	—	178	4	no	—
273	C₂×C₄	—	178	4	no	—
274	C₂×C₄	—	178	2	no	—
275	C₂×C₄	—	178	2	no	—
276	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
277	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
278	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
279	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
280	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
281	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
282	C₈	—	2	1	no	—
283	C₈	—	2	1	no	—
284	C₈	—	2	1	no	—
285	C₈	—	178	2	no	—
286	C₈	—	178	2	no	—
287	C₈	#15668057	1	1	yes	characteristic
288	C₈	#15668060	1	1	yes	characteristic
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	178	2	no	—
293	C₂²	—	178	2	no	—
294	C₂²	—	178	2	no	—
295	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	89	4	no	—
300	C₄	—	178	2	no	—
301	C₄	—	178	2	no	—
302	C₄	M₃₂×D₈₉	1	2	yes	—
303	C₄	#19152016	1	2	yes	—
304	C₄	#19152079	1	1	yes	characteristic
305	C₄	#19152364	1	1	yes	characteristic
306	C₂	—	4	1	no	—
307	C₂	—	356	2	no	—
308	C₂	#26029361	1	1	yes	characteristic
309	C₂	C₂×M₃₂×D₈₉	1	1	yes	characteristic
310	C₂	C₂×#19152016	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₈
Fitting	C₈₉×#2150	C₂
prime radical	C₂×#13083574	C₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2150	D₈₉
hypocenter	C₈₉	#33505
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
2. #26022154	C₂	semidirect
3. #26022083	C₂	semidirect
4. #26022082	C₂	semidirect
5. #26022051	C₂	semidirect
6. #26022050	C₂	semidirect
7. #26022006	C₂	semidirect
8. C₈₉×#2150	C₂	semidirect
125. Dic₁₇₈	M₃₂	semidirect
163. #2150	D₈₉	semidirect
164. C₈₉	#33505	semidirect

Maximal subgroups

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

subgroup	index
C₈×#13083574	2
#26022154	2
#26022083	2
#26022082	2
#26022051	2
#26022050	2
#26022006	2
C₈₉×#2150	2
#33505	89