Subgroups — #36708155

Order 22784.

484 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029227	C₂	1	1	yes	maximal, characteristic
2	#26022159	C₂	1	2	yes	maximal, complemented
3	#26022082	C₂	1	2	yes	maximal, complemented
4	#26022076	C₂	1	2	yes	maximal, complemented
5	#26022075	C₂	1	2	yes	maximal, complemented
6	#26022058	C₂	1	2	yes	maximal, complemented
7	#26022055	C₂	1	1	yes	maximal, characteristic, complemented
8	#26022009	C₂	1	2	yes	maximal, complemented
9	C₈₉×#2150	C₂	1	1	yes	maximal, characteristic, complemented
10	#19152132	C₄	1	1	yes	characteristic
11	C₂×#15667951	C₂²	1	2	yes	—
12	C₂×C₁₆×D₈₉	C₂²	1	2	yes	—
13	#19151728	C₄	1	1	yes	characteristic
14	C₈×Dic₁₇₈	C₂²	1	1	yes	characteristic
15	#19151677	C₄	1	1	yes	characteristic
16	#19151669	C₄	1	2	yes	—
17	C₈×D₃₅₆	C₂²	1	1	yes	characteristic
18	C₈×C₂²⋊D₈₉	C₂²	1	2	yes	—
19	#19151657	C₄	1	2	yes	—
20	#19151655	C₄	1	1	yes	characteristic
21	#19151552	C₂²	1	2	yes	—
22	C₄⋊#13083549	C₂²	1	1	yes	characteristic
23	C₄⋊#13083549	C₂²	1	1	yes	characteristic
24	#19151416	C₂²	1	2	yes	—
25	#19151416	C₂²	1	2	yes	—
26	C₂²⋊#13083549	C₂²	1	2	yes	—
27	#19151377	C₂²	1	2	yes	—
28	#19151377	C₂²	1	2	yes	—
29	#19151374	C₂²	1	2	yes	—
30	#19151373	C₂²	1	2	yes	—
31	C₁₆×Dic₈₉	C₂²	1	2	yes	—
32	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
33	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
35	C₄⋊C₁₆×C₈₉	C₂²	1	2	yes	—
36	C₄⋊C₁₆×C₈₉	C₂²	1	2	yes	—
37	C₂×#13083557	C₂×C₄	1	2	yes	—
38	C₂×C₈×D₈₉	C₂³	1	2	yes	—
39	#15668006	C₈	1	2	yes	—
40	#15668004	C₈	1	1	yes	characteristic
41	#15668000	C₈	1	1	yes	characteristic
42	#15667995	C₈	1	2	yes	—
43	#15667991	C₈	1	1	yes	characteristic
44	#15667986	C₈	1	1	yes	characteristic
45	C₄×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
46	C₄×D₃₅₆	C₂×C₄	1	1	yes	characteristic
47	C₄×C₂²⋊D₈₉	C₂×C₄	1	2	yes	—
48	#15667969	C₂×C₄	1	2	yes	—
49	#15667951	—	4	2	no	—
50	C₂×#13083549	C₂³	1	2	yes	—
51	C₂×#13083549	C₂³	1	2	yes	—
52	C₁₆×D₈₉	—	4	2	no	—
53	C₄⋊#10623494	C₂×C₄	1	1	yes	characteristic
54	C₄⋊#10623494	C₂×C₄	1	1	yes	characteristic
55	#15667941	C₂×C₄	1	2	yes	—
56	#15667941	C₂×C₄	1	2	yes	—
57	C₂²⋊#10623494	C₂×C₄	1	2	yes	—
58	#15667924	C₂×C₄	1	2	yes	—
59	#15667924	C₂×C₄	1	2	yes	—
60	#15667923	C₂×C₄	1	2	yes	—
61	#15667922	C₂×C₄	1	2	yes	—
62	C₈×Dic₈₉	C₂³	1	2	yes	—
63	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
66	M₃₂×C₈₉	—	4	1	no	—
67	M₃₂×C₈₉	—	4	1	no	—
68	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
69	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
70	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
74	C₂×Dic₁₇₈	C₂×C₈	1	1	yes	characteristic
75	C₂×D₃₅₆	C₂×C₈	1	1	yes	characteristic
76	C₂×C₂²⋊D₈₉	C₂×C₈	1	2	yes	—
77	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
78	#13083557	—	4	2	no	—
79	C₂×#10623494	C₂²×C₄	1	2	yes	—
80	C₂×#10623494	C₂²×C₄	1	2	yes	—
81	C₈×D₈₉	—	2	2	no	—
82	C₈×D₈₉	—	2	2	no	—
83	C₄⋊Dic₈₉	C₂×C₈	1	1	yes	characteristic
84	C₄⋊Dic₈₉	C₂×C₈	1	1	yes	characteristic
85	#13083553	C₂×C₈	1	2	yes	—
86	#13083553	C₂×C₈	1	2	yes	—
87	#13083553	C₂×C₈	1	2	yes	—
88	#13083553	C₂×C₈	1	2	yes	—
89	C₂²⋊Dic₈₉	C₂×C₈	1	2	yes	—
90	#13083551	C₂×C₈	1	2	yes	—
91	#13083551	C₂×C₈	1	2	yes	—
92	#13083551	C₂×C₈	1	2	yes	—
93	#13083551	C₂×C₈	1	2	yes	—
94	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
95	#13083549	—	2	2	no	—
96	#13083549	—	2	2	no	—
97	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
98	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
99	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₈₉	—	2	1	no	—
101	C₂×C₈×C₈₉	—	2	1	no	—
102	C₂×C₈×C₈₉	—	4	1	no	—
103	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
107	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
108	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
109	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
111	C₁₆×C₈₉	—	2	2	no	—
112	C₁₆×C₈₉	—	2	2	no	—
113	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
114	C₂²×D₈₉	C₂²×C₈	1	2	yes	—
115	C₂×C₄×C₈₉	—	2	1	no	—
116	C₂×C₄×C₈₉	—	2	1	no	—
117	C₂×C₄×C₈₉	—	4	1	no	—
118	C₂×C₄×C₈₉	—	4	1	no	—
119	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
123	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
125	C₂²⋊D₈₉	—	4	2	no	—
126	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
127	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
128	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
129	D₃₅₆	—	4	1	no	—
130	C₄×D₈₉	—	2	2	no	—
131	C₄×D₈₉	—	2	2	no	—
132	Dic₁₇₈	—	4	1	no	—
133	C₈×C₈₉	—	2	1	no	—
134	C₈×C₈₉	—	2	1	no	—
135	C₈×C₈₉	—	2	1	no	—
136	C₈×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
137	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
138	#10623494	—	2	2	no	—
139	#10623494	—	2	2	no	—
140	C₂²×C₈₉	—	2	1	no	—
141	C₂²×C₈₉	—	2	1	no	—
142	C₂²×C₈₉	—	2	1	no	—
143	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
144	C₂×D₈₉	—	2	2	no	—
145	C₂×D₈₉	—	2	2	no	—
146	C₂×D₈₉	—	2	2	no	—
147	C₄×C₈₉	—	2	1	no	—
148	C₄×C₈₉	—	2	1	no	—
149	C₄×C₈₉	—	2	1	no	—
150	C₄×C₈₉	—	2	1	no	—
151	C₄×C₈₉	—	2	1	no	—
152	C₄×C₈₉	#520	1	1	yes	characteristic
153	C₄×C₈₉	#568	1	1	yes	characteristic
154	Dic₈₉	—	2	2	no	—
155	Dic₈₉	—	2	2	no	—
156	Dic₈₉	—	2	2	no	—
157	#33509	—	89	1	no	maximal, Sylow
158	C₂×C₈₉	—	4	1	no	—
159	C₂×C₈₉	#2988	1	1	yes	characteristic
160	C₂×C₈₉	#3407	1	1	yes	characteristic
161	C₂×C₈₉	#3407	1	1	yes	characteristic
162	D₈₉	—	4	2	no	—
163	#2966	—	89	1	no	—
164	#2183	—	89	2	no	—
165	#2169	—	89	2	no	—
166	#2168	—	89	2	no	—
167	#2168	—	89	2	no	—
168	#2164	—	89	2	no	—
169	#2163	—	89	1	no	—
170	#2151	—	89	2	no	—
171	#2150	D₈₉	1	1	yes	characteristic, complemented
172	C₈₉	#33509	1	1	yes	characteristic, complemented, Sylow
173	#529	—	89	1	no	—
174	C₂×M₃₂	—	89	2	no	—
175	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
176	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
177	C₂²×C₁₆	—	89	2	no	—
178	#446	—	89	1	no	—
179	C₈×Q₈	—	89	1	no	—
180	#436	—	89	1	no	—
181	#435	—	89	2	no	—
182	C₈×D₄	—	89	2	no	—
183	C₈×D₄	—	89	1	no	—
184	#433	—	89	2	no	—
185	#432	—	89	1	no	—
186	#405	—	89	2	no	—
187	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
188	C₄⋊C₁₆	—	89	2	no	—
189	C₄⋊C₁₆	—	89	2	no	—
190	C₄⋊C₁₆	—	89	1	no	—
191	C₄⋊C₁₆	—	89	1	no	—
192	C₄⋊C₁₆	C₂×D₈₉	1	2	yes	—
193	C₄⋊C₁₆	C₂×D₈₉	1	2	yes	—
194	C₂²⋊C₁₆	—	89	2	no	—
195	C₂²⋊C₁₆	—	89	2	no	—
196	C₂²⋊C₁₆	—	89	2	no	—
197	#346	—	89	2	no	—
198	#346	—	89	2	no	—
199	C₄×C₁₆	—	89	2	no	—
200	C₂×M₁₆	—	89	2	no	—
201	C₂²×C₈	—	89	2	no	—
202	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
203	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
204	#126	—	89	2	no	—
205	#125	—	89	1	no	—
206	#124	—	89	1	no	—
207	#123	—	89	2	no	—
208	C₂²⋊Q₈	—	89	1	no	—
209	#121	—	89	1	no	—
210	C₄×Q₈	—	89	1	no	—
211	C₄×D₄	—	89	2	no	—
212	C₄×D₄	—	89	1	no	—
213	#117	—	89	2	no	—
214	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
215	M₃₂	—	4	1	no	—
216	M₃₂	—	4	1	no	—
217	M₃₂	—	356	2	no	—
218	C₂×C₁₆	—	89	2	no	—
219	C₂×C₁₆	—	89	2	no	—
220	C₂×C₁₆	—	356	2	no	—
221	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
222	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
223	C₄⋊C₈	—	89	2	no	—
224	C₄⋊C₈	—	89	2	no	—
225	C₄⋊C₈	—	89	1	no	—
226	C₄⋊C₈	—	89	1	no	—
227	C₂²⋊C₈	—	89	2	no	—
228	C₂²⋊C₈	—	89	2	no	—
229	C₂²⋊C₈	—	89	2	no	—
230	#97	—	89	2	no	—
231	#97	—	89	2	no	—
232	C₄×C₈	—	89	2	no	—
233	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
234	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
235	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
236	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
237	C₂×Q₈	—	89	1	no	—
238	C₂×D₄	—	89	2	no	—
239	C₂×D₄	—	89	1	no	—
240	C₂²×C₄	—	89	2	no	—
241	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
242	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
243	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
244	M₁₆	—	356	2	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	4	1	no	—
248	C₂×C₈	—	89	2	no	—
249	C₂×C₈	—	89	2	no	—
250	C₂×C₈	—	178	2	no	—
251	C₂×C₈	—	178	2	no	—
252	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
253	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
254	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
255	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
256	C₄⋊C₄	—	89	2	no	—
257	C₄⋊C₄	—	89	2	no	—
258	C₄⋊C₄	—	89	1	no	—
259	C₄⋊C₄	—	89	2	no	—
260	C₄⋊C₄	—	89	1	no	—
261	C₄⋊C₄	—	89	2	no	—
262	C₂²⋊C₄	—	89	2	no	—
263	C₂²⋊C₄	—	89	2	no	—
264	C₂²⋊C₄	—	89	2	no	—
265	C₂²⋊C₄	—	89	2	no	—
266	C₂²⋊C₄	—	89	2	no	—
267	C₄²	—	89	2	no	—
268	C₄²	C₈×D₈₉	1	1	yes	characteristic
269	C₄²	C₈×D₈₉	1	1	yes	characteristic
270	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
271	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
272	C₁₆	—	2	2	no	—
273	C₁₆	—	2	2	no	—
274	C₁₆	—	178	2	no	—
275	C₁₆	—	178	2	no	—
276	C₂³	—	89	2	no	—
277	C₂³	C₂×C₈×D₈₉	1	1	yes	characteristic
278	Q₈	—	356	1	no	—
279	D₄	—	356	1	no	—
280	D₄	—	356	2	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	—	4	1	no	—
285	C₂×C₄	—	89	2	no	—
286	C₂×C₄	—	89	2	no	—
287	C₂×C₄	—	89	2	no	—
288	C₂×C₄	—	178	2	no	—
289	C₂×C₄	—	178	2	no	—
290	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
291	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
292	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
293	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
294	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
295	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
296	C₈	—	2	1	no	—
297	C₈	—	2	1	no	—
298	C₈	—	2	1	no	—
299	C₈	—	178	2	no	—
300	C₈	—	178	2	no	—
301	C₈	#15668057	1	1	yes	characteristic
302	C₈	#15668060	1	1	yes	characteristic
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	178	2	no	—
307	C₂²	—	178	2	no	—
308	C₂²	—	178	2	no	—
309	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	178	2	no	—
316	C₄	—	178	2	no	—
317	C₄	—	178	2	no	—
318	C₄	#19152079	1	1	yes	characteristic
319	C₄	#19152364	1	1	yes	characteristic
320	C₂	—	4	1	no	—
321	C₂	—	356	2	no	—
322	C₂	#26029361	1	1	yes	characteristic
323	C₂	#26033354	1	1	yes	characteristic
324	C₂	#26033354	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	#15667986	C₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2150	D₈₉
hypocenter	C₈₉	#33509
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. #26022159	C₂	semidirect
3. #26022082	C₂	semidirect
4. #26022076	C₂	semidirect
5. #26022075	C₂	semidirect
6. #26022058	C₂	semidirect
7. #26022055	C₂	semidirect
8. #26022009	C₂	semidirect
9. C₈₉×#2150	C₂	semidirect
171. #2150	D₈₉	semidirect
172. C₈₉	#33509	semidirect

Maximal subgroups

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

subgroup	index
#26029227	2
#26022159	2
#26022082	2
#26022076	2
#26022075	2
#26022058	2
#26022055	2
#26022009	2
C₈₉×#2150	2
#33509	89