Subgroups — #36707975

Order 22784.

508 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26022161	C₂	1	2	yes	maximal, complemented
2	#26022084	C₂	1	2	yes	maximal, complemented
3	#26022078	C₂	1	2	yes	maximal, complemented
4	#26022078	C₂	1	2	yes	maximal, complemented
5	#26022059	C₂	1	2	yes	maximal, complemented
6	#26022055	C₂	1	1	yes	maximal, characteristic, complemented
7	#26021849	C₂	1	2	yes	maximal, complemented
8	C₄×#15667952	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#2966	C₂	1	1	yes	maximal, characteristic
10	C₂×#15667952	C₂²	1	2	yes	—
11	C₂×#15667952	C₂²	1	2	yes	—
12	C₂²×#13083549	C₂²	1	2	yes	—
13	C₄⋊#13083549	C₂²	1	2	yes	—
14	C₄⋊#13083549	C₂²	1	2	yes	—
15	C₄⋊#13083549	C₂²	1	2	yes	—
16	C₂²⋊#13083549	C₂²	1	1	yes	characteristic
17	C₂²⋊#13083549	C₂²	1	1	yes	characteristic
18	C₂²⋊#13083549	C₂²	1	2	yes	—
19	C₂²⋊#13083549	C₂²	1	2	yes	—
20	#19151374	C₂²	1	2	yes	—
21	#19151374	C₂²	1	2	yes	—
22	#19151374	C₂²	1	2	yes	—
23	C₄×#13083549	C₂²	1	2	yes	—
24	C₄×#13083549	C₂²	1	1	yes	characteristic
25	C₄×#13083549	C₂²	1	1	yes	characteristic
26	#529×C₈₉	C₄	1	1	yes	characteristic
27	#446×C₈₉	C₄	1	1	yes	characteristic
28	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
29	#436×C₈₉	C₄	1	1	yes	characteristic
30	#435×C₈₉	C₄	1	2	yes	—
31	C₈×D₄×C₈₉	C₂²	1	1	yes	characteristic
32	C₈×D₄×C₈₉	C₂²	1	2	yes	—
33	#433×C₈₉	C₄	1	2	yes	—
34	#432×C₈₉	C₄	1	1	yes	characteristic
35	#405×C₈₉	C₂²	1	2	yes	—
36	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
37	#15667952	—	2	4	no	—
38	#15667952	—	2	4	no	—
39	C₂×#13083549	—	2	4	no	—
40	C₂×#13083549	C₂³	1	2	yes	—
41	C₂×#13083549	C₂³	1	2	yes	—
42	C₂×#13083549	C₂³	1	2	yes	—
43	C₂×#13083549	C₂³	1	2	yes	—
44	C₂×M₁₆×C₈₉	C₂×C₄	1	2	yes	—
45	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
48	#126×C₈₉	C₈	1	2	yes	—
49	#125×C₈₉	C₈	1	1	yes	characteristic
50	#124×C₈₉	C₈	1	1	yes	characteristic
51	#123×C₈₉	C₈	1	2	yes	—
52	C₂²⋊Q₈×C₈₉	C₈	1	1	yes	characteristic
53	#121×C₈₉	C₈	1	1	yes	characteristic
54	C₄×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
55	C₄×D₄×C₈₉	C₂×C₄	1	2	yes	—
56	C₄×D₄×C₈₉	C₂×C₄	1	1	yes	characteristic
57	#117×C₈₉	C₂×C₄	1	2	yes	—
58	C₂×C₄²×C₈₉	C₂×C₄	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	C₄⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
62	C₄⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
63	C₂²⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
64	C₂²⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
65	C₂²⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
66	#97×C₈₉	C₂×C₄	1	2	yes	—
67	#97×C₈₉	C₂×C₄	1	2	yes	—
68	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
69	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
70	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₈₉	C₂³	1	2	yes	—
72	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
73	#13083549	—	2	2	no	—
74	#13083549	—	2	2	no	—
75	#13083549	—	2	2	no	—
76	#13083549	#41	1	4	yes	—
77	C₂×Q₈×C₈₉	C₂×C₈	1	1	yes	characteristic
78	C₂×D₄×C₈₉	C₂×C₈	1	2	yes	—
79	C₂×D₄×C₈₉	C₂×C₈	1	1	yes	characteristic
80	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
81	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
82	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
84	M₁₆×C₈₉	—	2	4	no	—
85	C₂×C₈×C₈₉	—	2	2	no	—
86	C₂×C₈×C₈₉	—	2	2	no	—
87	C₂×C₈×C₈₉	—	2	2	no	—
88	C₂×C₈×C₈₉	—	2	2	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	2	yes	—
92	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
93	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
95	C₄⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
96	C₄⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
97	C₄⋊C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
98	C₄⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
99	C₄⋊C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
100	C₄⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
101	C₂²⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
102	C₂²⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
103	C₂²⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
104	C₂²⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
105	C₂²⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
106	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
107	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
108	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
111	C₂³×C₈₉	C₂²×C₈	1	2	yes	—
112	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
113	Q₈×C₈₉	—	4	1	no	—
114	D₄×C₈₉	—	4	1	no	—
115	D₄×C₈₉	—	4	2	no	—
116	C₂×C₄×C₈₉	—	2	2	no	—
117	C₂×C₄×C₈₉	—	2	1	no	—
118	C₂×C₄×C₈₉	—	2	1	no	—
119	C₂×C₄×C₈₉	—	2	2	no	—
120	C₂×C₄×C₈₉	—	4	1	no	—
121	C₂×C₄×C₈₉	—	4	1	no	—
122	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
123	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
128	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
129	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
130	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
131	C₈×C₈₉	—	2	2	no	—
132	C₈×C₈₉	—	2	1	no	—
133	C₈×C₈₉	—	2	1	no	—
134	C₈×C₈₉	—	2	2	no	—
135	C₈×C₈₉	#131	1	2	yes	—
136	C₈×C₈₉	C₂×#41	1	2	yes	—
137	C₂²×C₈₉	—	2	1	no	—
138	C₂²×C₈₉	—	2	1	no	—
139	C₂²×C₈₉	—	2	2	no	—
140	C₂²×C₈₉	—	2	2	no	—
141	C₂²×C₈₉	—	2	1	no	—
142	C₂²×C₈₉	—	2	2	no	—
143	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
144	C₄×C₈₉	—	2	1	no	—
145	C₄×C₈₉	—	2	1	no	—
146	C₄×C₈₉	—	2	1	no	—
147	C₄×C₈₉	—	2	1	no	—
148	C₄×C₈₉	—	2	2	no	—
149	C₄×C₈₉	—	2	2	no	—
150	C₄×C₈₉	—	2	1	no	—
151	C₄×C₈₉	—	2	2	no	—
152	C₄×C₈₉	C₄×#41	1	1	yes	characteristic
153	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
154	#33481	—	89	1	no	maximal, Sylow
155	C₂×C₈₉	—	4	1	no	—
156	C₂×C₈₉	—	4	2	no	—
157	C₂×C₈₉	C₈×#41	1	1	yes	characteristic
158	C₂×C₈₉	#3407	1	1	yes	characteristic
159	C₂×C₈₉	#3407	1	1	yes	characteristic
160	#2966	D₈₉	1	1	yes	characteristic
161	#2183	—	89	2	no	—
162	#2169	—	89	2	no	—
163	#2168	—	89	2	no	—
164	#2168	—	89	2	no	—
165	#2164	—	89	2	no	—
166	#2163	—	89	1	no	—
167	#2108	—	89	2	no	—
168	C₄×M₃₂	—	89	1	no	—
169	C₈₉	#33481	1	1	yes	characteristic, complemented, Sylow
170	#529	Dic₈₉	1	1	yes	characteristic
171	C₂×M₃₂	—	89	2	no	—
172	C₂×M₃₂	—	89	2	no	—
173	C₂²×C₁₆	—	89	2	no	—
174	#446	Dic₈₉	1	1	yes	characteristic
175	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
176	#436	Dic₈₉	1	1	yes	characteristic
177	#435	Dic₈₉	1	2	yes	—
178	C₈×D₄	C₂×D₈₉	1	1	yes	characteristic
179	C₈×D₄	C₂×D₈₉	1	2	yes	—
180	#433	Dic₈₉	1	2	yes	—
181	#432	Dic₈₉	1	1	yes	characteristic
182	#405	C₂×D₈₉	1	2	yes	—
183	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
184	C₄⋊C₁₆	—	89	2	no	—
185	C₄⋊C₁₆	—	89	2	no	—
186	C₄⋊C₁₆	—	89	2	no	—
187	C₂²⋊C₁₆	—	89	1	no	—
188	C₂²⋊C₁₆	—	89	1	no	—
189	C₂²⋊C₁₆	—	89	2	no	—
190	C₂²⋊C₁₆	—	89	2	no	—
191	#346	—	89	2	no	—
192	#346	—	89	2	no	—
193	#346	—	89	2	no	—
194	C₄×C₁₆	—	89	1	no	—
195	C₄×C₁₆	—	89	1	no	—
196	C₄×C₁₆	—	89	2	no	—
197	C₂×M₁₆	C₂×Dic₈₉	1	2	yes	—
198	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
199	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
200	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
201	#126	#10623494	1	2	yes	—
202	#125	#10623494	1	1	yes	characteristic
203	#124	#10623494	1	1	yes	characteristic
204	#123	#10623494	1	2	yes	—
205	C₂²⋊Q₈	#10623494	1	1	yes	characteristic
206	#121	#10623494	1	1	yes	characteristic
207	C₄×Q₈	C₂×Dic₈₉	1	1	yes	characteristic
208	C₄×D₄	C₂×Dic₈₉	1	2	yes	—
209	C₄×D₄	C₂×Dic₈₉	1	1	yes	characteristic
210	#117	C₂×Dic₈₉	1	2	yes	—
211	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
212	M₃₂	—	178	4	no	—
213	M₃₂	—	178	4	no	—
214	C₂×C₁₆	—	89	2	no	—
215	C₂×C₁₆	—	89	2	no	—
216	C₂×C₁₆	—	89	2	no	—
217	C₂×C₁₆	—	89	2	no	—
218	C₂×C₁₆	—	178	4	no	—
219	C₄⋊C₈	C₂×Dic₈₉	1	1	yes	characteristic
220	C₄⋊C₈	C₂×Dic₈₉	1	1	yes	characteristic
221	C₄⋊C₈	C₂×Dic₈₉	1	2	yes	—
222	C₄⋊C₈	C₂×Dic₈₉	1	2	yes	—
223	C₂²⋊C₈	C₂×Dic₈₉	1	2	yes	—
224	C₂²⋊C₈	C₂×Dic₈₉	1	2	yes	—
225	C₂²⋊C₈	C₂×Dic₈₉	1	2	yes	—
226	#97	C₂×Dic₈₉	1	2	yes	—
227	#97	C₂×Dic₈₉	1	2	yes	—
228	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
229	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
230	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
231	C₄×C₈	C₂²×D₈₉	1	2	yes	—
232	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
233	C₂×Q₈	C₂×#10623494	1	1	yes	characteristic
234	C₂×D₄	C₂×#10623494	1	2	yes	—
235	C₂×D₄	C₂×#10623494	1	1	yes	characteristic
236	C₂²×C₄	C₂×#10623494	1	1	yes	characteristic
237	C₂²×C₄	C₂×#10623494	1	1	yes	characteristic
238	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
239	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
240	M₁₆	—	2	4	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
246	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
247	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
248	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
249	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
250	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
251	C₄⋊C₄	C₂×#10623494	1	2	yes	—
252	C₄⋊C₄	C₂×#10623494	1	2	yes	—
253	C₄⋊C₄	C₂×#10623494	1	1	yes	characteristic
254	C₄⋊C₄	C₂×#10623494	1	2	yes	—
255	C₄⋊C₄	C₂×#10623494	1	1	yes	characteristic
256	C₄⋊C₄	C₂×#10623494	1	2	yes	—
257	C₂²⋊C₄	C₂×#10623494	1	2	yes	—
258	C₂²⋊C₄	C₂×#10623494	1	2	yes	—
259	C₂²⋊C₄	C₂×#10623494	1	2	yes	—
260	C₂²⋊C₄	C₂×#10623494	1	2	yes	—
261	C₂²⋊C₄	C₂×#10623494	1	2	yes	—
262	C₄²	C₂×#10623494	1	1	yes	characteristic
263	C₄²	C₂×#10623494	1	1	yes	characteristic
264	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
265	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
266	C₄²	C₂²×Dic₈₉	1	2	yes	—
267	C₁₆	—	89	4	no	—
268	C₁₆	—	178	2	no	—
269	C₁₆	—	178	2	no	—
270	C₁₆	—	178	2	no	—
271	C₂³	C₂²×#10623494	1	2	yes	—
272	C₂³	C₂²×#10623494	1	1	yes	characteristic
273	Q₈	—	4	1	no	—
274	D₄	—	4	1	no	—
275	D₄	—	4	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
283	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
284	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
285	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
286	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
287	C₂×C₄	C₂²×#10623494	1	2	yes	—
288	C₂×C₄	C₂²×#10623494	1	2	yes	—
289	C₂×C₄	C₂²×#10623494	1	2	yes	—
290	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
291	C₈	—	2	2	no	—
292	C₈	—	2	1	no	—
293	C₈	—	2	1	no	—
294	C₈	—	2	2	no	—
295	C₈	#15668019	1	2	yes	—
296	C₈	#41×D₈₉	1	2	yes	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	2	no	—
302	C₂²	—	2	2	no	—
303	C₂²	C₂³×#10623494	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	2	2	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	2	2	no	—
311	C₄	—	2	2	no	—
312	C₄	#41×Dic₈₉	1	1	yes	characteristic
313	C₄	C₂×#15668019	1	1	yes	characteristic
314	C₂	—	4	1	no	—
315	C₂	—	4	2	no	—
316	C₂	#41×#10623494	1	1	yes	characteristic
317	C₂	#26033355	1	1	yes	characteristic
318	C₂	#26033355	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₈
Fitting	C₈₉×#2966	C₂
prime radical	#121×C₈₉	C₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2966	D₈₉
hypocenter	C₈₉	#33481
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ #2966

Complements

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

N	H ≅ G/N	type
1. #26022161	C₂	semidirect
2. #26022084	C₂	semidirect
3. #26022078	C₂	semidirect
4. #26022078	C₂	semidirect
5. #26022059	C₂	semidirect
6. #26022055	C₂	semidirect
7. #26021849	C₂	semidirect
8. C₄×#15667952	C₂	semidirect
169. C₈₉	#33481	semidirect

Maximal subgroups

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

subgroup	index
#26022161	2
#26022084	2
#26022078	2
#26022059	2
#26022055	2
#26021849	2
C₄×#15667952	2
C₈₉×#2966	2
#33481	89