Subgroups — #36537538

Order 22784.

674 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26031174	C₂	1	2	yes	maximal
2	C₂×#19152003	C₂	1	1	yes	maximal, characteristic, complemented
3	#26019765	C₂	1	4	yes	maximal, complemented
4	#26019641	C₂	1	4	yes	maximal, complemented
5	#26019000	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018938	C₂	1	2	yes	maximal, complemented
7	C₈₉×#2931	C₂	1	1	yes	maximal, characteristic
8	C₂×#15668012	C₂²	1	2	yes	—
9	#19152003	—	2	1	no	—
10	#19152003	—	2	1	no	—
11	#19152003	C₂²	1	2	yes	—
12	#19152003	C₂²	1	2	yes	—
13	#19151970	—	2	2	no	—
14	#19151950	—	2	2	no	—
15	#19151803	C₂²	1	8	yes	—
16	#19151736	—	2	4	no	—
17	#19151651	C₂²	1	2	yes	—
18	C₂×#15667946	—	2	1	no	—
19	C₂×#15667946	—	2	1	no	—
20	#19151351	C₂²	1	2	yes	—
21	#19151349	C₂²	1	2	yes	—
22	#19151343	C₂²	1	8	yes	—
23	C₂×#131×C₈₉	—	2	1	no	—
24	#532×C₈₉	C₂²	1	1	yes	characteristic
25	#436×C₈₉	—	2	1	no	—
26	#436×C₈₉	C₂²	1	2	yes	—
27	C₈×D₄×C₈₉	—	2	1	no	—
28	C₈×D₄×C₈₉	C₂²	1	2	yes	—
29	#423×C₈₉	C₂²	1	1	yes	characteristic
30	#408×C₈₉	—	2	1	no	—
31	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
32	C₂²×Dic₁₇₈	—	2	2	no	—
33	C₂×#13083562	—	2	8	no	—
34	#15668012	—	2	2	no	—
35	#15668012	—	2	2	no	—
36	#15668012	—	2	2	no	—
37	#15668012	C₂³	1	4	yes	—
38	C₂×C₄⋊Dic₈₉	—	2	2	no	—
39	C₂×C₄⋊Dic₈₉	—	2	2	no	—
40	#15667946	—	2	2	no	—
41	#15667946	—	2	2	no	—
42	#15667946	—	2	2	no	—
43	#15667946	—	2	2	no	—
44	C₄⋊#10623494	C₂³	1	4	yes	—
45	#15667935	—	2	8	no	—
46	C₂²⋊#10623494	—	2	2	no	—
47	C₄×#10623494	—	2	2	no	—
48	C₂×#41×C₈₉	—	2	1	no	—
49	#131×C₈₉	—	4	4	no	—
50	C₂×M₁₆×C₈₉	—	2	1	no	—
51	C₂×M₁₆×C₈₉	—	2	2	no	—
52	C₂²×C₈×C₈₉	—	2	2	no	—
53	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
55	C₄⋊Q₈×C₈₉	D₄	1	1	yes	characteristic
56	#127×C₈₉	D₄	1	1	yes	characteristic
57	#124×C₈₉	—	2	1	no	—
58	#121×C₈₉	—	2	1	no	—
59	#121×C₈₉	D₄	1	2	yes	—
60	C₄×D₄×C₈₉	—	2	1	no	—
61	C₄×D₄×C₈₉	C₂³	1	2	yes	—
62	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
63	C₄⋊C₈×C₈₉	—	2	1	no	—
64	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
65	C₂²⋊C₈×C₈₉	—	2	2	no	—
66	C₂²⋊C₈×C₈₉	—	2	2	no	—
67	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
68	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
69	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
71	C₂×Dic₁₇₈	—	2	4	no	—
72	C₂×Dic₁₇₈	—	4	2	no	—
73	C₂×Dic₁₇₈	—	8	4	no	—
74	C₂²×Dic₈₉	—	4	2	no	—
75	#13083562	—	8	8	no	—
76	C₂×#10623494	—	2	4	no	—
77	C₄⋊Dic₈₉	—	2	4	no	—
78	C₄⋊Dic₈₉	—	2	4	no	—
79	C₄⋊Dic₈₉	—	4	2	no	—
80	C₄⋊Dic₈₉	—	4	2	no	—
81	#41×C₈₉	—	4	2	no	—
82	#41×C₈₉	—	4	2	no	—
83	C₂×Q₈×C₈₉	—	2	1	no	—
84	C₂×D₄×C₈₉	—	2	2	no	—
85	C₂×D₄×C₈₉	—	2	1	no	—
86	C₂×D₄×C₈₉	—	2	1	no	—
87	C₂×D₄×C₈₉	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₈₉	—	2	2	no	—
89	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
92	M₁₆×C₈₉	—	4	2	no	—
93	M₁₆×C₈₉	—	4	4	no	—
94	C₂×C₈×C₈₉	—	2	2	no	—
95	C₂×C₈×C₈₉	—	2	2	no	—
96	C₂×C₈×C₈₉	—	4	4	no	—
97	C₂×C₈×C₈₉	SD₁₆	1	4	yes	complemented
98	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
102	C₄⋊C₄×C₈₉	—	2	1	no	—
103	C₄⋊C₄×C₈₉	C₂×D₄	1	2	yes	—
104	C₂²⋊C₄×C₈₉	—	2	2	no	—
105	C₂²⋊C₄×C₈₉	—	2	2	no	—
106	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₈₉	#41	1	1	yes	characteristic
109	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
110	C₂³×C₈₉	—	2	2	no	—
111	C₂³×C₈₉	#121	1	1	yes	characteristic
112	Q₈×C₈₉	—	4	2	no	—
113	D₄×C₈₉	—	2	4	no	—
114	D₄×C₈₉	—	4	4	no	—
115	D₄×C₈₉	—	4	2	no	—
116	D₄×C₈₉	—	4	2	no	—
117	C₂×C₄×C₈₉	—	2	2	no	—
118	C₂×C₄×C₈₉	—	2	2	no	—
119	C₂×C₄×C₈₉	—	2	2	no	—
120	C₂×C₄×C₈₉	—	4	2	no	—
121	C₂×C₄×C₈₉	—	4	2	no	—
122	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
123	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
124	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
125	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
126	C₂×C₄×C₈₉	#137	1	2	yes	—
127	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
130	C₂×Dic₈₉	—	4	4	no	—
131	C₂×Dic₈₉	—	8	4	no	—
132	Dic₁₇₈	—	4	8	no	—
133	Dic₁₇₈	—	8	4	no	—
134	C₈×C₈₉	—	2	1	no	—
135	C₈×C₈₉	—	2	1	no	—
136	C₈×C₈₉	—	4	2	no	—
137	C₈×C₈₉	C₂×SD₁₆	1	2	yes	—
138	C₈×C₈₉	C₂×SD₁₆	1	2	yes	—
139	#10623494	—	4	4	no	—
140	C₂²×C₈₉	—	2	1	no	—
141	C₂²×C₈₉	—	2	1	no	—
142	C₂²×C₈₉	—	4	2	no	—
143	C₂²×C₈₉	—	4	2	no	—
144	C₂²×C₈₉	—	4	2	no	—
145	C₂²×C₈₉	#461	1	2	yes	—
146	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
147	C₄×C₈₉	—	2	1	no	—
148	C₄×C₈₉	—	2	1	no	—
149	C₄×C₈₉	—	4	2	no	—
150	C₄×C₈₉	#461	1	2	yes	—
151	C₄×C₈₉	#473	1	2	yes	—
152	C₄×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
153	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
154	C₄×C₈₉	#576	1	2	yes	—
155	Dic₈₉	—	8	4	no	—
156	#19192	—	89	1	no	maximal, Sylow
157	C₂×C₈₉	—	2	2	no	—
158	C₂×C₈₉	—	8	2	no	—
159	C₂×C₈₉	C₂×#461	1	1	yes	characteristic
160	C₂×C₈₉	#3058	1	1	yes	characteristic
161	C₂×C₈₉	C₂×#473	1	1	yes	characteristic
162	#3206	—	89	2	no	—
163	C₂×#500	—	89	1	no	—
164	#2931	D₈₉	1	1	yes	characteristic
165	#1693	—	89	4	no	—
166	#1673	—	89	4	no	—
167	#1564	—	89	1	no	—
168	#1549	—	89	2	no	—
169	C₈₉	#19192	1	1	yes	characteristic, complemented, Sylow
170	C₂×#131	—	2	1	no	—
171	#532	C₂×D₈₉	1	1	yes	characteristic
172	C₂×C₄⋊Q₈	—	89	2	no	—
173	#500	—	89	2	no	—
174	#500	—	89	2	no	—
175	#500	—	178	1	no	—
176	#500	—	178	1	no	—
177	#492	—	178	2	no	—
178	#487	—	178	2	no	—
179	#461	—	89	8	no	—
180	#448	—	178	4	no	—
181	#436	—	2	1	no	—
182	#436	C₂×D₈₉	1	2	yes	—
183	C₈×D₄	—	2	1	no	—
184	C₈×D₄	C₂×D₈₉	1	2	yes	—
185	#431	—	89	2	no	—
186	C₂×#107	—	178	1	no	—
187	C₂×#107	—	178	1	no	—
188	#423	C₂×D₈₉	1	1	yes	characteristic
189	#408	—	2	1	no	—
190	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
191	#335	—	89	2	no	—
192	#334	—	89	2	no	—
193	#332	—	89	8	no	—
194	C₂×#41	—	2	1	no	—
195	C₂²×Q₈	—	178	2	no	—
196	C₂×SD₁₆	—	178	8	no	—
197	#131	—	4	4	no	—
198	C₂×M₁₆	—	2	1	no	—
199	C₂×M₁₆	—	2	2	no	—
200	C₂²×C₈	—	2	2	no	—
201	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
202	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
203	C₄⋊Q₈	—	89	4	no	—
204	C₄⋊Q₈	—	178	2	no	—
205	C₄⋊Q₈	—	178	2	no	—
206	C₄⋊Q₈	—	178	2	no	—
207	C₄⋊Q₈	D₃₅₆	1	1	yes	characteristic
208	#127	D₃₅₆	1	1	yes	characteristic
209	#124	—	2	1	no	—
210	#121	—	2	1	no	—
211	#121	D₃₅₆	1	2	yes	—
212	C₄×D₄	—	2	1	no	—
213	C₄×D₄	C₂²×D₈₉	1	2	yes	—
214	C₂×C₄⋊C₄	—	178	2	no	—
215	C₂×C₄⋊C₄	—	178	2	no	—
216	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
217	#107	—	178	2	no	—
218	#107	—	178	2	no	—
219	#107	—	178	2	no	—
220	#107	—	178	2	no	—
221	C₄⋊C₈	—	2	1	no	—
222	C₄⋊C₈	—	89	4	no	—
223	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
224	#103	—	178	8	no	—
225	C₂²⋊C₈	—	2	2	no	—
226	C₂²⋊C₈	—	2	2	no	—
227	C₂²⋊C₈	—	178	2	no	—
228	C₄×C₈	—	178	2	no	—
229	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
230	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
231	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
232	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
233	#41	—	4	2	no	—
234	#41	—	4	2	no	—
235	C₂×Q₈	—	2	1	no	—
236	C₂×Q₈	—	356	2	no	—
237	C₂×Q₈	—	178	4	no	—
238	C₂×Q₈	—	712	4	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	—	2	1	no	—
242	C₂×D₄	C₂×D₃₅₆	1	2	yes	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	356	2	no	—
245	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
246	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
247	C₂²×C₄	#13083574	1	1	yes	characteristic
248	SD₁₆	—	712	8	no	—
249	M₁₆	—	4	2	no	—
250	M₁₆	—	4	4	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	4	4	no	—
254	C₂×C₈	—	178	4	no	—
255	C₂×C₈	#13083562	1	4	yes	complemented
256	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
257	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
258	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
259	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	—	356	2	no	—
262	C₄⋊C₄	—	356	2	no	—
263	C₄⋊C₄	—	178	4	no	—
264	C₄⋊C₄	—	178	4	no	—
265	C₄⋊C₄	C₂×D₃₅₆	1	2	yes	—
266	C₂²⋊C₄	—	2	2	no	—
267	C₂²⋊C₄	—	2	2	no	—
268	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
269	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
270	C₄²	#13083574	1	1	yes	characteristic
271	C₄²	C₂³×D₈₉	1	1	yes	characteristic
272	C₂³	—	2	2	no	—
273	C₂³	#15667986	1	1	yes	characteristic
274	Q₈	—	4	2	no	—
275	Q₈	—	356	8	no	—
276	Q₈	—	712	4	no	—
277	D₄	—	2	4	no	—
278	D₄	—	4	4	no	—
279	D₄	—	4	2	no	—
280	D₄	—	4	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	356	4	no	—
287	C₂×C₄	—	712	4	no	—
288	C₂×C₄	#15667986	1	1	yes	characteristic
289	C₂×C₄	#15667986	1	1	yes	characteristic
290	C₂×C₄	#15667986	1	1	yes	characteristic
291	C₂×C₄	C₂×#13083562	1	2	yes	—
292	C₂×C₄	#15668042	1	2	yes	—
293	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
294	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
295	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
296	C₈	—	2	1	no	—
297	C₈	—	2	1	no	—
298	C₈	—	4	2	no	—
299	C₈	—	356	4	no	—
300	C₈	C₂×#13083562	1	2	yes	—
301	C₈	C₂×#13083562	1	2	yes	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	4	2	no	—
305	C₂²	—	4	2	no	—
306	C₂²	—	4	2	no	—
307	C₂²	#19151803	1	2	yes	—
308	C₂²	C₂×#15667986	1	1	yes	characteristic
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	—	4	2	no	—
312	C₄	—	712	4	no	—
313	C₄	#19151803	1	2	yes	—
314	C₄	#19151873	1	2	yes	—
315	C₄	C₂²×#13083562	1	1	yes	characteristic
316	C₄	C₂×#15668042	1	1	yes	characteristic
317	C₄	#19152403	1	2	yes	—
318	C₂	—	2	2	no	—
319	C₂	—	8	2	no	—
320	C₂	C₂×#19151803	1	1	yes	characteristic
321	C₂	#26029900	1	1	yes	characteristic
322	C₂	C₂×#19151873	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#15667986
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#2931	C₂
socle	C₂²×C₈₉	C₂×#121
prime radical	#532×C₈₉	C₂²
hypercenter	#2931	D₈₉
hypocenter	C₈₉	#19192
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931

Complements

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

N	H ≅ G/N	type
2. C₂×#19152003	C₂	semidirect
3. #26019765	C₂	semidirect
4. #26019641	C₂	semidirect
5. #26019000	C₂	semidirect
6. #26018938	C₂	semidirect
97. C₂×C₈×C₈₉	SD₁₆	semidirect
169. C₈₉	#19192	semidirect
255. C₂×C₈	#13083562	semidirect

Maximal subgroups

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

subgroup	index
#26031174	2
C₂×#19152003	2
#26019765	2
#26019641	2
#26019000	2
#26018938	2
C₈₉×#2931	2
#19192	89