Subgroups — #36707604

Order 22784.

682 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	C₂×#19152016	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#15667952	C₂	1	1	yes	maximal, characteristic, complemented
3	#26028946	C₂	1	1	yes	maximal, characteristic
4	#26022081	C₂	1	4	yes	maximal, complemented
5	#26022080	C₂	1	4	yes	maximal, complemented
6	#26022012	C₂	1	1	yes	maximal, characteristic, complemented
7	#26021867	C₂	1	1	yes	maximal, characteristic, complemented
8	#26021851	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#2112	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#15668018	C₂²	1	1	yes	characteristic
11	#19152109	C₄	1	1	yes	characteristic
12	#19152016	—	2	4	no	—
13	C₂×#15667952	—	2	4	no	—
14	C₂×#15667952	C₂²	1	2	yes	—
15	C₂×#15667952	C₂²	1	2	yes	—
16	C₂×#15667951	C₂²	1	2	yes	complemented
17	C₂²×#13083549	C₂²	1	1	yes	characteristic
18	C₂²×#13083549	C₂²	1	1	yes	characteristic
19	C₂×C₁₆×D₈₉	C₂²	1	2	yes	complemented
20	#19151669	C₄	1	4	yes	—
21	C₈×C₂²⋊D₈₉	C₂²	1	4	yes	—
22	#19151624	C₄	1	1	yes	characteristic
23	#19151564	C₄	1	1	yes	characteristic
24	#19151562	C₄	1	1	yes	characteristic
25	#19151553	C₂²	1	1	yes	characteristic
26	#19151416	C₂²	1	4	yes	complemented
27	#19151377	C₂²	1	4	yes	complemented
28	#19151373	C₂²	1	2	yes	complemented
29	C₁₆×Dic₈₉	C₂²	1	2	yes	complemented
30	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
31	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
32	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
33	C₂²⋊C₁₆×C₈₉	C₂²	1	4	yes	—
34	C₂×#13083574	C₂×C₄	1	1	yes	characteristic
35	#15668018	—	2	2	no	—
36	#15668018	—	2	2	no	—
37	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
38	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
39	C₂×#13083557	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₈₉	C₂³	1	2	yes	—
41	#15667995	C₈	1	2	yes	—
42	#15667991	C₈	1	2	yes	—
43	#15667986	C₈	1	2	yes	—
44	#15667981	C₈	1	2	yes	—
45	C₄×C₂²⋊D₈₉	C₂×C₄	1	4	yes	—
46	#15667971	C₂×C₄	1	1	yes	characteristic
47	#15667952	—	2	4	no	—
48	#15667952	—	2	4	no	—
49	#15667951	—	2	4	no	—
50	C₂×#13083549	—	2	2	no	—
51	C₂×#13083549	D₄	1	4	yes	complemented
52	C₂×#13083549	C₂³	1	2	yes	—
53	C₂×#13083549	C₂³	1	1	yes	characteristic
54	C₂×#13083549	C₂³	1	1	yes	characteristic
55	C₁₆×D₈₉	—	2	4	no	—
56	#15667941	C₂×C₄	1	4	yes	—
57	C₂²⋊#10623494	C₂×C₄	1	4	yes	—
58	#15667924	C₂×C₄	1	4	yes	—
59	#15667922	C₂×C₄	1	2	yes	—
60	C₈×Dic₈₉	C₂³	1	2	yes	—
61	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
62	C₂²×C₈×C₈₉	—	2	2	no	—
63	C₂²×C₈×C₈₉	—	2	2	no	—
64	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
65	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
66	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
67	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
68	M₃₂×C₈₉	—	2	2	no	—
69	M₃₂×C₈₉	—	2	2	no	—
70	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
71	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
72	#13083574	—	2	2	no	—
73	#13083574	—	2	2	no	—
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×C₂²⋊D₈₉	C₂×C₈	1	4	yes	—
78	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
79	#13083558	—	2	2	no	—
80	#13083558	—	2	2	no	—
81	#13083557	—	2	4	no	—
82	C₂×#10623494	C₂²×C₄	1	2	yes	—
83	C₂×#10623494	C₂²×C₄	1	2	yes	—
84	C₈×D₈₉	—	2	2	no	—
85	C₈×D₈₉	—	2	2	no	—
86	C₄⋊Dic₈₉	C₂×C₈	1	2	yes	—
87	#13083553	C₂×C₈	1	4	yes	—
88	C₂²⋊Dic₈₉	C₂×C₈	1	2	yes	—
89	C₂²⋊Dic₈₉	C₂×C₈	1	4	yes	—
90	#13083551	C₂×C₈	1	4	yes	—
91	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
92	#13083549	—	2	2	no	—
93	#13083549	C₂×D₄	1	2	yes	—
94	#13083549	C₂×D₄	1	2	yes	—
95	C₂⁴×C₈₉	C₂×C₈	1	1	yes	characteristic
96	C₂²×C₄×C₈₉	—	2	2	no	—
97	C₂²×C₄×C₈₉	—	2	2	no	—
98	C₂²×C₄×C₈₉	C₂×C₈	1	2	yes	—
99	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
100	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
101	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₈₉	—	2	2	no	—
103	C₂×C₈×C₈₉	—	2	4	no	—
104	C₂×C₈×C₈₉	—	2	2	no	—
105	C₂×C₈×C₈₉	—	2	2	no	—
106	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
107	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
109	C₂×C₈×C₈₉	#41	1	2	yes	—
110	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
111	C₁₆×C₈₉	—	2	2	no	—
112	C₁₆×C₈₉	—	2	2	no	—
113	C₂³×C₈₉	—	2	2	no	—
114	C₂³×C₈₉	—	4	2	no	—
115	C₂³×C₈₉	C₂²×C₈	1	2	yes	—
116	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
117	C₂²×D₈₉	C₂²×C₈	1	2	yes	—
118	C₂×C₄×C₈₉	—	2	2	no	—
119	C₂×C₄×C₈₉	—	2	2	no	—
120	C₂×C₄×C₈₉	—	2	2	no	—
121	C₂×C₄×C₈₉	—	2	4	no	—
122	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
124	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₄×C₈₉	#131	1	2	yes	—
126	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
127	C₂²⋊D₈₉	—	2	4	no	—
128	C₂²⋊D₈₉	—	4	4	no	—
129	C₂×Dic₈₉	C₂²×C₈	1	4	yes	—
130	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
131	D₃₅₆	—	2	2	no	—
132	C₄×D₈₉	—	2	2	no	—
133	C₄×D₈₉	—	2	2	no	—
134	Dic₁₇₈	—	2	2	no	—
135	C₈×C₈₉	—	2	2	no	—
136	C₈×C₈₉	C₄×D₄	1	2	yes	—
137	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
138	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
139	#10623494	—	2	2	no	—
140	#10623494	—	2	2	no	—
141	C₂²×C₈₉	—	2	2	no	—
142	C₂²×C₈₉	—	2	2	no	—
143	C₂²×C₈₉	—	2	2	no	—
144	C₂²×C₈₉	—	2	2	no	—
145	C₂²×C₈₉	—	4	4	no	—
146	C₂²×C₈₉	C₈×D₄	1	2	yes	—
147	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
148	C₂×D₈₉	—	2	4	no	—
149	C₂×D₈₉	—	2	2	no	—
150	C₄×C₈₉	—	2	2	no	—
151	C₄×C₈₉	C₈×D₄	1	2	yes	—
152	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
153	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
154	Dic₈₉	—	2	4	no	—
155	Dic₈₉	—	2	2	no	—
156	#33433	—	89	1	no	maximal, Sylow
157	C₂×C₈₉	—	2	2	no	—
158	C₂×C₈₉	—	4	2	no	—
159	C₂×C₈₉	C₂×C₈×D₄	1	1	yes	characteristic
160	C₂×C₈₉	#3407	1	2	yes	—
161	D₈₉	—	4	2	no	—
162	C₂×#504	—	89	1	no	—
163	C₂²×M₃₂	—	89	1	no	—
164	#2929	—	89	1	no	—
165	#2169	—	89	4	no	—
166	#2168	—	89	4	no	—
167	#2151	—	89	1	no	—
168	#2113	—	89	1	no	—
169	#2112	D₈₉	1	1	yes	characteristic, complemented
170	#2108	—	89	1	no	—
171	C₈₉	#33433	1	1	yes	characteristic, complemented, Sylow
172	C₂×#131	—	89	1	no	—
173	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
174	#525	—	89	1	no	—
175	#504	—	178	4	no	—
176	C₂×M₃₂	—	89	2	no	—
177	C₂×M₃₂	—	89	2	no	—
178	C₂×M₃₂	—	89	2	no	—
179	C₂×M₃₂	—	178	4	no	—
180	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
181	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
182	C₂²×C₁₆	—	89	1	no	—
183	C₂²×C₁₆	—	89	1	no	—
184	C₂²×C₁₆	—	89	2	no	—
185	#435	—	89	4	no	—
186	C₈×D₄	—	89	4	no	—
187	#424	—	89	1	no	—
188	#408	—	89	1	no	—
189	#407	—	89	1	no	—
190	#405	—	89	1	no	—
191	C₄⋊C₁₆	—	89	4	no	—
192	C₂²⋊C₁₆	—	89	4	no	—
193	C₂²⋊C₁₆	C₂×D₈₉	1	4	yes	—
194	#346	—	89	2	no	—
195	C₄×C₁₆	—	89	2	no	—
196	C₂×#41	—	89	1	no	—
197	C₂³×C₄	C₄×D₈₉	1	1	yes	characteristic
198	#131	—	178	2	no	—
199	#131	—	178	2	no	—
200	C₂×M₁₆	—	89	1	no	—
201	C₂×M₁₆	—	89	1	no	—
202	C₂×M₁₆	—	89	2	no	—
203	C₂²×C₈	—	2	2	no	—
204	C₂²×C₈	—	2	2	no	—
205	C₂²×C₈	—	89	2	no	—
206	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
207	C₂²×C₈	C₄×D₈₉	1	2	yes	—
208	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
209	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
210	#123	—	89	2	no	—
211	C₂²⋊Q₈	—	89	2	no	—
212	#121	—	89	2	no	—
213	#120	—	89	2	no	—
214	C₄×D₄	—	89	4	no	—
215	#117	—	89	1	no	—
216	M₃₂	—	2	2	no	—
217	M₃₂	—	2	2	no	—
218	M₃₂	—	178	4	no	—
219	M₃₂	—	178	4	no	—
220	M₃₂	—	178	4	no	—
221	C₂×C₁₆	—	89	4	no	—
222	C₂×C₁₆	—	89	1	no	—
223	C₂×C₁₆	—	89	1	no	—
224	C₂×C₁₆	—	89	2	no	—
225	C₂×C₁₆	—	178	4	no	—
226	C₂×C₁₆	—	178	2	no	—
227	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
228	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
229	C₄⋊C₈	—	89	4	no	—
230	C₂²⋊C₈	—	89	4	no	—
231	C₂²⋊C₈	—	89	4	no	—
232	#97	—	89	2	no	—
233	C₄×C₈	—	89	2	no	—
234	C₂⁴	C₈×D₈₉	1	1	yes	characteristic
235	#41	—	178	2	no	—
236	#41	—	178	2	no	—
237	C₂×Q₈	—	89	1	no	—
238	C₂×D₄	—	89	4	no	—
239	C₂×D₄	—	89	2	no	—
240	C₂×D₄	—	89	1	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	89	2	no	—
244	C₂²×C₄	C₈×D₈₉	1	2	yes	—
245	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
247	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
248	M₁₆	—	178	2	no	—
249	M₁₆	—	178	2	no	—
250	M₁₆	—	178	4	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	4	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	89	2	no	—
256	C₂×C₈	—	89	2	no	—
257	C₂×C₈	—	178	2	no	—
258	C₂×C₈	—	178	2	no	—
259	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
260	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
261	C₂×C₈	D₄×D₈₉	1	2	yes	—
262	C₂×C₈	#13083573	1	2	yes	—
263	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
264	C₄⋊C₄	—	89	2	no	—
265	C₄⋊C₄	—	89	4	no	—
266	C₂²⋊C₄	—	89	2	no	—
267	C₂²⋊C₄	—	89	4	no	—
268	C₂²⋊C₄	—	89	4	no	—
269	C₄²	—	89	2	no	—
270	C₁₆	—	2	2	no	—
271	C₁₆	—	2	2	no	—
272	C₁₆	—	89	2	no	—
273	C₁₆	—	89	2	no	—
274	C₁₆	—	178	2	no	—
275	C₂³	—	2	2	no	—
276	C₂³	—	4	2	no	—
277	C₂³	—	89	2	no	—
278	C₂³	C₂×C₈×D₈₉	1	2	yes	—
279	C₂³	C₂×C₈×D₈₉	1	1	yes	characteristic
280	Q₈	—	178	2	no	—
281	D₄	—	356	4	no	—
282	D₄	—	178	4	no	—
283	D₄	—	178	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	89	4	no	—
289	C₂×C₄	—	89	2	no	—
290	C₂×C₄	—	178	2	no	—
291	C₂×C₄	—	178	2	no	—
292	C₂×C₄	#15667972	1	2	yes	—
293	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
294	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
295	C₂×C₄	#15668020	1	2	yes	—
296	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
297	C₈	—	2	2	no	—
298	C₈	—	178	2	no	—
299	C₈	—	178	2	no	—
300	C₈	#15667972	1	2	yes	—
301	C₈	C₂×D₄×D₈₉	1	1	yes	characteristic
302	C₈	C₂×#13083573	1	1	yes	characteristic
303	C₂²	—	2	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	4	4	no	—
308	C₂²	—	178	2	no	—
309	C₂²	—	178	4	no	—
310	C₂²	#19151661	1	2	yes	—
311	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	178	2	no	—
314	C₄	—	178	4	no	—
315	C₄	#19151661	1	2	yes	—
316	C₄	C₂×#15667972	1	1	yes	characteristic
317	C₄	C₂×#15668020	1	1	yes	characteristic
318	C₂	—	2	2	no	—
319	C₂	—	4	2	no	—
320	C₂	—	356	2	no	—
321	C₂	C₂×#19151661	1	1	yes	characteristic
322	C₂	#26033354	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₈
Fitting	C₈₉×#2112	C₂
prime radical	#19152109	C₄
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2112	D₈₉
hypocenter	C₈₉	#33433
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ #2112

Complements

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

N	H ≅ G/N	type
1. C₂×#19152016	C₂	semidirect
2. C₂²×#15667952	C₂	semidirect
4. #26022081	C₂	semidirect
5. #26022080	C₂	semidirect
6. #26022012	C₂	semidirect
7. #26021867	C₂	semidirect
8. #26021851	C₂	semidirect
9. C₈₉×#2112	C₂	semidirect
16. C₂×#15667951	C₂²	semidirect
19. C₂×C₁₆×D₈₉	C₂²	semidirect
26. #19151416	C₂²	semidirect
27. #19151377	C₂²	semidirect
28. #19151373	C₂²	semidirect
29. C₁₆×Dic₈₉	C₂²	semidirect
51. C₂×#13083549	D₄	semidirect
169. #2112	D₈₉	semidirect
171. C₈₉	#33433	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19152016	2
C₂²×#15667952	2
#26028946	2
#26022081	2
#26022080	2
#26022012	2
#26021867	2
#26021851	2
C₈₉×#2112	2
#33433	89