Subgroups — #36707930

Order 22784.

498 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	#26029181	C₂	1	1	yes	maximal, characteristic
2	#26022161	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022160	C₂	1	2	yes	maximal, complemented
4	#26022160	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022159	C₂	1	2	yes	maximal, complemented
6	#26022154	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022009	C₂	1	2	yes	maximal
8	C₄⋊C₁₆×D₈₉	C₂	1	1	yes	maximal, characteristic
9	#26021847	C₂	1	2	yes	maximal
10	C₄×#15667951	C₂	1	1	yes	maximal, characteristic
11	Q₈×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152152	C₄	1	1	yes	characteristic
13	C₂×#15667951	C₂²	1	1	yes	characteristic
14	C₂×#15667951	C₂²	1	1	yes	characteristic
15	C₂×C₁₆×D₈₉	C₂²	1	2	yes	—
16	#19151729	C₄	1	1	yes	characteristic
17	#19151727	C₄	1	2	yes	—
18	#19151725	C₄	1	1	yes	characteristic
19	C₈×Dic₁₇₈	C₂²	1	1	yes	characteristic
20	C₈×Dic₁₇₈	C₂²	1	2	yes	—
21	#19151623	C₄	1	2	yes	—
22	#19151620	C₄	1	1	yes	characteristic
23	#19151552	C₂²	1	2	yes	—
24	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
25	C₄⋊#13083549	C₂²	1	1	yes	characteristic
26	C₄⋊#13083549	C₂²	1	2	yes	—
27	#19151416	C₂²	1	2	yes	—
28	#19151416	C₂²	1	2	yes	—
29	#19151416	C₂²	1	2	yes	—
30	#19151374	C₂²	1	2	yes	—
31	#19151373	C₂²	1	1	yes	characteristic
32	#19151373	C₂²	1	1	yes	characteristic
33	#19151373	C₂²	1	2	yes	—
34	C₄×#13083549	C₂²	1	1	yes	characteristic
35	C₁₆×Dic₈₉	C₂²	1	2	yes	—
36	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
37	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
38	C₄⋊C₁₆×C₈₉	C₂²	1	2	yes	—
39	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₈₉	C₂²	1	2	yes	—
41	C₂×#13083557	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×D₈₉	C₂³	1	1	yes	characteristic
44	#15668011	C₈	1	1	yes	characteristic
45	#15668010	C₈	1	1	yes	characteristic
46	#15668002	C₈	1	2	yes	—
47	#15667992	C₈	1	1	yes	characteristic
48	#15667990	C₈	1	2	yes	—
49	#15667988	C₈	1	1	yes	characteristic
50	C₄×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
51	C₄×Dic₁₇₈	C₂×C₄	1	2	yes	—
52	#15667969	C₂×C₄	1	2	yes	—
53	C₄²×D₈₉	C₂×C₄	1	1	yes	characteristic
54	#15667951	—	2	2	no	—
55	#15667951	Q₈	1	4	yes	complemented
56	C₂×#13083549	C₂³	1	1	yes	characteristic
57	C₂×#13083549	C₂³	1	2	yes	—
58	C₂×#13083549	C₂³	1	1	yes	characteristic
59	C₁₆×D₈₉	—	2	4	no	—
60	C₄⋊#10623494	C₂×C₄	1	1	yes	characteristic
61	C₄⋊#10623494	C₂×C₄	1	2	yes	—
62	#15667941	C₂×C₄	1	2	yes	—
63	#15667941	C₂×C₄	1	2	yes	—
64	#15667941	C₂×C₄	1	2	yes	—
65	#15667923	C₂×C₄	1	2	yes	—
66	#15667922	C₂×C₄	1	2	yes	—
67	C₄×#10623494	C₂×C₄	1	1	yes	characteristic
68	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
69	C₈×Dic₈₉	C₂³	1	2	yes	—
70	C₈×Dic₈₉	C₂³	1	1	yes	characteristic
71	C₄×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
72	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
73	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
75	C₄⋊C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
76	C₄⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
77	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₈₉	C₂³	1	2	yes	—
79	C₂×Dic₁₇₈	C₂×C₈	1	1	yes	characteristic
80	C₂×Dic₁₇₈	C₂×C₈	1	2	yes	—
81	C₂×C₄×D₈₉	C₂×C₈	1	1	yes	characteristic
82	C₂×C₄×D₈₉	C₂×C₈	1	1	yes	characteristic
83	C₂×C₄×D₈₉	C₂²×C₄	1	1	yes	characteristic
84	#13083557	—	2	4	no	—
85	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
86	C₂×#10623494	C₂²×C₄	1	2	yes	—
87	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
88	C₈×D₈₉	—	2	2	no	—
89	C₈×D₈₉	C₂×Q₈	1	2	yes	—
90	C₈×D₈₉	#41	1	2	yes	—
91	C₄⋊Dic₈₉	C₂×C₈	1	1	yes	characteristic
92	C₄⋊Dic₈₉	C₂×C₈	1	2	yes	—
93	#13083553	C₂×C₈	1	2	yes	—
94	#13083553	C₂×C₈	1	2	yes	—
95	#13083553	C₂×C₈	1	2	yes	—
96	#13083553	C₂×C₈	1	2	yes	—
97	#13083553	C₂×C₈	1	2	yes	—
98	#13083551	C₂×C₈	1	2	yes	—
99	#13083551	C₂×C₈	1	2	yes	—
100	C₄×Dic₈₉	C₂×C₈	1	1	yes	characteristic
101	C₄×Dic₈₉	C₂×C₈	1	1	yes	characteristic
102	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
103	C₄×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
104	#13083549	—	2	2	no	—
105	#13083549	—	2	1	no	—
106	#13083549	C₂×Q₈	1	2	yes	—
107	C₂×Q₈×C₈₉	C₂×C₈	1	1	yes	characteristic
108	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
110	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
111	C₄⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
112	C₄⋊C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
113	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
114	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
115	C₁₆×C₈₉	—	2	2	no	—
116	C₁₆×C₈₉	—	2	1	no	—
117	C₁₆×C₈₉	C₂×Q₈	1	2	yes	—
118	C₂²×D₈₉	C₂²×C₈	1	1	yes	characteristic
119	Q₈×C₈₉	—	2	2	no	—
120	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
122	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
123	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
125	C₂×Dic₈₉	C₂²×C₈	1	1	yes	characteristic
126	C₂×Dic₈₉	C₂²×C₈	1	1	yes	characteristic
127	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
128	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
129	C₂×Dic₈₉	C₂²×C₈	1	1	yes	characteristic
130	C₄×D₈₉	—	2	2	no	—
131	C₄×D₈₉	—	4	1	no	—
132	C₄×D₈₉	C₄×Q₈	1	2	yes	—
133	C₄×D₈₉	#131	1	2	yes	—
134	Dic₁₇₈	—	2	2	no	—
135	Dic₁₇₈	—	4	2	no	—
136	C₈×C₈₉	—	2	1	no	—
137	C₈×C₈₉	—	2	2	no	—
138	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
139	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
140	#10623494	—	2	1	no	—
141	#10623494	—	2	2	no	—
142	#10623494	C₄×Q₈	1	2	yes	—
143	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
144	C₂×D₈₉	—	2	1	no	—
145	C₂×D₈₉	—	2	1	no	—
146	C₂×D₈₉	C₈×Q₈	1	2	yes	—
147	C₄×C₈₉	—	2	2	no	—
148	C₄×C₈₉	—	2	2	no	—
149	C₄×C₈₉	—	2	1	no	—
150	C₄×C₈₉	#504	1	2	yes	—
151	C₄×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
152	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
153	Dic₈₉	—	2	2	no	—
154	Dic₈₉	—	2	1	no	—
155	Dic₈₉	—	2	1	no	—
156	Dic₈₉	—	2	2	no	—
157	Dic₈₉	C₈×Q₈	1	2	yes	—
158	#33476	—	89	1	no	maximal, Sylow
159	C₂×C₈₉	C₂×C₈×Q₈	1	1	yes	characteristic
160	C₂×C₈₉	C₂×#504	1	1	yes	characteristic
161	C₂×C₈₉	#3407	1	1	yes	characteristic
162	D₈₉	—	2	2	no	—
163	#2960	—	89	1	no	—
164	#2183	—	89	1	no	—
165	#2183	—	89	2	no	—
166	#2183	—	89	1	no	—
167	#2183	—	89	2	no	—
168	Q₈×C₁₆	—	89	1	no	—
169	Q₈×C₁₆	D₈₉	1	1	yes	characteristic, complemented
170	#2151	—	89	2	no	—
171	C₂×C₄⋊C₁₆	—	89	1	no	—
172	#2108	—	89	2	no	—
173	C₄×M₃₂	—	89	1	no	—
174	C₈₉	#33476	1	1	yes	characteristic, complemented, Sylow
175	#533	—	89	1	no	—
176	C₂×M₃₂	—	89	1	no	—
177	C₂×M₃₂	—	89	1	no	—
178	C₂²×C₁₆	—	89	2	no	—
179	#446	—	89	1	no	—
180	#446	—	89	1	no	—
181	#446	—	89	2	no	—
182	C₈×Q₈	—	89	1	no	—
183	C₈×Q₈	—	89	2	no	—
184	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
185	#424	—	89	2	no	—
186	#423	—	89	1	no	—
187	#405	—	89	2	no	—
188	C₂×C₄×C₈	—	89	1	no	—
189	C₄⋊C₁₆	—	89	2	no	—
190	C₄⋊C₁₆	—	89	2	no	—
191	C₄⋊C₁₆	—	89	2	no	—
192	C₄⋊C₁₆	—	89	1	no	—
193	C₄⋊C₁₆	—	89	2	no	—
194	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
195	C₄⋊C₁₆	C₂×D₈₉	1	2	yes	—
196	#346	—	89	1	no	—
197	#346	—	89	1	no	—
198	#346	—	89	2	no	—
199	#346	—	89	2	no	—
200	C₄×C₁₆	—	89	2	no	—
201	C₄×C₁₆	—	89	1	no	—
202	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
203	C₄×C₁₆	C₂×D₈₉	1	2	yes	—
204	C₂×M₁₆	—	89	2	no	—
205	C₂²×C₈	—	89	1	no	—
206	C₂²×C₈	—	89	1	no	—
207	C₄⋊Q₈	—	89	1	no	—
208	C₄⋊Q₈	—	89	1	no	—
209	#125	—	89	2	no	—
210	C₂²⋊Q₈	—	89	1	no	—
211	C₂²⋊Q₈	—	89	2	no	—
212	C₂²⋊Q₈	—	89	1	no	—
213	C₄×Q₈	—	89	2	no	—
214	C₄×Q₈	—	89	1	no	—
215	C₄×Q₈	C₄×D₈₉	1	1	yes	characteristic
216	#117	—	89	2	no	—
217	C₂×C₄²	—	89	1	no	—
218	M₃₂	—	89	4	no	—
219	M₃₂	—	178	2	no	—
220	C₂×C₁₆	—	89	1	no	—
221	C₂×C₁₆	—	89	1	no	—
222	C₂×C₁₆	—	89	2	no	—
223	C₂×C₁₆	—	178	4	no	—
224	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
225	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
226	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
227	C₄⋊C₈	—	89	2	no	—
228	C₄⋊C₈	—	89	2	no	—
229	C₄⋊C₈	—	89	2	no	—
230	C₄⋊C₈	—	89	1	no	—
231	C₄⋊C₈	—	89	2	no	—
232	C₄⋊C₈	C₄×D₈₉	1	1	yes	characteristic
233	C₄⋊C₈	C₄×D₈₉	1	2	yes	—
234	#97	—	89	2	no	—
235	#97	—	89	2	no	—
236	C₄×C₈	—	89	1	no	—
237	C₄×C₈	—	89	2	no	—
238	C₄×C₈	—	89	1	no	—
239	C₄×C₈	—	89	1	no	—
240	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
241	C₄×C₈	C₂²×D₈₉	1	2	yes	—
242	C₂×Q₈	—	89	2	no	—
243	C₂×Q₈	—	89	1	no	—
244	C₂×Q₈	C₈×D₈₉	1	1	yes	characteristic
245	C₂²×C₄	—	89	1	no	—
246	C₂²×C₄	—	89	1	no	—
247	C₂²×C₄	—	89	1	no	—
248	M₁₆	—	178	4	no	—
249	C₂×C₈	—	89	1	no	—
250	C₂×C₈	—	89	2	no	—
251	C₂×C₈	—	89	2	no	—
252	C₂×C₈	—	89	1	no	—
253	C₂×C₈	—	89	2	no	—
254	C₂×C₈	—	178	2	no	—
255	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
256	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
257	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
258	C₄⋊C₄	—	89	2	no	—
259	C₄⋊C₄	—	89	2	no	—
260	C₄⋊C₄	—	89	2	no	—
261	C₄⋊C₄	—	89	2	no	—
262	C₄⋊C₄	—	89	2	no	—
263	C₄⋊C₄	—	89	2	no	—
264	C₄⋊C₄	—	89	1	no	—
265	C₄⋊C₄	C₈×D₈₉	1	2	yes	—
266	C₄⋊C₄	C₈×D₈₉	1	1	yes	characteristic
267	C₂²⋊C₄	—	89	2	no	—
268	C₂²⋊C₄	—	89	2	no	—
269	C₄²	—	89	1	no	—
270	C₄²	—	89	1	no	—
271	C₄²	—	89	1	no	—
272	C₄²	—	89	2	no	—
273	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
274	C₄²	C₂×C₄×D₈₉	1	2	yes	—
275	C₁₆	—	2	2	no	—
276	C₁₆	—	2	1	no	—
277	C₁₆	—	89	2	no	—
278	C₁₆	—	178	2	no	—
279	C₁₆	—	178	1	no	—
280	C₁₆	Q₈×D₈₉	1	2	yes	—
281	C₂³	—	89	1	no	—
282	Q₈	—	2	2	no	—
283	Q₈	—	356	2	no	—
284	Q₈	—	178	2	no	—
285	C₂×C₄	—	89	1	no	—
286	C₂×C₄	—	89	2	no	—
287	C₂×C₄	—	89	2	no	—
288	C₂×C₄	—	89	1	no	—
289	C₂×C₄	—	89	1	no	—
290	C₂×C₄	—	89	2	no	—
291	C₂×C₄	—	89	2	no	—
292	C₂×C₄	—	356	1	no	—
293	C₂×C₄	—	178	2	no	—
294	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
295	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
296	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
297	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
298	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	2	2	no	—
301	C₈	—	89	2	no	—
302	C₈	—	178	1	no	—
303	C₈	—	178	2	no	—
304	C₈	C₂×Q₈×D₈₉	1	1	yes	characteristic
305	C₈	#41×D₈₉	1	1	yes	characteristic
306	C₂²	—	89	2	no	—
307	C₂²	—	178	1	no	—
308	C₂²	—	178	1	no	—
309	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
310	C₄	—	2	2	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	2	no	—
313	C₄	—	89	2	no	—
314	C₄	—	178	1	no	—
315	C₄	—	178	2	no	—
316	C₄	—	178	1	no	—
317	C₄	—	178	2	no	—
318	C₄	#19152014	1	2	yes	—
319	C₄	C₄×Q₈×D₈₉	1	1	yes	characteristic
320	C₄	#131×D₈₉	1	1	yes	characteristic
321	C₂	—	178	2	no	—
322	C₂	C₈×Q₈×D₈₉	1	1	yes	characteristic
323	C₂	C₂×#19152014	1	1	yes	characteristic
324	C₂	#26033353	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₈
Fitting	Q₈×C₁₆×C₈₉	C₂
prime radical	C₂²×D₈₉	C₂²×C₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	Q₈×C₁₆	D₈₉
hypocenter	C₈₉	#33476
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ Q₈×C₁₆

Complements

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

N	H ≅ G/N	type
2. #26022161	C₂	semidirect
3. #26022160	C₂	semidirect
4. #26022160	C₂	semidirect
5. #26022159	C₂	semidirect
6. #26022154	C₂	semidirect
11. Q₈×C₁₆×C₈₉	C₂	semidirect
55. #15667951	Q₈	semidirect
169. Q₈×C₁₆	D₈₉	semidirect
174. C₈₉	#33476	semidirect

Maximal subgroups

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

subgroup	index
#26029181	2
#26022161	2
#26022160	2
#26022159	2
#26022154	2
#26022009	2
C₄⋊C₁₆×D₈₉	2
#26021847	2
C₄×#15667951	2
Q₈×C₁₆×C₈₉	2
#33476	89