Subgroups — #36447353

Order 22784.

486 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₃₅₆	C₂	1	1	yes	maximal, characteristic
2	#26020241	C₂	1	2	yes	maximal
3	#26019011	C₂	1	1	yes	maximal, characteristic, complemented
4	#26018126	C₂	1	2	yes	maximal, complemented
5	C₈₉×#1567	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₈₉	C₄	1	2	yes	—
7	C₂×C₄×D₃₅₆	C₂²	1	1	yes	characteristic
8	C₈×D₃₅₆	—	2	1	no	—
9	C₈×D₃₅₆	—	2	1	no	—
10	C₈×D₃₅₆	—	2	1	no	—
11	C₈×D₃₅₆	—	2	1	no	—
12	#19151622	C₂²	1	2	yes	—
13	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
14	C₂×#15667928	—	2	2	no	—
15	C₂×#15667924	C₄	1	2	yes	—
16	#19151366	—	2	2	no	—
17	#19151351	—	2	1	no	—
18	#19151351	—	2	1	no	—
19	#423×C₈₉	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
21	#335×C₈₉	—	2	1	no	—
22	#335×C₈₉	—	2	1	no	—
23	C₂²×D₃₅₆	D₄	1	1	yes	characteristic
24	C₂²×C₄×D₈₉	C₂×C₄	1	2	yes	—
25	C₂×#13083558	—	2	2	no	—
26	C₂²×#10623494	C₂×C₄	1	2	yes	—
27	C₂×C₈×D₈₉	—	2	2	no	—
28	C₂×C₈×D₈₉	—	2	2	no	—
29	C₂×C₈×D₈₉	—	4	4	no	—
30	C₄×D₃₅₆	—	2	1	no	—
31	C₄×D₃₅₆	—	2	1	no	—
32	C₄×D₃₅₆	—	2	1	no	—
33	C₄×D₃₅₆	—	2	1	no	—
34	C₂×C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
35	C₂×#13083551	Q₈	1	2	yes	—
36	C₄⋊#10623494	—	2	1	no	—
37	C₄⋊#10623494	—	2	1	no	—
38	C₄⋊#10623494	—	2	2	no	—
39	C₄⋊#10623494	—	2	2	no	—
40	#15667928	—	4	4	no	—
41	#15667924	—	2	2	no	—
42	#15667924	—	2	2	no	—
43	C₂×M₁₆×C₈₉	—	2	2	no	—
44	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
47	C₄⋊C₈×C₈₉	—	2	2	no	—
48	C₄⋊C₈×C₈₉	—	2	2	no	—
49	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
50	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
51	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
52	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
53	C₂³×D₈₉	C₄⋊C₄	1	2	yes	—
54	C₂×D₃₅₆	—	2	1	no	—
55	C₂×D₃₅₆	—	2	1	no	—
56	C₂×D₃₅₆	—	2	1	no	—
57	C₂×D₃₅₆	—	2	1	no	—
58	C₂×D₃₅₆	D₈	1	2	yes	—
59	C₂×D₃₅₆	Q₁₆	1	2	yes	—
60	C₂²×Dic₈₉	C₄⋊C₄	1	2	yes	—
61	C₂×C₄×D₈₉	—	2	2	no	—
62	C₂×C₄×D₈₉	—	2	2	no	—
63	C₂×C₄×D₈₉	—	4	2	no	—
64	C₂×C₄×D₈₉	—	4	2	no	—
65	#13083558	—	4	2	no	—
66	#13083558	—	4	2	no	—
67	C₂×#10623494	—	2	2	no	—
68	C₂×#10623494	—	4	2	no	—
69	C₂×#10623494	—	4	2	no	—
70	C₈×D₈₉	—	4	4	no	—
71	C₈×D₈₉	—	4	4	no	—
72	C₄⋊Dic₈₉	—	2	1	no	—
73	C₄⋊Dic₈₉	—	2	1	no	—
74	#13083551	—	2	2	no	—
75	#13083551	—	2	2	no	—
76	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
79	M₁₆×C₈₉	—	4	2	no	—
80	M₁₆×C₈₉	—	4	2	no	—
81	C₂×C₈×C₈₉	—	2	2	no	—
82	C₂×C₈×C₈₉	—	4	1	no	—
83	C₂×C₈×C₈₉	—	4	2	no	—
84	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
88	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₈₉	#41	1	1	yes	characteristic
91	C₄²×C₈₉	#41	1	1	yes	characteristic
92	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂²×D₈₉	—	2	2	no	—
94	C₂²×D₈₉	—	2	2	no	—
95	C₂²×D₈₉	—	4	2	no	—
96	C₂²×D₈₉	—	4	2	no	—
97	C₂²×D₈₉	#107	1	4	yes	—
98	C₂×C₄×C₈₉	—	2	1	no	—
99	C₂×C₄×C₈₉	—	2	1	no	—
100	C₂×C₄×C₈₉	—	4	1	no	—
101	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
103	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
104	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
105	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
108	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
109	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
110	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
111	C₂×Dic₈₉	—	2	2	no	—
112	C₂×Dic₈₉	—	4	2	no	—
113	D₃₅₆	—	2	2	no	—
114	D₃₅₆	—	2	2	no	—
115	D₃₅₆	—	2	2	no	—
116	D₃₅₆	—	2	2	no	—
117	C₄×D₈₉	—	4	4	no	—
118	C₄×D₈₉	—	4	2	no	—
119	C₄×D₈₉	—	4	2	no	—
120	C₈×C₈₉	—	2	1	no	—
121	C₈×C₈₉	—	2	1	no	—
122	C₈×C₈₉	—	2	1	no	—
123	C₈×C₈₉	—	8	2	no	—
124	C₈×C₈₉	#100	1	2	yes	complemented
125	#10623494	—	4	2	no	—
126	#10623494	—	8	2	no	—
127	C₂²×C₈₉	—	2	1	no	—
128	C₂²×C₈₉	—	2	1	no	—
129	C₂²×C₈₉	#380	1	1	yes	characteristic
130	C₂²×C₈₉	C₂×#107	1	1	yes	characteristic
131	C₂²×C₈₉	#427	1	1	yes	characteristic
132	C₂×D₈₉	—	2	4	no	—
133	C₂×D₈₉	—	4	4	no	—
134	C₂×D₈₉	—	4	2	no	—
135	C₂×D₈₉	—	4	2	no	—
136	C₂×D₈₉	—	4	4	no	—
137	C₄×C₈₉	—	2	1	no	—
138	C₄×C₈₉	—	2	1	no	—
139	C₄×C₈₉	—	2	1	no	—
140	C₄×C₈₉	C₂×#100	1	1	yes	characteristic
141	C₄×C₈₉	#413	1	1	yes	characteristic
142	C₄×C₈₉	#466	1	1	yes	characteristic
143	C₄×C₈₉	#466	1	1	yes	characteristic
144	C₄×C₈₉	#467	1	1	yes	characteristic
145	C₄×C₈₉	#467	1	1	yes	characteristic
146	Dic₈₉	—	4	2	no	—
147	#8616	—	89	1	no	maximal, Sylow
148	C₂×C₈₉	—	2	1	no	—
149	C₂×C₈₉	—	2	1	no	—
150	C₂×C₈₉	#1800	1	1	yes	characteristic
151	C₂×C₈₉	#1808	1	1	yes	characteristic
152	C₂×C₈₉	#1813	1	1	yes	characteristic
153	D₈₉	—	4	4	no	—
154	C₂×C₈×D₄	—	89	1	no	—
155	#1800	—	89	2	no	—
156	#1567	—	89	1	no	—
157	#1567	D₈₉	1	1	yes	characteristic, complemented
158	#1292	—	89	2	no	—
159	C₈₉	#8616	1	1	yes	characteristic, complemented, Sylow
160	C₂³×C₈	—	89	2	no	—
161	C₂×C₄×D₄	—	89	1	no	—
162	C₈×D₄	—	178	1	no	—
163	C₈×D₄	—	178	1	no	—
164	C₈×D₄	—	178	1	no	—
165	C₈×D₄	—	178	1	no	—
166	#423	—	89	2	no	—
167	#423	C₂×D₈₉	1	2	yes	—
168	C₂×C₄⋊C₈	—	89	1	no	—
169	C₂×#100	—	178	2	no	—
170	C₂×C₂²⋊C₈	—	89	2	no	—
171	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
172	#343	—	178	2	no	—
173	#335	—	2	1	no	—
174	#335	—	2	1	no	—
175	#335	—	178	1	no	—
176	#335	—	178	1	no	—
177	C₂²×D₄	—	89	1	no	—
178	C₂³×C₄	—	89	2	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	178	2	no	—
181	C₂²×C₈	—	89	2	no	—
182	C₂²×C₈	—	356	4	no	—
183	C₂²×C₈	—	178	2	no	—
184	C₂²×C₈	—	178	2	no	—
185	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
186	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
187	C₄×D₄	—	178	1	no	—
188	C₄×D₄	—	178	1	no	—
189	C₄×D₄	—	178	1	no	—
190	C₄×D₄	—	178	1	no	—
191	C₂×C₄⋊C₄	—	89	1	no	—
192	C₂×C₂²⋊C₄	—	89	2	no	—
193	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
194	C₄⋊C₈	—	2	2	no	—
195	C₄⋊C₈	—	2	2	no	—
196	C₄⋊C₈	—	178	1	no	—
197	C₄⋊C₈	—	178	2	no	—
198	C₄⋊C₈	—	178	1	no	—
199	C₄⋊C₈	—	178	2	no	—
200	#100	—	356	4	no	—
201	C₂²⋊C₈	—	178	2	no	—
202	C₂²⋊C₈	—	178	2	no	—
203	C₄×C₈	D₃₅₆	1	1	yes	characteristic
204	C₄×C₈	D₃₅₆	1	1	yes	characteristic
205	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
206	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
207	C₂⁴	—	89	2	no	—
208	C₂×D₄	—	89	2	no	—
209	C₂×D₄	—	89	2	no	—
210	C₂×D₄	—	178	1	no	—
211	C₂×D₄	—	178	1	no	—
212	C₂×D₄	—	178	1	no	—
213	C₂×D₄	—	178	1	no	—
214	C₂²×C₄	—	89	2	no	—
215	C₂²×C₄	—	356	2	no	—
216	C₂²×C₄	—	356	2	no	—
217	C₂²×C₄	—	178	2	no	—
218	C₂²×C₄	—	178	2	no	—
219	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
220	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
221	C₂²×C₄	Q₈×D₈₉	1	1	yes	characteristic
222	M₁₆	—	4	2	no	—
223	M₁₆	—	4	2	no	—
224	M₁₆	—	356	2	no	—
225	M₁₆	—	356	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	4	1	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	—	356	4	no	—
230	C₂×C₈	—	356	4	no	—
231	C₂×C₈	—	356	2	no	—
232	C₂×C₈	—	356	2	no	—
233	C₂×C₈	—	178	2	no	—
234	C₂×C₈	#13083551	1	1	yes	characteristic
235	C₂×C₈	#13083551	1	1	yes	characteristic
236	C₂×C₈	#13083551	1	1	yes	characteristic
237	C₂×C₈	#13083551	1	1	yes	characteristic
238	C₄⋊C₄	—	178	1	no	—
239	C₄⋊C₄	—	178	1	no	—
240	C₂²⋊C₄	—	178	2	no	—
241	C₂²⋊C₄	—	178	2	no	—
242	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
243	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
244	C₄²	#13083573	1	1	yes	characteristic
245	C₄²	#13083575	1	1	yes	characteristic
246	C₂³	—	89	4	no	—
247	C₂³	—	356	2	no	—
248	C₂³	—	356	2	no	—
249	C₂³	—	178	2	no	—
250	C₂³	—	178	2	no	—
251	C₂³	C₄⋊C₄×D₈₉	1	1	yes	characteristic
252	D₄	—	178	2	no	—
253	D₄	—	178	2	no	—
254	D₄	—	178	2	no	—
255	D₄	—	178	2	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	356	4	no	—
260	C₂×C₄	—	356	2	no	—
261	C₂×C₄	—	356	2	no	—
262	C₂×C₄	—	356	2	no	—
263	C₂×C₄	—	178	2	no	—
264	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
265	C₂×C₄	#15667970	1	1	yes	characteristic
266	C₂×C₄	#15667983	1	1	yes	characteristic
267	C₂×C₄	#15667987	1	1	yes	characteristic
268	C₂×C₄	#15667988	1	1	yes	characteristic
269	C₂×C₄	#15667992	1	1	yes	characteristic
270	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
271	C₂×C₄	Q₁₆×D₈₉	1	1	yes	characteristic
272	C₂×C₄	#15668031	1	1	yes	characteristic
273	C₂×C₄	#15668035	1	1	yes	characteristic
274	C₈	—	2	1	no	—
275	C₈	—	2	1	no	—
276	C₈	—	2	1	no	—
277	C₈	—	8	2	no	—
278	C₈	—	356	2	no	—
279	C₈	—	712	2	no	—
280	C₈	#15667928	1	2	yes	complemented
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	356	4	no	—
284	C₂²	—	356	4	no	—
285	C₂²	—	356	2	no	—
286	C₂²	—	356	2	no	—
287	C₂²	—	178	4	no	—
288	C₂²	#19151455	1	1	yes	characteristic
289	C₂²	#107×D₈₉	1	1	yes	characteristic
290	C₂²	#19151635	1	1	yes	characteristic
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	356	2	no	—
295	C₄	C₂×#15667928	1	1	yes	characteristic
296	C₄	#19151580	1	1	yes	characteristic
297	C₄	#19151831	1	1	yes	characteristic
298	C₄	#19151835	1	1	yes	characteristic
299	C₄	#19151836	1	1	yes	characteristic
300	C₄	#19151840	1	1	yes	characteristic
301	C₂	—	2	1	no	—
302	C₂	—	2	1	no	—
303	C₂	—	356	4	no	—
304	C₂	#26020241	1	1	yes	characteristic
305	C₂	#26020278	1	1	yes	characteristic
306	C₂	#26020299	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19151455
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1567	C₂
socle	C₂²×C₈₉	#380
prime radical	C₂²×D₃₅₆	D₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1567	D₈₉
hypocenter	C₈₉	#8616
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1567

Complements

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

N	H ≅ G/N	type
3. #26019011	C₂	semidirect
4. #26018126	C₂	semidirect
5. C₈₉×#1567	C₂	semidirect
124. C₈×C₈₉	#100	semidirect
157. #1567	D₈₉	semidirect
159. C₈₉	#8616	semidirect
280. C₈	#15667928	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₃₅₆	2
#26020241	2
#26019011	2
#26018126	2
C₈₉×#1567	2
#8616	89