Subgroups — #36524945

Order 22784.

750 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030831	C₂	1	1	yes	maximal, characteristic
2	#26028965	C₂	1	2	yes	maximal, complemented
3	#26019138	C₂	1	4	yes	maximal, complemented
4	C₈×D₇₁₂	C₂	1	4	yes	maximal, complemented
5	#26019011	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018704	C₂	1	2	yes	maximal, complemented
7	C₂×C₈²×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15668018	—	2	2	no	—
9	#19152148	C₂²	1	2	yes	—
10	#19152003	C₄	1	1	yes	characteristic
11	#19151975	C₄	1	1	yes	characteristic
12	#19151945	C₄	1	2	yes	—
13	#19151834	C₄	1	4	yes	—
14	C₄×D₇₁₂	C₂²	1	4	yes	—
15	#19151677	—	2	2	no	—
16	#19151677	C₂²	1	4	yes	—
17	C₈×D₃₅₆	—	2	2	no	—
18	C₈×D₃₅₆	C₂²	1	4	yes	—
19	#19151622	C₂²	1	2	yes	—
20	#19151564	—	2	2	no	—
21	#19151351	C₂²	1	4	yes	—
22	#19151327	C₂²	1	8	yes	—
23	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
26	C₈²×C₈₉	C₂²	1	2	yes	—
27	C₈²×C₈₉	C₂²	1	2	yes	—
28	C₂×#13083574	—	2	2	no	—
29	C₂×D₇₁₂	C₂×C₄	1	4	yes	—
30	#15668018	—	4	8	no	—
31	C₂×#13083558	—	2	2	no	—
32	C₂×#13083557	—	2	4	no	—
33	C₂×C₈×D₈₉	—	2	4	no	—
34	#15668012	C₂×C₄	1	2	yes	—
35	#15668009	C₂×C₄	1	2	yes	—
36	#15668000	—	2	2	no	—
37	#15667986	—	2	2	no	—
38	#15667986	C₂×C₄	1	4	yes	—
39	C₄×D₃₅₆	—	2	2	no	—
40	C₄×D₃₅₆	C₂³	1	4	yes	—
41	#15667946	C₂×C₄	1	4	yes	—
42	C₄⋊#10623494	—	2	2	no	—
43	C₄⋊#10623494	C₂³	1	4	yes	—
44	#15667934	C₂×C₄	1	8	yes	—
45	#15667924	—	2	4	no	—
46	#15667924	—	2	4	no	—
47	C₂²×C₈×C₈₉	—	2	1	no	—
48	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
49	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
52	C₄×C₈×C₈₉	—	2	1	no	—
53	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
56	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
57	C₄×C₈×C₈₉	C₂³	1	2	yes	—
58	C₄×C₈×C₈₉	C₂³	1	2	yes	—
59	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
60	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
61	#13083574	—	4	4	no	—
62	#13083574	—	4	4	no	—
63	C₂×Dic₁₇₈	—	2	2	no	—
64	C₂×D₃₅₆	—	2	2	no	—
65	C₂×D₃₅₆	—	2	2	no	—
66	C₂×D₃₅₆	C₂²×C₄	1	4	yes	—
67	C₂×C₂²⋊D₈₉	—	2	4	no	—
68	C₂×C₄×D₈₉	—	2	4	no	—
69	D₇₁₂	—	2	8	no	—
70	#13083558	—	4	4	no	—
71	#13083557	—	4	8	no	—
72	C₂×#10623494	—	2	4	no	—
73	C₈×D₈₉	—	4	8	no	—
74	C₄⋊Dic₈₉	—	2	2	no	—
75	C₄⋊Dic₈₉	C₂²×C₄	1	4	yes	—
76	#13083551	—	2	4	no	—
77	#13083551	—	2	4	no	—
78	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
81	C₂×C₈×C₈₉	—	2	4	no	—
82	C₂×C₈×C₈₉	—	2	2	no	—
83	C₂×C₈×C₈₉	—	2	2	no	—
84	C₂×C₈×C₈₉	—	2	4	no	—
85	C₂×C₈×C₈₉	D₈	1	4	yes	complemented
86	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₈₉	#41	1	1	yes	characteristic
95	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
96	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
97	C₂²×D₈₉	—	2	4	no	—
98	C₂×C₄×C₈₉	—	2	2	no	—
99	C₂×C₄×C₈₉	—	2	2	no	—
100	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
101	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
104	C₂×C₄×C₈₉	#135	1	2	yes	—
105	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
108	C₂²⋊D₈₉	—	4	8	no	—
109	C₂×Dic₈₉	—	2	4	no	—
110	D₃₅₆	—	2	8	no	—
111	D₃₅₆	—	4	4	no	—
112	D₃₅₆	—	4	4	no	—
113	C₄×D₈₉	—	4	4	no	—
114	C₄×D₈₉	—	4	4	no	—
115	Dic₁₇₈	—	4	4	no	—
116	C₈×C₈₉	—	2	4	no	—
117	C₈×C₈₉	—	2	2	no	—
118	C₈×C₈₉	—	2	1	no	—
119	C₈×C₈₉	—	2	1	no	—
120	C₈×C₈₉	#131	1	4	yes	complemented
121	C₈×C₈₉	C₂×D₈	1	2	yes	—
122	C₈×C₈₉	C₂×D₈	1	2	yes	—
123	#10623494	—	4	4	no	—
124	C₂²×C₈₉	—	2	1	no	—
125	C₂²×C₈₉	—	2	1	no	—
126	C₂²×C₈₉	C₄×D₈	1	2	yes	—
127	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
128	C₂×D₈₉	—	4	4	no	—
129	C₂×D₈₉	—	4	4	no	—
130	C₂×D₈₉	—	4	4	no	—
131	C₄×C₈₉	—	2	1	no	—
132	C₄×C₈₉	—	2	1	no	—
133	C₄×C₈₉	C₄×D₈	1	2	yes	—
134	C₄×C₈₉	#443	1	2	yes	—
135	C₄×C₈₉	C₂×#131	1	2	yes	—
136	C₄×C₈₉	C₂²×D₈	1	1	yes	characteristic
137	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
138	Dic₈₉	—	4	4	no	—
139	#17956	—	89	1	no	maximal, Sylow
140	C₂×C₈₉	—	2	2	no	—
141	C₂×C₈₉	#2931	1	1	yes	characteristic
142	C₂×C₈₉	C₂×C₄×D₈	1	1	yes	characteristic
143	C₂×C₈₉	C₂×#443	1	1	yes	characteristic
144	D₈₉	—	8	4	no	—
145	#3169	—	89	1	no	—
146	#2931	—	89	2	no	—
147	#1589	—	89	4	no	—
148	C₈×D₈	—	89	4	no	—
149	#1567	—	89	1	no	—
150	#1492	—	89	2	no	—
151	C₂×C₈²	D₈₉	1	1	yes	characteristic, complemented
152	C₈₉	#17956	1	1	yes	characteristic, complemented, Sylow
153	C₂×#131	—	178	2	no	—
154	#532	—	89	2	no	—
155	#500	—	89	1	no	—
156	#493	—	89	1	no	—
157	#486	—	89	2	no	—
158	#466	—	89	4	no	—
159	C₄×D₈	—	89	4	no	—
160	#436	—	89	4	no	—
161	#436	—	178	2	no	—
162	C₈×D₄	—	89	4	no	—
163	C₈×D₄	—	178	2	no	—
164	#423	—	89	2	no	—
165	#408	—	178	2	no	—
166	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
167	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
168	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
169	#335	—	89	4	no	—
170	D₄⋊C₈	—	89	8	no	—
171	C₈²	C₂×D₈₉	1	2	yes	—
172	C₈²	C₂×D₈₉	1	2	yes	—
173	C₂×#41	—	178	2	no	—
174	C₂×D₈	—	89	4	no	—
175	#131	—	356	8	no	—
176	C₂×M₁₆	—	178	2	no	—
177	C₂×M₁₆	—	178	4	no	—
178	C₂²×C₈	—	2	1	no	—
179	C₂²×C₈	—	178	4	no	—
180	C₂²×C₈	C₄×D₈₉	1	2	yes	—
181	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
182	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
183	C₄⋊Q₈	—	89	2	no	—
184	#127	—	89	2	no	—
185	#124	—	178	2	no	—
186	#121	—	89	4	no	—
187	#121	—	178	2	no	—
188	C₄×D₄	—	89	4	no	—
189	C₄×D₄	—	178	2	no	—
190	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
191	#107	—	89	4	no	—
192	C₄⋊C₈	—	89	4	no	—
193	C₄⋊C₈	—	178	2	no	—
194	#102	—	89	8	no	—
195	C₂²⋊C₈	—	178	4	no	—
196	C₂²⋊C₈	—	178	4	no	—
197	C₄×C₈	—	2	1	no	—
198	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
199	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
200	C₄×C₈	D₃₅₆	1	1	yes	characteristic
201	C₄×C₈	D₃₅₆	1	1	yes	characteristic
202	C₄×C₈	C₂²×D₈₉	1	2	yes	—
203	C₄×C₈	C₂²×D₈₉	1	2	yes	—
204	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
205	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
206	#41	—	356	4	no	—
207	#41	—	356	4	no	—
208	C₂×Q₈	—	178	2	no	—
209	C₂×D₄	—	89	4	no	—
210	C₂×D₄	—	178	4	no	—
211	C₂×D₄	—	178	2	no	—
212	C₂×D₄	—	178	2	no	—
213	C₂²×C₄	—	178	4	no	—
214	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
216	C₂²×C₄	#13083574	1	1	yes	characteristic
217	D₈	—	178	8	no	—
218	M₁₆	—	356	4	no	—
219	M₁₆	—	356	8	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	356	8	no	—
225	C₂×C₈	—	178	4	no	—
226	C₂×C₈	D₇₁₂	1	4	yes	complemented
227	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
228	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
229	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
230	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
231	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
232	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
233	C₄⋊C₄	—	89	4	no	—
234	C₄⋊C₄	—	178	2	no	—
235	C₂²⋊C₄	—	178	4	no	—
236	C₂²⋊C₄	—	178	4	no	—
237	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
238	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
239	C₄²	#13083574	1	1	yes	characteristic
240	C₄²	C₂³×D₈₉	1	1	yes	characteristic
241	C₂³	—	178	4	no	—
242	C₂³	C₄×D₃₅₆	1	1	yes	characteristic
243	Q₈	—	356	4	no	—
244	D₄	—	356	8	no	—
245	D₄	—	356	4	no	—
246	D₄	—	356	4	no	—
247	D₄	—	178	8	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	356	4	no	—
251	C₂×C₄	—	356	4	no	—
252	C₂×C₄	—	178	4	no	—
253	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
254	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
255	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
256	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
257	C₂×C₄	#15668034	1	2	yes	—
258	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
259	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
260	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
261	C₈	—	2	4	no	—
262	C₈	—	2	2	no	—
263	C₈	—	2	1	no	—
264	C₈	—	2	1	no	—
265	C₈	—	356	4	no	—
266	C₈	#15668018	1	4	yes	complemented
267	C₈	C₂×D₇₁₂	1	2	yes	—
268	C₈	C₂×D₇₁₂	1	2	yes	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	356	4	no	—
272	C₂²	—	356	4	no	—
273	C₂²	—	356	4	no	—
274	C₂²	C₄×D₇₁₂	1	2	yes	—
275	C₂²	C₂×C₄×D₃₅₆	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	356	4	no	—
279	C₄	C₄×D₇₁₂	1	2	yes	—
280	C₄	#19151716	1	2	yes	—
281	C₄	C₂×#15668018	1	2	yes	—
282	C₄	C₂²×D₇₁₂	1	1	yes	characteristic
283	C₄	C₂×#15668034	1	1	yes	characteristic
284	C₂	—	2	2	no	—
285	C₂	—	712	4	no	—
286	C₂	#26028965	1	1	yes	characteristic
287	C₂	C₂×C₄×D₇₁₂	1	1	yes	characteristic
288	C₂	C₂×#19151716	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂×C₈²×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	#26030831	C₂
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	C₂×C₈²	D₈₉
hypocenter	C₈₉	#17956
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
2. #26028965	C₂	semidirect
3. #26019138	C₂	semidirect
4. C₈×D₇₁₂	C₂	semidirect
5. #26019011	C₂	semidirect
6. #26018704	C₂	semidirect
7. C₂×C₈²×C₈₉	C₂	semidirect
85. C₂×C₈×C₈₉	D₈	semidirect
120. C₈×C₈₉	#131	semidirect
151. C₂×C₈²	D₈₉	semidirect
152. C₈₉	#17956	semidirect
226. C₂×C₈	D₇₁₂	semidirect
266. C₈	#15668018	semidirect

Maximal subgroups

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

subgroup	index
#26030831	2
#26028965	2
#26019138	2
C₈×D₇₁₂	2
#26019011	2
#26018704	2
C₂×C₈²×C₈₉	2
#17956	89