Subgroups — #36470375

Order 22784.

450 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151552	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151419	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19151379	C₂	1	2	yes	maximal
4	#26018331	C₂	1	2	yes	maximal, complemented
5	C₂×#364×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#13083557	C₂²	1	1	yes	characteristic
7	C₂²×C₈×D₈₉	C₄	1	1	yes	characteristic
8	C₂×#15667969	C₄	1	1	yes	characteristic
9	C₂×#15667952	C₂²	1	2	yes	—
10	#19151552	—	2	2	no	—
11	#19151552	—	2	2	no	—
12	C₂×#15667923	C₄	1	1	yes	characteristic
13	C₂×#15667922	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₈₉	C₄	1	1	yes	characteristic
15	#19151419	—	2	2	no	—
16	#19151379	—	2	4	no	—
17	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
19	#364×C₈₉	—	2	2	no	—
20	C₂²×C₄×D₈₉	C₂×C₄	1	1	yes	characteristic
21	C₂×#13083557	—	2	1	no	—
22	C₂×#13083557	—	2	1	no	—
23	C₂×#13083557	—	2	1	no	—
24	C₂×#13083557	—	2	1	no	—
25	C₂×#13083557	D₄	1	2	yes	—
26	C₂×#13083557	Q₈	1	2	yes	—
27	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
28	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×D₈₉	—	2	2	no	—
30	C₂×C₈×D₈₉	—	2	2	no	—
31	C₂×C₈×D₈₉	C₈	1	4	yes	—
32	#15667969	—	2	2	no	—
33	#15667969	—	2	2	no	—
34	C₂×#13083553	—	2	1	no	—
35	C₂×#13083551	—	2	1	no	—
36	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
37	#15667952	—	2	2	no	—
38	#15667952	—	2	2	no	—
39	C₂×#13083549	—	2	2	no	—
40	#15667923	—	2	2	no	—
41	#15667922	—	2	1	no	—
42	#15667922	—	2	1	no	—
43	C₈×Dic₈₉	—	2	2	no	—
44	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
47	M₃₂×C₈₉	—	2	2	no	—
48	M₃₂×C₈₉	—	2	2	no	—
49	C₂×C₁₆×C₈₉	—	2	2	no	—
50	C₄×C₈×C₈₉	D₄	1	2	yes	—
51	C₄×C₈×C₈₉	D₄	1	2	yes	—
52	C₂³×D₈₉	M₁₆	1	1	yes	characteristic
53	C₂²×Dic₈₉	—	2	1	no	—
54	C₂²×Dic₈₉	M₁₆	1	1	yes	characteristic
55	C₂×C₄×D₈₉	—	2	1	no	—
56	C₂×C₄×D₈₉	—	2	1	no	—
57	C₂×C₄×D₈₉	—	2	1	no	—
58	C₂×C₄×D₈₉	—	2	1	no	—
59	C₂×C₄×D₈₉	C₄⋊C₄	1	2	yes	—
60	C₂×C₄×D₈₉	C₂×C₈	1	2	yes	—
61	#13083557	—	2	2	no	—
62	#13083557	—	2	2	no	—
63	#13083557	—	2	2	no	—
64	#13083557	—	2	2	no	—
65	C₂×#10623494	—	2	1	no	—
66	C₂×#10623494	—	2	1	no	—
67	C₂×#10623494	—	2	1	no	—
68	C₂×#10623494	—	2	1	no	—
69	C₂×#10623494	C₄⋊C₄	1	2	yes	—
70	C₂×#10623494	C₂×C₈	1	2	yes	—
71	C₈×D₈₉	—	2	8	no	—
72	#13083553	—	2	4	no	—
73	#13083551	—	2	4	no	—
74	C₄×Dic₈₉	—	2	1	no	—
75	C₄×Dic₈₉	M₁₆	1	2	yes	—
76	#13083549	—	4	2	no	—
77	C₂²×C₄×C₈₉	—	2	1	no	—
78	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
80	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
81	C₂×C₈×C₈₉	M₁₆	1	2	yes	—
82	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
86	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
87	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
88	C₄²×C₈₉	—	2	1	no	—
89	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
90	C₁₆×C₈₉	—	4	2	no	—
91	C₂³×C₈₉	C₂×M₁₆	1	1	yes	characteristic
92	C₂²×D₈₉	—	2	1	no	—
93	C₂²×D₈₉	—	2	1	no	—
94	C₂²×D₈₉	—	4	2	no	—
95	C₂²×D₈₉	C₄⋊C₈	1	2	yes	—
96	C₂×C₄×C₈₉	—	2	2	no	—
97	C₂×C₄×C₈₉	—	4	2	no	—
98	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
101	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
102	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
103	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
104	C₂×Dic₈₉	—	2	1	no	—
105	C₂×Dic₈₉	—	2	1	no	—
106	C₂×Dic₈₉	—	2	2	no	—
107	C₂×Dic₈₉	—	4	2	no	—
108	C₂×Dic₈₉	C₄⋊C₈	1	2	yes	—
109	C₄×D₈₉	—	2	2	no	—
110	C₄×D₈₉	—	2	2	no	—
111	C₄×D₈₉	—	2	2	no	—
112	C₄×D₈₉	—	2	2	no	—
113	C₈×C₈₉	C₂²⋊C₈	1	4	yes	—
114	C₈×C₈₉	#121	1	1	yes	characteristic
115	C₈×C₈₉	#121	1	1	yes	characteristic
116	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
117	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
118	#10623494	—	2	2	no	—
119	#10623494	—	2	2	no	—
120	C₂²×C₈₉	—	2	2	no	—
121	C₂²×C₈₉	#407	1	1	yes	characteristic
122	C₂²×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
123	C₂²×C₈₉	#432	1	1	yes	characteristic
124	C₂×D₈₉	—	2	2	no	—
125	C₂×D₈₉	—	4	2	no	—
126	C₂×D₈₉	—	4	2	no	—
127	C₂×D₈₉	—	4	2	no	—
128	C₄×C₈₉	—	4	2	no	—
129	C₄×C₈₉	#380	1	1	yes	characteristic
130	C₄×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
131	C₄×C₈₉	#423	1	1	yes	characteristic
132	C₄×C₈₉	#431	1	1	yes	characteristic
133	Dic₈₉	—	2	2	no	—
134	Dic₈₉	—	4	2	no	—
135	#12790	—	89	1	no	maximal, Sylow
136	C₂×C₈₉	—	2	1	no	—
137	C₂×C₈₉	—	2	1	no	—
138	C₂×C₈₉	#1797	1	1	yes	characteristic
139	C₂×C₈₉	#2166	1	2	yes	—
140	D₈₉	—	4	2	no	—
141	C₂×#405	—	89	1	no	—
142	C₂×#364	—	89	1	no	—
143	C₂×#364	D₈₉	1	1	yes	characteristic, complemented
144	C₂×#349	—	89	2	no	—
145	#1375	—	89	2	no	—
146	C₈₉	#12790	1	1	yes	characteristic, complemented, Sylow
147	C₂²×M₁₆	—	89	1	no	—
148	C₂³×C₈	—	89	1	no	—
149	C₂×#117	—	89	1	no	—
150	C₂×M₃₂	—	89	2	no	—
151	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
152	#405	—	178	2	no	—
153	#405	—	178	2	no	—
154	C₂×#97	—	89	1	no	—
155	C₂×#97	—	89	1	no	—
156	C₂×C₄×C₈	—	89	1	no	—
157	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
158	#364	—	2	2	no	—
159	#364	—	178	2	no	—
160	#349	—	178	4	no	—
161	C₂³×C₄	—	89	1	no	—
162	C₂×M₁₆	—	89	2	no	—
163	C₂×M₁₆	—	89	2	no	—
164	C₂×M₁₆	—	178	1	no	—
165	C₂×M₁₆	—	178	1	no	—
166	C₂×M₁₆	—	178	1	no	—
167	C₂×M₁₆	—	178	1	no	—
168	C₂²×C₈	—	89	1	no	—
169	C₂²×C₈	—	89	1	no	—
170	C₂²×C₈	—	89	4	no	—
171	C₂²×C₈	—	178	2	no	—
172	C₂²×C₈	—	178	2	no	—
173	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
174	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
175	#117	—	178	2	no	—
176	#117	—	178	2	no	—
177	C₂×C₄⋊C₄	—	178	1	no	—
178	C₂×C₂²⋊C₄	—	178	1	no	—
179	C₂×C₄²	—	89	1	no	—
180	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
181	M₃₂	—	2	2	no	—
182	M₃₂	—	2	2	no	—
183	M₃₂	—	178	2	no	—
184	M₃₂	—	178	2	no	—
185	C₂×C₁₆	—	2	2	no	—
186	C₂×C₁₆	—	178	2	no	—
187	#97	—	178	1	no	—
188	#97	—	178	1	no	—
189	#97	—	178	2	no	—
190	C₄×C₈	—	178	2	no	—
191	C₄×C₈	D₃₅₆	1	2	yes	—
192	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
193	C₂⁴	—	89	1	no	—
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	89	1	no	—
196	C₂²×C₄	—	89	2	no	—
197	C₂²×C₄	—	89	2	no	—
198	C₂²×C₄	—	178	1	no	—
199	C₂²×C₄	—	178	1	no	—
200	C₂²×C₄	—	178	1	no	—
201	C₂²×C₄	—	178	1	no	—
202	C₂²×C₄	—	178	1	no	—
203	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
204	M₁₆	—	178	2	no	—
205	M₁₆	—	178	2	no	—
206	M₁₆	—	178	2	no	—
207	M₁₆	—	178	2	no	—
208	C₂×C₈	—	89	2	no	—
209	C₂×C₈	—	89	2	no	—
210	C₂×C₈	—	178	1	no	—
211	C₂×C₈	—	178	1	no	—
212	C₂×C₈	—	178	1	no	—
213	C₂×C₈	—	178	1	no	—
214	C₂×C₈	—	178	8	no	—
215	C₂×C₈	#13083551	1	2	yes	—
216	C₂×C₈	C₈×D₈₉	1	2	yes	—
217	C₂×C₈	#13083557	1	2	yes	—
218	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
219	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
220	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
221	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
222	C₂×C₈	#13083573	1	1	yes	characteristic
223	C₂×C₈	#13083575	1	1	yes	characteristic
224	C₄⋊C₄	—	178	4	no	—
225	C₂²⋊C₄	—	178	4	no	—
226	C₄²	—	2	1	no	—
227	C₄²	—	89	2	no	—
228	C₄²	—	178	1	no	—
229	C₄²	#13083551	1	2	yes	—
230	C₁₆	—	4	2	no	—
231	C₁₆	—	356	2	no	—
232	C₂³	—	89	2	no	—
233	C₂³	—	356	2	no	—
234	C₂³	—	178	1	no	—
235	C₂³	—	178	1	no	—
236	C₂³	M₁₆×D₈₉	1	1	yes	characteristic
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	89	2	no	—
240	C₂×C₄	—	356	2	no	—
241	C₂×C₄	—	178	2	no	—
242	C₂×C₄	—	178	1	no	—
243	C₂×C₄	—	178	1	no	—
244	C₂×C₄	—	178	2	no	—
245	C₂×C₄	—	178	2	no	—
246	C₂×C₄	—	178	2	no	—
247	C₂×C₄	—	178	2	no	—
248	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
249	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
250	C₂×C₄	#15667970	1	1	yes	characteristic
251	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
252	C₂×C₄	M₁₆×D₈₉	1	1	yes	characteristic
253	C₂×C₄	C₂×#13083557	1	1	yes	characteristic
254	C₈	—	178	2	no	—
255	C₈	—	178	2	no	—
256	C₈	#15667924	1	4	yes	—
257	C₈	#15667983	1	1	yes	characteristic
258	C₈	#15667987	1	1	yes	characteristic
259	C₈	#15667988	1	1	yes	characteristic
260	C₈	#15667992	1	1	yes	characteristic
261	C₂²	—	2	2	no	—
262	C₂²	—	356	2	no	—
263	C₂²	—	356	2	no	—
264	C₂²	—	356	2	no	—
265	C₂²	—	178	2	no	—
266	C₂²	#19151561	1	1	yes	characteristic
267	C₂²	C₄⋊C₈×D₈₉	1	1	yes	characteristic
268	C₂²	#19151654	1	1	yes	characteristic
269	C₄	—	4	2	no	—
270	C₄	—	356	2	no	—
271	C₄	—	178	2	no	—
272	C₄	#19151455	1	1	yes	characteristic
273	C₄	C₂×#15667924	1	1	yes	characteristic
274	C₄	#19151620	1	1	yes	characteristic
275	C₄	#19151650	1	1	yes	characteristic
276	C₂	—	2	1	no	—
277	C₂	—	2	1	no	—
278	C₂	—	356	2	no	—
279	C₂	#26020229	1	1	yes	characteristic
280	C₂	#26022067	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#15667970
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×#364×C₈₉	C₂
socle	C₂²×C₈₉	#432
prime radical	C₂³×D₈₉	M₁₆
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×#364	D₈₉
hypocenter	C₈₉	#12790
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
2. C₂×#19151419	C₂	semidirect
4. #26018331	C₂	semidirect
5. C₂×#364×C₈₉	C₂	semidirect
143. C₂×#364	D₈₉	semidirect
146. C₈₉	#12790	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151552	2
C₂×#19151419	2
C₂×#19151379	2
#26018331	2
C₂×#364×C₈₉	2
#12790	89