Subgroups — C₈₉×#12944 #36390792

Order 22784.

388 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#444×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#2161	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#2145	C₂	1	1	yes	maximal, characteristic
4	C₈₉×#2143	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1946	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈₉×#1389	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1377	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135×C₈₉	C₂²	1	1	yes	characteristic
9	C₂×#131×C₈₉	—	2	1	no	—
10	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
11	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
12	#444×C₈₉	—	2	2	no	—
13	#444×C₈₉	—	2	2	no	—
14	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
16	#427×C₈₉	C₂²	1	1	yes	characteristic
17	#424×C₈₉	—	2	1	no	—
18	C₂×#104×C₈₉	—	2	1	no	—
19	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
20	#368×C₈₉	—	2	2	no	—
21	#360×C₈₉	C₄	1	4	yes	complemented
22	#359×C₈₉	—	2	2	no	—
23	#339×C₈₉	—	2	1	no	—
24	C₂×#41×C₈₉	—	2	1	no	—
25	#135×C₈₉	—	2	1	no	—
26	#135×C₈₉	—	2	1	no	—
27	#135×C₈₉	C₂×C₄	1	2	yes	—
28	#135×C₈₉	D₄	1	2	yes	—
29	C₂×Q₁₆×C₈₉	Q₈	1	1	yes	characteristic
30	C₂×SD₁₆×C₈₉	—	2	1	no	—
31	C₂×D₈×C₈₉	Q₈	1	1	yes	characteristic
32	#131×C₈₉	—	2	4	no	—
33	#131×C₈₉	—	4	2	no	—
34	C₂×M₁₆×C₈₉	—	2	1	no	—
35	C₂×M₁₆×C₈₉	—	2	1	no	—
36	C₂×M₁₆×C₈₉	—	4	1	no	—
37	C₂²×C₈×C₈₉	—	4	1	no	—
38	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
40	#117×C₈₉	—	2	1	no	—
41	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
42	M₃₂×C₈₉	—	2	1	no	—
43	M₃₂×C₈₉	—	2	1	no	—
44	M₃₂×C₈₉	—	2	1	no	—
45	M₃₂×C₈₉	C₂×C₄	1	2	yes	—
46	C₂×C₁₆×C₈₉	—	2	1	no	—
47	C₂×C₁₆×C₈₉	—	2	1	no	—
48	#108×C₈₉	—	2	1	no	—
49	#108×C₈₉	—	2	1	no	—
50	#108×C₈₉	—	2	1	no	—
51	#108×C₈₉	D₄	1	2	yes	—
52	#107×C₈₉	—	2	1	no	—
53	#106×C₈₉	C₂×C₄	1	2	yes	—
54	C₄⋊C₈×C₈₉	—	4	2	no	—
55	#104×C₈₉	—	2	4	no	—
56	#97×C₈₉	—	2	1	no	—
57	#97×C₈₉	—	2	1	no	—
58	#41×C₈₉	—	2	2	no	—
59	#41×C₈₉	—	2	2	no	—
60	#41×C₈₉	—	4	1	no	—
61	#41×C₈₉	—	4	1	no	—
62	C₂×Q₈×C₈₉	—	2	1	no	—
63	C₂×D₄×C₈₉	—	2	1	no	—
64	C₂×D₄×C₈₉	—	4	1	no	—
65	C₂²×C₄×C₈₉	—	2	1	no	—
66	C₂²×C₄×C₈₉	—	4	1	no	—
67	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
68	Q₁₆×C₈₉	—	2	1	no	—
69	Q₁₆×C₈₉	C₄⋊C₄	1	2	yes	—
70	SD₁₆×C₈₉	—	2	2	no	—
71	SD₁₆×C₈₉	—	2	2	no	—
72	D₈×C₈₉	—	2	1	no	—
73	D₈×C₈₉	C₄⋊C₄	1	2	yes	—
74	M₁₆×C₈₉	—	2	2	no	—
75	M₁₆×C₈₉	—	4	4	no	—
76	M₁₆×C₈₉	—	4	1	no	—
77	M₁₆×C₈₉	—	4	1	no	—
78	M₁₆×C₈₉	—	4	1	no	—
79	C₂×C₈×C₈₉	—	2	1	no	—
80	C₂×C₈×C₈₉	—	2	1	no	—
81	C₂×C₈×C₈₉	—	4	4	no	—
82	C₂×C₈×C₈₉	—	4	1	no	—
83	C₂×C₈×C₈₉	—	4	1	no	—
84	C₂×C₈×C₈₉	D₈	1	2	yes	—
85	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
89	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
90	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
91	C₄⋊C₄×C₈₉	—	2	2	no	—
92	C₂²⋊C₄×C₈₉	—	4	1	no	—
93	C₄²×C₈₉	—	2	1	no	—
94	C₄²×C₈₉	—	2	1	no	—
95	C₄²×C₈₉	—	4	1	no	—
96	C₁₆×C₈₉	—	2	2	no	—
97	C₁₆×C₈₉	—	4	1	no	—
98	C₂³×C₈₉	—	4	1	no	—
99	C₂³×C₈₉	#135	1	1	yes	characteristic
100	Q₈×C₈₉	—	2	2	no	—
101	Q₈×C₈₉	—	4	1	no	—
102	D₄×C₈₉	—	2	2	no	—
103	D₄×C₈₉	—	4	2	no	—
104	D₄×C₈₉	—	4	2	no	—
105	D₄×C₈₉	—	4	1	no	—
106	C₂×C₄×C₈₉	—	4	1	no	—
107	C₂×C₄×C₈₉	—	4	2	no	—
108	C₂×C₄×C₈₉	—	4	2	no	—
109	C₂×C₄×C₈₉	—	4	2	no	—
110	C₂×C₄×C₈₉	—	4	1	no	—
111	C₂×C₄×C₈₉	—	8	1	no	—
112	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
114	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
115	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
117	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
118	C₈×C₈₉	—	2	2	no	—
119	C₈×C₈₉	—	4	2	no	—
120	C₈×C₈₉	—	8	1	no	—
121	C₈×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
123	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
124	C₈×C₈₉	#123	1	1	yes	characteristic
125	C₂²×C₈₉	—	4	1	no	—
126	C₂²×C₈₉	—	4	2	no	—
127	C₂²×C₈₉	—	8	2	no	—
128	C₂²×C₈₉	C₄×D₈	1	1	yes	characteristic
129	C₂²×C₈₉	#452	1	1	yes	characteristic
130	C₂²×C₈₉	#478	1	1	yes	characteristic
131	C₄×C₈₉	—	4	2	no	—
132	C₄×C₈₉	—	8	2	no	—
133	C₄×C₈₉	—	8	1	no	—
134	C₄×C₈₉	#385	1	1	yes	characteristic
135	C₄×C₈₉	#440	1	1	yes	characteristic
136	C₄×C₈₉	#447	1	1	yes	characteristic
137	C₄×C₈₉	#474	1	1	yes	characteristic
138	#12944	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
139	C₂×C₈₉	—	2	1	no	—
140	C₂×C₈₉	—	2	1	no	—
141	C₂×C₈₉	—	2	1	no	—
142	C₂×C₈₉	—	8	2	no	—
143	C₂×C₈₉	#1862	1	1	yes	characteristic
144	C₂×#444	C₂×C₈₉	1	1	yes	characteristic
145	#2161	C₂×C₈₉	1	1	yes	characteristic, complemented
146	#2145	C₂×C₈₉	1	1	yes	characteristic
147	#2143	C₂×C₈₉	1	1	yes	characteristic
148	#1946	C₂×C₈₉	1	1	yes	characteristic, complemented
149	#1389	C₂×C₈₉	1	1	yes	characteristic, complemented
150	#1377	C₂×C₈₉	1	1	yes	characteristic, complemented
151	C₈₉	#12944	1	1	yes	characteristic, direct product factor, Sylow
152	C₂×#135	C₂²×C₈₉	1	1	yes	characteristic
153	C₂×#131	—	2	1	no	—
154	C₂×M₃₂	C₂²×C₈₉	1	1	yes	characteristic
155	C₂×M₃₂	C₂²×C₈₉	1	1	yes	characteristic
156	#444	—	2	2	no	—
157	#444	—	2	2	no	—
158	C₂×#108	C₂²×C₈₉	1	1	yes	characteristic
159	C₂×#108	C₂²×C₈₉	1	1	yes	characteristic
160	#427	C₂²×C₈₉	1	1	yes	characteristic
161	#424	—	2	1	no	—
162	C₂×#104	—	2	1	no	—
163	C₂×#97	C₂²×C₈₉	1	1	yes	characteristic
164	#368	—	2	2	no	—
165	#360	C₄×C₈₉	1	4	yes	complemented
166	#359	—	2	2	no	—
167	#339	—	2	1	no	—
168	C₂×#41	—	2	1	no	—
169	#135	—	2	1	no	—
170	#135	—	2	1	no	—
171	#135	C₂×C₄×C₈₉	1	2	yes	—
172	#135	D₄×C₈₉	1	2	yes	—
173	C₂×Q₁₆	Q₈×C₈₉	1	1	yes	characteristic
174	C₂×SD₁₆	—	2	1	no	—
175	C₂×D₈	Q₈×C₈₉	1	1	yes	characteristic
176	#131	—	2	4	no	—
177	#131	—	4	2	no	—
178	C₂×M₁₆	—	2	1	no	—
179	C₂×M₁₆	—	2	1	no	—
180	C₂×M₁₆	—	4	1	no	—
181	C₂²×C₈	—	4	1	no	—
182	C₂²×C₈	D₄×C₈₉	1	1	yes	characteristic
183	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
184	#117	—	2	1	no	—
185	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
186	M₃₂	—	2	1	no	—
187	M₃₂	—	2	1	no	—
188	M₃₂	—	2	1	no	—
189	M₃₂	C₂×C₄×C₈₉	1	2	yes	—
190	C₂×C₁₆	—	2	1	no	—
191	C₂×C₁₆	—	2	1	no	—
192	#108	—	2	1	no	—
193	#108	—	2	1	no	—
194	#108	—	2	1	no	—
195	#108	D₄×C₈₉	1	2	yes	—
196	#107	—	2	1	no	—
197	#106	C₂×C₄×C₈₉	1	2	yes	—
198	C₄⋊C₈	—	4	2	no	—
199	#104	—	2	4	no	—
200	#97	—	2	1	no	—
201	#97	—	2	1	no	—
202	#41	—	2	2	no	—
203	#41	—	2	2	no	—
204	#41	—	4	1	no	—
205	#41	—	4	1	no	—
206	C₂×Q₈	—	2	1	no	—
207	C₂×D₄	—	2	1	no	—
208	C₂×D₄	—	4	1	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	4	1	no	—
211	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
212	Q₁₆	—	2	1	no	—
213	Q₁₆	C₄⋊C₄×C₈₉	1	2	yes	—
214	SD₁₆	—	2	2	no	—
215	SD₁₆	—	2	2	no	—
216	D₈	—	2	1	no	—
217	D₈	C₄⋊C₄×C₈₉	1	2	yes	—
218	M₁₆	—	2	2	no	—
219	M₁₆	—	4	4	no	—
220	M₁₆	—	4	1	no	—
221	M₁₆	—	4	1	no	—
222	M₁₆	—	4	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	4	4	no	—
226	C₂×C₈	—	4	1	no	—
227	C₂×C₈	—	4	1	no	—
228	C₂×C₈	D₈×C₈₉	1	2	yes	—
229	C₂×C₈	C₂²×C₄×C₈₉	1	1	yes	characteristic
230	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
231	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
232	C₂×C₈	C₂×Q₈×C₈₉	1	1	yes	characteristic
233	C₂×C₈	#41×C₈₉	1	1	yes	characteristic
234	C₂×C₈	#41×C₈₉	1	1	yes	characteristic
235	C₄⋊C₄	—	2	2	no	—
236	C₂²⋊C₄	—	4	1	no	—
237	C₄²	—	2	1	no	—
238	C₄²	—	2	1	no	—
239	C₄²	—	4	1	no	—
240	C₁₆	—	2	2	no	—
241	C₁₆	—	4	1	no	—
242	C₂³	—	4	1	no	—
243	C₂³	#135×C₈₉	1	1	yes	characteristic
244	Q₈	—	2	2	no	—
245	Q₈	—	4	1	no	—
246	D₄	—	2	2	no	—
247	D₄	—	4	2	no	—
248	D₄	—	4	2	no	—
249	D₄	—	4	1	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	8	1	no	—
256	C₂×C₄	C₄×D₄×C₈₉	1	1	yes	characteristic
257	C₂×C₄	#120×C₈₉	1	1	yes	characteristic
258	C₂×C₄	C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
259	C₂×C₄	C₂×D₈×C₈₉	1	1	yes	characteristic
260	C₂×C₄	AGL(1,ℤ/8)×C₈₉	1	1	yes	characteristic
261	C₂×C₄	#137×C₈₉	1	1	yes	characteristic
262	C₈	—	2	2	no	—
263	C₈	—	4	2	no	—
264	C₈	—	8	1	no	—
265	C₈	C₂×C₄⋊C₄×C₈₉	1	1	yes	characteristic
266	C₈	C₄×D₄×C₈₉	1	1	yes	characteristic
267	C₈	C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
268	C₈	#123×C₈₉	1	1	yes	characteristic
269	C₂²	—	4	1	no	—
270	C₂²	—	4	2	no	—
271	C₂²	—	8	2	no	—
272	C₂²	C₄×D₈×C₈₉	1	1	yes	characteristic
273	C₂²	#452×C₈₉	1	1	yes	characteristic
274	C₂²	#478×C₈₉	1	1	yes	characteristic
275	C₄	—	4	2	no	—
276	C₄	—	8	1	no	—
277	C₄	—	8	2	no	—
278	C₄	#385×C₈₉	1	1	yes	characteristic
279	C₄	#440×C₈₉	1	1	yes	characteristic
280	C₄	#447×C₈₉	1	1	yes	characteristic
281	C₄	#474×C₈₉	1	1	yes	characteristic
282	C₂	—	2	1	no	—
283	C₂	—	2	1	no	—
284	C₂	—	2	1	no	—
285	C₂	—	8	2	no	—
286	C₂	C₈₉×#1862	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈₉	#447
commutator	C₂×C₈	C₂²×C₄×C₈₉
Frattini, cosocle	C₂²×C₈	C₂³×C₈₉
socle	C₂×C₈₉	#1862
prime radical	C₂×#135×C₈₉	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈₉ ◁ C₂²×C₄×C₈₉ ◁ C₂×#97×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈₉×#2161	C₂	semidirect
5. C₈₉×#1946	C₂	semidirect
6. C₈₉×#1389	C₂	semidirect
7. C₈₉×#1377	C₂	semidirect
21. #360×C₈₉	C₄	semidirect
138. #12944	C₈₉	direct (N × H)
145. #2161	C₂×C₈₉	semidirect
148. #1946	C₂×C₈₉	semidirect
149. #1389	C₂×C₈₉	semidirect
150. #1377	C₂×C₈₉	semidirect
151. C₈₉	#12944	direct (N × H)
165. #360	C₄×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#444×C₈₉	2
C₈₉×#2161	2
C₈₉×#2145	2
C₈₉×#2143	2
C₈₉×#1946	2
C₈₉×#1389	2
C₈₉×#1377	2
#12944	89

Subgroups — C89×#12944 #36390792

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#12944 #36390792