Subgroups — C₈₉×#12830 #36390678

Order 22784.

378 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	C₈₉×#2957	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#2151	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#2145	C₂	1	1	yes	maximal, characteristic
4	C₂×#359×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1833	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈₉×#1388	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1382	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135×C₈₉	C₂²	1	1	yes	characteristic
9	#568×C₈₉	—	2	1	no	—
10	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
12	#444×C₈₉	C₄	1	4	yes	complemented
13	#443×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	#421×C₈₉	—	2	1	no	—
18	#405×C₈₉	C₂²	1	1	yes	characteristic
19	C₄⋊C₁₆×C₈₉	—	2	2	no	—
20	#360×C₈₉	—	2	2	no	—
21	#359×C₈₉	—	2	2	no	—
22	#344×C₈₉	—	2	1	no	—
23	H(3,ℤ/4)×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	2	no	—
37	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
39	#117×C₈₉	—	2	1	no	—
40	#117×C₈₉	D₄	1	1	yes	characteristic
41	M₃₂×C₈₉	—	2	1	no	—
42	M₃₂×C₈₉	—	2	1	no	—
43	C₂×C₁₆×C₈₉	—	2	1	no	—
44	C₂×C₁₆×C₈₉	—	2	1	no	—
45	C₂×C₁₆×C₈₉	—	2	1	no	—
46	C₂×C₁₆×C₈₉	—	2	1	no	—
47	#108×C₈₉	—	2	1	no	—
48	#108×C₈₉	—	2	1	no	—
49	#108×C₈₉	—	2	1	no	—
50	#108×C₈₉	C₂×C₄	1	2	yes	—
51	#107×C₈₉	—	2	1	no	—
52	#106×C₈₉	—	2	1	no	—
53	#104×C₈₉	—	2	4	no	—
54	#97×C₈₉	C₂×C₄	1	2	yes	—
55	C₄×C₈×C₈₉	D₄	1	2	yes	—
56	#41×C₈₉	—	2	2	no	—
57	#41×C₈₉	—	2	2	no	—
58	#41×C₈₉	—	4	1	no	—
59	#41×C₈₉	—	4	1	no	—
60	C₂×Q₈×C₈₉	—	2	1	no	—
61	C₂×D₄×C₈₉	—	2	1	no	—
62	C₂×D₄×C₈₉	—	4	1	no	—
63	C₂²×C₄×C₈₉	—	4	1	no	—
64	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
65	Q₁₆×C₈₉	—	2	1	no	—
66	Q₁₆×C₈₉	C₄⋊C₄	1	2	yes	—
67	SD₁₆×C₈₉	—	2	2	no	—
68	SD₁₆×C₈₉	—	2	2	no	—
69	D₈×C₈₉	—	2	1	no	—
70	D₈×C₈₉	C₄⋊C₄	1	2	yes	—
71	M₁₆×C₈₉	—	2	1	no	—
72	M₁₆×C₈₉	—	2	1	no	—
73	M₁₆×C₈₉	—	2	2	no	—
74	M₁₆×C₈₉	—	4	4	no	—
75	M₁₆×C₈₉	—	4	1	no	—
76	M₁₆×C₈₉	—	4	1	no	—
77	M₁₆×C₈₉	—	4	1	no	—
78	C₂×C₈×C₈₉	—	4	4	no	—
79	C₂×C₈×C₈₉	—	4	1	no	—
80	C₂×C₈×C₈₉	—	4	1	no	—
81	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
82	C₂×C₈×C₈₉	C₂²×C₄	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	C₄⋊C₄×C₈₉	—	2	1	no	—
89	C₄⋊C₄×C₈₉	—	4	1	no	—
90	C₂²⋊C₄×C₈₉	—	2	1	no	—
91	C₂²⋊C₄×C₈₉	—	4	1	no	—
92	C₄²×C₈₉	—	4	1	no	—
93	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
94	C₁₆×C₈₉	—	4	1	no	—
95	C₁₆×C₈₉	—	4	1	no	—
96	C₂³×C₈₉	—	4	1	no	—
97	C₂³×C₈₉	#135	1	1	yes	characteristic
98	Q₈×C₈₉	—	2	2	no	—
99	Q₈×C₈₉	—	4	1	no	—
100	D₄×C₈₉	—	2	2	no	—
101	D₄×C₈₉	—	4	2	no	—
102	D₄×C₈₉	—	4	2	no	—
103	D₄×C₈₉	—	4	1	no	—
104	C₂×C₄×C₈₉	—	2	2	no	—
105	C₂×C₄×C₈₉	—	4	1	no	—
106	C₂×C₄×C₈₉	—	4	2	no	—
107	C₂×C₄×C₈₉	—	4	2	no	—
108	C₂×C₄×C₈₉	—	8	1	no	—
109	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
111	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
112	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
113	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
114	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
115	C₈×C₈₉	—	2	2	no	—
116	C₈×C₈₉	—	4	2	no	—
117	C₈×C₈₉	—	8	1	no	—
118	C₈×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₈×C₈₉	#117	1	1	yes	characteristic
120	C₈×C₈₉	#121	1	1	yes	characteristic
121	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂²×C₈₉	—	4	1	no	—
123	C₂²×C₈₉	—	4	2	no	—
124	C₂²×C₈₉	—	16	1	no	—
125	C₂²×C₈₉	#416	1	1	yes	characteristic
126	C₂²×C₈₉	#463	1	1	yes	characteristic
127	C₂²×C₈₉	#478	1	1	yes	characteristic
128	C₄×C₈₉	—	4	2	no	—
129	C₄×C₈₉	—	4	2	no	—
130	C₄×C₈₉	—	16	1	no	—
131	C₄×C₈₉	#380	1	1	yes	characteristic
132	C₄×C₈₉	#418	1	1	yes	characteristic
133	C₄×C₈₉	#463	1	1	yes	characteristic
134	C₄×C₈₉	#478	1	1	yes	characteristic
135	#12830	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂×C₈₉	—	2	1	no	—
137	C₂×C₈₉	—	2	1	no	—
138	C₂×C₈₉	—	2	1	no	—
139	C₂×C₈₉	—	8	2	no	—
140	C₂×C₈₉	#1804	1	1	yes	characteristic
141	#2957	C₂×C₈₉	1	1	yes	characteristic
142	#2151	C₂×C₈₉	1	1	yes	characteristic, complemented
143	#2145	C₂×C₈₉	1	1	yes	characteristic
144	C₂×#359	C₂×C₈₉	1	1	yes	characteristic
145	#1833	C₂×C₈₉	1	1	yes	characteristic, complemented
146	#1388	C₂×C₈₉	1	1	yes	characteristic, complemented
147	#1382	C₂×C₈₉	1	1	yes	characteristic, complemented
148	C₈₉	#12830	1	1	yes	characteristic, direct product factor, Sylow
149	C₂×#135	C₂²×C₈₉	1	1	yes	characteristic
150	#568	—	2	1	no	—
151	C₂×M₃₂	C₂²×C₈₉	1	1	yes	characteristic
152	C₂²×C₁₆	C₂²×C₈₉	1	1	yes	characteristic
153	#444	C₄×C₈₉	1	4	yes	complemented
154	#443	—	2	2	no	—
155	C₂×#108	C₂²×C₈₉	1	1	yes	characteristic
156	C₂×#108	C₂²×C₈₉	1	1	yes	characteristic
157	#427	C₂²×C₈₉	1	1	yes	characteristic
158	#421	—	2	1	no	—
159	#405	C₂²×C₈₉	1	1	yes	characteristic
160	C₄⋊C₁₆	—	2	2	no	—
161	#360	—	2	2	no	—
162	#359	—	2	2	no	—
163	#344	—	2	1	no	—
164	H(3,ℤ/4)	—	2	1	no	—
165	C₂×#41	—	2	1	no	—
166	#135	—	2	1	no	—
167	#135	—	2	1	no	—
168	#135	C₂×C₄×C₈₉	1	2	yes	—
169	#135	D₄×C₈₉	1	2	yes	—
170	C₂×Q₁₆	Q₈×C₈₉	1	1	yes	characteristic
171	C₂×SD₁₆	—	2	1	no	—
172	C₂×D₈	Q₈×C₈₉	1	1	yes	characteristic
173	#131	—	2	4	no	—
174	#131	—	4	2	no	—
175	C₂×M₁₆	—	2	1	no	—
176	C₂×M₁₆	—	2	1	no	—
177	C₂×M₁₆	—	4	2	no	—
178	C₂×M₁₆	D₄×C₈₉	1	1	yes	characteristic
179	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
180	#117	—	2	1	no	—
181	#117	D₄×C₈₉	1	1	yes	characteristic
182	M₃₂	—	2	1	no	—
183	M₃₂	—	2	1	no	—
184	C₂×C₁₆	—	2	1	no	—
185	C₂×C₁₆	—	2	1	no	—
186	C₂×C₁₆	—	2	1	no	—
187	C₂×C₁₆	—	2	1	no	—
188	#108	—	2	1	no	—
189	#108	—	2	1	no	—
190	#108	—	2	1	no	—
191	#108	C₂×C₄×C₈₉	1	2	yes	—
192	#107	—	2	1	no	—
193	#106	—	2	1	no	—
194	#104	—	2	4	no	—
195	#97	C₂×C₄×C₈₉	1	2	yes	—
196	C₄×C₈	D₄×C₈₉	1	2	yes	—
197	#41	—	2	2	no	—
198	#41	—	2	2	no	—
199	#41	—	4	1	no	—
200	#41	—	4	1	no	—
201	C₂×Q₈	—	2	1	no	—
202	C₂×D₄	—	2	1	no	—
203	C₂×D₄	—	4	1	no	—
204	C₂²×C₄	—	4	1	no	—
205	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
206	Q₁₆	—	2	1	no	—
207	Q₁₆	C₄⋊C₄×C₈₉	1	2	yes	—
208	SD₁₆	—	2	2	no	—
209	SD₁₆	—	2	2	no	—
210	D₈	—	2	1	no	—
211	D₈	C₄⋊C₄×C₈₉	1	2	yes	—
212	M₁₆	—	2	1	no	—
213	M₁₆	—	2	1	no	—
214	M₁₆	—	2	2	no	—
215	M₁₆	—	4	4	no	—
216	M₁₆	—	4	1	no	—
217	M₁₆	—	4	1	no	—
218	M₁₆	—	4	1	no	—
219	C₂×C₈	—	4	4	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	—	4	1	no	—
222	C₂×C₈	C₂²⋊C₄×C₈₉	1	2	yes	—
223	C₂×C₈	C₂²×C₄×C₈₉	1	1	yes	characteristic
224	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
225	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
226	C₂×C₈	C₂×Q₈×C₈₉	1	1	yes	characteristic
227	C₂×C₈	#41×C₈₉	1	1	yes	characteristic
228	C₂×C₈	#41×C₈₉	1	1	yes	characteristic
229	C₄⋊C₄	—	2	1	no	—
230	C₄⋊C₄	—	4	1	no	—
231	C₂²⋊C₄	—	2	1	no	—
232	C₂²⋊C₄	—	4	1	no	—
233	C₄²	—	4	1	no	—
234	C₄²	C₂²⋊C₄×C₈₉	1	2	yes	—
235	C₁₆	—	4	1	no	—
236	C₁₆	—	4	1	no	—
237	C₂³	—	4	1	no	—
238	C₂³	#135×C₈₉	1	1	yes	characteristic
239	Q₈	—	2	2	no	—
240	Q₈	—	4	1	no	—
241	D₄	—	2	2	no	—
242	D₄	—	4	2	no	—
243	D₄	—	4	2	no	—
244	D₄	—	4	1	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	8	1	no	—
250	C₂×C₄	C₂×C₂²⋊C₄×C₈₉	1	1	yes	characteristic
251	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
252	C₂×C₄	C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
253	C₂×C₄	#135×C₈₉	1	1	yes	characteristic
254	C₂×C₄	AGL(1,ℤ/8)×C₈₉	1	1	yes	characteristic
255	C₂×C₄	AGL(1,ℤ/8)×C₈₉	1	1	yes	characteristic
256	C₈	—	2	2	no	—
257	C₈	—	4	2	no	—
258	C₈	—	8	1	no	—
259	C₈	C₂×C₄⋊C₄×C₈₉	1	1	yes	characteristic
260	C₈	#117×C₈₉	1	1	yes	characteristic
261	C₈	#121×C₈₉	1	1	yes	characteristic
262	C₈	C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
263	C₂²	—	4	1	no	—
264	C₂²	—	4	2	no	—
265	C₂²	—	16	1	no	—
266	C₂²	#416×C₈₉	1	1	yes	characteristic
267	C₂²	#463×C₈₉	1	1	yes	characteristic
268	C₂²	#478×C₈₉	1	1	yes	characteristic
269	C₄	—	4	2	no	—
270	C₄	—	4	2	no	—
271	C₄	—	16	1	no	—
272	C₄	#380×C₈₉	1	1	yes	characteristic
273	C₄	#418×C₈₉	1	1	yes	characteristic
274	C₄	#463×C₈₉	1	1	yes	characteristic
275	C₄	#478×C₈₉	1	1	yes	characteristic
276	C₂	—	2	1	no	—
277	C₂	—	2	1	no	—
278	C₂	—	2	1	no	—
279	C₂	—	8	2	no	—
280	C₂	C₈₉×#1804	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈₉	#418
commutator	C₂×C₈	C₂²×C₄×C₈₉
Frattini, cosocle	C₂²×C₈	C₂³×C₈₉
socle	C₂×C₈₉	#1804
prime radical	C₂×#135×C₈₉	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈₉ ◁ C₂²×C₄×C₈₉ ◁ #405×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈₉×#2151	C₂	semidirect
5. C₈₉×#1833	C₂	semidirect
6. C₈₉×#1388	C₂	semidirect
7. C₈₉×#1382	C₂	semidirect
12. #444×C₈₉	C₄	semidirect
135. #12830	C₈₉	direct (N × H)
142. #2151	C₂×C₈₉	semidirect
145. #1833	C₂×C₈₉	semidirect
146. #1388	C₂×C₈₉	semidirect
147. #1382	C₂×C₈₉	semidirect
148. C₈₉	#12830	direct (N × H)
153. #444	C₄×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₈₉×#2957	2
C₈₉×#2151	2
C₈₉×#2145	2
C₂×#359×C₈₉	2
C₈₉×#1833	2
C₈₉×#1388	2
C₈₉×#1382	2
#12830	89

Subgroups — C89×#12830 #36390678

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#12830 #36390678