Subgroups — C₈₉×#21964 #36399812

Order 22784.

874 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#423×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#2285	C₂	1	2	yes	maximal
3	C₈₉×#1948	C₂	1	4	yes	maximal, complemented
4	C₈₉×#1947	C₂	1	4	yes	maximal, complemented
5	C₈₉×#1809	C₂	1	4	yes	maximal
6	C₂²×M₁₆×C₈₉	C₂²	1	2	yes	—
7	C₂×#117×C₈₉	—	2	2	no	—
8	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
9	C₂×#107×C₈₉	C₂²	1	4	yes	—
10	C₂×#106×C₈₉	C₂²	1	4	yes	—
11	#423×C₈₉	C₄	1	8	yes	complemented
12	C₂×C₄⋊C₈×C₈₉	C₂²	1	4	yes	—
13	#384×C₈₉	C₂²	1	2	yes	—
14	#384×C₈₉	C₂²	1	2	yes	—
15	#382×C₈₉	—	2	2	no	—
16	#380×C₈₉	—	2	2	no	—
17	#380×C₈₉	C₂²	1	4	yes	—
18	C₄×C₄⋊C₄×C₈₉	—	2	2	no	—
19	C₄×C₄⋊C₄×C₈₉	C₂²	1	4	yes	—
20	#340×C₈₉	C₂²	1	8	yes	—
21	C₂³×C₄×C₈₉	D₄	1	2	yes	—
22	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
23	C₂×M₁₆×C₈₉	—	4	2	no	—
24	C₂×M₁₆×C₈₉	—	4	2	no	—
25	C₂×M₁₆×C₈₉	C₂×C₄	1	8	yes	—
26	C₂²×C₈×C₈₉	C₂³	1	4	yes	—
27	#117×C₈₉	—	4	8	no	—
28	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
29	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
30	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
31	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
32	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
33	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
34	C₂×C₂²⋊C₄×C₈₉	—	2	4	no	—
35	C₂×C₄²×C₈₉	—	2	4	no	—
36	C₂×C₄²×C₈₉	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₈₉	Q₈	1	4	yes	—
40	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
41	#107×C₈₉	—	4	4	no	—
42	#106×C₈₉	—	4	4	no	—
43	C₄⋊C₈×C₈₉	C₂×C₄	1	16	yes	—
44	#95×C₈₉	—	2	4	no	—
45	#95×C₈₉	—	2	4	no	—
46	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₈₉	—	2	4	no	—
48	C₂²×C₄×C₈₉	—	2	4	no	—
49	C₂²×C₄×C₈₉	—	2	1	no	—
50	C₂²×C₄×C₈₉	—	2	1	no	—
51	C₂²×C₄×C₈₉	—	2	1	no	—
52	C₂²×C₄×C₈₉	—	2	1	no	—
53	C₂²×C₄×C₈₉	—	2	4	no	—
54	C₂²×C₄×C₈₉	—	2	4	no	—
55	C₂²×C₄×C₈₉	C₄⋊C₄	1	8	yes	—
56	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
61	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
62	M₁₆×C₈₉	—	4	8	no	—
63	C₂×C₈×C₈₉	—	2	4	no	—
64	C₂×C₈×C₈₉	—	2	4	no	—
65	C₂×C₈×C₈₉	C₂²×C₄	1	8	yes	—
66	C₄⋊C₄×C₈₉	—	2	4	no	—
67	C₄⋊C₄×C₈₉	—	2	4	no	—
68	C₄⋊C₄×C₈₉	—	2	4	no	—
69	C₄⋊C₄×C₈₉	—	2	4	no	—
70	C₄⋊C₄×C₈₉	—	4	4	no	—
71	C₄⋊C₄×C₈₉	—	4	4	no	—
72	C₄⋊C₄×C₈₉	—	4	4	no	—
73	C₄⋊C₄×C₈₉	—	4	4	no	—
74	C₂²⋊C₄×C₈₉	—	4	8	no	—
75	C₄²×C₈₉	—	4	8	no	—
76	C₄²×C₈₉	C₄⋊C₄	1	4	yes	—
77	C₄²×C₈₉	C₄⋊C₄	1	4	yes	—
78	C₄²×C₈₉	C₂²×C₄	1	4	yes	—
79	C₄²×C₈₉	C₂×Q₈	1	4	yes	—
80	C₂³×C₈₉	—	2	1	no	—
81	C₂³×C₈₉	—	2	1	no	—
82	C₂³×C₈₉	—	2	2	no	—
83	C₂³×C₈₉	—	2	2	no	—
84	C₂³×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
85	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
86	C₂×C₄×C₈₉	—	2	1	no	—
87	C₂×C₄×C₈₉	—	2	1	no	—
88	C₂×C₄×C₈₉	—	2	2	no	—
89	C₂×C₄×C₈₉	—	2	2	no	—
90	C₂×C₄×C₈₉	—	2	2	no	—
91	C₂×C₄×C₈₉	—	2	2	no	—
92	C₂×C₄×C₈₉	—	2	2	no	—
93	C₂×C₄×C₈₉	—	2	2	no	—
94	C₂×C₄×C₈₉	—	2	16	no	—
95	C₂×C₄×C₈₉	—	4	4	no	—
96	C₂×C₄×C₈₉	—	4	4	no	—
97	C₂×C₄×C₈₉	—	4	4	no	—
98	C₂×C₄×C₈₉	—	4	4	no	—
99	C₂×C₄×C₈₉	—	4	4	no	—
100	C₂×C₄×C₈₉	—	4	4	no	—
101	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	4	yes	—
103	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	4	yes	—
104	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	4	yes	—
105	C₂×C₄×C₈₉	#137	1	4	yes	—
106	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
108	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
109	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
110	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
111	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
112	C₈×C₈₉	—	2	8	no	—
113	C₂²×C₈₉	—	2	2	no	—
114	C₂²×C₈₉	—	2	4	no	—
115	C₂²×C₈₉	—	4	1	no	—
116	C₂²×C₈₉	—	4	2	no	—
117	C₂²×C₈₉	—	4	1	no	—
118	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
119	C₂²×C₈₉	#537	1	1	yes	characteristic
120	C₂²×C₈₉	#537	1	1	yes	characteristic
121	C₂²×C₈₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
122	C₂²×C₈₉	C₂×#137	1	2	yes	—
123	C₄×C₈₉	—	2	2	no	—
124	C₄×C₈₉	—	2	4	no	—
125	C₄×C₈₉	—	8	4	no	—
126	C₄×C₈₉	—	8	4	no	—
127	C₄×C₈₉	#428	1	8	yes	—
128	C₄×C₈₉	#518	1	1	yes	characteristic
129	C₄×C₈₉	#519	1	1	yes	characteristic
130	C₄×C₈₉	#544	1	1	yes	characteristic
131	C₄×C₈₉	#544	1	1	yes	characteristic
132	#21964	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₂×C₈₉	—	4	2	no	—
134	C₂×C₈₉	#2316	1	1	yes	characteristic
135	C₂×C₈₉	C₂×#428	1	2	yes	—
136	C₂×C₈₉	#3253	1	1	yes	characteristic
137	C₂×C₈₉	#3254	1	2	yes	—
138	C₂×C₈₉	#3255	1	1	yes	characteristic
139	C₂×#423	C₂×C₈₉	1	1	yes	characteristic
140	#2285	C₂×C₈₉	1	2	yes	—
141	#1948	C₂×C₈₉	1	4	yes	complemented
142	#1947	C₂×C₈₉	1	4	yes	complemented
143	#1809	C₂×C₈₉	1	4	yes	—
144	C₈₉	#21964	1	1	yes	characteristic, direct product factor, Sylow
145	C₂²×M₁₆	C₂²×C₈₉	1	2	yes	—
146	C₂×#117	—	2	2	no	—
147	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
148	C₂×#107	C₂²×C₈₉	1	4	yes	—
149	C₂×#106	C₂²×C₈₉	1	4	yes	—
150	#423	C₄×C₈₉	1	8	yes	complemented
151	C₂×C₄⋊C₈	C₂²×C₈₉	1	4	yes	—
152	#384	C₂²×C₈₉	1	2	yes	—
153	#384	C₂²×C₈₉	1	2	yes	—
154	#382	—	2	2	no	—
155	#380	—	2	2	no	—
156	#380	C₂²×C₈₉	1	4	yes	—
157	C₄×C₄⋊C₄	—	2	2	no	—
158	C₄×C₄⋊C₄	C₂²×C₈₉	1	4	yes	—
159	#340	C₂²×C₈₉	1	8	yes	—
160	C₂³×C₄	D₄×C₈₉	1	2	yes	—
161	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
162	C₂×M₁₆	—	4	2	no	—
163	C₂×M₁₆	—	4	2	no	—
164	C₂×M₁₆	C₂×C₄×C₈₉	1	8	yes	—
165	C₂²×C₈	C₂³×C₈₉	1	4	yes	—
166	#117	—	4	8	no	—
167	C₂×C₄⋊C₄	—	2	2	no	—
168	C₂×C₄⋊C₄	—	2	2	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
172	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
173	C₂×C₂²⋊C₄	—	2	4	no	—
174	C₂×C₄²	—	2	4	no	—
175	C₂×C₄²	C₂×C₄×C₈₉	1	4	yes	—
176	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
177	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
178	C₂×C₄²	Q₈×C₈₉	1	4	yes	—
179	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
180	#107	—	4	4	no	—
181	#106	—	4	4	no	—
182	C₄⋊C₈	C₂×C₄×C₈₉	1	16	yes	—
183	#95	—	2	4	no	—
184	#95	—	2	4	no	—
185	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
186	C₂²×C₄	—	2	4	no	—
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	C₄⋊C₄×C₈₉	1	8	yes	—
195	C₂²×C₄	C₂²×C₄×C₈₉	1	2	yes	—
196	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
197	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
198	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
199	C₂²×C₄	C₂×Q₈×C₈₉	1	2	yes	—
200	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
201	M₁₆	—	4	8	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	C₂²×C₄×C₈₉	1	8	yes	—
205	C₄⋊C₄	—	2	4	no	—
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	2	4	no	—
209	C₄⋊C₄	—	4	4	no	—
210	C₄⋊C₄	—	4	4	no	—
211	C₄⋊C₄	—	4	4	no	—
212	C₄⋊C₄	—	4	4	no	—
213	C₂²⋊C₄	—	4	8	no	—
214	C₄²	—	4	8	no	—
215	C₄²	C₄⋊C₄×C₈₉	1	4	yes	—
216	C₄²	C₄⋊C₄×C₈₉	1	4	yes	—
217	C₄²	C₂²×C₄×C₈₉	1	4	yes	—
218	C₄²	C₂×Q₈×C₈₉	1	4	yes	—
219	C₂³	—	2	1	no	—
220	C₂³	—	2	1	no	—
221	C₂³	—	2	2	no	—
222	C₂³	—	2	2	no	—
223	C₂³	C₂×C₄⋊C₄×C₈₉	1	2	yes	—
224	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	16	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	C₂×C₄⋊C₄×C₈₉	1	2	yes	—
241	C₂×C₄	C₂×C₄⋊C₄×C₈₉	1	4	yes	—
242	C₂×C₄	C₂×C₄⋊C₄×C₈₉	1	4	yes	—
243	C₂×C₄	AGL(1,ℤ/8)×C₈₉	1	4	yes	—
244	C₂×C₄	#137×C₈₉	1	4	yes	—
245	C₂×C₄	C₂³×C₄×C₈₉	1	1	yes	characteristic
246	C₂×C₄	C₂²×Q₈×C₈₉	1	1	yes	characteristic
247	C₂×C₄	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
248	C₂×C₄	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
249	C₂×C₄	2₋¹⁺⁴×C₈₉	1	1	yes	characteristic
250	C₂×C₄	2₋¹⁺⁴×C₈₉	1	1	yes	characteristic
251	C₈	—	2	8	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	4	1	no	—
255	C₂²	—	4	2	no	—
256	C₂²	—	4	1	no	—
257	C₂²	C₂²×C₄⋊C₄×C₈₉	1	1	yes	characteristic
258	C₂²	#537×C₈₉	1	1	yes	characteristic
259	C₂²	#537×C₈₉	1	1	yes	characteristic
260	C₂²	C₂×AGL(1,ℤ/8)×C₈₉	1	2	yes	—
261	C₂²	C₂×#137×C₈₉	1	2	yes	—
262	C₄	—	2	2	no	—
263	C₄	—	2	4	no	—
264	C₄	—	8	4	no	—
265	C₄	—	8	4	no	—
266	C₄	#428×C₈₉	1	8	yes	—
267	C₄	#518×C₈₉	1	1	yes	characteristic
268	C₄	#519×C₈₉	1	1	yes	characteristic
269	C₄	#544×C₈₉	1	1	yes	characteristic
270	C₄	#544×C₈₉	1	1	yes	characteristic
271	C₂	—	4	2	no	—
272	C₂	C₈₉×#2316	1	1	yes	characteristic
273	C₂	C₂×#428×C₈₉	1	2	yes	—
274	C₂	C₈₉×#3253	1	1	yes	characteristic
275	C₂	C₈₉×#3254	1	2	yes	—
276	C₂	C₈₉×#3255	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₈₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
prime radical	C₂⁴×C₈₉	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ C₂²×C₄²×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₈₉×#1948	C₂	semidirect
4. C₈₉×#1947	C₂	semidirect
11. #423×C₈₉	C₄	semidirect
132. #21964	C₈₉	direct (N × H)
141. #1948	C₂×C₈₉	semidirect
142. #1947	C₂×C₈₉	semidirect
144. C₈₉	#21964	direct (N × H)
150. #423	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×#423×C₈₉	2
C₈₉×#2285	2
C₈₉×#1948	2
C₈₉×#1947	2
C₈₉×#1809	2
#21964	89

Subgroups — C89×#21964 #36399812

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#21964 #36399812