Subgroups — C₈₉×#23751 #36401599

Order 22784.

664 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#430×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#2905	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#2102	C₂	1	8	yes	maximal, complemented
4	C₂×#344×C₈₉	C₂	1	2	yes	maximal
5	C₈₉×#1733	C₂	1	2	yes	maximal
6	C₂×H(3,ℤ/4)×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
8	C₂²×M₁₆×C₈₉	C₂²	1	2	yes	—
9	C₂²×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
10	C₂×#117×C₈₉	C₂²	1	2	yes	—
11	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
12	#430×C₈₉	—	2	2	no	—
13	#430×C₈₉	C₂²	1	4	yes	—
14	C₂×#108×C₈₉	—	2	2	no	—
15	#424×C₈₉	—	2	2	no	—
16	#424×C₈₉	—	2	2	no	—
17	#423×C₈₉	C₂²	1	4	yes	—
18	#344×C₈₉	C₂²	1	8	yes	—
19	#339×C₈₉	C₂²	1	8	yes	—
20	H(3,ℤ/4)×C₈₉	C₂²	1	4	yes	—
21	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
22	C₂×M₁₆×C₈₉	—	2	2	no	—
23	C₂×M₁₆×C₈₉	—	2	2	no	—
24	C₂×M₁₆×C₈₉	—	2	1	no	—
25	C₂×M₁₆×C₈₉	—	2	2	no	—
26	C₂×M₁₆×C₈₉	—	2	2	no	—
27	C₂×M₁₆×C₈₉	—	2	2	no	—
28	C₂×M₁₆×C₈₉	—	2	2	no	—
29	C₂×M₁₆×C₈₉	—	2	2	no	—
30	C₂×M₁₆×C₈₉	—	2	2	no	—
31	C₂×M₁₆×C₈₉	—	2	2	no	—
32	C₂×M₁₆×C₈₉	D₄	1	2	yes	—
33	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
34	C₂×M₁₆×C₈₉	C₂³	1	4	yes	—
35	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
36	C₂²×C₈×C₈₉	—	2	1	no	—
37	C₂²×C₈×C₈₉	—	2	2	no	—
38	C₂²×C₈×C₈₉	D₄	1	2	yes	—
39	#117×C₈₉	—	2	1	no	—
40	#117×C₈₉	—	2	1	no	—
41	#117×C₈₉	—	2	2	no	—
42	#117×C₈₉	—	2	2	no	—
43	#117×C₈₉	—	2	2	no	—
44	#117×C₈₉	Q₈	1	2	yes	—
45	#117×C₈₉	Q₈	1	2	yes	—
46	#117×C₈₉	C₂³	1	4	yes	—
47	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
48	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
49	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
50	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
51	C₂×C₄²×C₈₉	—	2	2	no	—
52	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
53	#108×C₈₉	—	2	8	no	—
54	C₄⋊C₈×C₈₉	—	2	8	no	—
55	C₂⁴×C₈₉	#41	1	1	yes	characteristic
56	C₂²×C₄×C₈₉	—	2	2	no	—
57	C₂²×C₄×C₈₉	—	4	2	no	—
58	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
63	C₂²×C₄×C₈₉	#41	1	2	yes	—
64	C₂²×C₄×C₈₉	#41	1	2	yes	—
65	C₂²×C₄×C₈₉	#41	1	2	yes	—
66	C₂²×C₄×C₈₉	#41	1	2	yes	—
67	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
68	M₁₆×C₈₉	—	2	4	no	—
69	M₁₆×C₈₉	—	2	4	no	—
70	M₁₆×C₈₉	—	2	4	no	—
71	M₁₆×C₈₉	—	2	4	no	—
72	M₁₆×C₈₉	—	2	4	no	—
73	M₁₆×C₈₉	—	4	1	no	—
74	M₁₆×C₈₉	—	4	1	no	—
75	M₁₆×C₈₉	—	4	2	no	—
76	M₁₆×C₈₉	—	4	2	no	—
77	C₂×C₈×C₈₉	—	2	2	no	—
78	C₂×C₈×C₈₉	—	2	2	no	—
79	C₂×C₈×C₈₉	—	2	2	no	—
80	C₂×C₈×C₈₉	—	2	4	no	—
81	C₂×C₈×C₈₉	—	4	2	no	—
82	C₂×C₈×C₈₉	—	4	2	no	—
83	C₂×C₈×C₈₉	—	4	2	no	—
84	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
85	C₄⋊C₄×C₈₉	—	2	2	no	—
86	C₄⋊C₄×C₈₉	—	2	2	no	—
87	C₄⋊C₄×C₈₉	—	2	4	no	—
88	C₄⋊C₄×C₈₉	—	4	2	no	—
89	C₂²⋊C₄×C₈₉	—	2	2	no	—
90	C₂²⋊C₄×C₈₉	—	2	2	no	—
91	C₂²⋊C₄×C₈₉	—	2	4	no	—
92	C₂²⋊C₄×C₈₉	—	4	2	no	—
93	C₄²×C₈₉	—	2	2	no	—
94	C₄²×C₈₉	—	2	4	no	—
95	C₄²×C₈₉	—	4	2	no	—
96	C₄²×C₈₉	C₂×Q₈	1	4	yes	—
97	C₂³×C₈₉	—	8	1	no	—
98	C₂³×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂³×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
100	C₂³×C₈₉	#126	1	2	yes	—
101	C₂³×C₈₉	#126	1	2	yes	—
102	C₂³×C₈₉	C₂×#41	1	1	yes	characteristic
103	C₂×C₄×C₈₉	—	2	4	no	—
104	C₂×C₄×C₈₉	—	4	4	no	—
105	C₂×C₄×C₈₉	—	8	2	no	—
106	C₂×C₄×C₈₉	—	8	4	no	—
107	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
108	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
110	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
111	C₂×C₄×C₈₉	#123	1	2	yes	—
112	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
113	C₂×C₄×C₈₉	#124	1	2	yes	—
114	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
115	C₂×C₄×C₈₉	#125	1	2	yes	—
116	C₂×C₄×C₈₉	#125	1	2	yes	—
117	C₂×C₄×C₈₉	#125	1	2	yes	—
118	C₂×C₄×C₈₉	#125	1	2	yes	—
119	C₂×C₄×C₈₉	#126	1	2	yes	—
120	C₂×C₄×C₈₉	#126	1	2	yes	—
121	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
123	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
124	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
125	C₈×C₈₉	—	2	4	no	—
126	C₈×C₈₉	—	4	2	no	—
127	C₈×C₈₉	—	4	4	no	—
128	C₂²×C₈₉	—	4	2	no	—
129	C₂²×C₈₉	—	4	1	no	—
130	C₂²×C₈₉	—	4	1	no	—
131	C₂²×C₈₉	—	4	1	no	—
132	C₂²×C₈₉	—	4	2	no	—
133	C₂²×C₈₉	#400	1	1	yes	characteristic
134	C₂²×C₈₉	#400	1	1	yes	characteristic
135	C₂²×C₈₉	#400	1	2	yes	—
136	C₂²×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
137	C₂²×C₈₉	C₂×#126	1	2	yes	—
138	C₄×C₈₉	—	4	4	no	—
139	C₄×C₈₉	—	8	4	no	—
140	C₄×C₈₉	#400	1	1	yes	characteristic
141	C₄×C₈₉	#400	1	1	yes	characteristic
142	C₄×C₈₉	#400	1	2	yes	—
143	C₄×C₈₉	C₂×#123	1	1	yes	characteristic
144	C₄×C₈₉	C₂×#124	1	1	yes	characteristic
145	C₄×C₈₉	C₂×#125	1	2	yes	—
146	#23751	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
147	C₂×C₈₉	—	2	1	no	—
148	C₂×C₈₉	—	2	2	no	—
149	C₂×C₈₉	—	2	1	no	—
150	C₂×C₈₉	—	2	2	no	—
151	C₂×C₈₉	—	2	1	no	—
152	C₂×C₈₉	C₂×#400	1	1	yes	characteristic
153	C₂×#430	C₂×C₈₉	1	1	yes	characteristic
154	#2905	C₂×C₈₉	1	1	yes	characteristic
155	#2102	C₂×C₈₉	1	8	yes	complemented
156	C₂×#344	C₂×C₈₉	1	2	yes	—
157	#1733	C₂×C₈₉	1	2	yes	—
158	C₂×H(3,ℤ/4)	C₂×C₈₉	1	1	yes	characteristic
159	C₈₉	#23751	1	1	yes	characteristic, direct product factor, Sylow
160	C₂²×M₁₆	C₂²×C₈₉	1	1	yes	characteristic
161	C₂²×M₁₆	C₂²×C₈₉	1	2	yes	—
162	C₂²×M₁₆	C₂²×C₈₉	1	1	yes	characteristic
163	C₂×#117	C₂²×C₈₉	1	2	yes	—
164	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
165	#430	—	2	2	no	—
166	#430	C₂²×C₈₉	1	4	yes	—
167	C₂×#108	—	2	2	no	—
168	#424	—	2	2	no	—
169	#424	—	2	2	no	—
170	#423	C₂²×C₈₉	1	4	yes	—
171	#344	C₂²×C₈₉	1	8	yes	—
172	#339	C₂²×C₈₉	1	8	yes	—
173	H(3,ℤ/4)	C₂²×C₈₉	1	4	yes	—
174	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	2	1	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	—	2	2	no	—
184	C₂×M₁₆	—	2	2	no	—
185	C₂×M₁₆	D₄×C₈₉	1	2	yes	—
186	C₂×M₁₆	C₂³×C₈₉	1	4	yes	—
187	C₂×M₁₆	C₂³×C₈₉	1	2	yes	—
188	C₂×M₁₆	C₂³×C₈₉	1	2	yes	—
189	C₂²×C₈	—	2	1	no	—
190	C₂²×C₈	—	2	2	no	—
191	C₂²×C₈	D₄×C₈₉	1	2	yes	—
192	#117	—	2	1	no	—
193	#117	—	2	1	no	—
194	#117	—	2	2	no	—
195	#117	—	2	2	no	—
196	#117	—	2	2	no	—
197	#117	Q₈×C₈₉	1	2	yes	—
198	#117	Q₈×C₈₉	1	2	yes	—
199	#117	C₂³×C₈₉	1	4	yes	—
200	C₂×C₄⋊C₄	—	2	1	no	—
201	C₂×C₄⋊C₄	—	2	2	no	—
202	C₂×C₂²⋊C₄	—	2	1	no	—
203	C₂×C₂²⋊C₄	—	2	2	no	—
204	C₂×C₄²	—	2	2	no	—
205	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
206	#108	—	2	8	no	—
207	C₄⋊C₈	—	2	8	no	—
208	C₂⁴	#41×C₈₉	1	1	yes	characteristic
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	4	2	no	—
211	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
213	C₂²×C₄	C₂×Q₈×C₈₉	1	1	yes	characteristic
214	C₂²×C₄	C₂×Q₈×C₈₉	1	1	yes	characteristic
215	C₂²×C₄	#41×C₈₉	1	1	yes	characteristic
216	C₂²×C₄	#41×C₈₉	1	2	yes	—
217	C₂²×C₄	#41×C₈₉	1	2	yes	—
218	C₂²×C₄	#41×C₈₉	1	2	yes	—
219	C₂²×C₄	#41×C₈₉	1	2	yes	—
220	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
221	M₁₆	—	2	4	no	—
222	M₁₆	—	2	4	no	—
223	M₁₆	—	2	4	no	—
224	M₁₆	—	2	4	no	—
225	M₁₆	—	2	4	no	—
226	M₁₆	—	4	1	no	—
227	M₁₆	—	4	1	no	—
228	M₁₆	—	4	2	no	—
229	M₁₆	—	4	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	4	2	no	—
235	C₂×C₈	—	4	2	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	C₂×D₄×C₈₉	1	4	yes	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	—	4	2	no	—
242	C₂²⋊C₄	—	2	2	no	—
243	C₂²⋊C₄	—	2	2	no	—
244	C₂²⋊C₄	—	2	4	no	—
245	C₂²⋊C₄	—	4	2	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	2	4	no	—
248	C₄²	—	4	2	no	—
249	C₄²	C₂×Q₈×C₈₉	1	4	yes	—
250	C₂³	—	8	1	no	—
251	C₂³	C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
252	C₂³	C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
253	C₂³	#126×C₈₉	1	2	yes	—
254	C₂³	#126×C₈₉	1	2	yes	—
255	C₂³	C₂×#41×C₈₉	1	1	yes	characteristic
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	—	8	2	no	—
259	C₂×C₄	—	8	4	no	—
260	C₂×C₄	C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
261	C₂×C₄	C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
262	C₂×C₄	#123×C₈₉	1	1	yes	characteristic
263	C₂×C₄	#123×C₈₉	1	1	yes	characteristic
264	C₂×C₄	#123×C₈₉	1	2	yes	—
265	C₂×C₄	#124×C₈₉	1	1	yes	characteristic
266	C₂×C₄	#124×C₈₉	1	2	yes	—
267	C₂×C₄	#124×C₈₉	1	1	yes	characteristic
268	C₂×C₄	#125×C₈₉	1	2	yes	—
269	C₂×C₄	#125×C₈₉	1	2	yes	—
270	C₂×C₄	#125×C₈₉	1	2	yes	—
271	C₂×C₄	#125×C₈₉	1	2	yes	—
272	C₂×C₄	#126×C₈₉	1	2	yes	—
273	C₂×C₄	#126×C₈₉	1	2	yes	—
274	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
275	C₂×C₄	C₂²×Q₈×C₈₉	1	1	yes	characteristic
276	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
277	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
278	C₈	—	2	4	no	—
279	C₈	—	4	2	no	—
280	C₈	—	4	4	no	—
281	C₂²	—	4	2	no	—
282	C₂²	—	4	1	no	—
283	C₂²	—	4	1	no	—
284	C₂²	—	4	1	no	—
285	C₂²	—	4	2	no	—
286	C₂²	#400×C₈₉	1	1	yes	characteristic
287	C₂²	#400×C₈₉	1	1	yes	characteristic
288	C₂²	#400×C₈₉	1	2	yes	—
289	C₂²	C₂×C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
290	C₂²	C₂×#126×C₈₉	1	2	yes	—
291	C₄	—	4	4	no	—
292	C₄	—	8	4	no	—
293	C₄	#400×C₈₉	1	1	yes	characteristic
294	C₄	#400×C₈₉	1	1	yes	characteristic
295	C₄	#400×C₈₉	1	2	yes	—
296	C₄	C₂×#123×C₈₉	1	1	yes	characteristic
297	C₄	C₂×#124×C₈₉	1	1	yes	characteristic
298	C₄	C₂×#125×C₈₉	1	2	yes	—
299	C₂	—	2	1	no	—
300	C₂	—	2	2	no	—
301	C₂	—	2	1	no	—
302	C₂	—	2	2	no	—
303	C₂	—	2	1	no	—
304	C₂	C₂×#400×C₈₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈₉	C₂×#124
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
socle	C₂×C₈₉	C₂×#400
prime radical	C₂⁴×C₈₉	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈₉ ◁ C₂³×C₄×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₈₉×#2102	C₂	semidirect
146. #23751	C₈₉	direct (N × H)
155. #2102	C₂×C₈₉	semidirect
159. C₈₉	#23751	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#430×C₈₉	2
C₈₉×#2905	2
C₈₉×#2102	2
C₂×#344×C₈₉	2
C₈₉×#1733	2
C₂×H(3,ℤ/4)×C₈₉	2
#23751	89

Subgroups — C89×#23751 #36401599

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#23751 #36401599