Subgroups — C₈₉×#29147 #36406995

Order 22784.

726 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#2862	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#2024	C₂	1	4	yes	maximal
4	C₈₉×#1960	C₂	1	2	yes	maximal
5	C₈₉×#1836	C₂	1	2	yes	maximal
6	C₈₉×#1805	C₂	1	4	yes	maximal
7	C₈₉×#1767	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
10	C₂×C₄×Q₈×C₈₉	C₂²	1	2	yes	—
11	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
12	C₂×#103×C₈₉	C₂²	1	8	yes	—
13	C₂×#97×C₈₉	C₂²	1	2	yes	—
14	#391×C₈₉	—	2	2	no	—
15	#391×C₈₉	—	2	2	no	—
16	#391×C₈₉	C₂²	1	2	yes	—
17	#389×C₈₉	—	2	2	no	—
18	#384×C₈₉	C₂²	1	1	yes	characteristic
19	#382×C₈₉	C₂²	1	2	yes	—
20	#382×C₈₉	C₂²	1	2	yes	—
21	C₄×C₄⋊C₄×C₈₉	—	2	2	no	—
22	C₄³×C₈₉	C₂²	1	1	yes	characteristic
23	#340×C₈₉	C₂²	1	8	yes	—
24	#336×C₈₉	C₂²	1	2	yes	—
25	C₂²×Q₈×C₈₉	—	2	2	no	—
26	C₂²×Q₈×C₈₉	C₂³	1	2	yes	—
27	C₂²×C₈×C₈₉	C₂³	1	4	yes	—
28	C₄⋊Q₈×C₈₉	—	2	2	no	—
29	C₄⋊Q₈×C₈₉	—	4	4	no	—
30	C₄⋊Q₈×C₈₉	Q₈	1	4	yes	complemented
31	C₄×Q₈×C₈₉	—	2	8	no	—
32	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
33	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
34	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
35	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
36	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
37	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
38	C₂×C₄²×C₈₉	—	2	2	no	—
39	C₂×C₄²×C₈₉	D₄	1	2	yes	—
40	C₂×C₄²×C₈₉	D₄	1	2	yes	—
41	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₈₉	—	2	4	no	—
45	#103×C₈₉	—	4	8	no	—
46	#97×C₈₉	—	2	4	no	—
47	#95×C₈₉	—	2	4	no	—
48	#95×C₈₉	—	2	4	no	—
49	#95×C₈₉	—	2	2	no	—
50	#95×C₈₉	—	2	2	no	—
51	C₂×Q₈×C₈₉	—	2	2	no	—
52	C₂×Q₈×C₈₉	—	2	4	no	—
53	C₂×Q₈×C₈₉	—	2	4	no	—
54	C₂×Q₈×C₈₉	—	4	4	no	—
55	C₂×Q₈×C₈₉	—	4	4	no	—
56	C₂×Q₈×C₈₉	—	4	4	no	—
57	C₂×Q₈×C₈₉	C₂×Q₈	1	4	yes	—
58	C₂²×C₄×C₈₉	—	2	2	no	—
59	C₂²×C₄×C₈₉	—	2	4	no	—
60	C₂²×C₄×C₈₉	—	2	2	no	—
61	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
66	C₂×C₈×C₈₉	—	2	4	no	—
67	C₂×C₈×C₈₉	—	2	4	no	—
68	C₂×C₈×C₈₉	—	2	4	no	—
69	C₄⋊C₄×C₈₉	—	2	8	no	—
70	C₄⋊C₄×C₈₉	—	2	4	no	—
71	C₄⋊C₄×C₈₉	—	2	8	no	—
72	C₄⋊C₄×C₈₉	—	4	4	no	—
73	C₄⋊C₄×C₈₉	—	4	4	no	—
74	C₄⋊C₄×C₈₉	—	4	4	no	—
75	C₄²×C₈₉	—	2	4	no	—
76	C₄²×C₈₉	—	2	2	no	—
77	C₄²×C₈₉	—	4	2	no	—
78	C₄²×C₈₉	—	4	4	no	—
79	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
80	C₄²×C₈₉	#41	1	2	yes	—
81	C₄²×C₈₉	#41	1	4	yes	—
82	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
83	Q₈×C₈₉	—	2	8	no	—
84	Q₈×C₈₉	—	4	4	no	—
85	Q₈×C₈₉	—	8	8	no	—
86	C₂×C₄×C₈₉	—	2	2	no	—
87	C₂×C₄×C₈₉	—	2	2	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	4	no	—
93	C₂×C₄×C₈₉	—	4	2	no	—
94	C₂×C₄×C₈₉	—	4	2	no	—
95	C₂×C₄×C₈₉	—	4	2	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	C₂²⋊Q₈	1	4	yes	—
101	C₂×C₄×C₈₉	#124	1	2	yes	—
102	C₂×C₄×C₈₉	#124	1	2	yes	—
103	C₂×C₄×C₈₉	#137	1	4	yes	—
104	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
105	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
106	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
107	C₂×C₄×C₈₉	H(5,𝔽₂)	1	2	yes	—
108	C₈×C₈₉	—	4	4	no	—
109	C₂²×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
110	C₂²×C₈₉	C₂×#124	1	1	yes	characteristic
111	C₂²×C₈₉	#535	1	1	yes	characteristic
112	C₂²×C₈₉	C₂×#137	1	2	yes	—
113	C₂²×C₈₉	#575	1	2	yes	—
114	C₄×C₈₉	—	2	4	no	—
115	C₄×C₈₉	—	2	2	no	—
116	C₄×C₈₉	—	2	2	no	—
117	C₄×C₈₉	—	2	4	no	—
118	C₄×C₈₉	—	4	4	no	—
119	C₄×C₈₉	—	8	2	no	—
120	C₄×C₈₉	—	8	4	no	—
121	C₄×C₈₉	#554	1	2	yes	—
122	C₄×C₈₉	#555	1	2	yes	—
123	#29147	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂×C₈₉	#2563	1	1	yes	characteristic
125	C₂×C₈₉	#3078	1	2	yes	—
126	C₂×C₈₉	#3135	1	1	yes	characteristic
127	C₂×C₈₉	#3139	1	1	yes	characteristic
128	C₂×C₈₉	#3215	1	2	yes	—
129	#2862	C₂×C₈₉	1	1	yes	characteristic
130	C₄×C₄⋊Q₈	C₂×C₈₉	1	1	yes	characteristic
131	#2024	C₂×C₈₉	1	4	yes	—
132	#1960	C₂×C₈₉	1	2	yes	—
133	#1836	C₂×C₈₉	1	2	yes	—
134	#1805	C₂×C₈₉	1	4	yes	—
135	#1767	C₂×C₈₉	1	1	yes	characteristic
136	C₈₉	#29147	1	1	yes	characteristic, direct product factor, Sylow
137	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
138	C₂×C₄⋊Q₈	C₂²×C₈₉	1	2	yes	—
139	C₂×C₄×Q₈	C₂²×C₈₉	1	2	yes	—
140	C₂×C₄⋊C₈	C₂²×C₈₉	1	2	yes	—
141	C₂×#103	C₂²×C₈₉	1	8	yes	—
142	C₂×#97	C₂²×C₈₉	1	2	yes	—
143	#391	—	2	2	no	—
144	#391	—	2	2	no	—
145	#391	C₂²×C₈₉	1	2	yes	—
146	#389	—	2	2	no	—
147	#384	C₂²×C₈₉	1	1	yes	characteristic
148	#382	C₂²×C₈₉	1	2	yes	—
149	#382	C₂²×C₈₉	1	2	yes	—
150	C₄×C₄⋊C₄	—	2	2	no	—
151	C₄³	C₂²×C₈₉	1	1	yes	characteristic
152	#340	C₂²×C₈₉	1	8	yes	—
153	#336	C₂²×C₈₉	1	2	yes	—
154	C₂²×Q₈	—	2	2	no	—
155	C₂²×Q₈	C₂³×C₈₉	1	2	yes	—
156	C₂²×C₈	C₂³×C₈₉	1	4	yes	—
157	C₄⋊Q₈	—	2	2	no	—
158	C₄⋊Q₈	—	4	4	no	—
159	C₄⋊Q₈	Q₈×C₈₉	1	4	yes	complemented
160	C₄×Q₈	—	2	8	no	—
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄⋊C₄	—	2	2	no	—
164	C₂×C₄⋊C₄	—	2	2	no	—
165	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
166	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
167	C₂×C₄²	—	2	2	no	—
168	C₂×C₄²	D₄×C₈₉	1	2	yes	—
169	C₂×C₄²	D₄×C₈₉	1	2	yes	—
170	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
171	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
172	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
173	C₄⋊C₈	—	2	4	no	—
174	#103	—	4	8	no	—
175	#97	—	2	4	no	—
176	#95	—	2	4	no	—
177	#95	—	2	4	no	—
178	#95	—	2	2	no	—
179	#95	—	2	2	no	—
180	C₂×Q₈	—	2	2	no	—
181	C₂×Q₈	—	2	4	no	—
182	C₂×Q₈	—	2	4	no	—
183	C₂×Q₈	—	4	4	no	—
184	C₂×Q₈	—	4	4	no	—
185	C₂×Q₈	—	4	4	no	—
186	C₂×Q₈	C₂×Q₈×C₈₉	1	4	yes	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
191	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
193	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
194	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
195	C₂×C₈	—	2	4	no	—
196	C₂×C₈	—	2	4	no	—
197	C₂×C₈	—	2	4	no	—
198	C₄⋊C₄	—	2	8	no	—
199	C₄⋊C₄	—	2	4	no	—
200	C₄⋊C₄	—	2	8	no	—
201	C₄⋊C₄	—	4	4	no	—
202	C₄⋊C₄	—	4	4	no	—
203	C₄⋊C₄	—	4	4	no	—
204	C₄²	—	2	4	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	4	2	no	—
207	C₄²	—	4	4	no	—
208	C₄²	C₂×Q₈×C₈₉	1	2	yes	—
209	C₄²	#41×C₈₉	1	2	yes	—
210	C₄²	#41×C₈₉	1	4	yes	—
211	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
212	Q₈	—	2	8	no	—
213	Q₈	—	4	4	no	—
214	Q₈	—	8	8	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	—	4	4	no	—
228	C₂×C₄	—	4	4	no	—
229	C₂×C₄	C₂²⋊Q₈×C₈₉	1	4	yes	—
230	C₂×C₄	#124×C₈₉	1	2	yes	—
231	C₂×C₄	#124×C₈₉	1	2	yes	—
232	C₂×C₄	#137×C₈₉	1	4	yes	—
233	C₂×C₄	C₂²×Q₈×C₈₉	1	1	yes	characteristic
234	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
235	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
236	C₂×C₄	H(5,𝔽₂)×C₈₉	1	2	yes	—
237	C₈	—	4	4	no	—
238	C₂²	C₂×C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
239	C₂²	C₂×#124×C₈₉	1	1	yes	characteristic
240	C₂²	#535×C₈₉	1	1	yes	characteristic
241	C₂²	C₂×#137×C₈₉	1	2	yes	—
242	C₂²	#575×C₈₉	1	2	yes	—
243	C₄	—	2	4	no	—
244	C₄	—	2	2	no	—
245	C₄	—	2	2	no	—
246	C₄	—	2	4	no	—
247	C₄	—	4	4	no	—
248	C₄	—	8	2	no	—
249	C₄	—	8	4	no	—
250	C₄	#554×C₈₉	1	2	yes	—
251	C₄	#555×C₈₉	1	2	yes	—
252	C₂	C₈₉×#2563	1	1	yes	characteristic
253	C₂	C₈₉×#3078	1	2	yes	—
254	C₂	C₈₉×#3135	1	1	yes	characteristic
255	C₂	C₈₉×#3139	1	1	yes	characteristic
256	C₂	C₈₉×#3215	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₈₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ C₄³×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
30. C₄⋊Q₈×C₈₉	Q₈	semidirect
123. #29147	C₈₉	direct (N × H)
136. C₈₉	#29147	direct (N × H)
159. C₄⋊Q₈	Q₈×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₈₉×#2862	2
C₄×C₄⋊Q₈×C₈₉	2
C₈₉×#2024	2
C₈₉×#1960	2
C₈₉×#1836	2
C₈₉×#1805	2
C₈₉×#1767	2
#29147	89

Subgroups — C89×#29147 #36406995

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#29147 #36406995