Subgroups — C₂²×#577×C₈₉ #36440933

Order 22784.

11118 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×2₋¹⁺⁴×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×H(5,𝔽₂)×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#577×C₈₉	C₂	1	48	yes	maximal, direct product factor
4	C₂²×#137×C₈₉	C₂	1	3	yes	maximal, complemented
5	C₂²×AGL(1,ℤ/8)×C₈₉	C₂	1	3	yes	maximal, complemented
6	C₂²×#135×C₈₉	C₂	1	3	yes	maximal, complemented
7	C₂³×SD₁₆×C₈₉	C₂	1	3	yes	maximal, complemented
8	C₂²×#131×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×2₋¹⁺⁴×C₈₉	—	2	12	no	—
10	C₂×2₋¹⁺⁴×C₈₉	C₂²	1	24	yes	complemented
11	C₂×H(5,𝔽₂)×C₈₉	—	2	12	no	—
12	C₂×H(5,𝔽₂)×C₈₉	C₂²	1	24	yes	complemented
13	C₂²×#41×C₈₉	—	2	1	no	—
14	C₂²×#41×C₈₉	—	2	3	no	—
15	C₂²×#41×C₈₉	C₂²	1	3	yes	—
16	C₂²×#41×C₈₉	C₂²	1	1	yes	characteristic
17	C₂²×#41×C₈₉	C₂²	1	3	yes	complemented
18	C₂³×Q₈×C₈₉	—	2	1	no	—
19	C₂³×Q₈×C₈₉	C₂²	1	3	yes	complemented
20	C₂³×D₄×C₈₉	—	2	3	no	—
21	C₂³×D₄×C₈₉	C₂²	1	3	yes	—
22	#577×C₈₉	C₂²	1	256	yes	direct product factor
23	C₂×#137×C₈₉	C₂²	1	72	yes	complemented
24	C₂×AGL(1,ℤ/8)×C₈₉	C₂²	1	72	yes	complemented
25	C₂×#135×C₈₉	C₂²	1	72	yes	complemented
26	C₂²×Q₁₆×C₈₉	C₂²	1	3	yes	complemented
27	C₂²×SD₁₆×C₈₉	C₂²	1	3	yes	complemented
28	C₂²×SD₁₆×C₈₉	C₂²	1	6	yes	complemented
29	C₂²×SD₁₆×C₈₉	C₂²	1	1	yes	characteristic, complemented
30	C₂²×SD₁₆×C₈₉	C₂²	1	72	yes	complemented
31	C₂²×D₈×C₈₉	C₂²	1	3	yes	complemented
32	C₂×#131×C₈₉	C₂²	1	24	yes	complemented
33	C₂²×M₁₆×C₈₉	C₂²	1	3	yes	complemented
34	C₂³×C₈×C₈₉	C₂²	1	3	yes	complemented
35	C₂⁵×C₈₉	—	2	3	no	—
36	2₋¹⁺⁴×C₈₉	—	2	96	no	—
37	2₋¹⁺⁴×C₈₉	C₂³	1	64	yes	complemented
38	H(5,𝔽₂)×C₈₉	—	2	96	no	—
39	H(5,𝔽₂)×C₈₉	C₂³	1	64	yes	complemented
40	C₂×#41×C₈₉	—	2	24	no	—
41	C₂×#41×C₈₉	—	2	72	no	—
42	C₂×#41×C₈₉	—	2	18	no	—
43	C₂×#41×C₈₉	—	2	18	no	—
44	C₂×#41×C₈₉	D₄	1	12	yes	—
45	C₂×#41×C₈₉	C₂³	1	36	yes	—
46	C₂×#41×C₈₉	C₂³	1	12	yes	—
47	C₂×#41×C₈₉	C₂³	1	36	yes	complemented
48	C₂²×Q₈×C₈₉	—	2	24	no	—
49	C₂²×Q₈×C₈₉	—	2	3	no	—
50	C₂²×Q₈×C₈₉	—	2	18	no	—
51	C₂²×Q₈×C₈₉	D₄	1	1	yes	characteristic
52	C₂²×Q₈×C₈₉	C₂³	1	3	yes	—
53	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
54	C₂²×Q₈×C₈₉	C₂³	1	36	yes	complemented
55	C₂²×D₄×C₈₉	—	2	3	no	—
56	C₂²×D₄×C₈₉	—	2	3	no	—
57	C₂²×D₄×C₈₉	—	2	72	no	—
58	C₂²×D₄×C₈₉	—	2	3	no	—
59	C₂²×D₄×C₈₉	—	2	18	no	—
60	C₂²×D₄×C₈₉	D₄	1	3	yes	—
61	C₂²×D₄×C₈₉	C₂³	1	1	yes	characteristic
62	C₂²×D₄×C₈₉	C₂³	1	3	yes	—
63	C₂²×D₄×C₈₉	C₂³	1	36	yes	—
64	C₂³×C₄×C₈₉	—	2	3	no	—
65	C₂³×C₄×C₈₉	—	2	3	no	—
66	C₂³×C₄×C₈₉	—	2	3	no	—
67	C₂³×C₄×C₈₉	C₂³	1	3	yes	—
68	#137×C₈₉	C₂³	1	192	yes	complemented
69	AGL(1,ℤ/8)×C₈₉	C₂³	1	192	yes	complemented
70	#135×C₈₉	C₂³	1	192	yes	complemented
71	C₂×Q₁₆×C₈₉	C₂³	1	36	yes	complemented
72	C₂×SD₁₆×C₈₉	C₂³	1	36	yes	complemented
73	C₂×SD₁₆×C₈₉	C₂³	1	72	yes	complemented
74	C₂×SD₁₆×C₈₉	C₂³	1	12	yes	complemented
75	C₂×SD₁₆×C₈₉	C₂³	1	192	yes	complemented
76	C₂×D₈×C₈₉	C₂³	1	36	yes	complemented
77	#131×C₈₉	C₂³	1	64	yes	complemented
78	C₂×M₁₆×C₈₉	C₂³	1	36	yes	complemented
79	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
80	C₂²×C₈×C₈₉	C₂³	1	3	yes	—
81	C₂²×C₈×C₈₉	C₂³	1	36	yes	complemented
82	C₂⁴×C₈₉	—	2	1	no	—
83	C₂⁴×C₈₉	—	2	3	no	—
84	C₂⁴×C₈₉	—	2	36	no	—
85	C₂⁴×C₈₉	—	8	12	no	—
86	C₂⁴×C₈₉	C₂×D₄	1	3	yes	—
87	#41×C₈₉	—	2	64	no	—
88	#41×C₈₉	—	2	192	no	—
89	#41×C₈₉	—	2	72	no	—
90	#41×C₈₉	—	2	72	no	—
91	#41×C₈₉	C₂×D₄	1	48	yes	—
92	#41×C₈₉	C₂⁴	1	48	yes	—
93	#41×C₈₉	C₂⁴	1	16	yes	—
94	#41×C₈₉	C₂⁴	1	48	yes	complemented
95	C₂×Q₈×C₈₉	—	2	64	no	—
96	C₂×Q₈×C₈₉	—	2	9	no	—
97	C₂×Q₈×C₈₉	—	2	36	no	—
98	C₂×Q₈×C₈₉	—	2	72	no	—
99	C₂×Q₈×C₈₉	—	2	3	no	—
100	C₂×Q₈×C₈₉	C₂×D₄	1	12	yes	—
101	C₂×Q₈×C₈₉	C₂⁴	1	18	yes	—
102	C₂×Q₈×C₈₉	C₂⁴	1	6	yes	—
103	C₂×Q₈×C₈₉	C₂⁴	1	48	yes	complemented
104	C₂×D₄×C₈₉	—	2	36	no	—
105	C₂×D₄×C₈₉	—	2	36	no	—
106	C₂×D₄×C₈₉	—	2	192	no	—
107	C₂×D₄×C₈₉	—	2	36	no	—
108	C₂×D₄×C₈₉	—	2	9	no	—
109	C₂×D₄×C₈₉	—	2	72	no	—
110	C₂×D₄×C₈₉	—	2	3	no	—
111	C₂×D₄×C₈₉	C₂×D₄	1	36	yes	—
112	C₂×D₄×C₈₉	C₂⁴	1	6	yes	—
113	C₂×D₄×C₈₉	C₂⁴	1	18	yes	—
114	C₂×D₄×C₈₉	C₂⁴	1	48	yes	—
115	C₂²×C₄×C₈₉	—	2	36	no	—
116	C₂²×C₄×C₈₉	—	2	1	no	—
117	C₂²×C₄×C₈₉	—	2	3	no	—
118	C₂²×C₄×C₈₉	—	2	36	no	—
119	C₂²×C₄×C₈₉	—	2	36	no	—
120	C₂²×C₄×C₈₉	C₂×D₄	1	18	yes	—
121	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
122	C₂²×C₄×C₈₉	C₂⁴	1	18	yes	—
123	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
124	Q₁₆×C₈₉	C₂⁴	1	48	yes	complemented
125	SD₁₆×C₈₉	C₂⁴	1	48	yes	complemented
126	SD₁₆×C₈₉	C₂⁴	1	96	yes	complemented
127	SD₁₆×C₈₉	C₂⁴	1	16	yes	complemented
128	D₈×C₈₉	C₂⁴	1	48	yes	complemented
129	M₁₆×C₈₉	C₂⁴	1	48	yes	complemented
130	C₂×C₈×C₈₉	C₂⁴	1	6	yes	—
131	C₂×C₈×C₈₉	C₂⁴	1	18	yes	—
132	C₂×C₈×C₈₉	C₂⁴	1	48	yes	complemented
133	C₂³×C₈₉	—	2	12	no	—
134	C₂³×C₈₉	—	2	6	no	—
135	C₂³×C₈₉	—	2	18	no	—
136	C₂³×C₈₉	—	2	48	no	—
137	C₂³×C₈₉	—	4	4	no	—
138	C₂³×C₈₉	—	4	12	no	—
139	C₂³×C₈₉	—	8	72	no	—
140	C₂³×C₈₉	C₂²×D₄	1	18	yes	—
141	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
142	Q₈×C₈₉	—	2	18	no	—
143	Q₈×C₈₉	—	2	48	no	—
144	Q₈×C₈₉	—	2	6	no	—
145	Q₈×C₈₉	C₂²×D₄	1	16	yes	—
146	Q₈×C₈₉	C₂⁵	1	12	yes	—
147	Q₈×C₈₉	C₂⁵	1	4	yes	—
148	D₄×C₈₉	—	2	48	no	—
149	D₄×C₈₉	—	2	48	no	—
150	D₄×C₈₉	—	2	48	no	—
151	D₄×C₈₉	—	2	18	no	—
152	D₄×C₈₉	—	2	6	no	—
153	D₄×C₈₉	C₂²×D₄	1	48	yes	—
154	D₄×C₈₉	C₂⁵	1	4	yes	—
155	D₄×C₈₉	C₂⁵	1	12	yes	—
156	C₂×C₄×C₈₉	—	2	6	no	—
157	C₂×C₄×C₈₉	—	2	18	no	—
158	C₂×C₄×C₈₉	—	2	48	no	—
159	C₂×C₄×C₈₉	—	2	48	no	—
160	C₂×C₄×C₈₉	—	2	48	no	—
161	C₂×C₄×C₈₉	C₂²×D₄	1	18	yes	—
162	C₂×C₄×C₈₉	C₂²×D₄	1	3	yes	—
163	C₂×C₄×C₈₉	C₂²×D₄	1	36	yes	—
164	C₂×C₄×C₈₉	C₂⁵	1	3	yes	—
165	C₂×C₄×C₈₉	C₂⁵	1	12	yes	—
166	C₈×C₈₉	C₂⁵	1	4	yes	—
167	C₈×C₈₉	C₂⁵	1	12	yes	—
168	C₂²×C₈₉	—	2	36	no	—
169	C₂²×C₈₉	—	2	4	no	—
170	C₂²×C₈₉	—	2	12	no	—
171	C₂²×C₈₉	—	4	12	no	—
172	C₂²×C₈₉	—	4	36	no	—
173	C₂²×C₈₉	—	8	48	no	—
174	C₂²×C₈₉	#577	1	4	yes	direct product factor
175	C₂²×C₈₉	C₂³×D₄	1	12	yes	—
176	C₂²×C₈₉	C₂³×D₄	1	3	yes	—
177	C₄×C₈₉	—	2	4	no	—
178	C₄×C₈₉	—	2	12	no	—
179	C₄×C₈₉	C₂³×D₄	1	12	yes	—
180	C₄×C₈₉	C₂³×D₄	1	3	yes	—
181	C₄×C₈₉	C₂⁶	1	1	yes	characteristic
182	C₂²×#577	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
183	C₂×C₈₉	—	2	12	no	—
184	C₂×C₈₉	—	4	4	no	—
185	C₂×C₈₉	—	4	12	no	—
186	C₂×C₈₉	C₂×#577	1	6	yes	direct product factor
187	C₂×C₈₉	C₂⁴×D₄	1	1	yes	characteristic
188	C₂²×2₋¹⁺⁴	C₂×C₈₉	1	1	yes	characteristic, complemented
189	C₂²×H(5,𝔽₂)	C₂×C₈₉	1	1	yes	characteristic, complemented
190	C₂×#577	C₂×C₈₉	1	48	yes	direct product factor
191	C₂²×#137	C₂×C₈₉	1	3	yes	complemented
192	C₂²×AGL(1,ℤ/8)	C₂×C₈₉	1	3	yes	complemented
193	C₂²×#135	C₂×C₈₉	1	3	yes	complemented
194	C₂³×SD₁₆	C₂×C₈₉	1	3	yes	complemented
195	C₂²×#131	C₂×C₈₉	1	1	yes	characteristic, complemented
196	C₈₉	C₂²×#577	1	1	yes	characteristic, direct product factor, Sylow
197	C₂×2₋¹⁺⁴	—	2	12	no	—
198	C₂×2₋¹⁺⁴	C₂²×C₈₉	1	24	yes	complemented
199	C₂×H(5,𝔽₂)	—	2	12	no	—
200	C₂×H(5,𝔽₂)	C₂²×C₈₉	1	24	yes	complemented
201	C₂²×#41	—	2	1	no	—
202	C₂²×#41	—	2	3	no	—
203	C₂²×#41	C₂²×C₈₉	1	3	yes	—
204	C₂²×#41	C₂²×C₈₉	1	1	yes	characteristic
205	C₂²×#41	C₂²×C₈₉	1	3	yes	complemented
206	C₂³×Q₈	—	2	1	no	—
207	C₂³×Q₈	C₂²×C₈₉	1	3	yes	complemented
208	C₂³×D₄	—	2	3	no	—
209	C₂³×D₄	C₂²×C₈₉	1	3	yes	—
210	#577	C₂²×C₈₉	1	256	yes	direct product factor
211	C₂×#137	C₂²×C₈₉	1	72	yes	complemented
212	C₂×AGL(1,ℤ/8)	C₂²×C₈₉	1	72	yes	complemented
213	C₂×#135	C₂²×C₈₉	1	72	yes	complemented
214	C₂²×Q₁₆	C₂²×C₈₉	1	3	yes	complemented
215	C₂²×SD₁₆	C₂²×C₈₉	1	3	yes	complemented
216	C₂²×SD₁₆	C₂²×C₈₉	1	6	yes	complemented
217	C₂²×SD₁₆	C₂²×C₈₉	1	1	yes	characteristic, complemented
218	C₂²×SD₁₆	C₂²×C₈₉	1	72	yes	complemented
219	C₂²×D₈	C₂²×C₈₉	1	3	yes	complemented
220	C₂×#131	C₂²×C₈₉	1	24	yes	complemented
221	C₂²×M₁₆	C₂²×C₈₉	1	3	yes	complemented
222	C₂³×C₈	C₂²×C₈₉	1	3	yes	complemented
223	C₂⁵	—	2	3	no	—
224	2₋¹⁺⁴	—	2	96	no	—
225	2₋¹⁺⁴	C₂³×C₈₉	1	64	yes	complemented
226	H(5,𝔽₂)	—	2	96	no	—
227	H(5,𝔽₂)	C₂³×C₈₉	1	64	yes	complemented
228	C₂×#41	—	2	24	no	—
229	C₂×#41	—	2	72	no	—
230	C₂×#41	—	2	18	no	—
231	C₂×#41	—	2	18	no	—
232	C₂×#41	D₄×C₈₉	1	12	yes	—
233	C₂×#41	C₂³×C₈₉	1	36	yes	—
234	C₂×#41	C₂³×C₈₉	1	12	yes	—
235	C₂×#41	C₂³×C₈₉	1	36	yes	complemented
236	C₂²×Q₈	—	2	24	no	—
237	C₂²×Q₈	—	2	3	no	—
238	C₂²×Q₈	—	2	18	no	—
239	C₂²×Q₈	D₄×C₈₉	1	1	yes	characteristic
240	C₂²×Q₈	C₂³×C₈₉	1	3	yes	—
241	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
242	C₂²×Q₈	C₂³×C₈₉	1	36	yes	complemented
243	C₂²×D₄	—	2	3	no	—
244	C₂²×D₄	—	2	3	no	—
245	C₂²×D₄	—	2	72	no	—
246	C₂²×D₄	—	2	3	no	—
247	C₂²×D₄	—	2	18	no	—
248	C₂²×D₄	D₄×C₈₉	1	3	yes	—
249	C₂²×D₄	C₂³×C₈₉	1	1	yes	characteristic
250	C₂²×D₄	C₂³×C₈₉	1	3	yes	—
251	C₂²×D₄	C₂³×C₈₉	1	36	yes	—
252	C₂³×C₄	—	2	3	no	—
253	C₂³×C₄	—	2	3	no	—
254	C₂³×C₄	—	2	3	no	—
255	C₂³×C₄	C₂³×C₈₉	1	3	yes	—
256	#137	C₂³×C₈₉	1	192	yes	complemented
257	AGL(1,ℤ/8)	C₂³×C₈₉	1	192	yes	complemented
258	#135	C₂³×C₈₉	1	192	yes	complemented
259	C₂×Q₁₆	C₂³×C₈₉	1	36	yes	complemented
260	C₂×SD₁₆	C₂³×C₈₉	1	36	yes	complemented
261	C₂×SD₁₆	C₂³×C₈₉	1	72	yes	complemented
262	C₂×SD₁₆	C₂³×C₈₉	1	12	yes	complemented
263	C₂×SD₁₆	C₂³×C₈₉	1	192	yes	complemented
264	C₂×D₈	C₂³×C₈₉	1	36	yes	complemented
265	#131	C₂³×C₈₉	1	64	yes	complemented
266	C₂×M₁₆	C₂³×C₈₉	1	36	yes	complemented
267	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
268	C₂²×C₈	C₂³×C₈₉	1	3	yes	—
269	C₂²×C₈	C₂³×C₈₉	1	36	yes	complemented
270	C₂⁴	—	2	1	no	—
271	C₂⁴	—	2	3	no	—
272	C₂⁴	—	2	36	no	—
273	C₂⁴	—	8	12	no	—
274	C₂⁴	C₂×D₄×C₈₉	1	3	yes	—
275	#41	—	2	64	no	—
276	#41	—	2	192	no	—
277	#41	—	2	72	no	—
278	#41	—	2	72	no	—
279	#41	C₂×D₄×C₈₉	1	48	yes	—
280	#41	C₂⁴×C₈₉	1	48	yes	—
281	#41	C₂⁴×C₈₉	1	16	yes	—
282	#41	C₂⁴×C₈₉	1	48	yes	complemented
283	C₂×Q₈	—	2	64	no	—
284	C₂×Q₈	—	2	9	no	—
285	C₂×Q₈	—	2	36	no	—
286	C₂×Q₈	—	2	72	no	—
287	C₂×Q₈	—	2	3	no	—
288	C₂×Q₈	C₂×D₄×C₈₉	1	12	yes	—
289	C₂×Q₈	C₂⁴×C₈₉	1	18	yes	—
290	C₂×Q₈	C₂⁴×C₈₉	1	6	yes	—
291	C₂×Q₈	C₂⁴×C₈₉	1	48	yes	complemented
292	C₂×D₄	—	2	36	no	—
293	C₂×D₄	—	2	36	no	—
294	C₂×D₄	—	2	192	no	—
295	C₂×D₄	—	2	36	no	—
296	C₂×D₄	—	2	9	no	—
297	C₂×D₄	—	2	72	no	—
298	C₂×D₄	—	2	3	no	—
299	C₂×D₄	C₂×D₄×C₈₉	1	36	yes	—
300	C₂×D₄	C₂⁴×C₈₉	1	6	yes	—
301	C₂×D₄	C₂⁴×C₈₉	1	18	yes	—
302	C₂×D₄	C₂⁴×C₈₉	1	48	yes	—
303	C₂²×C₄	—	2	1	no	—
304	C₂²×C₄	—	2	3	no	—
305	C₂²×C₄	—	2	36	no	—
306	C₂²×C₄	—	2	36	no	—
307	C₂²×C₄	—	2	36	no	—
308	C₂²×C₄	C₂×D₄×C₈₉	1	3	yes	—
309	C₂²×C₄	C₂×D₄×C₈₉	1	18	yes	—
310	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
311	C₂²×C₄	C₂⁴×C₈₉	1	18	yes	—
312	Q₁₆	C₂⁴×C₈₉	1	48	yes	complemented
313	SD₁₆	C₂⁴×C₈₉	1	48	yes	complemented
314	SD₁₆	C₂⁴×C₈₉	1	96	yes	complemented
315	SD₁₆	C₂⁴×C₈₉	1	16	yes	complemented
316	D₈	C₂⁴×C₈₉	1	48	yes	complemented
317	M₁₆	C₂⁴×C₈₉	1	48	yes	complemented
318	C₂×C₈	C₂⁴×C₈₉	1	6	yes	—
319	C₂×C₈	C₂⁴×C₈₉	1	18	yes	—
320	C₂×C₈	C₂⁴×C₈₉	1	48	yes	complemented
321	C₂³	—	2	12	no	—
322	C₂³	—	2	6	no	—
323	C₂³	—	2	18	no	—
324	C₂³	—	2	48	no	—
325	C₂³	—	4	4	no	—
326	C₂³	—	4	12	no	—
327	C₂³	—	8	72	no	—
328	C₂³	C₂²×D₄×C₈₉	1	18	yes	—
329	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
330	Q₈	—	2	18	no	—
331	Q₈	—	2	48	no	—
332	Q₈	—	2	6	no	—
333	Q₈	C₂²×D₄×C₈₉	1	16	yes	—
334	Q₈	C₂⁵×C₈₉	1	12	yes	—
335	Q₈	C₂⁵×C₈₉	1	4	yes	—
336	D₄	—	2	48	no	—
337	D₄	—	2	48	no	—
338	D₄	—	2	48	no	—
339	D₄	—	2	18	no	—
340	D₄	—	2	6	no	—
341	D₄	C₂²×D₄×C₈₉	1	48	yes	—
342	D₄	C₂⁵×C₈₉	1	4	yes	—
343	D₄	C₂⁵×C₈₉	1	12	yes	—
344	C₂×C₄	—	2	6	no	—
345	C₂×C₄	—	2	18	no	—
346	C₂×C₄	—	2	48	no	—
347	C₂×C₄	—	2	48	no	—
348	C₂×C₄	—	2	48	no	—
349	C₂×C₄	C₂²×D₄×C₈₉	1	18	yes	—
350	C₂×C₄	C₂²×D₄×C₈₉	1	3	yes	—
351	C₂×C₄	C₂²×D₄×C₈₉	1	36	yes	—
352	C₂×C₄	C₂⁵×C₈₉	1	3	yes	—
353	C₂×C₄	C₂⁵×C₈₉	1	12	yes	—
354	C₈	C₂⁵×C₈₉	1	4	yes	—
355	C₈	C₂⁵×C₈₉	1	12	yes	—
356	C₂²	—	2	36	no	—
357	C₂²	—	2	4	no	—
358	C₂²	—	2	12	no	—
359	C₂²	—	4	36	no	—
360	C₂²	—	4	12	no	—
361	C₂²	—	8	48	no	—
362	C₂²	#577×C₈₉	1	4	yes	direct product factor
363	C₂²	C₂³×D₄×C₈₉	1	12	yes	—
364	C₂²	C₂³×D₄×C₈₉	1	3	yes	—
365	C₄	—	2	4	no	—
366	C₄	—	2	12	no	—
367	C₄	C₂³×D₄×C₈₉	1	12	yes	—
368	C₄	C₂³×D₄×C₈₉	1	3	yes	—
369	C₄	C₂⁶×C₈₉	1	1	yes	characteristic
370	C₂	—	2	12	no	—
371	C₂	—	4	4	no	—
372	C₂	—	4	12	no	—
373	C₂	C₂×#577×C₈₉	1	6	yes	direct product factor
374	C₂	C₂⁴×D₄×C₈₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂²×D₄
commutator, Frattini, cosocle	C₄	C₂⁶×C₈₉
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ C₂²×#41×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×2₋¹⁺⁴×C₈₉	C₂	semidirect
2. C₂²×H(5,𝔽₂)×C₈₉	C₂	semidirect
3. C₂×#577×C₈₉	C₂	direct (N × H)
4. C₂²×#137×C₈₉	C₂	semidirect
5. C₂²×AGL(1,ℤ/8)×C₈₉	C₂	semidirect
6. C₂²×#135×C₈₉	C₂	semidirect
7. C₂³×SD₁₆×C₈₉	C₂	semidirect
8. C₂²×#131×C₈₉	C₂	semidirect
10. C₂×2₋¹⁺⁴×C₈₉	C₂²	semidirect
12. C₂×H(5,𝔽₂)×C₈₉	C₂²	semidirect
17. C₂²×#41×C₈₉	C₂²	semidirect
19. C₂³×Q₈×C₈₉	C₂²	semidirect
22. #577×C₈₉	C₂²	direct (N × H)
23. C₂×#137×C₈₉	C₂²	semidirect
24. C₂×AGL(1,ℤ/8)×C₈₉	C₂²	semidirect
25. C₂×#135×C₈₉	C₂²	semidirect
26. C₂²×Q₁₆×C₈₉	C₂²	semidirect
27. C₂²×SD₁₆×C₈₉	C₂²	semidirect
28. C₂²×SD₁₆×C₈₉	C₂²	semidirect
29. C₂²×SD₁₆×C₈₉	C₂²	semidirect
30. C₂²×SD₁₆×C₈₉	C₂²	semidirect
31. C₂²×D₈×C₈₉	C₂²	semidirect
32. C₂×#131×C₈₉	C₂²	semidirect
33. C₂²×M₁₆×C₈₉	C₂²	semidirect
34. C₂³×C₈×C₈₉	C₂²	semidirect
37. 2₋¹⁺⁴×C₈₉	C₂³	semidirect
39. H(5,𝔽₂)×C₈₉	C₂³	semidirect
47. C₂×#41×C₈₉	C₂³	semidirect
54. C₂²×Q₈×C₈₉	C₂³	semidirect
68. #137×C₈₉	C₂³	semidirect
69. AGL(1,ℤ/8)×C₈₉	C₂³	semidirect
70. #135×C₈₉	C₂³	semidirect
71. C₂×Q₁₆×C₈₉	C₂³	semidirect
72. C₂×SD₁₆×C₈₉	C₂³	semidirect
73. C₂×SD₁₆×C₈₉	C₂³	semidirect
74. C₂×SD₁₆×C₈₉	C₂³	semidirect
75. C₂×SD₁₆×C₈₉	C₂³	semidirect
76. C₂×D₈×C₈₉	C₂³	semidirect
77. #131×C₈₉	C₂³	semidirect
78. C₂×M₁₆×C₈₉	C₂³	semidirect
81. C₂²×C₈×C₈₉	C₂³	semidirect
94. #41×C₈₉	C₂⁴	semidirect
103. C₂×Q₈×C₈₉	C₂⁴	semidirect
124. Q₁₆×C₈₉	C₂⁴	semidirect
125. SD₁₆×C₈₉	C₂⁴	semidirect
126. SD₁₆×C₈₉	C₂⁴	semidirect
127. SD₁₆×C₈₉	C₂⁴	semidirect
128. D₈×C₈₉	C₂⁴	semidirect
129. M₁₆×C₈₉	C₂⁴	semidirect
132. C₂×C₈×C₈₉	C₂⁴	semidirect
174. C₂²×C₈₉	#577	direct (N × H)
182. C₂²×#577	C₈₉	direct (N × H)
186. C₂×C₈₉	C₂×#577	direct (N × H)
188. C₂²×2₋¹⁺⁴	C₂×C₈₉	semidirect
189. C₂²×H(5,𝔽₂)	C₂×C₈₉	semidirect
190. C₂×#577	C₂×C₈₉	direct (N × H)
191. C₂²×#137	C₂×C₈₉	semidirect
192. C₂²×AGL(1,ℤ/8)	C₂×C₈₉	semidirect
193. C₂²×#135	C₂×C₈₉	semidirect
194. C₂³×SD₁₆	C₂×C₈₉	semidirect
195. C₂²×#131	C₂×C₈₉	semidirect
196. C₈₉	C₂²×#577	direct (N × H)
198. C₂×2₋¹⁺⁴	C₂²×C₈₉	semidirect
200. C₂×H(5,𝔽₂)	C₂²×C₈₉	semidirect
205. C₂²×#41	C₂²×C₈₉	semidirect
207. C₂³×Q₈	C₂²×C₈₉	semidirect
210. #577	C₂²×C₈₉	direct (N × H)
211. C₂×#137	C₂²×C₈₉	semidirect
212. C₂×AGL(1,ℤ/8)	C₂²×C₈₉	semidirect
213. C₂×#135	C₂²×C₈₉	semidirect
214. C₂²×Q₁₆	C₂²×C₈₉	semidirect
215. C₂²×SD₁₆	C₂²×C₈₉	semidirect
216. C₂²×SD₁₆	C₂²×C₈₉	semidirect
217. C₂²×SD₁₆	C₂²×C₈₉	semidirect
218. C₂²×SD₁₆	C₂²×C₈₉	semidirect
219. C₂²×D₈	C₂²×C₈₉	semidirect
220. C₂×#131	C₂²×C₈₉	semidirect
221. C₂²×M₁₆	C₂²×C₈₉	semidirect
222. C₂³×C₈	C₂²×C₈₉	semidirect
225. 2₋¹⁺⁴	C₂³×C₈₉	semidirect
227. H(5,𝔽₂)	C₂³×C₈₉	semidirect
235. C₂×#41	C₂³×C₈₉	semidirect
242. C₂²×Q₈	C₂³×C₈₉	semidirect
256. #137	C₂³×C₈₉	semidirect
257. AGL(1,ℤ/8)	C₂³×C₈₉	semidirect
258. #135	C₂³×C₈₉	semidirect
259. C₂×Q₁₆	C₂³×C₈₉	semidirect
260. C₂×SD₁₆	C₂³×C₈₉	semidirect
261. C₂×SD₁₆	C₂³×C₈₉	semidirect
262. C₂×SD₁₆	C₂³×C₈₉	semidirect
263. C₂×SD₁₆	C₂³×C₈₉	semidirect
264. C₂×D₈	C₂³×C₈₉	semidirect
265. #131	C₂³×C₈₉	semidirect
266. C₂×M₁₆	C₂³×C₈₉	semidirect
269. C₂²×C₈	C₂³×C₈₉	semidirect
282. #41	C₂⁴×C₈₉	semidirect
291. C₂×Q₈	C₂⁴×C₈₉	semidirect
312. Q₁₆	C₂⁴×C₈₉	semidirect
313. SD₁₆	C₂⁴×C₈₉	semidirect
314. SD₁₆	C₂⁴×C₈₉	semidirect
315. SD₁₆	C₂⁴×C₈₉	semidirect
316. D₈	C₂⁴×C₈₉	semidirect
317. M₁₆	C₂⁴×C₈₉	semidirect
320. C₂×C₈	C₂⁴×C₈₉	semidirect
362. C₂²	#577×C₈₉	direct (N × H)
373. C₂	C₂×#577×C₈₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×2₋¹⁺⁴×C₈₉	2
C₂²×H(5,𝔽₂)×C₈₉	2
C₂×#577×C₈₉	2
C₂²×#137×C₈₉	2
C₂²×AGL(1,ℤ/8)×C₈₉	2
C₂²×#135×C₈₉	2
C₂³×SD₁₆×C₈₉	2
C₂²×#131×C₈₉	2
C₂²×#577	89

Subgroups — C22×#577×C89 #36440933

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#577×C₈₉ #36440933