Subgroups — C₈₉×#35527 #36413375

Order 22784.

6142 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×H(5,𝔽₂)×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#535×C₈₉	C₂	1	2	yes	maximal, complemented
3	C₂²×#127×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₉×#2433	C₂	1	8	yes	maximal, complemented
5	C₈₉×#2431	C₂	1	8	yes	maximal, complemented
6	C₈₉×#2403	C₂	1	8	yes	maximal, complemented
7	C₈₉×#2309	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈₉×#2297	C₂	1	2	yes	maximal, complemented
9	C₂×H(5,𝔽₂)×C₈₉	—	2	16	no	—
10	C₂×H(5,𝔽₂)×C₈₉	—	2	8	no	—
11	C₂²×#41×C₈₉	C₂²	1	2	yes	—
12	C₂²×#41×C₈₉	C₂²	1	4	yes	complemented
13	C₂³×D₄×C₈₉	C₂²	1	4	yes	—
14	C₂³×D₄×C₈₉	C₂²	1	1	yes	characteristic
15	C₂³×D₄×C₈₉	C₂²	1	4	yes	complemented
16	C₂³×D₄×C₈₉	C₂²	1	1	yes	characteristic, complemented
17	C₂³×D₄×C₈₉	C₂²	1	4	yes	complemented
18	#535×C₈₉	—	2	16	no	—
19	C₂×#127×C₈₉	—	2	8	no	—
20	C₂×#127×C₈₉	C₂²	1	8	yes	complemented
21	C₂×#127×C₈₉	C₂²	1	4	yes	complemented
22	C₂×#124×C₈₉	C₂²	1	4	yes	complemented
23	C₂×#121×C₈₉	C₂²	1	8	yes	complemented
24	C₂×#120×C₈₉	C₂²	1	8	yes	complemented
25	C₂×#120×C₈₉	C₂²	1	16	yes	complemented
26	C₂×#117×C₈₉	C₂²	1	2	yes	complemented
27	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic, complemented
28	#392×C₈₉	C₂²	1	32	yes	complemented
29	#391×C₈₉	C₂²	1	4	yes	complemented
30	#390×C₈₉	C₂²	1	4	yes	complemented
31	#390×C₈₉	C₂²	1	8	yes	complemented
32	#386×C₈₉	C₂²	1	8	yes	complemented
33	#386×C₈₉	C₂²	1	16	yes	complemented
34	#379×C₈₉	C₂²	1	4	yes	complemented
35	C₄×C₂²⋊C₄×C₈₉	C₂²	1	8	yes	complemented
36	C₂⁵×C₈₉	C₂³	1	4	yes	—
37	C₂⁵×C₈₉	C₂³	1	2	yes	—
38	C₂⁵×C₈₉	C₂³	1	2	yes	—
39	H(5,𝔽₂)×C₈₉	—	2	128	no	—
40	C₂×#41×C₈₉	—	2	16	no	—
41	C₂×#41×C₈₉	—	2	16	no	—
42	C₂×#41×C₈₉	—	2	8	no	—
43	C₂×#41×C₈₉	—	2	8	no	—
44	C₂×#41×C₈₉	—	2	16	no	—
45	C₂×#41×C₈₉	D₄	1	16	yes	complemented
46	C₂²×Q₈×C₈₉	C₂³	1	2	yes	—
47	C₂²×D₄×C₈₉	—	2	16	no	—
48	C₂²×D₄×C₈₉	—	2	32	no	—
49	C₂²×D₄×C₈₉	—	2	16	no	—
50	C₂²×D₄×C₈₉	—	2	16	no	—
51	C₂²×D₄×C₈₉	—	2	16	no	—
52	C₂²×D₄×C₈₉	—	2	8	no	—
53	C₂²×D₄×C₈₉	—	2	4	no	—
54	C₂²×D₄×C₈₉	—	2	8	no	—
55	C₂²×D₄×C₈₉	—	2	16	no	—
56	C₂²×D₄×C₈₉	—	4	16	no	—
57	C₂²×D₄×C₈₉	D₄	1	8	yes	complemented
58	C₂²×D₄×C₈₉	C₂³	1	8	yes	—
59	C₂²×D₄×C₈₉	C₂³	1	4	yes	—
60	C₂²×D₄×C₈₉	C₂³	1	16	yes	—
61	C₂²×D₄×C₈₉	C₂³	1	4	yes	—
62	C₂²×D₄×C₈₉	C₂³	1	4	yes	—
63	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
64	C₂²×D₄×C₈₉	C₂³	1	16	yes	—
65	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
66	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
67	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
68	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
69	#127×C₈₉	—	2	32	no	—
70	#127×C₈₉	—	2	16	no	—
71	#124×C₈₉	—	2	16	no	—
72	#121×C₈₉	—	2	32	no	—
73	#120×C₈₉	—	2	32	no	—
74	#120×C₈₉	—	4	32	no	—
75	#117×C₈₉	—	2	8	no	—
76	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
77	C₂×C₂²⋊C₄×C₈₉	C₂³	1	16	yes	—
78	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
79	C₂×C₂²⋊C₄×C₈₉	C₂³	1	16	yes	—
80	C₂×C₂²⋊C₄×C₈₉	C₂³	1	16	yes	—
81	C₂×C₄²×C₈₉	—	2	4	no	—
82	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
83	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
84	#95×C₈₉	C₂³	1	16	yes	—
85	C₂⁴×C₈₉	—	2	8	no	—
86	C₂⁴×C₈₉	—	2	16	no	—
87	C₂⁴×C₈₉	—	2	8	no	—
88	C₂⁴×C₈₉	—	2	4	no	—
89	C₂⁴×C₈₉	—	2	8	no	—
90	C₂⁴×C₈₉	—	4	8	no	—
91	C₂⁴×C₈₉	—	4	8	no	—
92	C₂⁴×C₈₉	—	4	16	no	—
93	C₂⁴×C₈₉	C₂×D₄	1	8	yes	—
94	C₂⁴×C₈₉	C₂⁴	1	4	yes	—
95	C₂⁴×C₈₉	C₂⁴	1	4	yes	—
96	C₂⁴×C₈₉	C₂⁴	1	4	yes	—
97	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
98	#41×C₈₉	—	2	128	no	—
99	#41×C₈₉	—	2	64	no	—
100	C₂×Q₈×C₈₉	—	2	4	no	—
101	C₂×Q₈×C₈₉	—	2	4	no	—
102	C₂×Q₈×C₈₉	C₂×D₄	1	8	yes	complemented
103	C₂×D₄×C₈₉	—	2	128	no	—
104	C₂×D₄×C₈₉	—	2	128	no	—
105	C₂×D₄×C₈₉	—	2	16	no	—
106	C₂×D₄×C₈₉	—	2	16	no	—
107	C₂×D₄×C₈₉	—	2	8	no	—
108	C₂×D₄×C₈₉	—	2	8	no	—
109	C₂×D₄×C₈₉	—	2	32	no	—
110	C₂×D₄×C₈₉	—	2	32	no	—
111	C₂×D₄×C₈₉	—	2	32	no	—
112	C₂×D₄×C₈₉	—	2	16	no	—
113	C₂×D₄×C₈₉	—	2	8	no	—
114	C₂×D₄×C₈₉	—	2	4	no	—
115	C₂×D₄×C₈₉	—	2	8	no	—
116	C₂×D₄×C₈₉	—	2	4	no	—
117	C₂×D₄×C₈₉	—	2	8	no	—
118	C₂×D₄×C₈₉	—	2	8	no	—
119	C₂×D₄×C₈₉	—	2	8	no	—
120	C₂×D₄×C₈₉	—	2	32	no	—
121	C₂×D₄×C₈₉	—	4	16	no	—
122	C₂×D₄×C₈₉	—	4	64	no	—
123	C₂×D₄×C₈₉	—	4	32	no	—
124	C₂×D₄×C₈₉	—	4	16	no	—
125	C₂×D₄×C₈₉	C₂×D₄	1	16	yes	—
126	C₂×D₄×C₈₉	C₂×D₄	1	16	yes	—
127	C₂×D₄×C₈₉	C₂×D₄	1	8	yes	complemented
128	C₂²×C₄×C₈₉	—	2	4	no	—
129	C₂²×C₄×C₈₉	—	2	8	no	—
130	C₂²×C₄×C₈₉	—	2	16	no	—
131	C₂²×C₄×C₈₉	—	2	8	no	—
132	C₂²×C₄×C₈₉	—	2	8	no	—
133	C₂²×C₄×C₈₉	—	2	8	no	—
134	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
135	C₂²×C₄×C₈₉	C₂×D₄	1	8	yes	—
136	C₂²×C₄×C₈₉	C₂×D₄	1	16	yes	complemented
137	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
138	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
139	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
140	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
141	C₄⋊C₄×C₈₉	—	2	8	no	—
142	C₂²⋊C₄×C₈₉	—	2	8	no	—
143	C₂²⋊C₄×C₈₉	—	2	32	no	—
144	C₂²⋊C₄×C₈₉	—	2	32	no	—
145	C₂²⋊C₄×C₈₉	—	4	16	no	—
146	C₄²×C₈₉	—	2	8	no	—
147	C₄²×C₈₉	—	2	8	no	—
148	C₂³×C₈₉	—	2	4	no	—
149	C₂³×C₈₉	—	2	32	no	—
150	C₂³×C₈₉	—	2	4	no	—
151	C₂³×C₈₉	—	2	8	no	—
152	C₂³×C₈₉	—	2	4	no	—
153	C₂³×C₈₉	—	2	16	no	—
154	C₂³×C₈₉	—	2	16	no	—
155	C₂³×C₈₉	—	2	16	no	—
156	C₂³×C₈₉	—	2	8	no	—
157	C₂³×C₈₉	—	2	4	no	—
158	C₂³×C₈₉	—	2	16	no	—
159	C₂³×C₈₉	—	4	16	no	—
160	C₂³×C₈₉	—	4	16	no	—
161	C₂³×C₈₉	—	4	8	no	—
162	C₂³×C₈₉	—	8	16	no	—
163	C₂³×C₈₉	—	8	16	no	—
164	C₂³×C₈₉	—	8	16	no	—
165	C₂³×C₈₉	—	8	32	no	—
166	C₂³×C₈₉	C₂²×D₄	1	8	yes	—
167	C₂³×C₈₉	C₂²×D₄	1	8	yes	—
168	C₂³×C₈₉	C₂²×D₄	1	4	yes	—
169	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
170	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
171	Q₈×C₈₉	—	2	16	no	—
172	D₄×C₈₉	—	2	64	no	—
173	D₄×C₈₉	—	2	32	no	—
174	D₄×C₈₉	—	2	32	no	—
175	D₄×C₈₉	—	2	16	no	—
176	D₄×C₈₉	—	4	64	no	—
177	D₄×C₈₉	—	4	16	no	—
178	D₄×C₈₉	—	8	32	no	—
179	C₂×C₄×C₈₉	—	2	32	no	—
180	C₂×C₄×C₈₉	—	2	16	no	—
181	C₂×C₄×C₈₉	—	2	4	no	—
182	C₂×C₄×C₈₉	—	2	4	no	—
183	C₂×C₄×C₈₉	—	2	4	no	—
184	C₂×C₄×C₈₉	—	2	4	no	—
185	C₂×C₄×C₈₉	—	2	32	no	—
186	C₂×C₄×C₈₉	—	2	8	no	—
187	C₂×C₄×C₈₉	—	2	16	no	—
188	C₂×C₄×C₈₉	#120	1	16	yes	complemented
189	C₂×C₄×C₈₉	C₂²×D₄	1	8	yes	—
190	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
191	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
192	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
193	C₂×C₄×C₈₉	H(5,𝔽₂)	1	8	yes	complemented
194	C₂²×C₈₉	—	2	4	no	—
195	C₂²×C₈₉	—	2	8	no	—
196	C₂²×C₈₉	—	2	8	no	—
197	C₂²×C₈₉	—	2	8	no	—
198	C₂²×C₈₉	—	2	16	no	—
199	C₂²×C₈₉	—	2	16	no	—
200	C₂²×C₈₉	—	4	8	no	—
201	C₂²×C₈₉	—	4	8	no	—
202	C₂²×C₈₉	—	4	16	no	—
203	C₂²×C₈₉	—	4	4	no	—
204	C₂²×C₈₉	—	4	8	no	—
205	C₂²×C₈₉	—	4	16	no	—
206	C₂²×C₈₉	—	8	16	no	—
207	C₂²×C₈₉	—	8	16	no	—
208	C₂²×C₈₉	—	8	32	no	—
209	C₂²×C₈₉	C₂×#120	1	4	yes	—
210	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
211	C₂²×C₈₉	C₂³×D₄	1	2	yes	—
212	C₂²×C₈₉	C₂×H(5,𝔽₂)	1	4	yes	—
213	C₄×C₈₉	—	2	8	no	—
214	C₄×C₈₉	—	2	8	no	—
215	C₄×C₈₉	—	4	8	no	—
216	C₄×C₈₉	C₂×#120	1	8	yes	—
217	#35527	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
218	C₂×C₈₉	—	2	4	no	—
219	C₂×C₈₉	—	4	8	no	—
220	C₂×C₈₉	—	4	8	no	—
221	C₂×C₈₉	—	8	4	no	—
222	C₂×C₈₉	C₂²×#120	1	1	yes	characteristic
223	C₂×C₈₉	#3484	1	2	yes	—
224	C₂×C₈₉	#3498	1	4	yes	—
225	C₂²×H(5,𝔽₂)	C₂×C₈₉	1	1	yes	characteristic, complemented
226	C₂×#535	C₂×C₈₉	1	2	yes	complemented
227	C₂²×#127	C₂×C₈₉	1	1	yes	characteristic, complemented
228	#2433	C₂×C₈₉	1	8	yes	complemented
229	#2431	C₂×C₈₉	1	8	yes	complemented
230	#2403	C₂×C₈₉	1	8	yes	complemented
231	#2309	C₂×C₈₉	1	1	yes	characteristic, complemented
232	#2297	C₂×C₈₉	1	2	yes	complemented
233	C₈₉	#35527	1	1	yes	characteristic, direct product factor, Sylow
234	C₂×H(5,𝔽₂)	—	2	16	no	—
235	C₂×H(5,𝔽₂)	—	2	8	no	—
236	C₂²×#41	C₂²×C₈₉	1	2	yes	—
237	C₂²×#41	C₂²×C₈₉	1	4	yes	complemented
238	C₂³×D₄	C₂²×C₈₉	1	4	yes	—
239	C₂³×D₄	C₂²×C₈₉	1	1	yes	characteristic
240	C₂³×D₄	C₂²×C₈₉	1	4	yes	complemented
241	C₂³×D₄	C₂²×C₈₉	1	1	yes	characteristic, complemented
242	C₂³×D₄	C₂²×C₈₉	1	4	yes	complemented
243	#535	—	2	16	no	—
244	C₂×#127	—	2	8	no	—
245	C₂×#127	C₂²×C₈₉	1	8	yes	complemented
246	C₂×#127	C₂²×C₈₉	1	4	yes	complemented
247	C₂×#124	C₂²×C₈₉	1	4	yes	complemented
248	C₂×#121	C₂²×C₈₉	1	8	yes	complemented
249	C₂×#120	C₂²×C₈₉	1	8	yes	complemented
250	C₂×#120	C₂²×C₈₉	1	16	yes	complemented
251	C₂×#117	C₂²×C₈₉	1	2	yes	complemented
252	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic, complemented
253	#392	C₂²×C₈₉	1	32	yes	complemented
254	#391	C₂²×C₈₉	1	4	yes	complemented
255	#390	C₂²×C₈₉	1	8	yes	complemented
256	#390	C₂²×C₈₉	1	4	yes	complemented
257	#386	C₂²×C₈₉	1	8	yes	complemented
258	#386	C₂²×C₈₉	1	16	yes	complemented
259	#379	C₂²×C₈₉	1	4	yes	complemented
260	C₄×C₂²⋊C₄	C₂²×C₈₉	1	8	yes	complemented
261	C₂⁵	C₂³×C₈₉	1	4	yes	—
262	C₂⁵	C₂³×C₈₉	1	2	yes	—
263	C₂⁵	C₂³×C₈₉	1	2	yes	—
264	H(5,𝔽₂)	—	2	128	no	—
265	C₂×#41	—	2	16	no	—
266	C₂×#41	—	2	16	no	—
267	C₂×#41	—	2	8	no	—
268	C₂×#41	—	2	8	no	—
269	C₂×#41	—	2	16	no	—
270	C₂×#41	D₄×C₈₉	1	16	yes	complemented
271	C₂²×Q₈	C₂³×C₈₉	1	2	yes	—
272	C₂²×D₄	—	2	16	no	—
273	C₂²×D₄	—	2	32	no	—
274	C₂²×D₄	—	2	16	no	—
275	C₂²×D₄	—	2	16	no	—
276	C₂²×D₄	—	2	16	no	—
277	C₂²×D₄	—	2	8	no	—
278	C₂²×D₄	—	2	4	no	—
279	C₂²×D₄	—	2	8	no	—
280	C₂²×D₄	—	2	16	no	—
281	C₂²×D₄	—	4	16	no	—
282	C₂²×D₄	D₄×C₈₉	1	8	yes	complemented
283	C₂²×D₄	C₂³×C₈₉	1	4	yes	—
284	C₂²×D₄	C₂³×C₈₉	1	8	yes	—
285	C₂²×D₄	C₂³×C₈₉	1	16	yes	—
286	C₂²×D₄	C₂³×C₈₉	1	4	yes	—
287	C₂²×D₄	C₂³×C₈₉	1	4	yes	—
288	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
289	C₂²×D₄	C₂³×C₈₉	1	16	yes	—
290	C₂³×C₄	C₂³×C₈₉	1	4	yes	—
291	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
292	C₂³×C₄	C₂³×C₈₉	1	4	yes	—
293	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
294	#127	—	2	32	no	—
295	#127	—	2	16	no	—
296	#124	—	2	16	no	—
297	#121	—	2	32	no	—
298	#120	—	2	32	no	—
299	#120	—	4	32	no	—
300	#117	—	2	8	no	—
301	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
302	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
303	C₂×C₂²⋊C₄	C₂³×C₈₉	1	16	yes	—
304	C₂×C₂²⋊C₄	C₂³×C₈₉	1	16	yes	—
305	C₂×C₂²⋊C₄	C₂³×C₈₉	1	16	yes	—
306	C₂×C₄²	—	2	4	no	—
307	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
308	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
309	#95	C₂³×C₈₉	1	16	yes	—
310	C₂⁴	—	2	8	no	—
311	C₂⁴	—	2	16	no	—
312	C₂⁴	—	2	8	no	—
313	C₂⁴	—	2	4	no	—
314	C₂⁴	—	2	8	no	—
315	C₂⁴	—	4	8	no	—
316	C₂⁴	—	4	8	no	—
317	C₂⁴	—	4	16	no	—
318	C₂⁴	C₂×D₄×C₈₉	1	8	yes	—
319	C₂⁴	C₂⁴×C₈₉	1	4	yes	—
320	C₂⁴	C₂⁴×C₈₉	1	4	yes	—
321	C₂⁴	C₂⁴×C₈₉	1	4	yes	—
322	C₂⁴	C₂⁴×C₈₉	1	1	yes	characteristic
323	#41	—	2	128	no	—
324	#41	—	2	64	no	—
325	C₂×Q₈	—	2	4	no	—
326	C₂×Q₈	—	2	4	no	—
327	C₂×Q₈	C₂×D₄×C₈₉	1	8	yes	complemented
328	C₂×D₄	—	2	128	no	—
329	C₂×D₄	—	2	128	no	—
330	C₂×D₄	—	2	16	no	—
331	C₂×D₄	—	2	16	no	—
332	C₂×D₄	—	2	8	no	—
333	C₂×D₄	—	2	8	no	—
334	C₂×D₄	—	2	32	no	—
335	C₂×D₄	—	2	32	no	—
336	C₂×D₄	—	2	32	no	—
337	C₂×D₄	—	2	16	no	—
338	C₂×D₄	—	2	8	no	—
339	C₂×D₄	—	2	4	no	—
340	C₂×D₄	—	2	8	no	—
341	C₂×D₄	—	2	4	no	—
342	C₂×D₄	—	2	8	no	—
343	C₂×D₄	—	2	8	no	—
344	C₂×D₄	—	2	8	no	—
345	C₂×D₄	—	2	32	no	—
346	C₂×D₄	—	4	16	no	—
347	C₂×D₄	—	4	64	no	—
348	C₂×D₄	—	4	32	no	—
349	C₂×D₄	—	4	16	no	—
350	C₂×D₄	C₂×D₄×C₈₉	1	16	yes	—
351	C₂×D₄	C₂×D₄×C₈₉	1	16	yes	—
352	C₂×D₄	C₂×D₄×C₈₉	1	8	yes	complemented
353	C₂²×C₄	—	2	4	no	—
354	C₂²×C₄	—	2	8	no	—
355	C₂²×C₄	—	2	16	no	—
356	C₂²×C₄	—	2	8	no	—
357	C₂²×C₄	—	2	8	no	—
358	C₂²×C₄	—	2	8	no	—
359	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
360	C₂²×C₄	C₂×D₄×C₈₉	1	8	yes	—
361	C₂²×C₄	C₂×D₄×C₈₉	1	16	yes	complemented
362	C₂²×C₄	C₂⁴×C₈₉	1	8	yes	—
363	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
364	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
365	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
366	C₄⋊C₄	—	2	8	no	—
367	C₂²⋊C₄	—	2	8	no	—
368	C₂²⋊C₄	—	2	32	no	—
369	C₂²⋊C₄	—	2	32	no	—
370	C₂²⋊C₄	—	4	16	no	—
371	C₄²	—	2	8	no	—
372	C₄²	—	2	8	no	—
373	C₂³	—	2	4	no	—
374	C₂³	—	2	32	no	—
375	C₂³	—	2	4	no	—
376	C₂³	—	2	8	no	—
377	C₂³	—	2	4	no	—
378	C₂³	—	2	16	no	—
379	C₂³	—	2	16	no	—
380	C₂³	—	2	16	no	—
381	C₂³	—	2	8	no	—
382	C₂³	—	2	4	no	—
383	C₂³	—	2	16	no	—
384	C₂³	—	4	16	no	—
385	C₂³	—	4	16	no	—
386	C₂³	—	4	8	no	—
387	C₂³	—	8	16	no	—
388	C₂³	—	8	16	no	—
389	C₂³	—	8	16	no	—
390	C₂³	—	8	32	no	—
391	C₂³	C₂²×D₄×C₈₉	1	8	yes	—
392	C₂³	C₂²×D₄×C₈₉	1	8	yes	—
393	C₂³	C₂²×D₄×C₈₉	1	4	yes	—
394	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
395	C₂³	C₂⁵×C₈₉	1	1	yes	characteristic
396	Q₈	—	2	16	no	—
397	D₄	—	2	64	no	—
398	D₄	—	2	32	no	—
399	D₄	—	2	32	no	—
400	D₄	—	2	16	no	—
401	D₄	—	4	64	no	—
402	D₄	—	4	16	no	—
403	D₄	—	8	32	no	—
404	C₂×C₄	—	2	32	no	—
405	C₂×C₄	—	2	16	no	—
406	C₂×C₄	—	2	4	no	—
407	C₂×C₄	—	2	4	no	—
408	C₂×C₄	—	2	4	no	—
409	C₂×C₄	—	2	4	no	—
410	C₂×C₄	—	2	32	no	—
411	C₂×C₄	—	2	8	no	—
412	C₂×C₄	—	2	16	no	—
413	C₂×C₄	#120×C₈₉	1	16	yes	complemented
414	C₂×C₄	C₂²×D₄×C₈₉	1	8	yes	—
415	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
416	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
417	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
418	C₂×C₄	H(5,𝔽₂)×C₈₉	1	8	yes	complemented
419	C₂²	—	2	4	no	—
420	C₂²	—	2	8	no	—
421	C₂²	—	2	8	no	—
422	C₂²	—	2	8	no	—
423	C₂²	—	2	16	no	—
424	C₂²	—	2	16	no	—
425	C₂²	—	4	8	no	—
426	C₂²	—	4	8	no	—
427	C₂²	—	4	16	no	—
428	C₂²	—	4	4	no	—
429	C₂²	—	4	8	no	—
430	C₂²	—	4	16	no	—
431	C₂²	—	8	16	no	—
432	C₂²	—	8	16	no	—
433	C₂²	—	8	32	no	—
434	C₂²	C₂×#120×C₈₉	1	4	yes	—
435	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
436	C₂²	C₂³×D₄×C₈₉	1	2	yes	—
437	C₂²	C₂×H(5,𝔽₂)×C₈₉	1	4	yes	—
438	C₄	—	2	8	no	—
439	C₄	—	2	8	no	—
440	C₄	—	4	8	no	—
441	C₄	C₂×#120×C₈₉	1	8	yes	—
442	C₂	—	2	4	no	—
443	C₂	—	4	8	no	—
444	C₂	—	4	8	no	—
445	C₂	—	8	4	no	—
446	C₂	C₂²×#120×C₈₉	1	1	yes	characteristic
447	C₂	C₈₉×#3484	1	2	yes	—
448	C₂	C₈₉×#3498	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₈₉
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×H(5,𝔽₂)×C₈₉	C₂	semidirect
2. C₂×#535×C₈₉	C₂	semidirect
3. C₂²×#127×C₈₉	C₂	semidirect
4. C₈₉×#2433	C₂	semidirect
5. C₈₉×#2431	C₂	semidirect
6. C₈₉×#2403	C₂	semidirect
7. C₈₉×#2309	C₂	semidirect
8. C₈₉×#2297	C₂	semidirect
12. C₂²×#41×C₈₉	C₂²	semidirect
15. C₂³×D₄×C₈₉	C₂²	semidirect
16. C₂³×D₄×C₈₉	C₂²	semidirect
17. C₂³×D₄×C₈₉	C₂²	semidirect
20. C₂×#127×C₈₉	C₂²	semidirect
21. C₂×#127×C₈₉	C₂²	semidirect
22. C₂×#124×C₈₉	C₂²	semidirect
23. C₂×#121×C₈₉	C₂²	semidirect
24. C₂×#120×C₈₉	C₂²	semidirect
25. C₂×#120×C₈₉	C₂²	semidirect
26. C₂×#117×C₈₉	C₂²	semidirect
27. C₂²×C₄²×C₈₉	C₂²	semidirect
28. #392×C₈₉	C₂²	semidirect
29. #391×C₈₉	C₂²	semidirect
30. #390×C₈₉	C₂²	semidirect
31. #390×C₈₉	C₂²	semidirect
32. #386×C₈₉	C₂²	semidirect
33. #386×C₈₉	C₂²	semidirect
34. #379×C₈₉	C₂²	semidirect
35. C₄×C₂²⋊C₄×C₈₉	C₂²	semidirect
45. C₂×#41×C₈₉	D₄	semidirect
57. C₂²×D₄×C₈₉	D₄	semidirect
102. C₂×Q₈×C₈₉	C₂×D₄	semidirect
127. C₂×D₄×C₈₉	C₂×D₄	semidirect
136. C₂²×C₄×C₈₉	C₂×D₄	semidirect
188. C₂×C₄×C₈₉	#120	semidirect
193. C₂×C₄×C₈₉	H(5,𝔽₂)	semidirect
217. #35527	C₈₉	direct (N × H)
225. C₂²×H(5,𝔽₂)	C₂×C₈₉	semidirect
226. C₂×#535	C₂×C₈₉	semidirect
227. C₂²×#127	C₂×C₈₉	semidirect
228. #2433	C₂×C₈₉	semidirect
229. #2431	C₂×C₈₉	semidirect
230. #2403	C₂×C₈₉	semidirect
231. #2309	C₂×C₈₉	semidirect
232. #2297	C₂×C₈₉	semidirect
233. C₈₉	#35527	direct (N × H)
237. C₂²×#41	C₂²×C₈₉	semidirect
240. C₂³×D₄	C₂²×C₈₉	semidirect
241. C₂³×D₄	C₂²×C₈₉	semidirect
242. C₂³×D₄	C₂²×C₈₉	semidirect
245. C₂×#127	C₂²×C₈₉	semidirect
246. C₂×#127	C₂²×C₈₉	semidirect
247. C₂×#124	C₂²×C₈₉	semidirect
248. C₂×#121	C₂²×C₈₉	semidirect
249. C₂×#120	C₂²×C₈₉	semidirect
250. C₂×#120	C₂²×C₈₉	semidirect
251. C₂×#117	C₂²×C₈₉	semidirect
252. C₂²×C₄²	C₂²×C₈₉	semidirect
253. #392	C₂²×C₈₉	semidirect
254. #391	C₂²×C₈₉	semidirect
255. #390	C₂²×C₈₉	semidirect
256. #390	C₂²×C₈₉	semidirect
257. #386	C₂²×C₈₉	semidirect
258. #386	C₂²×C₈₉	semidirect
259. #379	C₂²×C₈₉	semidirect
260. C₄×C₂²⋊C₄	C₂²×C₈₉	semidirect
270. C₂×#41	D₄×C₈₉	semidirect
282. C₂²×D₄	D₄×C₈₉	semidirect
327. C₂×Q₈	C₂×D₄×C₈₉	semidirect
352. C₂×D₄	C₂×D₄×C₈₉	semidirect
361. C₂²×C₄	C₂×D₄×C₈₉	semidirect
413. C₂×C₄	#120×C₈₉	semidirect
418. C₂×C₄	H(5,𝔽₂)×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×H(5,𝔽₂)×C₈₉	2
C₂×#535×C₈₉	2
C₂²×#127×C₈₉	2
C₈₉×#2433	2
C₈₉×#2431	2
C₈₉×#2403	2
C₈₉×#2309	2
C₈₉×#2297	2
#35527	89

Subgroups — C89×#35527 #36413375

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#35527 #36413375