Subgroups — C₈₉×#35224 #36413072

Order 22784.

2742 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#544×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₄⋊C₄×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#2848	C₂	1	4	yes	maximal
4	C₈₉×#2627	C₂	1	4	yes	maximal, complemented
5	C₈₉×#2442	C₂	1	8	yes	maximal, complemented
6	C₈₉×#2440	C₂	1	8	yes	maximal, complemented
7	C₈₉×#2317	C₂	1	2	yes	maximal, complemented
8	C₈₉×#2316	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#2280	C₂	1	2	yes	maximal
10	C₂⁴×C₄×C₈₉	C₂²	1	1	yes	characteristic
11	C₂⁴×C₄×C₈₉	C₂²	1	2	yes	—
12	#544×C₈₉	—	2	8	no	—
13	C₂×C₄⋊Q₈×C₈₉	C₂²	1	4	yes	complemented
14	C₂×#125×C₈₉	C₂²	1	4	yes	complemented
15	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	4	yes	—
16	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	8	yes	—
17	C₂×#117×C₈₉	C₂²	1	2	yes	—
18	C₂×#117×C₈₉	C₂²	1	2	yes	—
19	C₂²×C₄⋊C₄×C₈₉	—	2	8	no	—
20	C₂²×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
21	C₂²×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
22	C₂²×C₄⋊C₄×C₈₉	C₂²	1	8	yes	—
23	C₂²×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
24	#400×C₈₉	C₂²	1	8	yes	complemented
25	#398×C₈₉	C₂²	1	16	yes	complemented
26	#395×C₈₉	C₂²	1	8	yes	complemented
27	#389×C₈₉	C₂²	1	8	yes	complemented
28	#389×C₈₉	C₂²	1	8	yes	complemented
29	#385×C₈₉	C₂²	1	8	yes	—
30	#385×C₈₉	C₂²	1	16	yes	—
31	#384×C₈₉	C₂²	1	4	yes	complemented
32	#382×C₈₉	C₂²	1	4	yes	complemented
33	#381×C₈₉	C₂²	1	4	yes	—
34	#380×C₈₉	C₂²	1	4	yes	—
35	#380×C₈₉	C₂²	1	8	yes	—
36	#380×C₈₉	C₂²	1	4	yes	—
37	#380×C₈₉	C₂²	1	4	yes	—
38	#379×C₈₉	C₂²	1	4	yes	—
39	C₄×C₂²⋊C₄×C₈₉	C₂²	1	8	yes	—
40	C₂⁵×C₈₉	C₂³	1	1	yes	characteristic
41	C₂²×Q₈×C₈₉	C₂³	1	4	yes	—
42	C₂³×C₄×C₈₉	—	2	4	no	—
43	C₂³×C₄×C₈₉	—	2	4	no	—
44	C₂³×C₄×C₈₉	—	2	8	no	—
45	C₂³×C₄×C₈₉	—	4	8	no	—
46	C₂³×C₄×C₈₉	Q₈	1	8	yes	complemented
47	C₂³×C₄×C₈₉	C₂³	1	8	yes	—
48	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
49	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
50	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
51	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
52	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
53	C₄⋊Q₈×C₈₉	—	2	16	no	—
54	#125×C₈₉	—	2	16	no	—
55	C₂²⋊Q₈×C₈₉	—	2	16	no	—
56	C₂²⋊Q₈×C₈₉	—	4	16	no	—
57	#117×C₈₉	—	2	8	no	—
58	#117×C₈₉	—	2	8	no	—
59	C₂×C₄⋊C₄×C₈₉	—	2	8	no	—
60	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
61	C₂×C₄⋊C₄×C₈₉	—	2	32	no	—
62	C₂×C₄⋊C₄×C₈₉	D₄	1	8	yes	—
63	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
64	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
66	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
67	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
68	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
69	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
70	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
71	C₂×C₄⋊C₄×C₈₉	C₂³	1	16	yes	—
72	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
73	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
74	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
75	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
76	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
77	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
78	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
79	#95×C₈₉	C₂³	1	8	yes	—
80	#95×C₈₉	C₂³	1	8	yes	—
81	#95×C₈₉	C₂³	1	16	yes	—
82	#95×C₈₉	C₂³	1	8	yes	—
83	C₂⁴×C₈₉	—	2	2	no	—
84	C₂⁴×C₈₉	—	2	2	no	—
85	C₂⁴×C₈₉	—	2	4	no	—
86	C₂⁴×C₈₉	—	2	4	no	—
87	C₂⁴×C₈₉	C₂×Q₈	1	4	yes	—
88	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
89	C₂⁴×C₈₉	C₂⁴	1	2	yes	—
90	C₂×Q₈×C₈₉	—	2	8	no	—
91	C₂×Q₈×C₈₉	—	4	8	no	—
92	C₂²×C₄×C₈₉	—	2	2	no	—
93	C₂²×C₄×C₈₉	—	2	2	no	—
94	C₂²×C₄×C₈₉	—	2	32	no	—
95	C₂²×C₄×C₈₉	—	2	8	no	—
96	C₂²×C₄×C₈₉	—	2	8	no	—
97	C₂²×C₄×C₈₉	—	2	4	no	—
98	C₂²×C₄×C₈₉	—	2	16	no	—
99	C₂²×C₄×C₈₉	—	2	16	no	—
100	C₂²×C₄×C₈₉	—	4	4	no	—
101	C₂²×C₄×C₈₉	—	4	8	no	—
102	C₂²×C₄×C₈₉	—	4	4	no	—
103	C₂²×C₄×C₈₉	—	4	32	no	—
104	C₂²×C₄×C₈₉	C₂×D₄	1	8	yes	—
105	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
106	C₂²×C₄×C₈₉	C₂×Q₈	1	4	yes	—
107	C₂²×C₄×C₈₉	C₂×Q₈	1	16	yes	—
108	C₂²×C₄×C₈₉	C₂×Q₈	1	4	yes	—
109	C₂²×C₄×C₈₉	#41	1	4	yes	—
110	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
111	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
112	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
113	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
114	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
115	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
116	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
117	C₄⋊C₄×C₈₉	—	2	16	no	—
118	C₄⋊C₄×C₈₉	—	2	8	no	—
119	C₄⋊C₄×C₈₉	—	2	4	no	—
120	C₄⋊C₄×C₈₉	—	2	4	no	—
121	C₄⋊C₄×C₈₉	—	2	8	no	—
122	C₄⋊C₄×C₈₉	—	2	16	no	—
123	C₄⋊C₄×C₈₉	—	2	8	no	—
124	C₄⋊C₄×C₈₉	—	4	8	no	—
125	C₄⋊C₄×C₈₉	—	4	16	no	—
126	C₄⋊C₄×C₈₉	C₂×D₄	1	16	yes	—
127	C₂²⋊C₄×C₈₉	—	2	8	no	—
128	C₂²⋊C₄×C₈₉	—	2	8	no	—
129	C₂²⋊C₄×C₈₉	—	2	16	no	—
130	C₂²⋊C₄×C₈₉	—	2	16	no	—
131	C₄²×C₈₉	—	2	8	no	—
132	C₄²×C₈₉	—	2	8	no	—
133	C₂³×C₈₉	—	2	8	no	—
134	C₂³×C₈₉	—	2	8	no	—
135	C₂³×C₈₉	—	2	16	no	—
136	C₂³×C₈₉	—	2	8	no	—
137	C₂³×C₈₉	—	2	16	no	—
138	C₂³×C₈₉	—	2	2	no	—
139	C₂³×C₈₉	—	2	2	no	—
140	C₂³×C₈₉	—	2	2	no	—
141	C₂³×C₈₉	—	2	2	no	—
142	C₂³×C₈₉	—	2	2	no	—
143	C₂³×C₈₉	—	2	2	no	—
144	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
145	C₂³×C₈₉	C₂²×Q₈	1	2	yes	—
146	C₂³×C₈₉	C₂²×Q₈	1	4	yes	—
147	C₂³×C₈₉	C₂×#41	1	2	yes	—
148	C₂³×C₈₉	H(5,𝔽₂)	1	4	yes	—
149	C₂³×C₈₉	2₋¹⁺⁴	1	4	yes	—
150	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
151	Q₈×C₈₉	—	8	8	no	—
152	C₂×C₄×C₈₉	—	2	8	no	—
153	C₂×C₄×C₈₉	—	2	2	no	—
154	C₂×C₄×C₈₉	—	2	2	no	—
155	C₂×C₄×C₈₉	—	2	8	no	—
156	C₂×C₄×C₈₉	—	2	32	no	—
157	C₂×C₄×C₈₉	—	2	4	no	—
158	C₂×C₄×C₈₉	—	2	4	no	—
159	C₂×C₄×C₈₉	—	2	4	no	—
160	C₂×C₄×C₈₉	—	2	8	no	—
161	C₂×C₄×C₈₉	—	2	8	no	—
162	C₂×C₄×C₈₉	—	2	8	no	—
163	C₂×C₄×C₈₉	—	2	8	no	—
164	C₂×C₄×C₈₉	—	2	8	no	—
165	C₂×C₄×C₈₉	—	2	8	no	—
166	C₂×C₄×C₈₉	—	4	8	no	—
167	C₂×C₄×C₈₉	—	4	32	no	—
168	C₂×C₄×C₈₉	—	4	8	no	—
169	C₂×C₄×C₈₉	C₂²⋊Q₈	1	16	yes	complemented
170	C₂×C₄×C₈₉	C₂²×D₄	1	8	yes	—
171	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
172	C₂×C₄×C₈₉	C₂²×Q₈	1	4	yes	—
173	C₂×C₄×C₈₉	C₂²×Q₈	1	4	yes	—
174	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
175	C₂²×C₈₉	—	2	32	no	—
176	C₂²×C₈₉	—	2	4	no	—
177	C₂²×C₈₉	—	2	4	no	—
178	C₂²×C₈₉	—	2	4	no	—
179	C₂²×C₈₉	—	2	2	no	—
180	C₂²×C₈₉	—	2	4	no	—
181	C₂²×C₈₉	—	2	2	no	—
182	C₂²×C₈₉	—	2	4	no	—
183	C₂²×C₈₉	—	2	4	no	—
184	C₂²×C₈₉	—	2	4	no	—
185	C₂²×C₈₉	—	2	4	no	—
186	C₂²×C₈₉	C₂×C₂²⋊Q₈	1	4	yes	—
187	C₂²×C₈₉	#544	1	8	yes	complemented
188	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
189	C₂²×C₈₉	C₂³×Q₈	1	1	yes	characteristic
190	C₂²×C₈₉	C₂²×#41	1	1	yes	characteristic
191	C₂²×C₈₉	C₂×H(5,𝔽₂)	1	2	yes	—
192	C₂²×C₈₉	C₂×2₋¹⁺⁴	1	2	yes	—
193	C₄×C₈₉	—	2	4	no	—
194	C₄×C₈₉	—	2	8	no	—
195	C₄×C₈₉	—	4	4	no	—
196	C₄×C₈₉	—	4	4	no	—
197	C₄×C₈₉	—	4	4	no	—
198	C₄×C₈₉	—	4	8	no	—
199	C₄×C₈₉	C₂×C₂²⋊Q₈	1	8	yes	—
200	#35224	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
201	C₂×C₈₉	—	2	4	no	—
202	C₂×C₈₉	—	2	8	no	—
203	C₂×C₈₉	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
204	C₂×C₈₉	C₂×#544	1	2	yes	—
205	C₂×C₈₉	#3484	1	1	yes	characteristic
206	C₂×C₈₉	#3486	1	1	yes	characteristic
207	C₂×C₈₉	#3492	1	2	yes	—
208	C₂×#544	C₂×C₈₉	1	1	yes	characteristic, complemented
209	C₂³×C₄⋊C₄	C₂×C₈₉	1	1	yes	characteristic
210	#2848	C₂×C₈₉	1	4	yes	—
211	#2627	C₂×C₈₉	1	4	yes	complemented
212	#2442	C₂×C₈₉	1	8	yes	complemented
213	#2440	C₂×C₈₉	1	8	yes	complemented
214	#2317	C₂×C₈₉	1	2	yes	complemented
215	#2316	C₂×C₈₉	1	1	yes	characteristic, complemented
216	#2280	C₂×C₈₉	1	2	yes	—
217	C₈₉	#35224	1	1	yes	characteristic, direct product factor, Sylow
218	C₂⁴×C₄	C₂²×C₈₉	1	1	yes	characteristic
219	C₂⁴×C₄	C₂²×C₈₉	1	2	yes	—
220	#544	—	2	8	no	—
221	C₂×C₄⋊Q₈	C₂²×C₈₉	1	4	yes	complemented
222	C₂×#125	C₂²×C₈₉	1	4	yes	complemented
223	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	4	yes	—
224	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	8	yes	—
225	C₂×#117	C₂²×C₈₉	1	2	yes	—
226	C₂×#117	C₂²×C₈₉	1	2	yes	—
227	C₂²×C₄⋊C₄	—	2	8	no	—
228	C₂²×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
229	C₂²×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
230	C₂²×C₄⋊C₄	C₂²×C₈₉	1	8	yes	—
231	C₂²×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
232	#400	C₂²×C₈₉	1	8	yes	complemented
233	#398	C₂²×C₈₉	1	16	yes	complemented
234	#395	C₂²×C₈₉	1	8	yes	complemented
235	#389	C₂²×C₈₉	1	8	yes	complemented
236	#389	C₂²×C₈₉	1	8	yes	complemented
237	#385	C₂²×C₈₉	1	8	yes	—
238	#385	C₂²×C₈₉	1	16	yes	—
239	#384	C₂²×C₈₉	1	4	yes	complemented
240	#382	C₂²×C₈₉	1	4	yes	complemented
241	#381	C₂²×C₈₉	1	4	yes	—
242	#380	C₂²×C₈₉	1	4	yes	—
243	#380	C₂²×C₈₉	1	8	yes	—
244	#380	C₂²×C₈₉	1	4	yes	—
245	#380	C₂²×C₈₉	1	4	yes	—
246	#379	C₂²×C₈₉	1	4	yes	—
247	C₄×C₂²⋊C₄	C₂²×C₈₉	1	8	yes	—
248	C₂⁵	C₂³×C₈₉	1	1	yes	characteristic
249	C₂²×Q₈	C₂³×C₈₉	1	4	yes	—
250	C₂³×C₄	—	2	4	no	—
251	C₂³×C₄	—	2	4	no	—
252	C₂³×C₄	—	2	8	no	—
253	C₂³×C₄	—	4	8	no	—
254	C₂³×C₄	Q₈×C₈₉	1	8	yes	complemented
255	C₂³×C₄	C₂³×C₈₉	1	8	yes	—
256	C₂³×C₄	C₂³×C₈₉	1	4	yes	—
257	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
258	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
259	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
260	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
261	C₄⋊Q₈	—	2	16	no	—
262	#125	—	2	16	no	—
263	C₂²⋊Q₈	—	2	16	no	—
264	C₂²⋊Q₈	—	4	16	no	—
265	#117	—	2	8	no	—
266	#117	—	2	8	no	—
267	C₂×C₄⋊C₄	—	2	8	no	—
268	C₂×C₄⋊C₄	—	2	4	no	—
269	C₂×C₄⋊C₄	—	2	32	no	—
270	C₂×C₄⋊C₄	D₄×C₈₉	1	8	yes	—
271	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
272	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	—
273	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
274	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
275	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
276	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
277	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	—
278	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	—
279	C₂×C₄⋊C₄	C₂³×C₈₉	1	16	yes	—
280	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
281	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
282	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
283	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
284	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
285	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
286	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
287	#95	C₂³×C₈₉	1	8	yes	—
288	#95	C₂³×C₈₉	1	8	yes	—
289	#95	C₂³×C₈₉	1	16	yes	—
290	#95	C₂³×C₈₉	1	8	yes	—
291	C₂⁴	—	2	2	no	—
292	C₂⁴	—	2	2	no	—
293	C₂⁴	—	2	4	no	—
294	C₂⁴	—	2	4	no	—
295	C₂⁴	C₂×Q₈×C₈₉	1	4	yes	—
296	C₂⁴	C₂⁴×C₈₉	1	1	yes	characteristic
297	C₂⁴	C₂⁴×C₈₉	1	2	yes	—
298	C₂×Q₈	—	2	8	no	—
299	C₂×Q₈	—	4	8	no	—
300	C₂²×C₄	—	2	2	no	—
301	C₂²×C₄	—	2	2	no	—
302	C₂²×C₄	—	2	32	no	—
303	C₂²×C₄	—	2	8	no	—
304	C₂²×C₄	—	2	8	no	—
305	C₂²×C₄	—	2	4	no	—
306	C₂²×C₄	—	2	16	no	—
307	C₂²×C₄	—	2	16	no	—
308	C₂²×C₄	—	4	4	no	—
309	C₂²×C₄	—	4	8	no	—
310	C₂²×C₄	—	4	4	no	—
311	C₂²×C₄	—	4	32	no	—
312	C₂²×C₄	C₂×D₄×C₈₉	1	8	yes	—
313	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
314	C₂²×C₄	C₂×Q₈×C₈₉	1	4	yes	—
315	C₂²×C₄	C₂×Q₈×C₈₉	1	16	yes	—
316	C₂²×C₄	C₂×Q₈×C₈₉	1	4	yes	—
317	C₂²×C₄	#41×C₈₉	1	4	yes	—
318	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
319	C₂²×C₄	C₂⁴×C₈₉	1	8	yes	—
320	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
321	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
322	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
323	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
324	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
325	C₄⋊C₄	—	2	16	no	—
326	C₄⋊C₄	—	2	8	no	—
327	C₄⋊C₄	—	2	4	no	—
328	C₄⋊C₄	—	2	4	no	—
329	C₄⋊C₄	—	2	8	no	—
330	C₄⋊C₄	—	2	16	no	—
331	C₄⋊C₄	—	2	8	no	—
332	C₄⋊C₄	—	4	8	no	—
333	C₄⋊C₄	—	4	16	no	—
334	C₄⋊C₄	C₂×D₄×C₈₉	1	16	yes	—
335	C₂²⋊C₄	—	2	8	no	—
336	C₂²⋊C₄	—	2	8	no	—
337	C₂²⋊C₄	—	2	16	no	—
338	C₂²⋊C₄	—	2	16	no	—
339	C₄²	—	2	8	no	—
340	C₄²	—	2	8	no	—
341	C₂³	—	2	8	no	—
342	C₂³	—	2	8	no	—
343	C₂³	—	2	16	no	—
344	C₂³	—	2	8	no	—
345	C₂³	—	2	16	no	—
346	C₂³	—	2	2	no	—
347	C₂³	—	2	2	no	—
348	C₂³	—	2	2	no	—
349	C₂³	—	2	2	no	—
350	C₂³	—	2	2	no	—
351	C₂³	—	2	2	no	—
352	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
353	C₂³	C₂²×Q₈×C₈₉	1	2	yes	—
354	C₂³	C₂²×Q₈×C₈₉	1	4	yes	—
355	C₂³	C₂×#41×C₈₉	1	2	yes	—
356	C₂³	H(5,𝔽₂)×C₈₉	1	4	yes	—
357	C₂³	2₋¹⁺⁴×C₈₉	1	4	yes	—
358	C₂³	C₂⁵×C₈₉	1	1	yes	characteristic
359	Q₈	—	8	8	no	—
360	C₂×C₄	—	2	8	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	2	8	no	—
364	C₂×C₄	—	2	32	no	—
365	C₂×C₄	—	2	4	no	—
366	C₂×C₄	—	2	4	no	—
367	C₂×C₄	—	2	4	no	—
368	C₂×C₄	—	2	8	no	—
369	C₂×C₄	—	2	8	no	—
370	C₂×C₄	—	2	8	no	—
371	C₂×C₄	—	2	8	no	—
372	C₂×C₄	—	2	8	no	—
373	C₂×C₄	—	2	8	no	—
374	C₂×C₄	—	4	8	no	—
375	C₂×C₄	—	4	32	no	—
376	C₂×C₄	—	4	8	no	—
377	C₂×C₄	C₂²⋊Q₈×C₈₉	1	16	yes	complemented
378	C₂×C₄	C₂²×D₄×C₈₉	1	8	yes	—
379	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
380	C₂×C₄	C₂²×Q₈×C₈₉	1	4	yes	—
381	C₂×C₄	C₂²×Q₈×C₈₉	1	4	yes	—
382	C₂×C₄	C₂×#41×C₈₉	1	4	yes	—
383	C₂²	—	2	32	no	—
384	C₂²	—	2	4	no	—
385	C₂²	—	2	4	no	—
386	C₂²	—	2	4	no	—
387	C₂²	—	2	2	no	—
388	C₂²	—	2	4	no	—
389	C₂²	—	2	2	no	—
390	C₂²	—	2	4	no	—
391	C₂²	—	2	4	no	—
392	C₂²	—	2	4	no	—
393	C₂²	—	2	4	no	—
394	C₂²	C₂×C₂²⋊Q₈×C₈₉	1	4	yes	—
395	C₂²	#544×C₈₉	1	8	yes	complemented
396	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
397	C₂²	C₂³×Q₈×C₈₉	1	1	yes	characteristic
398	C₂²	C₂²×#41×C₈₉	1	1	yes	characteristic
399	C₂²	C₂×H(5,𝔽₂)×C₈₉	1	2	yes	—
400	C₂²	C₂×2₋¹⁺⁴×C₈₉	1	2	yes	—
401	C₄	—	2	4	no	—
402	C₄	—	2	8	no	—
403	C₄	—	4	4	no	—
404	C₄	—	4	4	no	—
405	C₄	—	4	4	no	—
406	C₄	—	4	8	no	—
407	C₄	C₂×C₂²⋊Q₈×C₈₉	1	8	yes	—
408	C₂	—	2	4	no	—
409	C₂	—	2	8	no	—
410	C₂	C₂²×C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
411	C₂	C₂×#544×C₈₉	1	2	yes	—
412	C₂	C₈₉×#3484	1	1	yes	characteristic
413	C₂	C₈₉×#3486	1	1	yes	characteristic
414	C₂	C₈₉×#3492	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₈₉
prime radical	C₂⁵×C₈₉	C₂³
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. C₂×#544×C₈₉	C₂	semidirect
4. C₈₉×#2627	C₂	semidirect
5. C₈₉×#2442	C₂	semidirect
6. C₈₉×#2440	C₂	semidirect
7. C₈₉×#2317	C₂	semidirect
8. C₈₉×#2316	C₂	semidirect
13. C₂×C₄⋊Q₈×C₈₉	C₂²	semidirect
14. C₂×#125×C₈₉	C₂²	semidirect
24. #400×C₈₉	C₂²	semidirect
25. #398×C₈₉	C₂²	semidirect
26. #395×C₈₉	C₂²	semidirect
27. #389×C₈₉	C₂²	semidirect
28. #389×C₈₉	C₂²	semidirect
31. #384×C₈₉	C₂²	semidirect
32. #382×C₈₉	C₂²	semidirect
46. C₂³×C₄×C₈₉	Q₈	semidirect
169. C₂×C₄×C₈₉	C₂²⋊Q₈	semidirect
187. C₂²×C₈₉	#544	semidirect
200. #35224	C₈₉	direct (N × H)
208. C₂×#544	C₂×C₈₉	semidirect
211. #2627	C₂×C₈₉	semidirect
212. #2442	C₂×C₈₉	semidirect
213. #2440	C₂×C₈₉	semidirect
214. #2317	C₂×C₈₉	semidirect
215. #2316	C₂×C₈₉	semidirect
217. C₈₉	#35224	direct (N × H)
221. C₂×C₄⋊Q₈	C₂²×C₈₉	semidirect
222. C₂×#125	C₂²×C₈₉	semidirect
232. #400	C₂²×C₈₉	semidirect
233. #398	C₂²×C₈₉	semidirect
234. #395	C₂²×C₈₉	semidirect
235. #389	C₂²×C₈₉	semidirect
236. #389	C₂²×C₈₉	semidirect
239. #384	C₂²×C₈₉	semidirect
240. #382	C₂²×C₈₉	semidirect
254. C₂³×C₄	Q₈×C₈₉	semidirect
377. C₂×C₄	C₂²⋊Q₈×C₈₉	semidirect
395. C₂²	#544×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#544×C₈₉	2
C₂³×C₄⋊C₄×C₈₉	2
C₈₉×#2848	2
C₈₉×#2627	2
C₈₉×#2442	2
C₈₉×#2440	2
C₈₉×#2317	2
C₈₉×#2316	2
C₈₉×#2280	2
#35224	89

Subgroups — C89×#35224 #36413072

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#35224 #36413072