Subgroups — C₈₉×#24315 #36402163

Order 22784.

966 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#3502	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#3078	C₂	1	2	yes	maximal
3	C₄×C₄⋊Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₈₉×#2060	C₂	1	2	yes	maximal
5	C₈₉×#2028	C₂	1	2	yes	maximal
6	C₈₉×#1994	C₂	1	2	yes	maximal
7	C₈₉×#1886	C₂	1	2	yes	maximal
8	C₈₉×#1857	C₂	1	2	yes	maximal
9	C₈₉×#1801	C₂	1	1	yes	maximal, characteristic
10	Q₈²×C₈₉	—	2	2	no	—
11	#557×C₈₉	—	2	2	no	—
12	#557×C₈₉	—	2	4	no	—
13	#544×C₈₉	—	2	2	no	—
14	#544×C₈₉	C₂²	1	2	yes	—
15	#533×C₈₉	—	2	2	no	—
16	#533×C₈₉	C₂²	1	2	yes	—
17	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
18	#519×C₈₉	C₂²	1	2	yes	—
19	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
21	#479×C₈₉	—	2	4	no	—
22	#477×C₈₉	—	2	2	no	—
23	#475×C₈₉	—	2	2	no	—
24	#428×C₈₉	C₂²	1	4	yes	—
25	#423×C₈₉	C₂²	1	2	yes	—
26	#419×C₈₉	C₂²	1	4	yes	—
27	C₂×#101×C₈₉	C₂²	1	2	yes	—
28	#391×C₈₉	C₂²	1	1	yes	characteristic
29	#391×C₈₉	C₂²	1	1	yes	characteristic
30	#389×C₈₉	—	2	2	no	—
31	#384×C₈₉	C₂²	1	1	yes	characteristic
32	C₄×C₄⋊C₄×C₈₉	—	2	2	no	—
33	C₄³×C₈₉	C₂²	1	1	yes	characteristic
34	#343×C₈₉	C₂²	1	2	yes	—
35	#339×C₈₉	C₂²	1	4	yes	—
36	#339×C₈₉	C₂²	1	4	yes	—
37	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
38	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
39	C₂×M₁₆×C₈₉	C₂³	1	4	yes	—
40	C₄⋊Q₈×C₈₉	—	2	2	no	—
41	C₄⋊Q₈×C₈₉	—	2	2	no	—
42	C₄⋊Q₈×C₈₉	—	2	8	no	—
43	C₄⋊Q₈×C₈₉	—	2	2	no	—
44	C₄⋊Q₈×C₈₉	—	2	4	no	—
45	C₄⋊Q₈×C₈₉	D₄	1	2	yes	—
46	C₄⋊Q₈×C₈₉	D₄	1	2	yes	—
47	C₄⋊Q₈×C₈₉	Q₈	1	4	yes	complemented
48	#125×C₈₉	—	2	4	no	—
49	#125×C₈₉	—	2	8	no	—
50	#125×C₈₉	—	2	4	no	—
51	#125×C₈₉	D₄	1	4	yes	—
52	C₂²⋊Q₈×C₈₉	—	2	2	no	—
53	C₂²⋊Q₈×C₈₉	—	2	4	no	—
54	C₂²⋊Q₈×C₈₉	—	2	2	no	—
55	C₄×Q₈×C₈₉	—	2	4	no	—
56	C₄×Q₈×C₈₉	—	2	4	no	—
57	C₄×Q₈×C₈₉	—	2	4	no	—
58	C₄×Q₈×C₈₉	—	2	4	no	—
59	C₄×Q₈×C₈₉	—	2	2	no	—
60	C₄×Q₈×C₈₉	—	2	2	no	—
61	C₄×Q₈×C₈₉	—	2	2	no	—
62	C₄×Q₈×C₈₉	—	2	2	no	—
63	C₄×Q₈×C₈₉	—	2	2	no	—
64	C₄×Q₈×C₈₉	—	2	2	no	—
65	#117×C₈₉	—	2	4	no	—
66	#117×C₈₉	C₂³	1	4	yes	—
67	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
68	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
69	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
70	C₂×C₄⋊C₄×C₈₉	D₄	1	2	yes	—
71	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
72	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
73	C₂×C₄²×C₈₉	—	2	1	no	—
74	C₂×C₄²×C₈₉	—	2	1	no	—
75	C₂×C₄²×C₈₉	—	2	1	no	—
76	C₂×C₄²×C₈₉	D₄	1	2	yes	—
77	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
79	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
80	#107×C₈₉	—	2	4	no	—
81	#106×C₈₉	—	2	4	no	—
82	C₄⋊C₈×C₈₉	—	2	2	no	—
83	C₄⋊C₈×C₈₉	—	2	2	no	—
84	#103×C₈₉	—	2	4	no	—
85	#103×C₈₉	—	2	4	no	—
86	#101×C₈₉	—	2	4	no	—
87	#95×C₈₉	—	2	2	no	—
88	#95×C₈₉	—	2	2	no	—
89	C₂×Q₈×C₈₉	—	2	4	no	—
90	C₂×Q₈×C₈₉	—	2	2	no	—
91	C₂×Q₈×C₈₉	—	2	2	no	—
92	C₂×Q₈×C₈₉	—	2	2	no	—
93	C₂×Q₈×C₈₉	—	2	1	no	—
94	C₂×Q₈×C₈₉	—	2	2	no	—
95	C₂×Q₈×C₈₉	—	2	2	no	—
96	C₂×Q₈×C₈₉	—	2	2	no	—
97	C₂×Q₈×C₈₉	C₂×Q₈	1	2	yes	—
98	C₂×Q₈×C₈₉	C₂×Q₈	1	2	yes	—
99	C₂×Q₈×C₈₉	#41	1	2	yes	—
100	C₂²×C₄×C₈₉	—	2	1	no	—
101	C₂²×C₄×C₈₉	—	2	1	no	—
102	C₂²×C₄×C₈₉	—	2	1	no	—
103	C₂²×C₄×C₈₉	—	2	1	no	—
104	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
106	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
107	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
109	M₁₆×C₈₉	—	2	4	no	—
110	M₁₆×C₈₉	—	2	4	no	—
111	C₂×C₈×C₈₉	—	2	4	no	—
112	C₄⋊C₄×C₈₉	—	2	4	no	—
113	C₄⋊C₄×C₈₉	—	2	4	no	—
114	C₄⋊C₄×C₈₉	—	2	4	no	—
115	C₄⋊C₄×C₈₉	—	2	8	no	—
116	C₄⋊C₄×C₈₉	—	2	4	no	—
117	C₄⋊C₄×C₈₉	—	2	4	no	—
118	C₄⋊C₄×C₈₉	—	2	2	no	—
119	C₄⋊C₄×C₈₉	—	2	2	no	—
120	C₄⋊C₄×C₈₉	—	2	4	no	—
121	C₄⋊C₄×C₈₉	—	2	8	no	—
122	C₄⋊C₄×C₈₉	—	2	2	no	—
123	C₄⋊C₄×C₈₉	—	2	2	no	—
124	C₄⋊C₄×C₈₉	—	4	4	no	—
125	C₄⋊C₄×C₈₉	—	4	2	no	—
126	C₄⋊C₄×C₈₉	C₂×D₄	1	4	yes	—
127	C₄⋊C₄×C₈₉	C₂×D₄	1	4	yes	—
128	C₂²⋊C₄×C₈₉	—	2	4	no	—
129	C₄²×C₈₉	—	2	4	no	—
130	C₄²×C₈₉	—	2	8	no	—
131	C₄²×C₈₉	—	2	2	no	—
132	C₄²×C₈₉	—	2	2	no	—
133	C₄²×C₈₉	—	2	4	no	—
134	C₄²×C₈₉	—	2	1	no	—
135	C₄²×C₈₉	—	2	2	no	—
136	C₄²×C₈₉	—	4	2	no	—
137	C₄²×C₈₉	—	4	1	no	—
138	C₄²×C₈₉	—	4	1	no	—
139	C₄²×C₈₉	C₂×D₄	1	2	yes	—
140	C₄²×C₈₉	C₂×D₄	1	2	yes	—
141	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
142	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
143	Q₈×C₈₉	—	2	4	no	—
144	Q₈×C₈₉	—	2	4	no	—
145	Q₈×C₈₉	—	2	4	no	—
146	Q₈×C₈₉	—	4	2	no	—
147	Q₈×C₈₉	—	4	4	no	—
148	Q₈×C₈₉	—	8	2	no	—
149	Q₈×C₈₉	—	8	4	no	—
150	C₂×C₄×C₈₉	—	2	2	no	—
151	C₂×C₄×C₈₉	—	2	2	no	—
152	C₂×C₄×C₈₉	—	2	2	no	—
153	C₂×C₄×C₈₉	—	2	2	no	—
154	C₂×C₄×C₈₉	—	2	2	no	—
155	C₂×C₄×C₈₉	—	2	2	no	—
156	C₂×C₄×C₈₉	—	2	2	no	—
157	C₂×C₄×C₈₉	—	2	4	no	—
158	C₂×C₄×C₈₉	—	2	4	no	—
159	C₂×C₄×C₈₉	—	2	2	no	—
160	C₂×C₄×C₈₉	—	2	2	no	—
161	C₂×C₄×C₈₉	—	4	1	no	—
162	C₂×C₄×C₈₉	—	4	1	no	—
163	C₂×C₄×C₈₉	—	4	1	no	—
164	C₂×C₄×C₈₉	—	4	1	no	—
165	C₂×C₄×C₈₉	—	4	1	no	—
166	C₂×C₄×C₈₉	—	4	2	no	—
167	C₂×C₄×C₈₉	—	4	2	no	—
168	C₂×C₄×C₈₉	—	4	1	no	—
169	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
170	C₂×C₄×C₈₉	#120	1	2	yes	—
171	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
172	C₂×C₄×C₈₉	C₂²⋊Q₈	1	4	yes	—
173	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
174	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
175	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
176	C₂×C₄×C₈₉	2₋¹⁺⁴	1	2	yes	—
177	C₈×C₈₉	—	4	4	no	—
178	C₂²×C₈₉	—	2	1	no	—
179	C₂²×C₈₉	—	2	1	no	—
180	C₂²×C₈₉	C₂×#120	1	1	yes	characteristic
181	C₂²×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
182	C₂²×C₈₉	#536	1	1	yes	characteristic
183	C₄×C₈₉	—	2	4	no	—
184	C₄×C₈₉	—	2	4	no	—
185	C₄×C₈₉	—	2	1	no	—
186	C₄×C₈₉	—	2	2	no	—
187	C₄×C₈₉	—	4	2	no	—
188	C₄×C₈₉	—	4	2	no	—
189	C₄×C₈₉	—	4	4	no	—
190	C₄×C₈₉	—	4	2	no	—
191	C₄×C₈₉	—	4	4	no	—
192	C₄×C₈₉	—	8	1	no	—
193	C₄×C₈₉	#456	1	2	yes	—
194	C₄×C₈₉	#548	1	2	yes	—
195	C₄×C₈₉	D₄×Q₈	1	2	yes	—
196	#24315	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
197	C₂×C₈₉	—	2	1	no	—
198	C₂×C₈₉	—	2	1	no	—
199	C₂×C₈₉	#2409	1	1	yes	characteristic
200	C₂×C₈₉	C₂×#456	1	1	yes	characteristic
201	C₂×C₈₉	#3018	1	1	yes	characteristic
202	#3502	C₂×C₈₉	1	1	yes	characteristic
203	#3078	C₂×C₈₉	1	2	yes	—
204	C₄×C₄⋊Q₈	C₂×C₈₉	1	1	yes	characteristic
205	#2060	C₂×C₈₉	1	2	yes	—
206	#2028	C₂×C₈₉	1	2	yes	—
207	#1994	C₂×C₈₉	1	2	yes	—
208	#1886	C₂×C₈₉	1	2	yes	—
209	#1857	C₂×C₈₉	1	2	yes	—
210	#1801	C₂×C₈₉	1	1	yes	characteristic
211	C₈₉	#24315	1	1	yes	characteristic, direct product factor, Sylow
212	Q₈²	—	2	2	no	—
213	#557	—	2	2	no	—
214	#557	—	2	4	no	—
215	#544	—	2	2	no	—
216	#544	C₂²×C₈₉	1	2	yes	—
217	#533	—	2	2	no	—
218	#533	C₂²×C₈₉	1	2	yes	—
219	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
220	#519	C₂²×C₈₉	1	2	yes	—
221	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
222	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
223	#479	—	2	4	no	—
224	#477	—	2	2	no	—
225	#475	—	2	2	no	—
226	#428	C₂²×C₈₉	1	4	yes	—
227	#423	C₂²×C₈₉	1	2	yes	—
228	#419	C₂²×C₈₉	1	4	yes	—
229	C₂×#101	C₂²×C₈₉	1	2	yes	—
230	#391	C₂²×C₈₉	1	1	yes	characteristic
231	#391	C₂²×C₈₉	1	1	yes	characteristic
232	#389	—	2	2	no	—
233	#384	C₂²×C₈₉	1	1	yes	characteristic
234	C₄×C₄⋊C₄	—	2	2	no	—
235	C₄³	C₂²×C₈₉	1	1	yes	characteristic
236	#343	C₂²×C₈₉	1	2	yes	—
237	#339	C₂²×C₈₉	1	4	yes	—
238	#339	C₂²×C₈₉	1	4	yes	—
239	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
240	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
241	C₂×M₁₆	C₂³×C₈₉	1	4	yes	—
242	C₄⋊Q₈	—	2	2	no	—
243	C₄⋊Q₈	—	2	2	no	—
244	C₄⋊Q₈	—	2	8	no	—
245	C₄⋊Q₈	—	2	2	no	—
246	C₄⋊Q₈	—	2	4	no	—
247	C₄⋊Q₈	D₄×C₈₉	1	2	yes	—
248	C₄⋊Q₈	D₄×C₈₉	1	2	yes	—
249	C₄⋊Q₈	Q₈×C₈₉	1	4	yes	complemented
250	#125	—	2	4	no	—
251	#125	—	2	8	no	—
252	#125	—	2	4	no	—
253	#125	D₄×C₈₉	1	4	yes	—
254	C₂²⋊Q₈	—	2	2	no	—
255	C₂²⋊Q₈	—	2	4	no	—
256	C₂²⋊Q₈	—	2	2	no	—
257	C₄×Q₈	—	2	4	no	—
258	C₄×Q₈	—	2	4	no	—
259	C₄×Q₈	—	2	4	no	—
260	C₄×Q₈	—	2	4	no	—
261	C₄×Q₈	—	2	2	no	—
262	C₄×Q₈	—	2	2	no	—
263	C₄×Q₈	—	2	2	no	—
264	C₄×Q₈	—	2	2	no	—
265	C₄×Q₈	—	2	2	no	—
266	C₄×Q₈	—	2	2	no	—
267	#117	—	2	4	no	—
268	#117	C₂³×C₈₉	1	4	yes	—
269	C₂×C₄⋊C₄	—	2	2	no	—
270	C₂×C₄⋊C₄	—	2	1	no	—
271	C₂×C₄⋊C₄	—	2	2	no	—
272	C₂×C₄⋊C₄	D₄×C₈₉	1	2	yes	—
273	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
274	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
275	C₂×C₄²	—	2	1	no	—
276	C₂×C₄²	—	2	1	no	—
277	C₂×C₄²	—	2	1	no	—
278	C₂×C₄²	D₄×C₈₉	1	2	yes	—
279	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
280	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
281	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
282	#107	—	2	4	no	—
283	#106	—	2	4	no	—
284	C₄⋊C₈	—	2	2	no	—
285	C₄⋊C₈	—	2	2	no	—
286	#103	—	2	4	no	—
287	#103	—	2	4	no	—
288	#101	—	2	4	no	—
289	#95	—	2	2	no	—
290	#95	—	2	2	no	—
291	C₂×Q₈	—	2	4	no	—
292	C₂×Q₈	—	2	2	no	—
293	C₂×Q₈	—	2	2	no	—
294	C₂×Q₈	—	2	2	no	—
295	C₂×Q₈	—	2	1	no	—
296	C₂×Q₈	—	2	2	no	—
297	C₂×Q₈	—	2	2	no	—
298	C₂×Q₈	—	2	2	no	—
299	C₂×Q₈	C₂×Q₈×C₈₉	1	2	yes	—
300	C₂×Q₈	C₂×Q₈×C₈₉	1	2	yes	—
301	C₂×Q₈	#41×C₈₉	1	2	yes	—
302	C₂²×C₄	—	2	1	no	—
303	C₂²×C₄	—	2	1	no	—
304	C₂²×C₄	—	2	1	no	—
305	C₂²×C₄	—	2	1	no	—
306	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
307	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
308	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
309	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
310	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
311	M₁₆	—	2	4	no	—
312	M₁₆	—	2	4	no	—
313	C₂×C₈	—	2	4	no	—
314	C₄⋊C₄	—	2	4	no	—
315	C₄⋊C₄	—	2	4	no	—
316	C₄⋊C₄	—	2	4	no	—
317	C₄⋊C₄	—	2	8	no	—
318	C₄⋊C₄	—	2	4	no	—
319	C₄⋊C₄	—	2	4	no	—
320	C₄⋊C₄	—	2	2	no	—
321	C₄⋊C₄	—	2	2	no	—
322	C₄⋊C₄	—	2	4	no	—
323	C₄⋊C₄	—	2	8	no	—
324	C₄⋊C₄	—	2	2	no	—
325	C₄⋊C₄	—	2	2	no	—
326	C₄⋊C₄	—	4	4	no	—
327	C₄⋊C₄	—	4	2	no	—
328	C₄⋊C₄	C₂×D₄×C₈₉	1	4	yes	—
329	C₄⋊C₄	C₂×D₄×C₈₉	1	4	yes	—
330	C₂²⋊C₄	—	2	4	no	—
331	C₄²	—	2	4	no	—
332	C₄²	—	2	8	no	—
333	C₄²	—	2	2	no	—
334	C₄²	—	2	2	no	—
335	C₄²	—	2	4	no	—
336	C₄²	—	2	1	no	—
337	C₄²	—	2	2	no	—
338	C₄²	—	4	2	no	—
339	C₄²	—	4	1	no	—
340	C₄²	—	4	1	no	—
341	C₄²	C₂×D₄×C₈₉	1	2	yes	—
342	C₄²	C₂×D₄×C₈₉	1	2	yes	—
343	C₄²	C₂×Q₈×C₈₉	1	2	yes	—
344	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
345	Q₈	—	2	4	no	—
346	Q₈	—	2	4	no	—
347	Q₈	—	2	4	no	—
348	Q₈	—	4	4	no	—
349	Q₈	—	4	2	no	—
350	Q₈	—	8	2	no	—
351	Q₈	—	8	4	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	4	no	—
360	C₂×C₄	—	2	4	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	4	1	no	—
364	C₂×C₄	—	4	1	no	—
365	C₂×C₄	—	4	1	no	—
366	C₂×C₄	—	4	1	no	—
367	C₂×C₄	—	4	1	no	—
368	C₂×C₄	—	4	2	no	—
369	C₂×C₄	—	4	2	no	—
370	C₂×C₄	—	4	1	no	—
371	C₂×C₄	#120×C₈₉	1	1	yes	characteristic
372	C₂×C₄	#120×C₈₉	1	2	yes	—
373	C₂×C₄	#120×C₈₉	1	1	yes	characteristic
374	C₂×C₄	C₂²⋊Q₈×C₈₉	1	4	yes	—
375	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
376	C₂×C₄	C₂²×Q₈×C₈₉	1	1	yes	characteristic
377	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
378	C₂×C₄	2₋¹⁺⁴×C₈₉	1	2	yes	—
379	C₈	—	4	4	no	—
380	C₂²	—	2	1	no	—
381	C₂²	—	2	1	no	—
382	C₂²	C₂×#120×C₈₉	1	1	yes	characteristic
383	C₂²	C₂×C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
384	C₂²	#536×C₈₉	1	1	yes	characteristic
385	C₄	—	2	4	no	—
386	C₄	—	2	4	no	—
387	C₄	—	2	1	no	—
388	C₄	—	2	2	no	—
389	C₄	—	4	2	no	—
390	C₄	—	4	2	no	—
391	C₄	—	4	4	no	—
392	C₄	—	4	2	no	—
393	C₄	—	4	4	no	—
394	C₄	—	8	1	no	—
395	C₄	#456×C₈₉	1	2	yes	—
396	C₄	#548×C₈₉	1	2	yes	—
397	C₄	D₄×Q₈×C₈₉	1	2	yes	—
398	C₂	—	2	1	no	—
399	C₂	—	2	1	no	—
400	C₂	C₈₉×#2409	1	1	yes	characteristic
401	C₂	C₂×#456×C₈₉	1	1	yes	characteristic
402	C₂	C₈₉×#3018	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₈₉	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
prime radical	C₂³×C₈₉	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₂×C₄²×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
47. C₄⋊Q₈×C₈₉	Q₈	semidirect
196. #24315	C₈₉	direct (N × H)
211. C₈₉	#24315	direct (N × H)
249. 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×#3502	2
C₈₉×#3078	2
C₄×C₄⋊Q₈×C₈₉	2
C₈₉×#2060	2
C₈₉×#2028	2
C₈₉×#1994	2
C₈₉×#1886	2
C₈₉×#1857	2
C₈₉×#1801	2
#24315	89

Subgroups — C89×#24315 #36402163

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#24315 #36402163