Subgroups — C₈₉×#35174 #36413022

Order 22784.

2794 conjugacy classes of subgroups.

The subgroup lattice has 408 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	C₂²×C₄×D₄×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#2848	C₂	1	4	yes	maximal, complemented
4	C₈₉×#2626	C₂	1	4	yes	maximal, complemented
5	C₈₉×#2454	C₂	1	8	yes	maximal, complemented
6	C₈₉×#2452	C₂	1	8	yes	maximal, complemented
7	C₈₉×#2316	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈₉×#2309	C₂	1	2	yes	maximal
9	C₈₉×#2296	C₂	1	2	yes	maximal, complemented
10	C₂³×D₄×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	complemented
17	C₂×C₄×D₄×C₈₉	—	2	8	no	—
18	C₂×C₄×D₄×C₈₉	C₂²	1	8	yes	—
19	C₂×#117×C₈₉	C₂²	1	2	yes	—
20	C₂×#117×C₈₉	C₂²	1	2	yes	—
21	C₂²×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
22	C₂²×C₂²⋊C₄×C₈₉	C₂²	1	2	yes	—
23	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
24	#399×C₈₉	C₂²	1	8	yes	complemented
25	#396×C₈₉	C₂²	1	16	yes	complemented
26	#393×C₈₉	C₂²	1	8	yes	complemented
27	#390×C₈₉	C₂²	1	4	yes	—
28	#390×C₈₉	C₂²	1	8	yes	—
29	#390×C₈₉	C₂²	1	4	yes	—
30	#389×C₈₉	C₂²	1	4	yes	complemented
31	#389×C₈₉	C₂²	1	4	yes	complemented
32	#389×C₈₉	C₂²	1	8	yes	complemented
33	#385×C₈₉	C₂²	1	4	yes	—
34	#385×C₈₉	C₂²	1	4	yes	—
35	#385×C₈₉	C₂²	1	16	yes	complemented
36	#380×C₈₉	C₂²	1	4	yes	—
37	#380×C₈₉	C₂²	1	8	yes	complemented
38	#379×C₈₉	C₂²	1	8	yes	—
39	C₄×C₄⋊C₄×C₈₉	C₂²	1	8	yes	complemented
40	C₂⁵×C₈₉	C₂³	1	2	yes	—
41	C₂²×Q₈×C₈₉	C₂³	1	4	yes	—
42	C₂²×D₄×C₈₉	—	2	4	no	—
43	C₂²×D₄×C₈₉	—	2	4	no	—
44	C₂²×D₄×C₈₉	Q₈	1	8	yes	complemented
45	C₂²×D₄×C₈₉	C₂³	1	4	yes	—
46	C₂³×C₄×C₈₉	—	2	8	no	—
47	C₂³×C₄×C₈₉	—	4	8	no	—
48	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
49	C₂³×C₄×C₈₉	C₂³	1	8	yes	—
50	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
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	C₄×D₄×C₈₉	—	2	32	no	—
58	#117×C₈₉	—	2	8	no	—
59	#117×C₈₉	—	2	8	no	—
60	C₂×C₄⋊C₄×C₈₉	D₄	1	8	yes	—
61	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
62	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
63	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
64	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
65	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
67	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
68	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
69	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
70	C₂×C₂²⋊C₄×C₈₉	—	2	8	no	—
71	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
72	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
73	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
74	C₂×C₂²⋊C₄×C₈₉	C₂³	1	16	yes	—
75	C₂×C₂²⋊C₄×C₈₉	C₂³	1	16	yes	—
76	C₂×C₄²×C₈₉	—	2	4	no	—
77	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
78	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
79	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
80	#95×C₈₉	C₂³	1	8	yes	—
81	#95×C₈₉	C₂³	1	8	yes	—
82	#95×C₈₉	C₂³	1	8	yes	—
83	C₂⁴×C₈₉	—	2	4	no	—
84	C₂⁴×C₈₉	—	2	4	no	—
85	C₂⁴×C₈₉	—	4	4	no	—
86	C₂⁴×C₈₉	—	4	4	no	—
87	C₂⁴×C₈₉	C₂×Q₈	1	8	yes	—
88	C₂⁴×C₈₉	C₂⁴	1	4	yes	—
89	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
90	C₂×Q₈×C₈₉	—	2	8	no	—
91	C₂×Q₈×C₈₉	—	4	8	no	—
92	C₂×D₄×C₈₉	—	2	32	no	—
93	C₂×D₄×C₈₉	—	2	8	no	—
94	C₂×D₄×C₈₉	—	2	8	no	—
95	C₂×D₄×C₈₉	C₂×Q₈	1	16	yes	—
96	C₂²×C₄×C₈₉	—	2	2	no	—
97	C₂²×C₄×C₈₉	—	2	4	no	—
98	C₂²×C₄×C₈₉	—	2	2	no	—
99	C₂²×C₄×C₈₉	—	2	16	no	—
100	C₂²×C₄×C₈₉	—	2	16	no	—
101	C₂²×C₄×C₈₉	—	4	4	no	—
102	C₂²×C₄×C₈₉	—	4	8	no	—
103	C₂²×C₄×C₈₉	—	4	2	no	—
104	C₂²×C₄×C₈₉	—	4	32	no	—
105	C₂²×C₄×C₈₉	C₂×D₄	1	8	yes	—
106	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
107	C₂²×C₄×C₈₉	C₂×Q₈	1	4	yes	—
108	C₂²×C₄×C₈₉	#41	1	4	yes	—
109	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
110	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
111	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
112	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
113	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
114	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
115	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
116	C₄⋊C₄×C₈₉	—	2	4	no	—
117	C₄⋊C₄×C₈₉	—	2	4	no	—
118	C₄⋊C₄×C₈₉	—	2	8	no	—
119	C₄⋊C₄×C₈₉	—	2	16	no	—
120	C₄⋊C₄×C₈₉	—	2	8	no	—
121	C₄⋊C₄×C₈₉	—	2	16	no	—
122	C₄⋊C₄×C₈₉	—	4	8	no	—
123	C₄⋊C₄×C₈₉	—	4	8	no	—
124	C₄⋊C₄×C₈₉	C₂×D₄	1	16	yes	—
125	C₂²⋊C₄×C₈₉	—	2	8	no	—
126	C₂²⋊C₄×C₈₉	—	2	8	no	—
127	C₂²⋊C₄×C₈₉	—	2	16	no	—
128	C₂²⋊C₄×C₈₉	—	4	16	no	—
129	C₂²⋊C₄×C₈₉	—	4	16	no	—
130	C₄²×C₈₉	—	2	8	no	—
131	C₄²×C₈₉	—	2	8	no	—
132	C₄²×C₈₉	—	2	8	no	—
133	C₂³×C₈₉	—	2	2	no	—
134	C₂³×C₈₉	—	2	2	no	—
135	C₂³×C₈₉	—	2	16	no	—
136	C₂³×C₈₉	—	2	4	no	—
137	C₂³×C₈₉	—	2	4	no	—
138	C₂³×C₈₉	—	2	8	no	—
139	C₂³×C₈₉	—	2	8	no	—
140	C₂³×C₈₉	—	4	16	no	—
141	C₂³×C₈₉	—	4	8	no	—
142	C₂³×C₈₉	—	4	16	no	—
143	C₂³×C₈₉	—	4	8	no	—
144	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
145	C₂³×C₈₉	C₂²×Q₈	1	8	yes	—
146	C₂³×C₈₉	C₂²×Q₈	1	2	yes	—
147	C₂³×C₈₉	C₂×#41	1	2	yes	—
148	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
149	Q₈×C₈₉	—	8	8	no	—
150	D₄×C₈₉	—	2	32	no	—
151	C₂×C₄×C₈₉	—	2	8	no	—
152	C₂×C₄×C₈₉	—	2	2	no	—
153	C₂×C₄×C₈₉	—	2	8	no	—
154	C₂×C₄×C₈₉	—	2	2	no	—
155	C₂×C₄×C₈₉	—	2	2	no	—
156	C₂×C₄×C₈₉	—	2	4	no	—
157	C₂×C₄×C₈₉	—	2	4	no	—
158	C₂×C₄×C₈₉	—	2	8	no	—
159	C₂×C₄×C₈₉	—	2	8	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	4	no	—
164	C₂×C₄×C₈₉	—	4	8	no	—
165	C₂×C₄×C₈₉	—	4	32	no	—
166	C₂×C₄×C₈₉	—	4	4	no	—
167	C₂×C₄×C₈₉	C₂²⋊Q₈	1	16	yes	complemented
168	C₂×C₄×C₈₉	C₂²×D₄	1	8	yes	—
169	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
170	C₂×C₄×C₈₉	C₂²×Q₈	1	4	yes	—
171	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
172	C₂×C₄×C₈₉	H(5,𝔽₂)	1	4	yes	—
173	C₂²×C₈₉	—	2	4	no	—
174	C₂²×C₈₉	—	2	2	no	—
175	C₂²×C₈₉	—	2	2	no	—
176	C₂²×C₈₉	—	2	4	no	—
177	C₂²×C₈₉	—	2	8	no	—
178	C₂²×C₈₉	—	2	16	no	—
179	C₂²×C₈₉	—	4	32	no	—
180	C₂²×C₈₉	—	4	8	no	—
181	C₂²×C₈₉	—	4	4	no	—
182	C₂²×C₈₉	—	4	4	no	—
183	C₂²×C₈₉	C₂×C₂²⋊Q₈	1	4	yes	—
184	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
185	C₂²×C₈₉	C₂³×Q₈	1	1	yes	characteristic
186	C₂²×C₈₉	C₂²×#41	1	1	yes	characteristic
187	C₂²×C₈₉	C₂×H(5,𝔽₂)	1	2	yes	—
188	C₂²×C₈₉	#585	1	2	yes	—
189	C₄×C₈₉	—	2	4	no	—
190	C₄×C₈₉	—	2	8	no	—
191	C₄×C₈₉	—	4	8	no	—
192	C₄×C₈₉	—	4	4	no	—
193	C₄×C₈₉	—	4	4	no	—
194	C₄×C₈₉	—	4	2	no	—
195	C₄×C₈₉	C₂×C₂²⋊Q₈	1	8	yes	—
196	#35174	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
197	C₂×C₈₉	—	2	4	no	—
198	C₂×C₈₉	—	4	8	no	—
199	C₂×C₈₉	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
200	C₂×C₈₉	#3460	1	2	yes	—
201	C₂×C₈₉	#3484	1	1	yes	characteristic
202	C₂×C₈₉	#3487	1	1	yes	characteristic
203	C₂×C₈₉	#3501	1	2	yes	—
204	C₂×#544	C₂×C₈₉	1	1	yes	characteristic, complemented
205	C₂²×C₄×D₄	C₂×C₈₉	1	1	yes	characteristic
206	#2848	C₂×C₈₉	1	4	yes	complemented
207	#2626	C₂×C₈₉	1	4	yes	complemented
208	#2454	C₂×C₈₉	1	8	yes	complemented
209	#2452	C₂×C₈₉	1	8	yes	complemented
210	#2316	C₂×C₈₉	1	1	yes	characteristic, complemented
211	#2309	C₂×C₈₉	1	2	yes	—
212	#2296	C₂×C₈₉	1	2	yes	complemented
213	C₈₉	#35174	1	1	yes	characteristic, direct product factor, Sylow
214	C₂³×D₄	C₂²×C₈₉	1	1	yes	characteristic
215	C₂⁴×C₄	C₂²×C₈₉	1	2	yes	—
216	#544	—	2	8	no	—
217	C₂×C₄⋊Q₈	C₂²×C₈₉	1	4	yes	complemented
218	C₂×#125	C₂²×C₈₉	1	4	yes	complemented
219	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	4	yes	—
220	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	8	yes	complemented
221	C₂×C₄×D₄	—	2	8	no	—
222	C₂×C₄×D₄	C₂²×C₈₉	1	8	yes	—
223	C₂×#117	C₂²×C₈₉	1	2	yes	—
224	C₂×#117	C₂²×C₈₉	1	2	yes	—
225	C₂²×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
226	C₂²×C₂²⋊C₄	C₂²×C₈₉	1	2	yes	—
227	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
228	#399	C₂²×C₈₉	1	8	yes	complemented
229	#396	C₂²×C₈₉	1	16	yes	complemented
230	#393	C₂²×C₈₉	1	8	yes	complemented
231	#390	C₂²×C₈₉	1	4	yes	—
232	#390	C₂²×C₈₉	1	8	yes	—
233	#390	C₂²×C₈₉	1	4	yes	—
234	#389	C₂²×C₈₉	1	8	yes	complemented
235	#389	C₂²×C₈₉	1	4	yes	complemented
236	#389	C₂²×C₈₉	1	4	yes	complemented
237	#385	C₂²×C₈₉	1	4	yes	—
238	#385	C₂²×C₈₉	1	4	yes	—
239	#385	C₂²×C₈₉	1	16	yes	complemented
240	#380	C₂²×C₈₉	1	4	yes	—
241	#380	C₂²×C₈₉	1	8	yes	complemented
242	#379	C₂²×C₈₉	1	8	yes	—
243	C₄×C₄⋊C₄	C₂²×C₈₉	1	8	yes	complemented
244	C₂⁵	C₂³×C₈₉	1	2	yes	—
245	C₂²×Q₈	C₂³×C₈₉	1	4	yes	—
246	C₂²×D₄	—	2	4	no	—
247	C₂²×D₄	—	2	4	no	—
248	C₂²×D₄	Q₈×C₈₉	1	8	yes	complemented
249	C₂²×D₄	C₂³×C₈₉	1	4	yes	—
250	C₂³×C₄	—	2	8	no	—
251	C₂³×C₄	—	4	8	no	—
252	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
253	C₂³×C₄	C₂³×C₈₉	1	8	yes	—
254	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
255	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
256	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
257	C₄⋊Q₈	—	2	16	no	—
258	#125	—	2	16	no	—
259	C₂²⋊Q₈	—	2	16	no	—
260	C₂²⋊Q₈	—	4	16	no	—
261	C₄×D₄	—	2	32	no	—
262	#117	—	2	8	no	—
263	#117	—	2	8	no	—
264	C₂×C₄⋊C₄	D₄×C₈₉	1	8	yes	—
265	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
266	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
267	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
268	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
269	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	—
270	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	—
271	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	—
272	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	—
273	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
274	C₂×C₂²⋊C₄	—	2	8	no	—
275	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
276	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
277	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
278	C₂×C₂²⋊C₄	C₂³×C₈₉	1	16	yes	—
279	C₂×C₂²⋊C₄	C₂³×C₈₉	1	16	yes	—
280	C₂×C₄²	—	2	4	no	—
281	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
282	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
283	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
284	#95	C₂³×C₈₉	1	8	yes	—
285	#95	C₂³×C₈₉	1	8	yes	—
286	#95	C₂³×C₈₉	1	8	yes	—
287	C₂⁴	—	2	4	no	—
288	C₂⁴	—	2	4	no	—
289	C₂⁴	—	4	4	no	—
290	C₂⁴	—	4	4	no	—
291	C₂⁴	C₂×Q₈×C₈₉	1	8	yes	—
292	C₂⁴	C₂⁴×C₈₉	1	4	yes	—
293	C₂⁴	C₂⁴×C₈₉	1	1	yes	characteristic
294	C₂×Q₈	—	2	8	no	—
295	C₂×Q₈	—	4	8	no	—
296	C₂×D₄	—	2	32	no	—
297	C₂×D₄	—	2	8	no	—
298	C₂×D₄	—	2	8	no	—
299	C₂×D₄	C₂×Q₈×C₈₉	1	16	yes	—
300	C₂²×C₄	—	2	2	no	—
301	C₂²×C₄	—	2	4	no	—
302	C₂²×C₄	—	2	2	no	—
303	C₂²×C₄	—	2	16	no	—
304	C₂²×C₄	—	2	16	no	—
305	C₂²×C₄	—	4	4	no	—
306	C₂²×C₄	—	4	8	no	—
307	C₂²×C₄	—	4	2	no	—
308	C₂²×C₄	—	4	32	no	—
309	C₂²×C₄	C₂×D₄×C₈₉	1	8	yes	—
310	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
311	C₂²×C₄	C₂×Q₈×C₈₉	1	4	yes	—
312	C₂²×C₄	#41×C₈₉	1	4	yes	—
313	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
314	C₂²×C₄	C₂⁴×C₈₉	1	8	yes	—
315	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
316	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
317	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
318	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
319	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
320	C₄⋊C₄	—	2	4	no	—
321	C₄⋊C₄	—	2	4	no	—
322	C₄⋊C₄	—	2	8	no	—
323	C₄⋊C₄	—	2	16	no	—
324	C₄⋊C₄	—	2	8	no	—
325	C₄⋊C₄	—	2	16	no	—
326	C₄⋊C₄	—	4	8	no	—
327	C₄⋊C₄	—	4	8	no	—
328	C₄⋊C₄	C₂×D₄×C₈₉	1	16	yes	—
329	C₂²⋊C₄	—	2	8	no	—
330	C₂²⋊C₄	—	2	8	no	—
331	C₂²⋊C₄	—	2	16	no	—
332	C₂²⋊C₄	—	4	16	no	—
333	C₂²⋊C₄	—	4	16	no	—
334	C₄²	—	2	8	no	—
335	C₄²	—	2	8	no	—
336	C₄²	—	2	8	no	—
337	C₂³	—	2	2	no	—
338	C₂³	—	2	2	no	—
339	C₂³	—	2	16	no	—
340	C₂³	—	2	4	no	—
341	C₂³	—	2	4	no	—
342	C₂³	—	2	8	no	—
343	C₂³	—	2	8	no	—
344	C₂³	—	4	16	no	—
345	C₂³	—	4	8	no	—
346	C₂³	—	4	16	no	—
347	C₂³	—	4	8	no	—
348	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
349	C₂³	C₂²×Q₈×C₈₉	1	8	yes	—
350	C₂³	C₂²×Q₈×C₈₉	1	2	yes	—
351	C₂³	C₂×#41×C₈₉	1	2	yes	—
352	C₂³	C₂⁵×C₈₉	1	1	yes	characteristic
353	Q₈	—	8	8	no	—
354	D₄	—	2	32	no	—
355	C₂×C₄	—	2	8	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	8	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	2	4	no	—
361	C₂×C₄	—	2	4	no	—
362	C₂×C₄	—	2	8	no	—
363	C₂×C₄	—	2	8	no	—
364	C₂×C₄	—	2	8	no	—
365	C₂×C₄	—	2	8	no	—
366	C₂×C₄	—	2	8	no	—
367	C₂×C₄	—	2	4	no	—
368	C₂×C₄	—	4	8	no	—
369	C₂×C₄	—	4	32	no	—
370	C₂×C₄	—	4	4	no	—
371	C₂×C₄	C₂²⋊Q₈×C₈₉	1	16	yes	complemented
372	C₂×C₄	C₂²×D₄×C₈₉	1	8	yes	—
373	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
374	C₂×C₄	C₂²×Q₈×C₈₉	1	4	yes	—
375	C₂×C₄	C₂×#41×C₈₉	1	4	yes	—
376	C₂×C₄	H(5,𝔽₂)×C₈₉	1	4	yes	—
377	C₂²	—	2	4	no	—
378	C₂²	—	2	2	no	—
379	C₂²	—	2	2	no	—
380	C₂²	—	2	4	no	—
381	C₂²	—	2	8	no	—
382	C₂²	—	2	16	no	—
383	C₂²	—	4	32	no	—
384	C₂²	—	4	8	no	—
385	C₂²	—	4	4	no	—
386	C₂²	—	4	4	no	—
387	C₂²	C₂×C₂²⋊Q₈×C₈₉	1	4	yes	—
388	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
389	C₂²	C₂³×Q₈×C₈₉	1	1	yes	characteristic
390	C₂²	C₂²×#41×C₈₉	1	1	yes	characteristic
391	C₂²	C₂×H(5,𝔽₂)×C₈₉	1	2	yes	—
392	C₂²	#585×C₈₉	1	2	yes	—
393	C₄	—	2	4	no	—
394	C₄	—	2	8	no	—
395	C₄	—	4	8	no	—
396	C₄	—	4	4	no	—
397	C₄	—	4	2	no	—
398	C₄	—	4	4	no	—
399	C₄	C₂×C₂²⋊Q₈×C₈₉	1	8	yes	—
400	C₂	—	2	4	no	—
401	C₂	—	4	8	no	—
402	C₂	C₂²×C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
403	C₂	C₈₉×#3460	1	2	yes	—
404	C₂	C₈₉×#3484	1	1	yes	characteristic
405	C₂	C₈₉×#3487	1	1	yes	characteristic
406	C₂	C₈₉×#3501	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₈₉
prime radical	C₂³×D₄×C₈₉	C₂²
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
3. C₈₉×#2848	C₂	semidirect
4. C₈₉×#2626	C₂	semidirect
5. C₈₉×#2454	C₂	semidirect
6. C₈₉×#2452	C₂	semidirect
7. C₈₉×#2316	C₂	semidirect
9. C₈₉×#2296	C₂	semidirect
13. C₂×C₄⋊Q₈×C₈₉	C₂²	semidirect
14. C₂×#125×C₈₉	C₂²	semidirect
16. C₂×C₂²⋊Q₈×C₈₉	C₂²	semidirect
24. #399×C₈₉	C₂²	semidirect
25. #396×C₈₉	C₂²	semidirect
26. #393×C₈₉	C₂²	semidirect
30. #389×C₈₉	C₂²	semidirect
31. #389×C₈₉	C₂²	semidirect
32. #389×C₈₉	C₂²	semidirect
35. #385×C₈₉	C₂²	semidirect
37. #380×C₈₉	C₂²	semidirect
39. C₄×C₄⋊C₄×C₈₉	C₂²	semidirect
44. C₂²×D₄×C₈₉	Q₈	semidirect
167. C₂×C₄×C₈₉	C₂²⋊Q₈	semidirect
196. #35174	C₈₉	direct (N × H)
204. C₂×#544	C₂×C₈₉	semidirect
206. #2848	C₂×C₈₉	semidirect
207. #2626	C₂×C₈₉	semidirect
208. #2454	C₂×C₈₉	semidirect
209. #2452	C₂×C₈₉	semidirect
210. #2316	C₂×C₈₉	semidirect
212. #2296	C₂×C₈₉	semidirect
213. C₈₉	#35174	direct (N × H)
217. C₂×C₄⋊Q₈	C₂²×C₈₉	semidirect
218. C₂×#125	C₂²×C₈₉	semidirect
220. C₂×C₂²⋊Q₈	C₂²×C₈₉	semidirect
228. #399	C₂²×C₈₉	semidirect
229. #396	C₂²×C₈₉	semidirect
230. #393	C₂²×C₈₉	semidirect
234. #389	C₂²×C₈₉	semidirect
235. #389	C₂²×C₈₉	semidirect
236. #389	C₂²×C₈₉	semidirect
239. #385	C₂²×C₈₉	semidirect
241. #380	C₂²×C₈₉	semidirect
243. C₄×C₄⋊C₄	C₂²×C₈₉	semidirect
248. C₂²×D₄	Q₈×C₈₉	semidirect
371. C₂×C₄	C₂²⋊Q₈×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
C₂²×C₄×D₄×C₈₉	2
C₈₉×#2848	2
C₈₉×#2626	2
C₈₉×#2454	2
C₈₉×#2452	2
C₈₉×#2316	2
C₈₉×#2309	2
C₈₉×#2296	2
#35174	89

Subgroups — C89×#35174 #36413022

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#35174 #36413022