Subgroups — C₂×C₈₉×#2437 #36412495

Order 22784.

4926 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂²⋊Q₈×C₈₉	C₂	1	2	yes	maximal, complemented
2	C₂³×C₂²⋊C₄×C₈₉	C₂	1	2	yes	maximal
3	C₈₉×#2437	C₂	1	16	yes	maximal, direct product factor
4	C₂×#399×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#396×C₈₉	C₂	1	2	yes	maximal, complemented
6	C₂×#393×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#385×C₈₉	C₂	1	2	yes	maximal, complemented
8	C₂×#385×C₈₉	C₂	1	2	yes	maximal, complemented
9	C₂×#380×C₈₉	C₂	1	2	yes	maximal, complemented
10	C₂×#379×C₈₉	C₂	1	1	yes	maximal, characteristic
11	C₂⁶×C₈₉	C₂²	1	1	yes	characteristic
12	C₂³×Q₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
13	C₂⁴×C₄×C₈₉	C₂²	1	2	yes	—
14	C₂⁴×C₄×C₈₉	C₂²	1	2	yes	—
15	C₂×C₂²⋊Q₈×C₈₉	—	2	16	no	—
16	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	16	yes	complemented
17	C₂²×C₄⋊C₄×C₈₉	C₂²	1	2	yes	complemented
18	C₂²×C₄⋊C₄×C₈₉	C₂²	1	2	yes	complemented
19	C₂²×C₄⋊C₄×C₈₉	C₂²	1	2	yes	complemented
20	C₂²×C₄⋊C₄×C₈₉	C₂²	1	2	yes	complemented
21	C₂²×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic, complemented
22	C₂²×C₂²⋊C₄×C₈₉	—	2	16	no	—
23	C₂²×C₂²⋊C₄×C₈₉	C₂²	1	2	yes	—
24	C₂²×C₂²⋊C₄×C₈₉	C₂²	1	2	yes	—
25	C₂²×C₂²⋊C₄×C₈₉	C₂²	1	16	yes	—
26	C₂²×C₂²⋊C₄×C₈₉	C₂²	1	2	yes	—
27	C₂²×C₂²⋊C₄×C₈₉	C₂²	1	2	yes	—
28	C₂²×C₂²⋊C₄×C₈₉	C₂²	1	2	yes	—
29	C₂²×C₂²⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
30	C₂²×C₂²⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
31	C₂²×C₂²⋊C₄×C₈₉	C₂²	1	2	yes	—
32	#399×C₈₉	C₂²	1	8	yes	complemented
33	#396×C₈₉	C₂²	1	16	yes	complemented
34	#393×C₈₉	C₂²	1	8	yes	complemented
35	#385×C₈₉	C₂²	1	16	yes	complemented
36	#385×C₈₉	C₂²	1	16	yes	complemented
37	#380×C₈₉	C₂²	1	16	yes	complemented
38	#379×C₈₉	C₂²	1	8	yes	—
39	C₂×#95×C₈₉	C₂²	1	1	yes	characteristic, complemented
40	C₂×#95×C₈₉	C₂²	1	2	yes	complemented
41	C₂×#95×C₈₉	C₂²	1	2	yes	complemented
42	C₂×#95×C₈₉	C₂²	1	1	yes	characteristic, complemented
43	C₂⁵×C₈₉	—	2	8	no	—
44	C₂⁵×C₈₉	—	2	8	no	—
45	C₂⁵×C₈₉	—	4	2	no	—
46	C₂⁵×C₈₉	—	4	2	no	—
47	C₂⁵×C₈₉	Q₈	1	8	yes	complemented
48	C₂⁵×C₈₉	C₂³	1	1	yes	characteristic
49	C₂⁵×C₈₉	C₂³	1	4	yes	—
50	C₂⁵×C₈₉	C₂³	1	2	yes	—
51	C₂²×Q₈×C₈₉	—	2	8	no	—
52	C₂²×Q₈×C₈₉	—	4	2	no	—
53	C₂²×Q₈×C₈₉	C₂³	1	4	yes	complemented
54	C₂³×C₄×C₈₉	—	2	8	no	—
55	C₂³×C₄×C₈₉	—	2	8	no	—
56	C₂³×C₄×C₈₉	—	2	8	no	—
57	C₂³×C₄×C₈₉	—	2	8	no	—
58	C₂³×C₄×C₈₉	—	4	4	no	—
59	C₂³×C₄×C₈₉	—	4	4	no	—
60	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
61	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
62	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
63	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
64	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
65	C₂³×C₄×C₈₉	C₂³	1	8	yes	—
66	C₂³×C₄×C₈₉	C₂³	1	8	yes	—
67	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
68	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
69	C₂²⋊Q₈×C₈₉	—	2	64	no	—
70	C₂×C₄⋊C₄×C₈₉	—	2	8	no	—
71	C₂×C₄⋊C₄×C₈₉	—	2	8	no	—
72	C₂×C₄⋊C₄×C₈₉	—	2	8	no	—
73	C₂×C₄⋊C₄×C₈₉	—	4	4	no	—
74	C₂×C₄⋊C₄×C₈₉	—	4	2	no	—
75	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	complemented
76	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	complemented
77	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	complemented
78	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	complemented
79	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	complemented
80	C₂×C₂²⋊C₄×C₈₉	—	2	8	no	—
81	C₂×C₂²⋊C₄×C₈₉	—	2	8	no	—
82	C₂×C₂²⋊C₄×C₈₉	—	2	8	no	—
83	C₂×C₂²⋊C₄×C₈₉	—	2	8	no	—
84	C₂×C₂²⋊C₄×C₈₉	—	2	8	no	—
85	C₂×C₂²⋊C₄×C₈₉	—	2	64	no	—
86	C₂×C₂²⋊C₄×C₈₉	—	4	2	no	—
87	C₂×C₂²⋊C₄×C₈₉	—	4	2	no	—
88	C₂×C₂²⋊C₄×C₈₉	D₄	1	16	yes	—
89	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
90	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
91	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
92	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
93	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
94	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
95	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
96	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
97	#95×C₈₉	C₂³	1	4	yes	complemented
98	#95×C₈₉	C₂³	1	8	yes	complemented
99	#95×C₈₉	C₂³	1	8	yes	complemented
100	#95×C₈₉	C₂³	1	4	yes	complemented
101	C₂⁴×C₈₉	—	2	4	no	—
102	C₂⁴×C₈₉	—	2	4	no	—
103	C₂⁴×C₈₉	—	2	4	no	—
104	C₂⁴×C₈₉	—	2	4	no	—
105	C₂⁴×C₈₉	—	2	64	no	—
106	C₂⁴×C₈₉	—	2	16	no	—
107	C₂⁴×C₈₉	—	2	4	no	—
108	C₂⁴×C₈₉	—	2	16	no	—
109	C₂⁴×C₈₉	—	2	4	no	—
110	C₂⁴×C₈₉	—	2	2	no	—
111	C₂⁴×C₈₉	—	2	2	no	—
112	C₂⁴×C₈₉	—	4	16	no	—
113	C₂⁴×C₈₉	—	4	4	no	—
114	C₂⁴×C₈₉	—	4	4	no	—
115	C₂⁴×C₈₉	—	4	16	no	—
116	C₂⁴×C₈₉	—	4	32	no	—
117	C₂⁴×C₈₉	—	4	16	no	—
118	C₂⁴×C₈₉	C₂×D₄	1	8	yes	—
119	C₂⁴×C₈₉	C₂×Q₈	1	4	yes	—
120	C₂⁴×C₈₉	C₂×Q₈	1	8	yes	—
121	C₂⁴×C₈₉	C₂×Q₈	1	16	yes	complemented
122	C₂⁴×C₈₉	#41	1	8	yes	—
123	C₂⁴×C₈₉	C₂⁴	1	2	yes	—
124	C₂⁴×C₈₉	C₂⁴	1	4	yes	—
125	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
126	C₂×Q₈×C₈₉	—	2	16	no	—
127	C₂×Q₈×C₈₉	—	4	4	no	—
128	C₂×Q₈×C₈₉	—	4	16	no	—
129	C₂²×C₄×C₈₉	—	2	4	no	—
130	C₂²×C₄×C₈₉	—	2	4	no	—
131	C₂²×C₄×C₈₉	—	2	4	no	—
132	C₂²×C₄×C₈₉	—	2	4	no	—
133	C₂²×C₄×C₈₉	—	2	4	no	—
134	C₂²×C₄×C₈₉	—	2	4	no	—
135	C₂²×C₄×C₈₉	—	2	4	no	—
136	C₂²×C₄×C₈₉	—	2	4	no	—
137	C₂²×C₄×C₈₉	—	2	4	no	—
138	C₂²×C₄×C₈₉	—	2	16	no	—
139	C₂²×C₄×C₈₉	—	2	4	no	—
140	C₂²×C₄×C₈₉	—	2	4	no	—
141	C₂²×C₄×C₈₉	—	2	16	no	—
142	C₂²×C₄×C₈₉	—	2	4	no	—
143	C₂²×C₄×C₈₉	—	2	4	no	—
144	C₂²×C₄×C₈₉	—	2	2	no	—
145	C₂²×C₄×C₈₉	—	2	2	no	—
146	C₂²×C₄×C₈₉	—	2	4	no	—
147	C₂²×C₄×C₈₉	—	2	4	no	—
148	C₂²×C₄×C₈₉	—	2	16	no	—
149	C₂²×C₄×C₈₉	—	2	16	no	—
150	C₂²×C₄×C₈₉	—	4	8	no	—
151	C₂²×C₄×C₈₉	—	4	8	no	—
152	C₂²×C₄×C₈₉	—	4	32	no	—
153	C₂²×C₄×C₈₉	—	4	32	no	—
154	C₂²×C₄×C₈₉	C₂×D₄	1	8	yes	—
155	C₂²×C₄×C₈₉	C₂×D₄	1	8	yes	—
156	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
157	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
158	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
159	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
160	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
161	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
162	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
163	C₄⋊C₄×C₈₉	—	2	16	no	—
164	C₄⋊C₄×C₈₉	—	2	16	no	—
165	C₄⋊C₄×C₈₉	—	2	16	no	—
166	C₄⋊C₄×C₈₉	—	4	8	no	—
167	C₄⋊C₄×C₈₉	—	4	4	no	—
168	C₂²⋊C₄×C₈₉	—	2	16	no	—
169	C₂²⋊C₄×C₈₉	—	2	16	no	—
170	C₂²⋊C₄×C₈₉	—	2	16	no	—
171	C₂²⋊C₄×C₈₉	—	2	16	no	—
172	C₂²⋊C₄×C₈₉	—	2	16	no	—
173	C₂²⋊C₄×C₈₉	—	4	4	no	—
174	C₂²⋊C₄×C₈₉	—	4	4	no	—
175	C₂²⋊C₄×C₈₉	C₂×D₄	1	32	yes	—
176	C₂³×C₈₉	—	2	16	no	—
177	C₂³×C₈₉	—	2	16	no	—
178	C₂³×C₈₉	—	2	16	no	—
179	C₂³×C₈₉	—	2	16	no	—
180	C₂³×C₈₉	—	2	16	no	—
181	C₂³×C₈₉	—	2	16	no	—
182	C₂³×C₈₉	—	2	4	no	—
183	C₂³×C₈₉	—	2	4	no	—
184	C₂³×C₈₉	—	2	4	no	—
185	C₂³×C₈₉	—	2	4	no	—
186	C₂³×C₈₉	—	2	16	no	—
187	C₂³×C₈₉	—	2	64	no	—
188	C₂³×C₈₉	—	2	8	no	—
189	C₂³×C₈₉	—	2	4	no	—
190	C₂³×C₈₉	—	2	4	no	—
191	C₂³×C₈₉	—	2	2	no	—
192	C₂³×C₈₉	—	2	2	no	—
193	C₂³×C₈₉	—	4	16	no	—
194	C₂³×C₈₉	—	4	128	no	—
195	C₂³×C₈₉	—	4	16	no	—
196	C₂³×C₈₉	—	4	32	no	—
197	C₂³×C₈₉	—	4	16	no	—
198	C₂³×C₈₉	—	4	4	no	—
199	C₂³×C₈₉	—	4	8	no	—
200	C₂³×C₈₉	—	4	4	no	—
201	C₂³×C₈₉	C₂²⋊Q₈	1	32	yes	complemented
202	C₂³×C₈₉	C₂²×D₄	1	8	yes	—
203	C₂³×C₈₉	C₂²×D₄	1	4	yes	—
204	C₂³×C₈₉	C₂²×Q₈	1	4	yes	—
205	C₂³×C₈₉	C₂²×Q₈	1	8	yes	—
206	C₂³×C₈₉	C₂²×Q₈	1	2	yes	—
207	C₂³×C₈₉	C₂×#41	1	8	yes	—
208	C₂³×C₈₉	C₂×#41	1	4	yes	—
209	C₂³×C₈₉	H(5,𝔽₂)	1	2	yes	—
210	C₂³×C₈₉	H(5,𝔽₂)	1	2	yes	—
211	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
212	Q₈×C₈₉	—	4	16	no	—
213	C₂×C₄×C₈₉	—	2	16	no	—
214	C₂×C₄×C₈₉	—	2	16	no	—
215	C₂×C₄×C₈₉	—	2	4	no	—
216	C₂×C₄×C₈₉	—	2	4	no	—
217	C₂×C₄×C₈₉	—	2	4	no	—
218	C₂×C₄×C₈₉	—	2	4	no	—
219	C₂×C₄×C₈₉	—	2	4	no	—
220	C₂×C₄×C₈₉	—	2	4	no	—
221	C₂×C₄×C₈₉	—	2	4	no	—
222	C₂×C₄×C₈₉	—	2	4	no	—
223	C₂×C₄×C₈₉	—	2	4	no	—
224	C₂×C₄×C₈₉	—	2	4	no	—
225	C₂×C₄×C₈₉	—	2	4	no	—
226	C₂×C₄×C₈₉	—	2	4	no	—
227	C₂×C₄×C₈₉	—	2	4	no	—
228	C₂×C₄×C₈₉	—	2	2	no	—
229	C₂×C₄×C₈₉	—	2	2	no	—
230	C₂×C₄×C₈₉	—	2	4	no	—
231	C₂×C₄×C₈₉	—	2	4	no	—
232	C₂×C₄×C₈₉	—	4	32	no	—
233	C₂×C₄×C₈₉	—	4	32	no	—
234	C₂×C₄×C₈₉	—	4	8	no	—
235	C₂×C₄×C₈₉	—	4	4	no	—
236	C₂×C₄×C₈₉	—	4	8	no	—
237	C₂×C₄×C₈₉	—	4	8	no	—
238	C₂×C₄×C₈₉	—	4	4	no	—
239	C₂×C₄×C₈₉	C₂²×D₄	1	8	yes	—
240	C₂×C₄×C₈₉	C₂²×D₄	1	8	yes	—
241	C₂²×C₈₉	—	2	8	no	—
242	C₂²×C₈₉	—	2	8	no	—
243	C₂²×C₈₉	—	2	64	no	—
244	C₂²×C₈₉	—	2	8	no	—
245	C₂²×C₈₉	—	2	8	no	—
246	C₂²×C₈₉	—	2	8	no	—
247	C₂²×C₈₉	—	2	8	no	—
248	C₂²×C₈₉	—	2	8	no	—
249	C₂²×C₈₉	—	2	4	no	—
250	C₂²×C₈₉	—	4	64	no	—
251	C₂²×C₈₉	—	4	16	no	—
252	C₂²×C₈₉	—	4	2	no	—
253	C₂²×C₈₉	—	4	4	no	—
254	C₂²×C₈₉	—	4	2	no	—
255	C₂²×C₈₉	C₂×C₂²⋊Q₈	1	8	yes	—
256	C₂²×C₈₉	C₂×C₂²⋊Q₈	1	16	yes	complemented
257	C₂²×C₈₉	#543	1	4	yes	—
258	C₂²×C₈₉	D₄²	1	4	yes	—
259	C₂²×C₈₉	#546	1	8	yes	—
260	C₂²×C₈₉	#551	1	4	yes	—
261	C₂²×C₈₉	C₂³×D₄	1	2	yes	—
262	C₂²×C₈₉	C₂³×Q₈	1	1	yes	characteristic
263	C₂²×C₈₉	C₂²×#41	1	2	yes	—
264	C₂²×C₈₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
265	C₂²×C₈₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
266	C₄×C₈₉	—	2	8	no	—
267	C₄×C₈₉	—	2	8	no	—
268	C₄×C₈₉	—	4	4	no	—
269	C₄×C₈₉	—	4	2	no	—
270	C₄×C₈₉	—	4	4	no	—
271	C₄×C₈₉	—	4	4	no	—
272	C₄×C₈₉	—	4	2	no	—
273	C₂×#2437	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
274	C₂×C₈₉	—	2	16	no	—
275	C₂×C₈₉	—	4	4	no	—
276	C₂×C₈₉	#2437	1	8	yes	direct product factor
277	C₂×C₈₉	C₂²×C₂²⋊Q₈	1	2	yes	—
278	C₂×C₈₉	C₂×#543	1	1	yes	characteristic
279	C₂×C₈₉	C₂×D₄²	1	1	yes	characteristic
280	C₂×C₈₉	C₂×#546	1	2	yes	—
281	C₂×C₈₉	C₂×#551	1	1	yes	characteristic
282	C₂²×C₂²⋊Q₈	C₂×C₈₉	1	2	yes	complemented
283	C₂³×C₂²⋊C₄	C₂×C₈₉	1	2	yes	—
284	#2437	C₂×C₈₉	1	16	yes	direct product factor
285	C₂×#399	C₂×C₈₉	1	1	yes	characteristic, complemented
286	C₂×#396	C₂×C₈₉	1	2	yes	complemented
287	C₂×#393	C₂×C₈₉	1	1	yes	characteristic, complemented
288	C₂×#385	C₂×C₈₉	1	2	yes	complemented
289	C₂×#385	C₂×C₈₉	1	2	yes	complemented
290	C₂×#380	C₂×C₈₉	1	2	yes	complemented
291	C₂×#379	C₂×C₈₉	1	1	yes	characteristic
292	C₈₉	C₂×#2437	1	1	yes	characteristic, direct product factor, Sylow
293	C₂⁶	C₂²×C₈₉	1	1	yes	characteristic
294	C₂³×Q₈	C₂²×C₈₉	1	1	yes	characteristic, complemented
295	C₂⁴×C₄	C₂²×C₈₉	1	2	yes	—
296	C₂⁴×C₄	C₂²×C₈₉	1	2	yes	—
297	C₂×C₂²⋊Q₈	—	2	16	no	—
298	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	16	yes	complemented
299	C₂²×C₄⋊C₄	C₂²×C₈₉	1	2	yes	complemented
300	C₂²×C₄⋊C₄	C₂²×C₈₉	1	2	yes	complemented
301	C₂²×C₄⋊C₄	C₂²×C₈₉	1	2	yes	complemented
302	C₂²×C₄⋊C₄	C₂²×C₈₉	1	2	yes	complemented
303	C₂²×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic, complemented
304	C₂²×C₂²⋊C₄	—	2	16	no	—
305	C₂²×C₂²⋊C₄	C₂²×C₈₉	1	2	yes	—
306	C₂²×C₂²⋊C₄	C₂²×C₈₉	1	2	yes	—
307	C₂²×C₂²⋊C₄	C₂²×C₈₉	1	16	yes	—
308	C₂²×C₂²⋊C₄	C₂²×C₈₉	1	2	yes	—
309	C₂²×C₂²⋊C₄	C₂²×C₈₉	1	2	yes	—
310	C₂²×C₂²⋊C₄	C₂²×C₈₉	1	2	yes	—
311	C₂²×C₂²⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
312	C₂²×C₂²⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
313	C₂²×C₂²⋊C₄	C₂²×C₈₉	1	2	yes	—
314	#399	C₂²×C₈₉	1	8	yes	complemented
315	#396	C₂²×C₈₉	1	16	yes	complemented
316	#393	C₂²×C₈₉	1	8	yes	complemented
317	#385	C₂²×C₈₉	1	16	yes	complemented
318	#385	C₂²×C₈₉	1	16	yes	complemented
319	#380	C₂²×C₈₉	1	16	yes	complemented
320	#379	C₂²×C₈₉	1	8	yes	—
321	C₂×#95	C₂²×C₈₉	1	1	yes	characteristic, complemented
322	C₂×#95	C₂²×C₈₉	1	2	yes	complemented
323	C₂×#95	C₂²×C₈₉	1	2	yes	complemented
324	C₂×#95	C₂²×C₈₉	1	1	yes	characteristic, complemented
325	C₂⁵	—	2	8	no	—
326	C₂⁵	—	2	8	no	—
327	C₂⁵	—	4	2	no	—
328	C₂⁵	—	4	2	no	—
329	C₂⁵	Q₈×C₈₉	1	8	yes	complemented
330	C₂⁵	C₂³×C₈₉	1	1	yes	characteristic
331	C₂⁵	C₂³×C₈₉	1	4	yes	—
332	C₂⁵	C₂³×C₈₉	1	2	yes	—
333	C₂²×Q₈	—	2	8	no	—
334	C₂²×Q₈	—	4	2	no	—
335	C₂²×Q₈	C₂³×C₈₉	1	4	yes	complemented
336	C₂³×C₄	—	2	8	no	—
337	C₂³×C₄	—	2	8	no	—
338	C₂³×C₄	—	2	8	no	—
339	C₂³×C₄	—	2	8	no	—
340	C₂³×C₄	—	4	4	no	—
341	C₂³×C₄	—	4	4	no	—
342	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
343	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
344	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
345	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
346	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
347	C₂³×C₄	C₂³×C₈₉	1	8	yes	—
348	C₂³×C₄	C₂³×C₈₉	1	8	yes	—
349	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
350	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
351	C₂²⋊Q₈	—	2	64	no	—
352	C₂×C₄⋊C₄	—	2	8	no	—
353	C₂×C₄⋊C₄	—	2	8	no	—
354	C₂×C₄⋊C₄	—	2	8	no	—
355	C₂×C₄⋊C₄	—	4	4	no	—
356	C₂×C₄⋊C₄	—	4	2	no	—
357	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	complemented
358	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	complemented
359	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	complemented
360	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	complemented
361	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	complemented
362	C₂×C₂²⋊C₄	—	2	8	no	—
363	C₂×C₂²⋊C₄	—	2	8	no	—
364	C₂×C₂²⋊C₄	—	2	8	no	—
365	C₂×C₂²⋊C₄	—	2	8	no	—
366	C₂×C₂²⋊C₄	—	2	8	no	—
367	C₂×C₂²⋊C₄	—	2	64	no	—
368	C₂×C₂²⋊C₄	—	4	2	no	—
369	C₂×C₂²⋊C₄	—	4	2	no	—
370	C₂×C₂²⋊C₄	D₄×C₈₉	1	16	yes	—
371	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
372	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
373	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
374	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
375	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
376	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
377	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
378	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
379	#95	C₂³×C₈₉	1	4	yes	complemented
380	#95	C₂³×C₈₉	1	8	yes	complemented
381	#95	C₂³×C₈₉	1	8	yes	complemented
382	#95	C₂³×C₈₉	1	4	yes	complemented
383	C₂⁴	—	2	4	no	—
384	C₂⁴	—	2	4	no	—
385	C₂⁴	—	2	4	no	—
386	C₂⁴	—	2	4	no	—
387	C₂⁴	—	2	64	no	—
388	C₂⁴	—	2	16	no	—
389	C₂⁴	—	2	4	no	—
390	C₂⁴	—	2	16	no	—
391	C₂⁴	—	2	4	no	—
392	C₂⁴	—	2	2	no	—
393	C₂⁴	—	2	2	no	—
394	C₂⁴	—	4	16	no	—
395	C₂⁴	—	4	4	no	—
396	C₂⁴	—	4	4	no	—
397	C₂⁴	—	4	16	no	—
398	C₂⁴	—	4	32	no	—
399	C₂⁴	—	4	16	no	—
400	C₂⁴	C₂×D₄×C₈₉	1	8	yes	—
401	C₂⁴	C₂×Q₈×C₈₉	1	4	yes	—
402	C₂⁴	C₂×Q₈×C₈₉	1	8	yes	—
403	C₂⁴	C₂×Q₈×C₈₉	1	16	yes	complemented
404	C₂⁴	#41×C₈₉	1	8	yes	—
405	C₂⁴	C₂⁴×C₈₉	1	2	yes	—
406	C₂⁴	C₂⁴×C₈₉	1	4	yes	—
407	C₂⁴	C₂⁴×C₈₉	1	1	yes	characteristic
408	C₂×Q₈	—	2	16	no	—
409	C₂×Q₈	—	4	4	no	—
410	C₂×Q₈	—	4	16	no	—
411	C₂²×C₄	—	2	4	no	—
412	C₂²×C₄	—	2	4	no	—
413	C₂²×C₄	—	2	4	no	—
414	C₂²×C₄	—	2	4	no	—
415	C₂²×C₄	—	2	4	no	—
416	C₂²×C₄	—	2	4	no	—
417	C₂²×C₄	—	2	4	no	—
418	C₂²×C₄	—	2	4	no	—
419	C₂²×C₄	—	2	4	no	—
420	C₂²×C₄	—	2	16	no	—
421	C₂²×C₄	—	2	4	no	—
422	C₂²×C₄	—	2	4	no	—
423	C₂²×C₄	—	2	16	no	—
424	C₂²×C₄	—	2	4	no	—
425	C₂²×C₄	—	2	4	no	—
426	C₂²×C₄	—	2	2	no	—
427	C₂²×C₄	—	2	2	no	—
428	C₂²×C₄	—	2	4	no	—
429	C₂²×C₄	—	2	4	no	—
430	C₂²×C₄	—	2	16	no	—
431	C₂²×C₄	—	2	16	no	—
432	C₂²×C₄	—	4	8	no	—
433	C₂²×C₄	—	4	8	no	—
434	C₂²×C₄	—	4	32	no	—
435	C₂²×C₄	—	4	32	no	—
436	C₂²×C₄	C₂×D₄×C₈₉	1	8	yes	—
437	C₂²×C₄	C₂×D₄×C₈₉	1	8	yes	—
438	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
439	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
440	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
441	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
442	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
443	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
444	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
445	C₄⋊C₄	—	2	16	no	—
446	C₄⋊C₄	—	2	16	no	—
447	C₄⋊C₄	—	2	16	no	—
448	C₄⋊C₄	—	4	8	no	—
449	C₄⋊C₄	—	4	4	no	—
450	C₂²⋊C₄	—	2	16	no	—
451	C₂²⋊C₄	—	2	16	no	—
452	C₂²⋊C₄	—	2	16	no	—
453	C₂²⋊C₄	—	2	16	no	—
454	C₂²⋊C₄	—	2	16	no	—
455	C₂²⋊C₄	—	4	4	no	—
456	C₂²⋊C₄	—	4	4	no	—
457	C₂²⋊C₄	C₂×D₄×C₈₉	1	32	yes	—
458	C₂³	—	2	16	no	—
459	C₂³	—	2	16	no	—
460	C₂³	—	2	16	no	—
461	C₂³	—	2	16	no	—
462	C₂³	—	2	16	no	—
463	C₂³	—	2	16	no	—
464	C₂³	—	2	4	no	—
465	C₂³	—	2	4	no	—
466	C₂³	—	2	4	no	—
467	C₂³	—	2	4	no	—
468	C₂³	—	2	16	no	—
469	C₂³	—	2	64	no	—
470	C₂³	—	2	8	no	—
471	C₂³	—	2	4	no	—
472	C₂³	—	2	4	no	—
473	C₂³	—	2	2	no	—
474	C₂³	—	2	2	no	—
475	C₂³	—	4	16	no	—
476	C₂³	—	4	128	no	—
477	C₂³	—	4	16	no	—
478	C₂³	—	4	32	no	—
479	C₂³	—	4	16	no	—
480	C₂³	—	4	4	no	—
481	C₂³	—	4	8	no	—
482	C₂³	—	4	4	no	—
483	C₂³	C₂²⋊Q₈×C₈₉	1	32	yes	complemented
484	C₂³	C₂²×D₄×C₈₉	1	8	yes	—
485	C₂³	C₂²×D₄×C₈₉	1	4	yes	—
486	C₂³	C₂²×Q₈×C₈₉	1	4	yes	—
487	C₂³	C₂²×Q₈×C₈₉	1	8	yes	—
488	C₂³	C₂²×Q₈×C₈₉	1	2	yes	—
489	C₂³	C₂×#41×C₈₉	1	8	yes	—
490	C₂³	C₂×#41×C₈₉	1	4	yes	—
491	C₂³	H(5,𝔽₂)×C₈₉	1	2	yes	—
492	C₂³	H(5,𝔽₂)×C₈₉	1	2	yes	—
493	C₂³	C₂⁵×C₈₉	1	1	yes	characteristic
494	Q₈	—	4	16	no	—
495	C₂×C₄	—	2	16	no	—
496	C₂×C₄	—	2	16	no	—
497	C₂×C₄	—	2	4	no	—
498	C₂×C₄	—	2	4	no	—
499	C₂×C₄	—	2	4	no	—
500	C₂×C₄	—	2	4	no	—
501	C₂×C₄	—	2	4	no	—
502	C₂×C₄	—	2	4	no	—
503	C₂×C₄	—	2	4	no	—
504	C₂×C₄	—	2	4	no	—
505	C₂×C₄	—	2	4	no	—
506	C₂×C₄	—	2	4	no	—
507	C₂×C₄	—	2	4	no	—
508	C₂×C₄	—	2	4	no	—
509	C₂×C₄	—	2	4	no	—
510	C₂×C₄	—	2	2	no	—
511	C₂×C₄	—	2	2	no	—
512	C₂×C₄	—	2	4	no	—
513	C₂×C₄	—	2	4	no	—
514	C₂×C₄	—	4	32	no	—
515	C₂×C₄	—	4	32	no	—
516	C₂×C₄	—	4	8	no	—
517	C₂×C₄	—	4	4	no	—
518	C₂×C₄	—	4	8	no	—
519	C₂×C₄	—	4	8	no	—
520	C₂×C₄	—	4	4	no	—
521	C₂×C₄	C₂²×D₄×C₈₉	1	8	yes	—
522	C₂×C₄	C₂²×D₄×C₈₉	1	8	yes	—
523	C₂²	—	2	8	no	—
524	C₂²	—	2	8	no	—
525	C₂²	—	2	64	no	—
526	C₂²	—	2	8	no	—
527	C₂²	—	2	8	no	—
528	C₂²	—	2	8	no	—
529	C₂²	—	2	8	no	—
530	C₂²	—	2	8	no	—
531	C₂²	—	2	4	no	—
532	C₂²	—	4	64	no	—
533	C₂²	—	4	16	no	—
534	C₂²	—	4	2	no	—
535	C₂²	—	4	4	no	—
536	C₂²	—	4	2	no	—
537	C₂²	C₂×C₂²⋊Q₈×C₈₉	1	8	yes	—
538	C₂²	C₂×C₂²⋊Q₈×C₈₉	1	16	yes	complemented
539	C₂²	#543×C₈₉	1	4	yes	—
540	C₂²	D₄²×C₈₉	1	4	yes	—
541	C₂²	#546×C₈₉	1	8	yes	—
542	C₂²	#551×C₈₉	1	4	yes	—
543	C₂²	C₂³×D₄×C₈₉	1	2	yes	—
544	C₂²	C₂³×Q₈×C₈₉	1	1	yes	characteristic
545	C₂²	C₂²×#41×C₈₉	1	2	yes	—
546	C₂²	C₂×H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
547	C₂²	C₂×H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
548	C₄	—	2	8	no	—
549	C₄	—	2	8	no	—
550	C₄	—	4	4	no	—
551	C₄	—	4	2	no	—
552	C₄	—	4	4	no	—
553	C₄	—	4	4	no	—
554	C₄	—	4	2	no	—
555	C₂	—	2	16	no	—
556	C₂	—	4	4	no	—
557	C₂	C₈₉×#2437	1	8	yes	direct product factor
558	C₂	C₂²×C₂²⋊Q₈×C₈₉	1	2	yes	—
559	C₂	C₂×#543×C₈₉	1	1	yes	characteristic
560	C₂	C₂×D₄²×C₈₉	1	1	yes	characteristic
561	C₂	C₂×#546×C₈₉	1	2	yes	—
562	C₂	C₂×#551×C₈₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₈₉	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₈₉
prime radical	C₂⁶×C₈₉	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₂²⋊Q₈×C₈₉	C₂	semidirect
3. C₈₉×#2437	C₂	direct (N × H)
4. C₂×#399×C₈₉	C₂	semidirect
5. C₂×#396×C₈₉	C₂	semidirect
6. C₂×#393×C₈₉	C₂	semidirect
7. C₂×#385×C₈₉	C₂	semidirect
8. C₂×#385×C₈₉	C₂	semidirect
9. C₂×#380×C₈₉	C₂	semidirect
12. C₂³×Q₈×C₈₉	C₂²	semidirect
16. C₂×C₂²⋊Q₈×C₈₉	C₂²	semidirect
17. C₂²×C₄⋊C₄×C₈₉	C₂²	semidirect
18. C₂²×C₄⋊C₄×C₈₉	C₂²	semidirect
19. C₂²×C₄⋊C₄×C₈₉	C₂²	semidirect
20. C₂²×C₄⋊C₄×C₈₉	C₂²	semidirect
21. C₂²×C₄⋊C₄×C₈₉	C₂²	semidirect
32. #399×C₈₉	C₂²	semidirect
33. #396×C₈₉	C₂²	semidirect
34. #393×C₈₉	C₂²	semidirect
35. #385×C₈₉	C₂²	semidirect
36. #385×C₈₉	C₂²	semidirect
37. #380×C₈₉	C₂²	semidirect
39. C₂×#95×C₈₉	C₂²	semidirect
40. C₂×#95×C₈₉	C₂²	semidirect
41. C₂×#95×C₈₉	C₂²	semidirect
42. C₂×#95×C₈₉	C₂²	semidirect
47. C₂⁵×C₈₉	Q₈	semidirect
53. C₂²×Q₈×C₈₉	C₂³	semidirect
75. C₂×C₄⋊C₄×C₈₉	C₂³	semidirect
76. C₂×C₄⋊C₄×C₈₉	C₂³	semidirect
77. C₂×C₄⋊C₄×C₈₉	C₂³	semidirect
78. C₂×C₄⋊C₄×C₈₉	C₂³	semidirect
79. C₂×C₄⋊C₄×C₈₉	C₂³	semidirect
97. #95×C₈₉	C₂³	semidirect
98. #95×C₈₉	C₂³	semidirect
99. #95×C₈₉	C₂³	semidirect
100. #95×C₈₉	C₂³	semidirect
121. C₂⁴×C₈₉	C₂×Q₈	semidirect
201. C₂³×C₈₉	C₂²⋊Q₈	semidirect
256. C₂²×C₈₉	C₂×C₂²⋊Q₈	semidirect
273. C₂×#2437	C₈₉	direct (N × H)
276. C₂×C₈₉	#2437	direct (N × H)
282. C₂²×C₂²⋊Q₈	C₂×C₈₉	semidirect
284. #2437	C₂×C₈₉	direct (N × H)
285. C₂×#399	C₂×C₈₉	semidirect
286. C₂×#396	C₂×C₈₉	semidirect
287. C₂×#393	C₂×C₈₉	semidirect
288. C₂×#385	C₂×C₈₉	semidirect
289. C₂×#385	C₂×C₈₉	semidirect
290. C₂×#380	C₂×C₈₉	semidirect
292. C₈₉	C₂×#2437	direct (N × H)
294. C₂³×Q₈	C₂²×C₈₉	semidirect
298. C₂×C₂²⋊Q₈	C₂²×C₈₉	semidirect
299. C₂²×C₄⋊C₄	C₂²×C₈₉	semidirect
300. C₂²×C₄⋊C₄	C₂²×C₈₉	semidirect
301. C₂²×C₄⋊C₄	C₂²×C₈₉	semidirect
302. C₂²×C₄⋊C₄	C₂²×C₈₉	semidirect
303. C₂²×C₄⋊C₄	C₂²×C₈₉	semidirect
314. #399	C₂²×C₈₉	semidirect
315. #396	C₂²×C₈₉	semidirect
316. #393	C₂²×C₈₉	semidirect
317. #385	C₂²×C₈₉	semidirect
318. #385	C₂²×C₈₉	semidirect
319. #380	C₂²×C₈₉	semidirect
321. C₂×#95	C₂²×C₈₉	semidirect
322. C₂×#95	C₂²×C₈₉	semidirect
323. C₂×#95	C₂²×C₈₉	semidirect
324. C₂×#95	C₂²×C₈₉	semidirect
329. C₂⁵	Q₈×C₈₉	semidirect
335. C₂²×Q₈	C₂³×C₈₉	semidirect
357. C₂×C₄⋊C₄	C₂³×C₈₉	semidirect
358. C₂×C₄⋊C₄	C₂³×C₈₉	semidirect
359. C₂×C₄⋊C₄	C₂³×C₈₉	semidirect
360. C₂×C₄⋊C₄	C₂³×C₈₉	semidirect
361. C₂×C₄⋊C₄	C₂³×C₈₉	semidirect
379. #95	C₂³×C₈₉	semidirect
380. #95	C₂³×C₈₉	semidirect
381. #95	C₂³×C₈₉	semidirect
382. #95	C₂³×C₈₉	semidirect
403. C₂⁴	C₂×Q₈×C₈₉	semidirect
483. C₂³	C₂²⋊Q₈×C₈₉	semidirect
538. C₂²	C₂×C₂²⋊Q₈×C₈₉	semidirect
557. C₂	C₈₉×#2437	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×C₂²⋊Q₈×C₈₉	2
C₂³×C₂²⋊C₄×C₈₉	2
C₈₉×#2437	2
C₂×#399×C₈₉	2
C₂×#396×C₈₉	2
C₂×#393×C₈₉	2
C₂×#385×C₈₉	2
C₂×#380×C₈₉	2
C₂×#379×C₈₉	2
C₂×#2437	89

Subgroups — C2×C89×#2437 #36412495

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈₉×#2437 #36412495