Subgroups — C₂×C₈₉×#3218 #36438634

Order 22784.

2146 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#529×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#3218	C₂	1	16	yes	maximal, direct product factor
4	C₂×#497×C₈₉	C₂	1	2	yes	maximal, complemented
5	C₂×#491×C₈₉	C₂	1	2	yes	maximal, complemented
6	C₂×#462×C₈₉	C₂	1	2	yes	maximal, complemented
7	C₂×#461×C₈₉	C₂	1	2	yes	maximal, complemented
8	C₂×#451×C₈₉	C₂	1	2	yes	maximal, complemented
9	C₂×#448×C₈₉	C₂	1	2	yes	maximal
10	C₂×#432×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂³×Q₈×C₈₉	C₂²	1	2	yes	—
12	C₂²×Q₁₆×C₈₉	C₂²	1	4	yes	—
13	C₂²×SD₁₆×C₈₉	C₂²	1	4	yes	—
14	C₂×C₄⋊Q₈×C₈₉	—	2	1	no	—
15	C₂×C₄⋊Q₈×C₈₉	—	2	1	no	—
16	C₂×C₄⋊Q₈×C₈₉	—	2	4	no	—
17	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
18	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
19	C₂×C₄⋊Q₈×C₈₉	C₂²	1	8	yes	complemented
20	#529×C₈₉	—	2	4	no	—
21	#529×C₈₉	C₂²	1	8	yes	—
22	C₂×#126×C₈₉	—	2	1	no	—
23	C₂×#125×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×#124×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×#123×C₈₉	—	2	1	no	—
26	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×#121×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
29	C₂×C₄×D₄×C₈₉	—	2	1	no	—
30	C₂×C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
31	C₂×#117×C₈₉	—	2	1	no	—
32	C₂²×C₄⋊C₄×C₈₉	—	2	1	no	—
33	C₂²×C₄⋊C₄×C₈₉	—	2	1	no	—
34	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
35	#497×C₈₉	C₂²	1	16	yes	complemented
36	#491×C₈₉	C₂²	1	16	yes	complemented
37	#462×C₈₉	C₂²	1	16	yes	complemented
38	#461×C₈₉	C₂²	1	16	yes	complemented
39	#451×C₈₉	C₂²	1	16	yes	complemented
40	#448×C₈₉	C₂²	1	16	yes	—
41	#432×C₈₉	C₂²	1	8	yes	complemented
42	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
43	C₂×#103×C₈₉	C₂²	1	4	yes	—
44	C₂×#103×C₈₉	C₂²	1	4	yes	—
45	C₂×C₂²⋊C₈×C₈₉	C₂²	1	2	yes	—
46	C₂×#97×C₈₉	C₂²	1	2	yes	—
47	C₂²×Q₈×C₈₉	—	2	4	no	—
48	C₂²×Q₈×C₈₉	—	2	2	no	—
49	C₂²×Q₈×C₈₉	—	4	8	no	—
50	C₂²×Q₈×C₈₉	C₂³	1	4	yes	—
51	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
52	C₂²×Q₈×C₈₉	C₂³	1	8	yes	—
53	C₂²×D₄×C₈₉	—	2	1	no	—
54	C₂²×D₄×C₈₉	C₂³	1	1	yes	characteristic
55	C₂³×C₄×C₈₉	—	2	2	no	—
56	C₂³×C₄×C₈₉	—	2	1	no	—
57	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
58	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
59	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
60	C₂×Q₁₆×C₈₉	—	4	8	no	—
61	C₂×Q₁₆×C₈₉	C₂³	1	16	yes	—
62	C₂×SD₁₆×C₈₉	—	4	8	no	—
63	C₂×SD₁₆×C₈₉	C₂³	1	16	yes	—
64	C₂²×C₈×C₈₉	C₂³	1	4	yes	—
65	C₄⋊Q₈×C₈₉	—	2	4	no	—
66	C₄⋊Q₈×C₈₉	—	2	4	no	—
67	C₄⋊Q₈×C₈₉	—	2	8	no	—
68	C₄⋊Q₈×C₈₉	—	2	8	no	—
69	C₄⋊Q₈×C₈₉	C₂³	1	8	yes	—
70	C₄⋊Q₈×C₈₉	C₂³	1	8	yes	—
71	#126×C₈₉	—	2	8	no	—
72	#125×C₈₉	—	2	2	no	—
73	#125×C₈₉	C₂³	1	4	yes	—
74	#124×C₈₉	—	2	2	no	—
75	#124×C₈₉	C₂³	1	4	yes	—
76	#123×C₈₉	—	2	8	no	—
77	C₂²⋊Q₈×C₈₉	—	2	2	no	—
78	C₂²⋊Q₈×C₈₉	C₂³	1	4	yes	—
79	#121×C₈₉	—	2	2	no	—
80	#121×C₈₉	C₂³	1	4	yes	—
81	C₄×Q₈×C₈₉	—	2	2	no	—
82	C₄×Q₈×C₈₉	C₂³	1	4	yes	—
83	C₄×D₄×C₈₉	—	2	8	no	—
84	C₄×D₄×C₈₉	—	2	2	no	—
85	C₄×D₄×C₈₉	C₂³	1	4	yes	—
86	#117×C₈₉	—	2	8	no	—
87	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
88	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
89	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
90	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
91	C₂×C₄⋊C₄×C₈₉	—	2	8	no	—
92	C₂×C₄⋊C₄×C₈₉	—	2	8	no	—
93	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
94	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
95	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
96	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
97	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
98	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
99	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
100	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
101	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
102	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
103	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
104	C₂×C₄²×C₈₉	—	2	1	no	—
105	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
106	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
107	C₂×C₄²×C₈₉	D₄	1	4	yes	—
108	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
109	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
110	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
111	C₄⋊C₈×C₈₉	C₂³	1	8	yes	—
112	#103×C₈₉	C₂³	1	16	yes	—
113	#103×C₈₉	C₂³	1	16	yes	—
114	C₂²⋊C₈×C₈₉	C₂³	1	8	yes	—
115	#97×C₈₉	C₂³	1	8	yes	—
116	C₂⁴×C₈₉	—	2	1	no	—
117	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
118	C₂×Q₈×C₈₉	—	2	1	no	—
119	C₂×Q₈×C₈₉	—	2	8	no	—
120	C₂×Q₈×C₈₉	—	2	4	no	—
121	C₂×Q₈×C₈₉	—	2	8	no	—
122	C₂×Q₈×C₈₉	—	2	4	no	—
123	C₂×Q₈×C₈₉	—	2	8	no	—
124	C₂×Q₈×C₈₉	—	4	16	no	—
125	C₂×Q₈×C₈₉	—	4	8	no	—
126	C₂×Q₈×C₈₉	—	4	16	no	—
127	C₂×Q₈×C₈₉	C₂⁴	1	8	yes	—
128	C₂×Q₈×C₈₉	C₂⁴	1	2	yes	—
129	C₂×D₄×C₈₉	—	2	4	no	—
130	C₂×D₄×C₈₉	—	2	1	no	—
131	C₂×D₄×C₈₉	—	2	4	no	—
132	C₂×D₄×C₈₉	—	4	4	no	—
133	C₂×D₄×C₈₉	C₂⁴	1	2	yes	—
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	8	no	—
138	C₂²×C₄×C₈₉	—	2	1	no	—
139	C₂²×C₄×C₈₉	—	2	1	no	—
140	C₂²×C₄×C₈₉	—	2	1	no	—
141	C₂²×C₄×C₈₉	—	2	4	no	—
142	C₂²×C₄×C₈₉	—	2	4	no	—
143	C₂²×C₄×C₈₉	—	4	8	no	—
144	C₂²×C₄×C₈₉	—	4	4	no	—
145	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
146	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
147	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
148	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
149	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
150	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
151	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
152	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
154	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
155	Q₁₆×C₈₉	—	4	16	no	—
156	SD₁₆×C₈₉	—	4	16	no	—
157	C₂×C₈×C₈₉	—	2	8	no	—
158	C₂×C₈×C₈₉	C₂⁴	1	8	yes	—
159	C₄⋊C₄×C₈₉	—	2	8	no	—
160	C₄⋊C₄×C₈₉	—	2	8	no	—
161	C₄⋊C₄×C₈₉	—	2	8	no	—
162	C₄⋊C₄×C₈₉	—	2	8	no	—
163	C₄⋊C₄×C₈₉	—	2	4	no	—
164	C₄⋊C₄×C₈₉	—	2	4	no	—
165	C₄⋊C₄×C₈₉	—	2	4	no	—
166	C₄⋊C₄×C₈₉	—	2	4	no	—
167	C₄⋊C₄×C₈₉	—	2	1	no	—
168	C₄⋊C₄×C₈₉	—	2	1	no	—
169	C₄⋊C₄×C₈₉	C₂⁴	1	2	yes	—
170	C₄⋊C₄×C₈₉	C₂⁴	1	2	yes	—
171	C₂²⋊C₄×C₈₉	—	2	4	no	—
172	C₂²⋊C₄×C₈₉	—	2	4	no	—
173	C₂²⋊C₄×C₈₉	—	2	4	no	—
174	C₂²⋊C₄×C₈₉	—	2	4	no	—
175	C₂²⋊C₄×C₈₉	—	2	4	no	—
176	C₄²×C₈₉	—	2	4	no	—
177	C₄²×C₈₉	C₂×D₄	1	4	yes	—
178	C₄²×C₈₉	C₂×D₄	1	2	yes	—
179	C₄²×C₈₉	C₂×D₄	1	2	yes	—
180	C₄²×C₈₉	C₂×D₄	1	4	yes	—
181	C₄²×C₈₉	C₂⁴	1	2	yes	—
182	C₄²×C₈₉	C₂⁴	1	2	yes	—
183	C₂³×C₈₉	—	2	2	no	—
184	C₂³×C₈₉	—	2	2	no	—
185	C₂³×C₈₉	—	2	2	no	—
186	C₂³×C₈₉	—	4	2	no	—
187	C₂³×C₈₉	—	4	2	no	—
188	C₂³×C₈₉	—	4	2	no	—
189	C₂³×C₈₉	#137	1	4	yes	—
190	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
191	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
192	Q₈×C₈₉	—	2	8	no	—
193	Q₈×C₈₉	—	2	4	no	—
194	Q₈×C₈₉	—	2	8	no	—
195	Q₈×C₈₉	—	4	16	no	—
196	Q₈×C₈₉	—	4	2	no	—
197	Q₈×C₈₉	—	4	8	no	—
198	D₄×C₈₉	—	2	4	no	—
199	D₄×C₈₉	—	4	8	no	—
200	D₄×C₈₉	—	4	2	no	—
201	C₂×C₄×C₈₉	—	2	2	no	—
202	C₂×C₄×C₈₉	—	2	8	no	—
203	C₂×C₄×C₈₉	—	2	2	no	—
204	C₂×C₄×C₈₉	—	2	2	no	—
205	C₂×C₄×C₈₉	—	2	2	no	—
206	C₂×C₄×C₈₉	—	2	4	no	—
207	C₂×C₄×C₈₉	—	2	8	no	—
208	C₂×C₄×C₈₉	—	2	4	no	—
209	C₂×C₄×C₈₉	—	2	2	no	—
210	C₂×C₄×C₈₉	—	2	2	no	—
211	C₂×C₄×C₈₉	—	2	2	no	—
212	C₂×C₄×C₈₉	—	2	8	no	—
213	C₂×C₄×C₈₉	—	4	16	no	—
214	C₂×C₄×C₈₉	—	4	8	no	—
215	C₂×C₄×C₈₉	—	4	2	no	—
216	C₂×C₄×C₈₉	—	4	2	no	—
217	C₂×C₄×C₈₉	—	4	2	no	—
218	C₂×C₄×C₈₉	—	4	8	no	—
219	C₂×C₄×C₈₉	#137	1	4	yes	—
220	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
221	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
222	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
223	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
224	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
225	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
226	C₂×C₄×C₈₉	H(5,𝔽₂)	1	4	yes	—
227	C₂×C₄×C₈₉	C₂⁵	1	1	yes	characteristic
228	C₈×C₈₉	—	2	8	no	—
229	C₂²×C₈₉	—	2	2	no	—
230	C₂²×C₈₉	—	2	8	no	—
231	C₂²×C₈₉	—	2	2	no	—
232	C₂²×C₈₉	—	4	2	no	—
233	C₂²×C₈₉	—	4	2	no	—
234	C₂²×C₈₉	—	4	2	no	—
235	C₂²×C₈₉	—	8	4	no	—
236	C₂²×C₈₉	#535	1	2	yes	—
237	C₂²×C₈₉	C₂×#137	1	4	yes	—
238	C₂²×C₈₉	C₂×#137	1	2	yes	—
239	C₂²×C₈₉	C₂×#137	1	2	yes	—
240	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
241	C₄×C₈₉	—	2	2	no	—
242	C₄×C₈₉	—	2	2	no	—
243	C₄×C₈₉	—	2	2	no	—
244	C₄×C₈₉	—	2	2	no	—
245	C₄×C₈₉	—	4	8	no	—
246	C₄×C₈₉	—	4	2	no	—
247	C₄×C₈₉	—	4	2	no	—
248	C₄×C₈₉	—	4	2	no	—
249	C₄×C₈₉	#535	1	2	yes	—
250	C₄×C₈₉	C₂×#137	1	4	yes	—
251	C₄×C₈₉	C₂×H(5,𝔽₂)	1	2	yes	—
252	C₂×#3218	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
253	C₂×C₈₉	—	2	4	no	—
254	C₂×C₈₉	—	8	2	no	—
255	C₂×C₈₉	#3218	1	4	yes	direct product factor
256	C₂×C₈₉	C₂×#535	1	1	yes	characteristic
257	C₂×C₈₉	C₂²×#137	1	1	yes	characteristic
258	C₂×C₈₉	C₂²×#137	1	1	yes	characteristic
259	C₂²×C₄⋊Q₈	C₂×C₈₉	1	1	yes	characteristic, complemented
260	C₂×#529	C₂×C₈₉	1	1	yes	characteristic
261	#3218	C₂×C₈₉	1	16	yes	direct product factor
262	C₂×#497	C₂×C₈₉	1	2	yes	complemented
263	C₂×#491	C₂×C₈₉	1	2	yes	complemented
264	C₂×#462	C₂×C₈₉	1	2	yes	complemented
265	C₂×#461	C₂×C₈₉	1	2	yes	complemented
266	C₂×#451	C₂×C₈₉	1	2	yes	complemented
267	C₂×#448	C₂×C₈₉	1	2	yes	—
268	C₂×#432	C₂×C₈₉	1	1	yes	characteristic, complemented
269	C₈₉	C₂×#3218	1	1	yes	characteristic, direct product factor, Sylow
270	C₂³×Q₈	C₂²×C₈₉	1	2	yes	—
271	C₂²×Q₁₆	C₂²×C₈₉	1	4	yes	—
272	C₂²×SD₁₆	C₂²×C₈₉	1	4	yes	—
273	C₂×C₄⋊Q₈	—	2	1	no	—
274	C₂×C₄⋊Q₈	—	2	1	no	—
275	C₂×C₄⋊Q₈	—	2	4	no	—
276	C₂×C₄⋊Q₈	C₂²×C₈₉	1	2	yes	—
277	C₂×C₄⋊Q₈	C₂²×C₈₉	1	2	yes	—
278	C₂×C₄⋊Q₈	C₂²×C₈₉	1	8	yes	complemented
279	#529	—	2	4	no	—
280	#529	C₂²×C₈₉	1	8	yes	—
281	C₂×#126	—	2	1	no	—
282	C₂×#125	C₂²×C₈₉	1	1	yes	characteristic
283	C₂×#124	C₂²×C₈₉	1	1	yes	characteristic
284	C₂×#123	—	2	1	no	—
285	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
286	C₂×#121	C₂²×C₈₉	1	1	yes	characteristic
287	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
288	C₂×C₄×D₄	—	2	1	no	—
289	C₂×C₄×D₄	C₂²×C₈₉	1	1	yes	characteristic
290	C₂×#117	—	2	1	no	—
291	C₂²×C₄⋊C₄	—	2	1	no	—
292	C₂²×C₄⋊C₄	—	2	1	no	—
293	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
294	#497	C₂²×C₈₉	1	16	yes	complemented
295	#491	C₂²×C₈₉	1	16	yes	complemented
296	#462	C₂²×C₈₉	1	16	yes	complemented
297	#461	C₂²×C₈₉	1	16	yes	complemented
298	#451	C₂²×C₈₉	1	16	yes	complemented
299	#448	C₂²×C₈₉	1	16	yes	—
300	#432	C₂²×C₈₉	1	8	yes	complemented
301	C₂×C₄⋊C₈	C₂²×C₈₉	1	2	yes	—
302	C₂×#103	C₂²×C₈₉	1	4	yes	—
303	C₂×#103	C₂²×C₈₉	1	4	yes	—
304	C₂×C₂²⋊C₈	C₂²×C₈₉	1	2	yes	—
305	C₂×#97	C₂²×C₈₉	1	2	yes	—
306	C₂²×Q₈	—	2	2	no	—
307	C₂²×Q₈	—	2	4	no	—
308	C₂²×Q₈	—	4	8	no	—
309	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
310	C₂²×Q₈	C₂³×C₈₉	1	4	yes	—
311	C₂²×Q₈	C₂³×C₈₉	1	8	yes	—
312	C₂²×D₄	—	2	1	no	—
313	C₂²×D₄	C₂³×C₈₉	1	1	yes	characteristic
314	C₂³×C₄	—	2	2	no	—
315	C₂³×C₄	—	2	1	no	—
316	C₂³×C₄	D₄×C₈₉	1	1	yes	characteristic
317	C₂³×C₄	D₄×C₈₉	1	1	yes	characteristic
318	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
319	C₂×Q₁₆	—	4	8	no	—
320	C₂×Q₁₆	C₂³×C₈₉	1	16	yes	—
321	C₂×SD₁₆	—	4	8	no	—
322	C₂×SD₁₆	C₂³×C₈₉	1	16	yes	—
323	C₂²×C₈	C₂³×C₈₉	1	4	yes	—
324	C₄⋊Q₈	—	2	4	no	—
325	C₄⋊Q₈	—	2	4	no	—
326	C₄⋊Q₈	—	2	8	no	—
327	C₄⋊Q₈	—	2	8	no	—
328	C₄⋊Q₈	C₂³×C₈₉	1	8	yes	—
329	C₄⋊Q₈	C₂³×C₈₉	1	8	yes	—
330	#126	—	2	8	no	—
331	#125	—	2	2	no	—
332	#125	C₂³×C₈₉	1	4	yes	—
333	#124	—	2	2	no	—
334	#124	C₂³×C₈₉	1	4	yes	—
335	#123	—	2	8	no	—
336	C₂²⋊Q₈	—	2	2	no	—
337	C₂²⋊Q₈	C₂³×C₈₉	1	4	yes	—
338	#121	—	2	2	no	—
339	#121	C₂³×C₈₉	1	4	yes	—
340	C₄×Q₈	—	2	2	no	—
341	C₄×Q₈	C₂³×C₈₉	1	4	yes	—
342	C₄×D₄	—	2	8	no	—
343	C₄×D₄	—	2	2	no	—
344	C₄×D₄	C₂³×C₈₉	1	4	yes	—
345	#117	—	2	8	no	—
346	C₂×C₄⋊C₄	—	2	2	no	—
347	C₂×C₄⋊C₄	—	2	2	no	—
348	C₂×C₄⋊C₄	—	2	2	no	—
349	C₂×C₄⋊C₄	—	2	2	no	—
350	C₂×C₄⋊C₄	—	2	8	no	—
351	C₂×C₄⋊C₄	—	2	8	no	—
352	C₂×C₄⋊C₄	—	2	1	no	—
353	C₂×C₄⋊C₄	—	2	1	no	—
354	C₂×C₄⋊C₄	—	2	1	no	—
355	C₂×C₄⋊C₄	—	2	1	no	—
356	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
357	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
358	C₂×C₂²⋊C₄	—	2	1	no	—
359	C₂×C₂²⋊C₄	—	2	1	no	—
360	C₂×C₂²⋊C₄	—	2	1	no	—
361	C₂×C₂²⋊C₄	—	2	1	no	—
362	C₂×C₂²⋊C₄	—	2	1	no	—
363	C₂×C₄²	—	2	1	no	—
364	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
365	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
366	C₂×C₄²	D₄×C₈₉	1	4	yes	—
367	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
368	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
369	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
370	C₄⋊C₈	C₂³×C₈₉	1	8	yes	—
371	#103	C₂³×C₈₉	1	16	yes	—
372	#103	C₂³×C₈₉	1	16	yes	—
373	C₂²⋊C₈	C₂³×C₈₉	1	8	yes	—
374	#97	C₂³×C₈₉	1	8	yes	—
375	C₂⁴	—	2	1	no	—
376	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
377	C₂×Q₈	—	2	1	no	—
378	C₂×Q₈	—	2	8	no	—
379	C₂×Q₈	—	2	4	no	—
380	C₂×Q₈	—	2	8	no	—
381	C₂×Q₈	—	2	4	no	—
382	C₂×Q₈	—	2	8	no	—
383	C₂×Q₈	—	4	16	no	—
384	C₂×Q₈	—	4	8	no	—
385	C₂×Q₈	—	4	16	no	—
386	C₂×Q₈	C₂⁴×C₈₉	1	8	yes	—
387	C₂×Q₈	C₂⁴×C₈₉	1	2	yes	—
388	C₂×D₄	—	2	4	no	—
389	C₂×D₄	—	2	1	no	—
390	C₂×D₄	—	2	4	no	—
391	C₂×D₄	—	4	4	no	—
392	C₂×D₄	C₂⁴×C₈₉	1	2	yes	—
393	C₂²×C₄	—	2	4	no	—
394	C₂²×C₄	—	2	4	no	—
395	C₂²×C₄	—	2	4	no	—
396	C₂²×C₄	—	2	1	no	—
397	C₂²×C₄	—	2	1	no	—
398	C₂²×C₄	—	2	1	no	—
399	C₂²×C₄	—	2	8	no	—
400	C₂²×C₄	—	2	4	no	—
401	C₂²×C₄	—	2	4	no	—
402	C₂²×C₄	—	4	8	no	—
403	C₂²×C₄	—	4	4	no	—
404	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
405	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
406	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
407	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
408	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
409	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
410	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
411	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
412	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
413	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
414	Q₁₆	—	4	16	no	—
415	SD₁₆	—	4	16	no	—
416	C₂×C₈	—	2	8	no	—
417	C₂×C₈	C₂⁴×C₈₉	1	8	yes	—
418	C₄⋊C₄	—	2	8	no	—
419	C₄⋊C₄	—	2	8	no	—
420	C₄⋊C₄	—	2	8	no	—
421	C₄⋊C₄	—	2	8	no	—
422	C₄⋊C₄	—	2	4	no	—
423	C₄⋊C₄	—	2	4	no	—
424	C₄⋊C₄	—	2	4	no	—
425	C₄⋊C₄	—	2	4	no	—
426	C₄⋊C₄	—	2	1	no	—
427	C₄⋊C₄	—	2	1	no	—
428	C₄⋊C₄	C₂⁴×C₈₉	1	2	yes	—
429	C₄⋊C₄	C₂⁴×C₈₉	1	2	yes	—
430	C₂²⋊C₄	—	2	4	no	—
431	C₂²⋊C₄	—	2	4	no	—
432	C₂²⋊C₄	—	2	4	no	—
433	C₂²⋊C₄	—	2	4	no	—
434	C₂²⋊C₄	—	2	4	no	—
435	C₄²	—	2	4	no	—
436	C₄²	C₂×D₄×C₈₉	1	4	yes	—
437	C₄²	C₂×D₄×C₈₉	1	2	yes	—
438	C₄²	C₂×D₄×C₈₉	1	2	yes	—
439	C₄²	C₂×D₄×C₈₉	1	4	yes	—
440	C₄²	C₂⁴×C₈₉	1	2	yes	—
441	C₄²	C₂⁴×C₈₉	1	2	yes	—
442	C₂³	—	2	2	no	—
443	C₂³	—	2	2	no	—
444	C₂³	—	2	2	no	—
445	C₂³	—	4	2	no	—
446	C₂³	—	4	2	no	—
447	C₂³	—	4	2	no	—
448	C₂³	#137×C₈₉	1	4	yes	—
449	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
450	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
451	Q₈	—	2	8	no	—
452	Q₈	—	2	4	no	—
453	Q₈	—	2	8	no	—
454	Q₈	—	4	16	no	—
455	Q₈	—	4	2	no	—
456	Q₈	—	4	8	no	—
457	D₄	—	2	4	no	—
458	D₄	—	4	8	no	—
459	D₄	—	4	2	no	—
460	C₂×C₄	—	2	2	no	—
461	C₂×C₄	—	2	8	no	—
462	C₂×C₄	—	2	2	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	2	2	no	—
465	C₂×C₄	—	2	4	no	—
466	C₂×C₄	—	2	8	no	—
467	C₂×C₄	—	2	4	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	2	8	no	—
472	C₂×C₄	—	4	16	no	—
473	C₂×C₄	—	4	8	no	—
474	C₂×C₄	—	4	2	no	—
475	C₂×C₄	—	4	2	no	—
476	C₂×C₄	—	4	2	no	—
477	C₂×C₄	—	4	8	no	—
478	C₂×C₄	#137×C₈₉	1	4	yes	—
479	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
480	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
481	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
482	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
483	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
484	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
485	C₂×C₄	H(5,𝔽₂)×C₈₉	1	4	yes	—
486	C₂×C₄	C₂⁵×C₈₉	1	1	yes	characteristic
487	C₈	—	2	8	no	—
488	C₂²	—	2	2	no	—
489	C₂²	—	2	2	no	—
490	C₂²	—	2	8	no	—
491	C₂²	—	4	2	no	—
492	C₂²	—	4	2	no	—
493	C₂²	—	4	2	no	—
494	C₂²	—	8	4	no	—
495	C₂²	#535×C₈₉	1	2	yes	—
496	C₂²	C₂×#137×C₈₉	1	4	yes	—
497	C₂²	C₂×#137×C₈₉	1	2	yes	—
498	C₂²	C₂×#137×C₈₉	1	2	yes	—
499	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
500	C₄	—	2	2	no	—
501	C₄	—	2	2	no	—
502	C₄	—	2	2	no	—
503	C₄	—	2	2	no	—
504	C₄	—	4	2	no	—
505	C₄	—	4	2	no	—
506	C₄	—	4	8	no	—
507	C₄	—	4	2	no	—
508	C₄	#535×C₈₉	1	2	yes	—
509	C₄	C₂×#137×C₈₉	1	4	yes	—
510	C₄	C₂×H(5,𝔽₂)×C₈₉	1	2	yes	—
511	C₂	—	2	4	no	—
512	C₂	—	8	2	no	—
513	C₂	C₈₉×#3218	1	4	yes	direct product factor
514	C₂	C₂×#535×C₈₉	1	1	yes	characteristic
515	C₂	C₂²×#137×C₈₉	1	1	yes	characteristic
516	C₂	C₂²×#137×C₈₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₈₉
prime radical	C₂×#121×C₈₉	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ C₂²×C₄²×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₄⋊Q₈×C₈₉	C₂	semidirect
3. C₈₉×#3218	C₂	direct (N × H)
4. C₂×#497×C₈₉	C₂	semidirect
5. C₂×#491×C₈₉	C₂	semidirect
6. C₂×#462×C₈₉	C₂	semidirect
7. C₂×#461×C₈₉	C₂	semidirect
8. C₂×#451×C₈₉	C₂	semidirect
10. C₂×#432×C₈₉	C₂	semidirect
19. C₂×C₄⋊Q₈×C₈₉	C₂²	semidirect
35. #497×C₈₉	C₂²	semidirect
36. #491×C₈₉	C₂²	semidirect
37. #462×C₈₉	C₂²	semidirect
38. #461×C₈₉	C₂²	semidirect
39. #451×C₈₉	C₂²	semidirect
41. #432×C₈₉	C₂²	semidirect
252. C₂×#3218	C₈₉	direct (N × H)
255. C₂×C₈₉	#3218	direct (N × H)
259. C₂²×C₄⋊Q₈	C₂×C₈₉	semidirect
261. #3218	C₂×C₈₉	direct (N × H)
262. C₂×#497	C₂×C₈₉	semidirect
263. C₂×#491	C₂×C₈₉	semidirect
264. C₂×#462	C₂×C₈₉	semidirect
265. C₂×#461	C₂×C₈₉	semidirect
266. C₂×#451	C₂×C₈₉	semidirect
268. C₂×#432	C₂×C₈₉	semidirect
269. C₈₉	C₂×#3218	direct (N × H)
278. C₂×C₄⋊Q₈	C₂²×C₈₉	semidirect
294. #497	C₂²×C₈₉	semidirect
295. #491	C₂²×C₈₉	semidirect
296. #462	C₂²×C₈₉	semidirect
297. #461	C₂²×C₈₉	semidirect
298. #451	C₂²×C₈₉	semidirect
300. #432	C₂²×C₈₉	semidirect
513. C₂	C₈₉×#3218	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×C₄⋊Q₈×C₈₉	2
C₂×#529×C₈₉	2
C₈₉×#3218	2
C₂×#497×C₈₉	2
C₂×#491×C₈₉	2
C₂×#462×C₈₉	2
C₂×#461×C₈₉	2
C₂×#451×C₈₉	2
C₂×#448×C₈₉	2
C₂×#432×C₈₉	2
C₂×#3218	89

Subgroups — C2×C89×#3218 #36438634

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈₉×#3218 #36438634