Subgroups — C₂×C₈₉×#3123 #36438400

Order 22784.

2210 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#541×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#518×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#3123	C₂	1	16	yes	maximal, direct product factor
4	C₂×#485×C₈₉	C₂	1	2	yes	maximal, complemented
5	C₂×#478×C₈₉	C₂	1	2	yes	maximal, complemented
6	C₂×#461×C₈₉	C₂	1	2	yes	maximal, complemented
7	C₂×#450×C₈₉	C₂	1	2	yes	maximal
8	C₂×#440×C₈₉	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×SD₁₆×C₈₉	C₂	1	2	yes	maximal, complemented
10	C₂×#433×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂³×D₄×C₈₉	C₂²	1	1	yes	characteristic
12	C₂²×SD₁₆×C₈₉	C₂²	1	2	yes	—
13	C₂²×SD₁₆×C₈₉	C₂²	1	2	yes	—
14	#541×C₈₉	—	2	4	no	—
15	#541×C₈₉	C₂²	1	8	yes	complemented
16	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
17	C₂×#126×C₈₉	—	2	2	no	—
18	C₂×#125×C₈₉	—	2	2	no	—
19	C₂×#125×C₈₉	C₂²	1	1	yes	characteristic, complemented
20	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	2	yes	—
21	#518×C₈₉	—	2	4	no	—
22	#518×C₈₉	C₂²	1	8	yes	—
23	C₂×C₄×Q₈×C₈₉	C₂²	1	2	yes	complemented
24	C₂×C₄×D₄×C₈₉	—	2	2	no	—
25	C₂×C₄×D₄×C₈₉	C₂²	1	2	yes	—
26	C₂×C₄×D₄×C₈₉	C₂²	1	2	yes	—
27	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
29	C₂²×C₂²⋊C₄×C₈₉	—	2	2	no	—
30	#485×C₈₉	C₂²	1	16	yes	complemented
31	#478×C₈₉	C₂²	1	16	yes	complemented
32	#461×C₈₉	C₂²	1	16	yes	complemented
33	#450×C₈₉	C₂²	1	16	yes	—
34	#440×C₈₉	C₂²	1	16	yes	complemented
35	C₄×SD₁₆×C₈₉	C₂²	1	16	yes	complemented
36	#433×C₈₉	C₂²	1	8	yes	complemented
37	C₂×#107×C₈₉	C₂²	1	2	yes	complemented
38	C₂×#106×C₈₉	C₂²	1	2	yes	complemented
39	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	complemented
40	C₂×#103×C₈₉	C₂²	1	2	yes	complemented
41	C₂×#103×C₈₉	C₂²	1	2	yes	complemented
42	C₂×#102×C₈₉	C₂²	1	2	yes	—
43	C₂×#102×C₈₉	C₂²	1	2	yes	—
44	C₂×C₂²⋊C₈×C₈₉	C₂²	1	2	yes	—
45	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic, complemented
46	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
47	C₂⁵×C₈₉	—	2	1	no	—
48	C₂²×Q₈×C₈₉	C₂³	1	2	yes	—
49	C₂²×D₄×C₈₉	—	2	8	no	—
50	C₂²×D₄×C₈₉	—	2	1	no	—
51	C₂²×D₄×C₈₉	—	2	2	no	—
52	C₂²×D₄×C₈₉	C₂³	1	4	yes	—
53	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
54	C₂³×C₄×C₈₉	—	2	2	no	—
55	C₂³×C₄×C₈₉	—	2	2	no	—
56	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
57	C₂×SD₁₆×C₈₉	—	2	8	no	—
58	C₂×SD₁₆×C₈₉	—	2	8	no	—
59	C₂×SD₁₆×C₈₉	C₂³	1	8	yes	—
60	C₂×SD₁₆×C₈₉	C₂³	1	8	yes	—
61	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
62	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
63	C₄⋊Q₈×C₈₉	—	2	2	no	—
64	C₄⋊Q₈×C₈₉	C₂³	1	4	yes	complemented
65	#126×C₈₉	—	2	16	no	—
66	#125×C₈₉	—	2	16	no	—
67	#125×C₈₉	—	2	2	no	—
68	#125×C₈₉	C₂³	1	4	yes	complemented
69	C₂²⋊Q₈×C₈₉	—	2	4	no	—
70	C₂²⋊Q₈×C₈₉	C₂³	1	8	yes	—
71	C₄×Q₈×C₈₉	—	2	4	no	—
72	C₄×Q₈×C₈₉	C₂³	1	8	yes	complemented
73	C₄×D₄×C₈₉	—	2	16	no	—
74	C₄×D₄×C₈₉	—	2	4	no	—
75	C₄×D₄×C₈₉	—	2	4	no	—
76	C₄×D₄×C₈₉	C₂³	1	8	yes	—
77	C₄×D₄×C₈₉	C₂³	1	8	yes	—
78	#117×C₈₉	—	2	2	no	—
79	#117×C₈₉	D₄	1	4	yes	—
80	#117×C₈₉	C₂³	1	4	yes	—
81	#117×C₈₉	C₂³	1	4	yes	—
82	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
83	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
84	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
85	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
86	C₂×C₄⋊C₄×C₈₉	D₄	1	2	yes	—
87	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
88	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
89	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
90	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
91	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
92	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
93	C₂×C₂²⋊C₄×C₈₉	—	2	16	no	—
94	C₂×C₂²⋊C₄×C₈₉	D₄	1	2	yes	—
95	C₂×C₄²×C₈₉	—	2	1	no	—
96	C₂×C₄²×C₈₉	—	2	2	no	—
97	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
98	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
99	#107×C₈₉	C₂³	1	8	yes	complemented
100	#106×C₈₉	C₂³	1	8	yes	complemented
101	C₄⋊C₈×C₈₉	C₂³	1	8	yes	complemented
102	#103×C₈₉	C₂³	1	8	yes	complemented
103	#103×C₈₉	C₂³	1	8	yes	complemented
104	#102×C₈₉	C₂³	1	8	yes	—
105	#102×C₈₉	C₂³	1	8	yes	—
106	C₂²⋊C₈×C₈₉	C₂³	1	8	yes	—
107	#97×C₈₉	C₂³	1	4	yes	complemented
108	C₄×C₈×C₈₉	C₂³	1	4	yes	complemented
109	C₂⁴×C₈₉	—	2	8	no	—
110	C₂⁴×C₈₉	—	2	2	no	—
111	C₂⁴×C₈₉	—	2	4	no	—
112	C₂⁴×C₈₉	—	8	4	no	—
113	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
114	C₂×Q₈×C₈₉	—	2	2	no	—
115	C₂×Q₈×C₈₉	—	2	4	no	—
116	C₂×Q₈×C₈₉	—	2	4	no	—
117	C₂×Q₈×C₈₉	C₂⁴	1	4	yes	—
118	C₂×D₄×C₈₉	—	2	8	no	—
119	C₂×D₄×C₈₉	—	2	16	no	—
120	C₂×D₄×C₈₉	—	2	16	no	—
121	C₂×D₄×C₈₉	—	2	4	no	—
122	C₂×D₄×C₈₉	—	2	2	no	—
123	C₂×D₄×C₈₉	—	2	4	no	—
124	C₂×D₄×C₈₉	#41	1	8	yes	—
125	C₂×D₄×C₈₉	C₂⁴	1	4	yes	—
126	C₂²×C₄×C₈₉	—	2	4	no	—
127	C₂²×C₄×C₈₉	—	2	2	no	—
128	C₂²×C₄×C₈₉	—	2	2	no	—
129	C₂²×C₄×C₈₉	—	2	2	no	—
130	C₂²×C₄×C₈₉	—	2	8	no	—
131	C₂²×C₄×C₈₉	—	2	8	no	—
132	C₂²×C₄×C₈₉	—	8	4	no	—
133	C₂²×C₄×C₈₉	—	8	4	no	—
134	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
135	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
136	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
137	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
138	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
139	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
140	SD₁₆×C₈₉	—	2	16	no	—
141	SD₁₆×C₈₉	—	2	16	no	—
142	C₂×C₈×C₈₉	—	2	4	no	—
143	C₂×C₈×C₈₉	—	2	4	no	—
144	C₂×C₈×C₈₉	C₂⁴	1	4	yes	—
145	C₂×C₈×C₈₉	C₂⁴	1	4	yes	—
146	C₄⋊C₄×C₈₉	—	2	8	no	—
147	C₄⋊C₄×C₈₉	—	2	8	no	—
148	C₄⋊C₄×C₈₉	—	2	8	no	—
149	C₄⋊C₄×C₈₉	—	2	8	no	—
150	C₄⋊C₄×C₈₉	—	2	2	no	—
151	C₄⋊C₄×C₈₉	—	2	2	no	—
152	C₄⋊C₄×C₈₉	C₂×D₄	1	8	yes	—
153	C₄⋊C₄×C₈₉	C₂⁴	1	4	yes	—
154	C₄⋊C₄×C₈₉	C₂⁴	1	4	yes	—
155	C₂²⋊C₄×C₈₉	—	2	4	no	—
156	C₂²⋊C₄×C₈₉	—	2	8	no	—
157	C₂²⋊C₄×C₈₉	—	2	8	no	—
158	C₂²⋊C₄×C₈₉	—	2	8	no	—
159	C₂²⋊C₄×C₈₉	C₂×D₄	1	8	yes	—
160	C₄²×C₈₉	—	2	4	no	—
161	C₄²×C₈₉	—	2	8	no	—
162	C₄²×C₈₉	C₂×D₄	1	2	yes	—
163	C₄²×C₈₉	C₂×D₄	1	2	yes	—
164	C₄²×C₈₉	C₂⁴	1	2	yes	—
165	C₄²×C₈₉	C₂⁴	1	2	yes	—
166	C₂³×C₈₉	—	2	16	no	—
167	C₂³×C₈₉	—	2	2	no	—
168	C₂³×C₈₉	—	2	8	no	—
169	C₂³×C₈₉	—	2	4	no	—
170	C₂³×C₈₉	—	2	4	no	—
171	C₂³×C₈₉	—	4	4	no	—
172	C₂³×C₈₉	—	4	4	no	—
173	C₂³×C₈₉	—	8	8	no	—
174	C₂³×C₈₉	—	8	16	no	—
175	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
176	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
177	Q₈×C₈₉	—	2	4	no	—
178	Q₈×C₈₉	—	2	4	no	—
179	Q₈×C₈₉	—	4	4	no	—
180	D₄×C₈₉	—	2	16	no	—
181	D₄×C₈₉	—	2	8	no	—
182	D₄×C₈₉	—	2	4	no	—
183	D₄×C₈₉	C₂×#41	1	8	yes	—
184	C₂×C₄×C₈₉	—	2	2	no	—
185	C₂×C₄×C₈₉	—	2	4	no	—
186	C₂×C₄×C₈₉	—	2	4	no	—
187	C₂×C₄×C₈₉	—	2	4	no	—
188	C₂×C₄×C₈₉	—	2	4	no	—
189	C₂×C₄×C₈₉	—	2	4	no	—
190	C₂×C₄×C₈₉	—	2	4	no	—
191	C₂×C₄×C₈₉	—	2	4	no	—
192	C₂×C₄×C₈₉	—	4	4	no	—
193	C₂×C₄×C₈₉	—	4	4	no	—
194	C₂×C₄×C₈₉	—	4	4	no	—
195	C₂×C₄×C₈₉	—	8	8	no	—
196	C₂×C₄×C₈₉	—	8	8	no	—
197	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
198	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
199	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
200	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
201	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
202	C₂×C₄×C₈₉	C₂⁵	1	1	yes	characteristic
203	C₈×C₈₉	—	2	4	no	—
204	C₈×C₈₉	—	2	4	no	—
205	C₂²×C₈₉	—	2	2	no	—
206	C₂²×C₈₉	—	2	8	no	—
207	C₂²×C₈₉	—	2	4	no	—
208	C₂²×C₈₉	—	4	4	no	—
209	C₂²×C₈₉	—	4	16	no	—
210	C₂²×C₈₉	—	4	4	no	—
211	C₂²×C₈₉	—	4	4	no	—
212	C₂²×C₈₉	—	8	16	no	—
213	C₂²×C₈₉	#525	1	2	yes	—
214	C₂²×C₈₉	#577	1	4	yes	—
215	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
216	C₄×C₈₉	—	2	2	no	—
217	C₄×C₈₉	—	2	4	no	—
218	C₄×C₈₉	—	2	4	no	—
219	C₄×C₈₉	—	4	4	no	—
220	C₄×C₈₉	—	4	4	no	—
221	C₄×C₈₉	—	4	4	no	—
222	C₄×C₈₉	#525	1	2	yes	—
223	C₄×C₈₉	C₂²×#41	1	2	yes	—
224	C₂×#3123	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
225	C₂×C₈₉	—	4	2	no	—
226	C₂×C₈₉	—	4	8	no	—
227	C₂×C₈₉	#3123	1	4	yes	direct product factor
228	C₂×C₈₉	C₂×#525	1	1	yes	characteristic
229	C₂×C₈₉	C₂×#577	1	2	yes	—
230	C₂×#541	C₂×C₈₉	1	1	yes	characteristic, complemented
231	C₂×#518	C₂×C₈₉	1	1	yes	characteristic
232	#3123	C₂×C₈₉	1	16	yes	direct product factor
233	C₂×#485	C₂×C₈₉	1	2	yes	complemented
234	C₂×#478	C₂×C₈₉	1	2	yes	complemented
235	C₂×#461	C₂×C₈₉	1	2	yes	complemented
236	C₂×#450	C₂×C₈₉	1	2	yes	—
237	C₂×#440	C₂×C₈₉	1	2	yes	complemented
238	C₂×C₄×SD₁₆	C₂×C₈₉	1	2	yes	complemented
239	C₂×#433	C₂×C₈₉	1	1	yes	characteristic, complemented
240	C₈₉	C₂×#3123	1	1	yes	characteristic, direct product factor, Sylow
241	C₂³×D₄	C₂²×C₈₉	1	1	yes	characteristic
242	C₂²×SD₁₆	C₂²×C₈₉	1	2	yes	—
243	C₂²×SD₁₆	C₂²×C₈₉	1	2	yes	—
244	#541	—	2	4	no	—
245	#541	C₂²×C₈₉	1	8	yes	complemented
246	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic, complemented
247	C₂×#126	—	2	2	no	—
248	C₂×#125	—	2	2	no	—
249	C₂×#125	C₂²×C₈₉	1	1	yes	characteristic, complemented
250	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	2	yes	—
251	#518	—	2	4	no	—
252	#518	C₂²×C₈₉	1	8	yes	—
253	C₂×C₄×Q₈	C₂²×C₈₉	1	2	yes	complemented
254	C₂×C₄×D₄	—	2	2	no	—
255	C₂×C₄×D₄	C₂²×C₈₉	1	2	yes	—
256	C₂×C₄×D₄	C₂²×C₈₉	1	2	yes	—
257	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
258	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
259	C₂²×C₂²⋊C₄	—	2	2	no	—
260	#485	C₂²×C₈₉	1	16	yes	complemented
261	#478	C₂²×C₈₉	1	16	yes	complemented
262	#461	C₂²×C₈₉	1	16	yes	complemented
263	#450	C₂²×C₈₉	1	16	yes	—
264	#440	C₂²×C₈₉	1	16	yes	complemented
265	C₄×SD₁₆	C₂²×C₈₉	1	16	yes	complemented
266	#433	C₂²×C₈₉	1	8	yes	complemented
267	C₂×#107	C₂²×C₈₉	1	2	yes	complemented
268	C₂×#106	C₂²×C₈₉	1	2	yes	complemented
269	C₂×C₄⋊C₈	C₂²×C₈₉	1	2	yes	complemented
270	C₂×#103	C₂²×C₈₉	1	2	yes	complemented
271	C₂×#103	C₂²×C₈₉	1	2	yes	complemented
272	C₂×#102	C₂²×C₈₉	1	2	yes	—
273	C₂×#102	C₂²×C₈₉	1	2	yes	—
274	C₂×C₂²⋊C₈	C₂²×C₈₉	1	2	yes	—
275	C₂×#97	C₂²×C₈₉	1	1	yes	characteristic, complemented
276	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic, complemented
277	C₂⁵	—	2	1	no	—
278	C₂²×Q₈	C₂³×C₈₉	1	2	yes	—
279	C₂²×D₄	—	2	8	no	—
280	C₂²×D₄	—	2	1	no	—
281	C₂²×D₄	—	2	2	no	—
282	C₂²×D₄	C₂³×C₈₉	1	4	yes	—
283	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
284	C₂³×C₄	—	2	2	no	—
285	C₂³×C₄	—	2	2	no	—
286	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
287	C₂×SD₁₆	—	2	8	no	—
288	C₂×SD₁₆	—	2	8	no	—
289	C₂×SD₁₆	C₂³×C₈₉	1	8	yes	—
290	C₂×SD₁₆	C₂³×C₈₉	1	8	yes	—
291	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
292	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
293	C₄⋊Q₈	—	2	2	no	—
294	C₄⋊Q₈	C₂³×C₈₉	1	4	yes	complemented
295	#126	—	2	16	no	—
296	#125	—	2	16	no	—
297	#125	—	2	2	no	—
298	#125	C₂³×C₈₉	1	4	yes	complemented
299	C₂²⋊Q₈	—	2	4	no	—
300	C₂²⋊Q₈	C₂³×C₈₉	1	8	yes	—
301	C₄×Q₈	—	2	4	no	—
302	C₄×Q₈	C₂³×C₈₉	1	8	yes	complemented
303	C₄×D₄	—	2	16	no	—
304	C₄×D₄	—	2	4	no	—
305	C₄×D₄	—	2	4	no	—
306	C₄×D₄	C₂³×C₈₉	1	8	yes	—
307	C₄×D₄	C₂³×C₈₉	1	8	yes	—
308	#117	—	2	2	no	—
309	#117	D₄×C₈₉	1	4	yes	—
310	#117	C₂³×C₈₉	1	4	yes	—
311	#117	C₂³×C₈₉	1	4	yes	—
312	C₂×C₄⋊C₄	—	2	2	no	—
313	C₂×C₄⋊C₄	—	2	2	no	—
314	C₂×C₄⋊C₄	—	2	2	no	—
315	C₂×C₄⋊C₄	—	2	2	no	—
316	C₂×C₄⋊C₄	D₄×C₈₉	1	2	yes	—
317	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
318	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
319	C₂×C₂²⋊C₄	—	2	1	no	—
320	C₂×C₂²⋊C₄	—	2	2	no	—
321	C₂×C₂²⋊C₄	—	2	2	no	—
322	C₂×C₂²⋊C₄	—	2	2	no	—
323	C₂×C₂²⋊C₄	—	2	16	no	—
324	C₂×C₂²⋊C₄	D₄×C₈₉	1	2	yes	—
325	C₂×C₄²	—	2	1	no	—
326	C₂×C₄²	—	2	2	no	—
327	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
328	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
329	#107	C₂³×C₈₉	1	8	yes	complemented
330	#106	C₂³×C₈₉	1	8	yes	complemented
331	C₄⋊C₈	C₂³×C₈₉	1	8	yes	complemented
332	#103	C₂³×C₈₉	1	8	yes	complemented
333	#103	C₂³×C₈₉	1	8	yes	complemented
334	#102	C₂³×C₈₉	1	8	yes	—
335	#102	C₂³×C₈₉	1	8	yes	—
336	C₂²⋊C₈	C₂³×C₈₉	1	8	yes	—
337	#97	C₂³×C₈₉	1	4	yes	complemented
338	C₄×C₈	C₂³×C₈₉	1	4	yes	complemented
339	C₂⁴	—	2	8	no	—
340	C₂⁴	—	2	2	no	—
341	C₂⁴	—	2	4	no	—
342	C₂⁴	—	8	4	no	—
343	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
344	C₂×Q₈	—	2	2	no	—
345	C₂×Q₈	—	2	4	no	—
346	C₂×Q₈	—	2	4	no	—
347	C₂×Q₈	C₂⁴×C₈₉	1	4	yes	—
348	C₂×D₄	—	2	8	no	—
349	C₂×D₄	—	2	16	no	—
350	C₂×D₄	—	2	16	no	—
351	C₂×D₄	—	2	4	no	—
352	C₂×D₄	—	2	2	no	—
353	C₂×D₄	—	2	4	no	—
354	C₂×D₄	#41×C₈₉	1	8	yes	—
355	C₂×D₄	C₂⁴×C₈₉	1	4	yes	—
356	C₂²×C₄	—	2	4	no	—
357	C₂²×C₄	—	2	2	no	—
358	C₂²×C₄	—	2	2	no	—
359	C₂²×C₄	—	2	2	no	—
360	C₂²×C₄	—	2	8	no	—
361	C₂²×C₄	—	2	8	no	—
362	C₂²×C₄	—	8	4	no	—
363	C₂²×C₄	—	8	4	no	—
364	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
365	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
366	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
367	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
368	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
369	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
370	SD₁₆	—	2	16	no	—
371	SD₁₆	—	2	16	no	—
372	C₂×C₈	—	2	4	no	—
373	C₂×C₈	—	2	4	no	—
374	C₂×C₈	C₂⁴×C₈₉	1	4	yes	—
375	C₂×C₈	C₂⁴×C₈₉	1	4	yes	—
376	C₄⋊C₄	—	2	8	no	—
377	C₄⋊C₄	—	2	8	no	—
378	C₄⋊C₄	—	2	8	no	—
379	C₄⋊C₄	—	2	8	no	—
380	C₄⋊C₄	—	2	2	no	—
381	C₄⋊C₄	—	2	2	no	—
382	C₄⋊C₄	C₂×D₄×C₈₉	1	8	yes	—
383	C₄⋊C₄	C₂⁴×C₈₉	1	4	yes	—
384	C₄⋊C₄	C₂⁴×C₈₉	1	4	yes	—
385	C₂²⋊C₄	—	2	4	no	—
386	C₂²⋊C₄	—	2	8	no	—
387	C₂²⋊C₄	—	2	8	no	—
388	C₂²⋊C₄	—	2	8	no	—
389	C₂²⋊C₄	C₂×D₄×C₈₉	1	8	yes	—
390	C₄²	—	2	4	no	—
391	C₄²	—	2	8	no	—
392	C₄²	C₂×D₄×C₈₉	1	2	yes	—
393	C₄²	C₂×D₄×C₈₉	1	2	yes	—
394	C₄²	C₂⁴×C₈₉	1	2	yes	—
395	C₄²	C₂⁴×C₈₉	1	2	yes	—
396	C₂³	—	2	16	no	—
397	C₂³	—	2	2	no	—
398	C₂³	—	2	8	no	—
399	C₂³	—	2	4	no	—
400	C₂³	—	2	4	no	—
401	C₂³	—	4	4	no	—
402	C₂³	—	4	4	no	—
403	C₂³	—	8	8	no	—
404	C₂³	—	8	16	no	—
405	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
406	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
407	Q₈	—	2	4	no	—
408	Q₈	—	2	4	no	—
409	Q₈	—	4	4	no	—
410	D₄	—	2	16	no	—
411	D₄	—	2	8	no	—
412	D₄	—	2	4	no	—
413	D₄	C₂×#41×C₈₉	1	8	yes	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	2	4	no	—
416	C₂×C₄	—	2	4	no	—
417	C₂×C₄	—	2	4	no	—
418	C₂×C₄	—	2	4	no	—
419	C₂×C₄	—	2	4	no	—
420	C₂×C₄	—	2	4	no	—
421	C₂×C₄	—	2	4	no	—
422	C₂×C₄	—	4	4	no	—
423	C₂×C₄	—	4	4	no	—
424	C₂×C₄	—	4	4	no	—
425	C₂×C₄	—	8	8	no	—
426	C₂×C₄	—	8	8	no	—
427	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
428	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
429	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
430	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
431	C₂×C₄	C₂×#41×C₈₉	1	4	yes	—
432	C₂×C₄	C₂⁵×C₈₉	1	1	yes	characteristic
433	C₈	—	2	4	no	—
434	C₈	—	2	4	no	—
435	C₂²	—	2	4	no	—
436	C₂²	—	2	2	no	—
437	C₂²	—	2	8	no	—
438	C₂²	—	4	4	no	—
439	C₂²	—	4	4	no	—
440	C₂²	—	4	4	no	—
441	C₂²	—	4	16	no	—
442	C₂²	—	8	16	no	—
443	C₂²	#525×C₈₉	1	2	yes	—
444	C₂²	#577×C₈₉	1	4	yes	—
445	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
446	C₄	—	2	4	no	—
447	C₄	—	2	4	no	—
448	C₄	—	2	2	no	—
449	C₄	—	4	4	no	—
450	C₄	—	4	4	no	—
451	C₄	—	4	4	no	—
452	C₄	#525×C₈₉	1	2	yes	—
453	C₄	C₂²×#41×C₈₉	1	2	yes	—
454	C₂	—	4	2	no	—
455	C₂	—	4	8	no	—
456	C₂	C₈₉×#3123	1	4	yes	direct product factor
457	C₂	C₂×#525×C₈₉	1	1	yes	characteristic
458	C₂	C₂×#577×C₈₉	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₈₉
prime radical	C₂³×D₄×C₈₉	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ C₂×#117×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#541×C₈₉	C₂	semidirect
3. C₈₉×#3123	C₂	direct (N × H)
4. C₂×#485×C₈₉	C₂	semidirect
5. C₂×#478×C₈₉	C₂	semidirect
6. C₂×#461×C₈₉	C₂	semidirect
8. C₂×#440×C₈₉	C₂	semidirect
9. C₂×C₄×SD₁₆×C₈₉	C₂	semidirect
10. C₂×#433×C₈₉	C₂	semidirect
15. #541×C₈₉	C₂²	semidirect
16. C₂×C₄⋊Q₈×C₈₉	C₂²	semidirect
19. C₂×#125×C₈₉	C₂²	semidirect
23. C₂×C₄×Q₈×C₈₉	C₂²	semidirect
30. #485×C₈₉	C₂²	semidirect
31. #478×C₈₉	C₂²	semidirect
32. #461×C₈₉	C₂²	semidirect
34. #440×C₈₉	C₂²	semidirect
35. C₄×SD₁₆×C₈₉	C₂²	semidirect
36. #433×C₈₉	C₂²	semidirect
37. C₂×#107×C₈₉	C₂²	semidirect
38. C₂×#106×C₈₉	C₂²	semidirect
39. C₂×C₄⋊C₈×C₈₉	C₂²	semidirect
40. C₂×#103×C₈₉	C₂²	semidirect
41. C₂×#103×C₈₉	C₂²	semidirect
45. C₂×#97×C₈₉	C₂²	semidirect
46. C₂×C₄×C₈×C₈₉	C₂²	semidirect
64. C₄⋊Q₈×C₈₉	C₂³	semidirect
68. #125×C₈₉	C₂³	semidirect
72. C₄×Q₈×C₈₉	C₂³	semidirect
99. #107×C₈₉	C₂³	semidirect
100. #106×C₈₉	C₂³	semidirect
101. C₄⋊C₈×C₈₉	C₂³	semidirect
102. #103×C₈₉	C₂³	semidirect
103. #103×C₈₉	C₂³	semidirect
107. #97×C₈₉	C₂³	semidirect
108. C₄×C₈×C₈₉	C₂³	semidirect
224. C₂×#3123	C₈₉	direct (N × H)
227. C₂×C₈₉	#3123	direct (N × H)
230. C₂×#541	C₂×C₈₉	semidirect
232. #3123	C₂×C₈₉	direct (N × H)
233. C₂×#485	C₂×C₈₉	semidirect
234. C₂×#478	C₂×C₈₉	semidirect
235. C₂×#461	C₂×C₈₉	semidirect
237. C₂×#440	C₂×C₈₉	semidirect
238. C₂×C₄×SD₁₆	C₂×C₈₉	semidirect
239. C₂×#433	C₂×C₈₉	semidirect
240. C₈₉	C₂×#3123	direct (N × H)
245. #541	C₂²×C₈₉	semidirect
246. C₂×C₄⋊Q₈	C₂²×C₈₉	semidirect
249. C₂×#125	C₂²×C₈₉	semidirect
253. C₂×C₄×Q₈	C₂²×C₈₉	semidirect
260. #485	C₂²×C₈₉	semidirect
261. #478	C₂²×C₈₉	semidirect
262. #461	C₂²×C₈₉	semidirect
264. #440	C₂²×C₈₉	semidirect
265. C₄×SD₁₆	C₂²×C₈₉	semidirect
266. #433	C₂²×C₈₉	semidirect
267. C₂×#107	C₂²×C₈₉	semidirect
268. C₂×#106	C₂²×C₈₉	semidirect
269. C₂×C₄⋊C₈	C₂²×C₈₉	semidirect
270. C₂×#103	C₂²×C₈₉	semidirect
271. C₂×#103	C₂²×C₈₉	semidirect
275. C₂×#97	C₂²×C₈₉	semidirect
276. C₂×C₄×C₈	C₂²×C₈₉	semidirect
294. C₄⋊Q₈	C₂³×C₈₉	semidirect
298. #125	C₂³×C₈₉	semidirect
302. C₄×Q₈	C₂³×C₈₉	semidirect
329. #107	C₂³×C₈₉	semidirect
330. #106	C₂³×C₈₉	semidirect
331. C₄⋊C₈	C₂³×C₈₉	semidirect
332. #103	C₂³×C₈₉	semidirect
333. #103	C₂³×C₈₉	semidirect
337. #97	C₂³×C₈₉	semidirect
338. C₄×C₈	C₂³×C₈₉	semidirect
456. C₂	C₈₉×#3123	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#541×C₈₉	2
C₂×#518×C₈₉	2
C₈₉×#3123	2
C₂×#485×C₈₉	2
C₂×#478×C₈₉	2
C₂×#461×C₈₉	2
C₂×#450×C₈₉	2
C₂×#440×C₈₉	2
C₂×C₄×SD₁₆×C₈₉	2
C₂×#433×C₈₉	2
C₂×#3123	89

Subgroups — C2×C89×#3123 #36438400

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈₉×#3123 #36438400