Subgroups — C₂×C₈₉×#3187 #36438577

Order 22784.

2698 conjugacy classes of subgroups.

The subgroup lattice has 576 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	1	yes	maximal, characteristic, complemented
2	C₂²×#121×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#3187	C₂	1	16	yes	maximal, direct product factor
4	C₂×#481×C₈₉	C₂	1	2	yes	maximal, complemented
5	C₂×#483×C₈₉	C₂	1	2	yes	maximal, complemented
6	C₂×#465×C₈₉	C₂	1	4	yes	maximal, complemented
7	C₂×#448×C₈₉	C₂	1	2	yes	maximal, complemented
8	C₂×#450×C₈₉	C₂	1	2	yes	maximal, complemented
9	C₂²×C₂²⋊C₈×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×Q₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
11	C₂³×D₄×C₈₉	—	2	1	no	—
12	C₂³×D₄×C₈₉	C₂²	1	1	yes	characteristic
13	C₂⁴×C₄×C₈₉	C₂²	1	1	yes	characteristic
14	C₂²×SD₁₆×C₈₉	C₂²	1	4	yes	complemented
15	C₂³×C₈×C₈₉	C₂²	1	2	yes	complemented
16	C₂×C₂²⋊Q₈×C₈₉	—	2	1	no	—
17	C₂×C₂²⋊Q₈×C₈₉	—	2	1	no	—
18	C₂×C₂²⋊Q₈×C₈₉	—	2	4	no	—
19	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	2	yes	—
20	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	2	yes	complemented
21	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	8	yes	complemented
22	C₂×#121×C₈₉	—	2	1	no	—
23	C₂×#121×C₈₉	—	2	4	no	—
24	C₂×#121×C₈₉	—	2	1	no	—
25	C₂×#121×C₈₉	C₂²	1	2	yes	—
26	C₂×#121×C₈₉	C₂²	1	8	yes	—
27	C₂×#121×C₈₉	C₂²	1	2	yes	—
28	C₂²×C₄⋊C₄×C₈₉	—	2	1	no	—
29	C₂²×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
30	C₂²×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic, complemented
31	C₂²×C₂²⋊C₄×C₈₉	—	2	1	no	—
32	C₂²×C₂²⋊C₄×C₈₉	—	2	1	no	—
33	#483×C₈₉	C₂²	1	16	yes	complemented
34	#481×C₈₉	C₂²	1	16	yes	complemented
35	#465×C₈₉	C₂²	1	32	yes	complemented
36	#450×C₈₉	C₂²	1	16	yes	complemented
37	#448×C₈₉	C₂²	1	16	yes	complemented
38	C₂×#106×C₈₉	C₂²	1	4	yes	complemented
39	C₂×#103×C₈₉	C₂²	1	4	yes	complemented
40	C₂×#102×C₈₉	C₂²	1	4	yes	complemented
41	C₂×C₂²⋊C₈×C₈₉	C₂²	1	2	yes	—
42	C₂×C₂²⋊C₈×C₈₉	C₂²	1	8	yes	complemented
43	C₂×C₂²⋊C₈×C₈₉	C₂²	1	2	yes	complemented
44	C₂⁵×C₈₉	—	2	1	no	—
45	C₂⁵×C₈₉	D₄	1	1	yes	characteristic
46	C₂²×Q₈×C₈₉	—	2	2	no	—
47	C₂²×Q₈×C₈₉	—	2	1	no	—
48	C₂²×Q₈×C₈₉	—	4	4	no	—
49	C₂²×Q₈×C₈₉	C₂³	1	2	yes	—
50	C₂²×Q₈×C₈₉	C₂³	1	4	yes	complemented
51	C₂²×D₄×C₈₉	—	2	8	no	—
52	C₂²×D₄×C₈₉	—	2	2	no	—
53	C₂²×D₄×C₈₉	—	2	2	no	—
54	C₂²×D₄×C₈₉	—	2	2	no	—
55	C₂²×D₄×C₈₉	—	2	1	no	—
56	C₂²×D₄×C₈₉	—	4	4	no	—
57	C₂²×D₄×C₈₉	—	4	8	no	—
58	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
59	C₂²×D₄×C₈₉	C₂³	1	4	yes	—
60	C₂³×C₄×C₈₉	—	2	1	no	—
61	C₂³×C₄×C₈₉	—	2	1	no	—
62	C₂³×C₄×C₈₉	—	2	1	no	—
63	C₂³×C₄×C₈₉	—	4	4	no	—
64	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
65	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
66	C₂³×C₄×C₈₉	D₄	1	4	yes	—
67	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
68	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
69	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
70	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
71	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
72	C₂×SD₁₆×C₈₉	—	4	8	no	—
73	C₂×SD₁₆×C₈₉	C₂³	1	16	yes	complemented
74	C₂²×C₈×C₈₉	—	4	4	no	—
75	C₂²×C₈×C₈₉	C₂³	1	4	yes	—
76	C₂²×C₈×C₈₉	C₂³	1	8	yes	complemented
77	C₂²⋊Q₈×C₈₉	—	2	4	no	—
78	C₂²⋊Q₈×C₈₉	—	2	8	no	—
79	C₂²⋊Q₈×C₈₉	—	2	8	no	—
80	C₂²⋊Q₈×C₈₉	—	2	4	no	—
81	C₂²⋊Q₈×C₈₉	C₂³	1	8	yes	—
82	C₂²⋊Q₈×C₈₉	C₂³	1	8	yes	complemented
83	#121×C₈₉	—	2	4	no	—
84	#121×C₈₉	—	2	4	no	—
85	#121×C₈₉	—	2	8	no	—
86	#121×C₈₉	—	2	8	no	—
87	#121×C₈₉	C₂³	1	8	yes	—
88	#121×C₈₉	C₂³	1	8	yes	—
89	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
90	C₂×C₄⋊C₄×C₈₉	—	2	8	no	—
91	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
92	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
93	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
94	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
95	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
96	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
97	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
98	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
99	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	complemented
100	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
101	C₂×C₂²⋊C₄×C₈₉	—	2	8	no	—
102	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
103	C₂×C₂²⋊C₄×C₈₉	—	2	8	no	—
104	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
105	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
106	#106×C₈₉	C₂³	1	16	yes	complemented
107	#103×C₈₉	C₂³	1	16	yes	complemented
108	#102×C₈₉	C₂³	1	16	yes	complemented
109	C₂²⋊C₈×C₈₉	C₂³	1	8	yes	—
110	C₂²⋊C₈×C₈₉	C₂³	1	8	yes	complemented
111	C₂⁴×C₈₉	—	2	4	no	—
112	C₂⁴×C₈₉	—	2	2	no	—
113	C₂⁴×C₈₉	—	4	2	no	—
114	C₂⁴×C₈₉	—	4	2	no	—
115	C₂⁴×C₈₉	—	4	4	no	—
116	C₂⁴×C₈₉	—	4	4	no	—
117	C₂⁴×C₈₉	—	4	4	no	—
118	C₂⁴×C₈₉	SD₁₆	1	8	yes	complemented
119	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
120	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
121	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
122	C₂⁴×C₈₉	C₂×D₄	1	4	yes	—
123	C₂×Q₈×C₈₉	—	2	2	no	—
124	C₂×Q₈×C₈₉	—	2	4	no	—
125	C₂×Q₈×C₈₉	—	2	8	no	—
126	C₂×Q₈×C₈₉	—	4	8	no	—
127	C₂×Q₈×C₈₉	—	4	4	no	—
128	C₂×Q₈×C₈₉	—	4	8	no	—
129	C₂×Q₈×C₈₉	C₂⁴	1	4	yes	—
130	C₂×D₄×C₈₉	—	2	8	no	—
131	C₂×D₄×C₈₉	—	2	8	no	—
132	C₂×D₄×C₈₉	—	2	4	no	—
133	C₂×D₄×C₈₉	—	2	2	no	—
134	C₂×D₄×C₈₉	—	2	8	no	—
135	C₂×D₄×C₈₉	—	4	8	no	—
136	C₂×D₄×C₈₉	—	4	4	no	—
137	C₂×D₄×C₈₉	—	4	8	no	—
138	C₂×D₄×C₈₉	—	4	8	no	—
139	C₂×D₄×C₈₉	—	4	8	no	—
140	C₂×D₄×C₈₉	—	4	32	no	—
141	C₂×D₄×C₈₉	C₂⁴	1	4	yes	—
142	C₂²×C₄×C₈₉	—	2	4	no	—
143	C₂²×C₄×C₈₉	—	2	4	no	—
144	C₂²×C₄×C₈₉	—	2	4	no	—
145	C₂²×C₄×C₈₉	—	2	4	no	—
146	C₂²×C₄×C₈₉	—	2	4	no	—
147	C₂²×C₄×C₈₉	—	2	4	no	—
148	C₂²×C₄×C₈₉	—	2	2	no	—
149	C₂²×C₄×C₈₉	—	2	2	no	—
150	C₂²×C₄×C₈₉	—	2	2	no	—
151	C₂²×C₄×C₈₉	—	4	8	no	—
152	C₂²×C₄×C₈₉	—	4	8	no	—
153	C₂²×C₄×C₈₉	—	4	2	no	—
154	C₂²×C₄×C₈₉	—	4	2	no	—
155	C₂²×C₄×C₈₉	—	4	2	no	—
156	C₂²×C₄×C₈₉	—	4	4	no	—
157	C₂²×C₄×C₈₉	—	4	4	no	—
158	C₂²×C₄×C₈₉	—	4	4	no	—
159	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
160	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
161	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
162	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
166	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
167	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
168	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
169	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
170	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
171	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
172	SD₁₆×C₈₉	—	4	16	no	—
173	C₂×C₈×C₈₉	—	2	8	no	—
174	C₂×C₈×C₈₉	—	4	8	no	—
175	C₂×C₈×C₈₉	C₂⁴	1	8	yes	—
176	C₄⋊C₄×C₈₉	—	2	8	no	—
177	C₄⋊C₄×C₈₉	—	2	8	no	—
178	C₄⋊C₄×C₈₉	—	2	4	no	—
179	C₄⋊C₄×C₈₉	—	2	2	no	—
180	C₄⋊C₄×C₈₉	—	2	2	no	—
181	C₄⋊C₄×C₈₉	—	2	4	no	—
182	C₄⋊C₄×C₈₉	C₂⁴	1	4	yes	—
183	C₄⋊C₄×C₈₉	C₂⁴	1	4	yes	—
184	C₂²⋊C₄×C₈₉	—	2	8	no	—
185	C₂²⋊C₄×C₈₉	—	2	8	no	—
186	C₂²⋊C₄×C₈₉	—	2	8	no	—
187	C₂²⋊C₄×C₈₉	—	2	8	no	—
188	C₂³×C₈₉	—	2	2	no	—
189	C₂³×C₈₉	—	2	2	no	—
190	C₂³×C₈₉	—	2	2	no	—
191	C₂³×C₈₉	—	2	2	no	—
192	C₂³×C₈₉	—	2	2	no	—
193	C₂³×C₈₉	—	2	2	no	—
194	C₂³×C₈₉	—	2	4	no	—
195	C₂³×C₈₉	—	4	4	no	—
196	C₂³×C₈₉	—	4	16	no	—
197	C₂³×C₈₉	—	4	4	no	—
198	C₂³×C₈₉	—	4	8	no	—
199	C₂³×C₈₉	—	4	8	no	—
200	C₂³×C₈₉	—	4	4	no	—
201	C₂³×C₈₉	—	4	8	no	—
202	C₂³×C₈₉	—	4	4	no	—
203	C₂³×C₈₉	—	4	4	no	—
204	C₂³×C₈₉	—	8	16	no	—
205	C₂³×C₈₉	C₂×SD₁₆	1	4	yes	—
206	C₂³×C₈₉	C₂×SD₁₆	1	4	yes	—
207	C₂³×C₈₉	C₂×SD₁₆	1	4	yes	—
208	C₂³×C₈₉	C₂×SD₁₆	1	16	yes	complemented
209	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
210	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
211	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
212	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
213	Q₈×C₈₉	—	2	8	no	—
214	Q₈×C₈₉	—	4	8	no	—
215	Q₈×C₈₉	—	4	4	no	—
216	D₄×C₈₉	—	2	8	no	—
217	D₄×C₈₉	—	4	8	no	—
218	D₄×C₈₉	—	4	16	no	—
219	D₄×C₈₉	—	4	16	no	—
220	D₄×C₈₉	—	4	4	no	—
221	C₂×C₄×C₈₉	—	2	2	no	—
222	C₂×C₄×C₈₉	—	2	2	no	—
223	C₂×C₄×C₈₉	—	2	2	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	4	no	—
229	C₂×C₄×C₈₉	—	2	4	no	—
230	C₂×C₄×C₈₉	—	2	4	no	—
231	C₂×C₄×C₈₉	—	2	4	no	—
232	C₂×C₄×C₈₉	—	2	4	no	—
233	C₂×C₄×C₈₉	—	4	4	no	—
234	C₂×C₄×C₈₉	—	4	4	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	—	4	8	no	—
240	C₂×C₄×C₈₉	—	4	4	no	—
241	C₂×C₄×C₈₉	—	4	4	no	—
242	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
243	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
244	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
245	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
246	C₂×C₄×C₈₉	H(5,𝔽₂)	1	4	yes	—
247	C₂×C₄×C₈₉	C₂⁵	1	1	yes	characteristic
248	C₈×C₈₉	—	2	8	no	—
249	C₂²×C₈₉	—	2	8	no	—
250	C₂²×C₈₉	—	2	8	no	—
251	C₂²×C₈₉	—	2	8	no	—
252	C₂²×C₈₉	—	2	2	no	—
253	C₂²×C₈₉	—	2	2	no	—
254	C₂²×C₈₉	—	2	2	no	—
255	C₂²×C₈₉	—	2	2	no	—
256	C₂²×C₈₉	—	2	2	no	—
257	C₂²×C₈₉	—	2	2	no	—
258	C₂²×C₈₉	—	4	16	no	—
259	C₂²×C₈₉	—	4	4	no	—
260	C₂²×C₈₉	—	4	4	no	—
261	C₂²×C₈₉	—	4	4	no	—
262	C₂²×C₈₉	—	8	16	no	—
263	C₂²×C₈₉	—	8	8	no	—
264	C₂²×C₈₉	#534	1	2	yes	—
265	C₂²×C₈₉	C₂²×SD₁₆	1	4	yes	—
266	C₂²×C₈₉	C₂²×SD₁₆	1	4	yes	—
267	C₂²×C₈₉	C₂²×SD₁₆	1	4	yes	—
268	C₂²×C₈₉	C₂²×SD₁₆	1	2	yes	—
269	C₂²×C₈₉	#575	1	2	yes	—
270	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
271	C₄×C₈₉	—	2	2	no	—
272	C₄×C₈₉	—	2	2	no	—
273	C₄×C₈₉	—	2	2	no	—
274	C₄×C₈₉	—	4	4	no	—
275	C₄×C₈₉	—	4	4	no	—
276	C₄×C₈₉	—	4	4	no	—
277	C₄×C₈₉	#534	1	2	yes	—
278	C₄×C₈₉	C₂×H(5,𝔽₂)	1	2	yes	—
279	C₂×#3187	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
280	C₂×C₈₉	—	2	4	no	—
281	C₂×C₈₉	—	2	4	no	—
282	C₂×C₈₉	—	2	4	no	—
283	C₂×C₈₉	—	8	4	no	—
284	C₂×C₈₉	#3187	1	4	yes	direct product factor
285	C₂×C₈₉	C₂×#534	1	1	yes	characteristic
286	C₂×C₈₉	C₂³×SD₁₆	1	1	yes	characteristic
287	C₂×C₈₉	C₂×#575	1	1	yes	characteristic
288	C₂²×C₂²⋊Q₈	C₂×C₈₉	1	1	yes	characteristic, complemented
289	C₂²×#121	C₂×C₈₉	1	1	yes	characteristic
290	#3187	C₂×C₈₉	1	16	yes	direct product factor
291	C₂×#481	C₂×C₈₉	1	2	yes	complemented
292	C₂×#483	C₂×C₈₉	1	2	yes	complemented
293	C₂×#465	C₂×C₈₉	1	4	yes	complemented
294	C₂×#448	C₂×C₈₉	1	2	yes	complemented
295	C₂×#450	C₂×C₈₉	1	2	yes	complemented
296	C₂²×C₂²⋊C₈	C₂×C₈₉	1	1	yes	characteristic, complemented
297	C₈₉	C₂×#3187	1	1	yes	characteristic, direct product factor, Sylow
298	C₂³×Q₈	C₂²×C₈₉	1	1	yes	characteristic, complemented
299	C₂³×D₄	—	2	1	no	—
300	C₂³×D₄	C₂²×C₈₉	1	1	yes	characteristic
301	C₂⁴×C₄	C₂²×C₈₉	1	1	yes	characteristic
302	C₂²×SD₁₆	C₂²×C₈₉	1	4	yes	complemented
303	C₂³×C₈	C₂²×C₈₉	1	2	yes	complemented
304	C₂×C₂²⋊Q₈	—	2	1	no	—
305	C₂×C₂²⋊Q₈	—	2	1	no	—
306	C₂×C₂²⋊Q₈	—	2	4	no	—
307	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	2	yes	—
308	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	2	yes	complemented
309	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	8	yes	complemented
310	C₂×#121	—	2	1	no	—
311	C₂×#121	—	2	1	no	—
312	C₂×#121	—	2	4	no	—
313	C₂×#121	C₂²×C₈₉	1	2	yes	—
314	C₂×#121	C₂²×C₈₉	1	8	yes	—
315	C₂×#121	C₂²×C₈₉	1	2	yes	—
316	C₂²×C₄⋊C₄	—	2	1	no	—
317	C₂²×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
318	C₂²×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic, complemented
319	C₂²×C₂²⋊C₄	—	2	1	no	—
320	C₂²×C₂²⋊C₄	—	2	1	no	—
321	#483	C₂²×C₈₉	1	16	yes	complemented
322	#481	C₂²×C₈₉	1	16	yes	complemented
323	#465	C₂²×C₈₉	1	32	yes	complemented
324	#450	C₂²×C₈₉	1	16	yes	complemented
325	#448	C₂²×C₈₉	1	16	yes	complemented
326	C₂×#106	C₂²×C₈₉	1	4	yes	complemented
327	C₂×#103	C₂²×C₈₉	1	4	yes	complemented
328	C₂×#102	C₂²×C₈₉	1	4	yes	complemented
329	C₂×C₂²⋊C₈	C₂²×C₈₉	1	2	yes	—
330	C₂×C₂²⋊C₈	C₂²×C₈₉	1	8	yes	complemented
331	C₂×C₂²⋊C₈	C₂²×C₈₉	1	2	yes	complemented
332	C₂⁵	—	2	1	no	—
333	C₂⁵	D₄×C₈₉	1	1	yes	characteristic
334	C₂²×Q₈	—	2	1	no	—
335	C₂²×Q₈	—	2	2	no	—
336	C₂²×Q₈	—	4	4	no	—
337	C₂²×Q₈	C₂³×C₈₉	1	2	yes	—
338	C₂²×Q₈	C₂³×C₈₉	1	4	yes	complemented
339	C₂²×D₄	—	2	2	no	—
340	C₂²×D₄	—	2	2	no	—
341	C₂²×D₄	—	2	8	no	—
342	C₂²×D₄	—	2	1	no	—
343	C₂²×D₄	—	2	2	no	—
344	C₂²×D₄	—	4	4	no	—
345	C₂²×D₄	—	4	8	no	—
346	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
347	C₂²×D₄	C₂³×C₈₉	1	4	yes	—
348	C₂³×C₄	—	2	1	no	—
349	C₂³×C₄	—	2	1	no	—
350	C₂³×C₄	—	2	1	no	—
351	C₂³×C₄	—	4	4	no	—
352	C₂³×C₄	D₄×C₈₉	1	1	yes	characteristic
353	C₂³×C₄	D₄×C₈₉	1	1	yes	characteristic
354	C₂³×C₄	D₄×C₈₉	1	4	yes	—
355	C₂³×C₄	D₄×C₈₉	1	1	yes	characteristic
356	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
357	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
358	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
359	C₂³×C₄	C₂³×C₈₉	1	4	yes	—
360	C₂×SD₁₆	—	4	8	no	—
361	C₂×SD₁₆	C₂³×C₈₉	1	16	yes	complemented
362	C₂²×C₈	—	4	4	no	—
363	C₂²×C₈	C₂³×C₈₉	1	4	yes	—
364	C₂²×C₈	C₂³×C₈₉	1	8	yes	complemented
365	C₂²⋊Q₈	—	2	4	no	—
366	C₂²⋊Q₈	—	2	8	no	—
367	C₂²⋊Q₈	—	2	8	no	—
368	C₂²⋊Q₈	—	2	4	no	—
369	C₂²⋊Q₈	C₂³×C₈₉	1	8	yes	—
370	C₂²⋊Q₈	C₂³×C₈₉	1	8	yes	complemented
371	#121	—	2	4	no	—
372	#121	—	2	4	no	—
373	#121	—	2	8	no	—
374	#121	—	2	8	no	—
375	#121	C₂³×C₈₉	1	8	yes	—
376	#121	C₂³×C₈₉	1	8	yes	—
377	C₂×C₄⋊C₄	—	2	2	no	—
378	C₂×C₄⋊C₄	—	2	8	no	—
379	C₂×C₄⋊C₄	—	2	2	no	—
380	C₂×C₄⋊C₄	—	2	1	no	—
381	C₂×C₄⋊C₄	—	2	2	no	—
382	C₂×C₄⋊C₄	—	2	2	no	—
383	C₂×C₄⋊C₄	—	2	1	no	—
384	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
385	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
386	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
387	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	complemented
388	C₂×C₂²⋊C₄	—	2	2	no	—
389	C₂×C₂²⋊C₄	—	2	8	no	—
390	C₂×C₂²⋊C₄	—	2	2	no	—
391	C₂×C₂²⋊C₄	—	2	2	no	—
392	C₂×C₂²⋊C₄	—	2	2	no	—
393	C₂×C₂²⋊C₄	—	2	8	no	—
394	#106	C₂³×C₈₉	1	16	yes	complemented
395	#103	C₂³×C₈₉	1	16	yes	complemented
396	#102	C₂³×C₈₉	1	16	yes	complemented
397	C₂²⋊C₈	C₂³×C₈₉	1	8	yes	—
398	C₂²⋊C₈	C₂³×C₈₉	1	8	yes	complemented
399	C₂⁴	—	2	4	no	—
400	C₂⁴	—	2	2	no	—
401	C₂⁴	—	4	2	no	—
402	C₂⁴	—	4	2	no	—
403	C₂⁴	—	4	4	no	—
404	C₂⁴	—	4	4	no	—
405	C₂⁴	—	4	4	no	—
406	C₂⁴	SD₁₆×C₈₉	1	8	yes	complemented
407	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
408	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
409	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
410	C₂⁴	C₂×D₄×C₈₉	1	4	yes	—
411	C₂×Q₈	—	2	2	no	—
412	C₂×Q₈	—	2	4	no	—
413	C₂×Q₈	—	2	8	no	—
414	C₂×Q₈	—	4	8	no	—
415	C₂×Q₈	—	4	4	no	—
416	C₂×Q₈	—	4	8	no	—
417	C₂×Q₈	C₂⁴×C₈₉	1	4	yes	—
418	C₂×D₄	—	2	8	no	—
419	C₂×D₄	—	2	8	no	—
420	C₂×D₄	—	2	4	no	—
421	C₂×D₄	—	2	2	no	—
422	C₂×D₄	—	2	8	no	—
423	C₂×D₄	—	4	8	no	—
424	C₂×D₄	—	4	4	no	—
425	C₂×D₄	—	4	8	no	—
426	C₂×D₄	—	4	8	no	—
427	C₂×D₄	—	4	8	no	—
428	C₂×D₄	—	4	32	no	—
429	C₂×D₄	C₂⁴×C₈₉	1	4	yes	—
430	C₂²×C₄	—	2	4	no	—
431	C₂²×C₄	—	2	4	no	—
432	C₂²×C₄	—	2	4	no	—
433	C₂²×C₄	—	2	2	no	—
434	C₂²×C₄	—	2	4	no	—
435	C₂²×C₄	—	2	2	no	—
436	C₂²×C₄	—	2	4	no	—
437	C₂²×C₄	—	2	4	no	—
438	C₂²×C₄	—	2	2	no	—
439	C₂²×C₄	—	4	8	no	—
440	C₂²×C₄	—	4	8	no	—
441	C₂²×C₄	—	4	2	no	—
442	C₂²×C₄	—	4	2	no	—
443	C₂²×C₄	—	4	2	no	—
444	C₂²×C₄	—	4	4	no	—
445	C₂²×C₄	—	4	4	no	—
446	C₂²×C₄	—	4	4	no	—
447	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
448	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
449	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
450	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
451	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
452	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
453	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
454	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
455	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
456	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
457	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
458	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
459	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
460	SD₁₆	—	4	16	no	—
461	C₂×C₈	—	2	8	no	—
462	C₂×C₈	—	4	8	no	—
463	C₂×C₈	C₂⁴×C₈₉	1	8	yes	—
464	C₄⋊C₄	—	2	8	no	—
465	C₄⋊C₄	—	2	8	no	—
466	C₄⋊C₄	—	2	4	no	—
467	C₄⋊C₄	—	2	2	no	—
468	C₄⋊C₄	—	2	2	no	—
469	C₄⋊C₄	—	2	4	no	—
470	C₄⋊C₄	C₂⁴×C₈₉	1	4	yes	—
471	C₄⋊C₄	C₂⁴×C₈₉	1	4	yes	—
472	C₂²⋊C₄	—	2	8	no	—
473	C₂²⋊C₄	—	2	8	no	—
474	C₂²⋊C₄	—	2	8	no	—
475	C₂²⋊C₄	—	2	8	no	—
476	C₂³	—	2	2	no	—
477	C₂³	—	2	2	no	—
478	C₂³	—	2	2	no	—
479	C₂³	—	2	2	no	—
480	C₂³	—	2	2	no	—
481	C₂³	—	2	2	no	—
482	C₂³	—	2	4	no	—
483	C₂³	—	4	4	no	—
484	C₂³	—	4	16	no	—
485	C₂³	—	4	4	no	—
486	C₂³	—	4	8	no	—
487	C₂³	—	4	8	no	—
488	C₂³	—	4	4	no	—
489	C₂³	—	4	8	no	—
490	C₂³	—	4	4	no	—
491	C₂³	—	4	4	no	—
492	C₂³	—	8	16	no	—
493	C₂³	C₂×SD₁₆×C₈₉	1	4	yes	—
494	C₂³	C₂×SD₁₆×C₈₉	1	4	yes	—
495	C₂³	C₂×SD₁₆×C₈₉	1	4	yes	—
496	C₂³	C₂×SD₁₆×C₈₉	1	16	yes	complemented
497	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
498	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
499	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
500	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
501	Q₈	—	2	8	no	—
502	Q₈	—	4	8	no	—
503	Q₈	—	4	4	no	—
504	D₄	—	2	8	no	—
505	D₄	—	4	8	no	—
506	D₄	—	4	16	no	—
507	D₄	—	4	16	no	—
508	D₄	—	4	4	no	—
509	C₂×C₄	—	2	2	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	—	2	4	no	—
515	C₂×C₄	—	2	4	no	—
516	C₂×C₄	—	2	4	no	—
517	C₂×C₄	—	2	4	no	—
518	C₂×C₄	—	2	4	no	—
519	C₂×C₄	—	2	4	no	—
520	C₂×C₄	—	2	4	no	—
521	C₂×C₄	—	4	4	no	—
522	C₂×C₄	—	4	4	no	—
523	C₂×C₄	—	4	4	no	—
524	C₂×C₄	—	4	8	no	—
525	C₂×C₄	—	4	8	no	—
526	C₂×C₄	—	4	4	no	—
527	C₂×C₄	—	4	8	no	—
528	C₂×C₄	—	4	4	no	—
529	C₂×C₄	—	4	4	no	—
530	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
531	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
532	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
533	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
534	C₂×C₄	H(5,𝔽₂)×C₈₉	1	4	yes	—
535	C₂×C₄	C₂⁵×C₈₉	1	1	yes	characteristic
536	C₈	—	2	8	no	—
537	C₂²	—	2	2	no	—
538	C₂²	—	2	2	no	—
539	C₂²	—	2	2	no	—
540	C₂²	—	2	8	no	—
541	C₂²	—	2	8	no	—
542	C₂²	—	2	8	no	—
543	C₂²	—	2	2	no	—
544	C₂²	—	2	2	no	—
545	C₂²	—	2	2	no	—
546	C₂²	—	4	16	no	—
547	C₂²	—	4	4	no	—
548	C₂²	—	4	4	no	—
549	C₂²	—	4	4	no	—
550	C₂²	—	8	16	no	—
551	C₂²	—	8	8	no	—
552	C₂²	#534×C₈₉	1	2	yes	—
553	C₂²	C₂²×SD₁₆×C₈₉	1	4	yes	—
554	C₂²	C₂²×SD₁₆×C₈₉	1	4	yes	—
555	C₂²	C₂²×SD₁₆×C₈₉	1	4	yes	—
556	C₂²	C₂²×SD₁₆×C₈₉	1	2	yes	—
557	C₂²	#575×C₈₉	1	2	yes	—
558	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
559	C₄	—	2	2	no	—
560	C₄	—	2	2	no	—
561	C₄	—	2	2	no	—
562	C₄	—	4	4	no	—
563	C₄	—	4	4	no	—
564	C₄	—	4	4	no	—
565	C₄	#534×C₈₉	1	2	yes	—
566	C₄	C₂×H(5,𝔽₂)×C₈₉	1	2	yes	—
567	C₂	—	2	4	no	—
568	C₂	—	2	4	no	—
569	C₂	—	2	4	no	—
570	C₂	—	8	4	no	—
571	C₂	C₈₉×#3187	1	4	yes	direct product factor
572	C₂	C₂×#534×C₈₉	1	1	yes	characteristic
573	C₂	C₂³×SD₁₆×C₈₉	1	1	yes	characteristic
574	C₂	C₂×#575×C₈₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₈₉
prime radical	C₂²×#121×C₈₉	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ C₂⁴×C₄×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×#3187	C₂	direct (N × H)
4. C₂×#481×C₈₉	C₂	semidirect
5. C₂×#483×C₈₉	C₂	semidirect
6. C₂×#465×C₈₉	C₂	semidirect
7. C₂×#448×C₈₉	C₂	semidirect
8. C₂×#450×C₈₉	C₂	semidirect
9. C₂²×C₂²⋊C₈×C₈₉	C₂	semidirect
10. C₂³×Q₈×C₈₉	C₂²	semidirect
14. C₂²×SD₁₆×C₈₉	C₂²	semidirect
15. C₂³×C₈×C₈₉	C₂²	semidirect
20. C₂×C₂²⋊Q₈×C₈₉	C₂²	semidirect
21. C₂×C₂²⋊Q₈×C₈₉	C₂²	semidirect
30. C₂²×C₄⋊C₄×C₈₉	C₂²	semidirect
33. #483×C₈₉	C₂²	semidirect
34. #481×C₈₉	C₂²	semidirect
35. #465×C₈₉	C₂²	semidirect
36. #450×C₈₉	C₂²	semidirect
37. #448×C₈₉	C₂²	semidirect
38. C₂×#106×C₈₉	C₂²	semidirect
39. C₂×#103×C₈₉	C₂²	semidirect
40. C₂×#102×C₈₉	C₂²	semidirect
42. C₂×C₂²⋊C₈×C₈₉	C₂²	semidirect
43. C₂×C₂²⋊C₈×C₈₉	C₂²	semidirect
50. C₂²×Q₈×C₈₉	C₂³	semidirect
73. C₂×SD₁₆×C₈₉	C₂³	semidirect
76. C₂²×C₈×C₈₉	C₂³	semidirect
82. C₂²⋊Q₈×C₈₉	C₂³	semidirect
99. C₂×C₄⋊C₄×C₈₉	C₂³	semidirect
106. #106×C₈₉	C₂³	semidirect
107. #103×C₈₉	C₂³	semidirect
108. #102×C₈₉	C₂³	semidirect
110. C₂²⋊C₈×C₈₉	C₂³	semidirect
118. C₂⁴×C₈₉	SD₁₆	semidirect
208. C₂³×C₈₉	C₂×SD₁₆	semidirect
279. C₂×#3187	C₈₉	direct (N × H)
284. C₂×C₈₉	#3187	direct (N × H)
288. C₂²×C₂²⋊Q₈	C₂×C₈₉	semidirect
290. #3187	C₂×C₈₉	direct (N × H)
291. C₂×#481	C₂×C₈₉	semidirect
292. C₂×#483	C₂×C₈₉	semidirect
293. C₂×#465	C₂×C₈₉	semidirect
294. C₂×#448	C₂×C₈₉	semidirect
295. C₂×#450	C₂×C₈₉	semidirect
296. C₂²×C₂²⋊C₈	C₂×C₈₉	semidirect
297. C₈₉	C₂×#3187	direct (N × H)
298. C₂³×Q₈	C₂²×C₈₉	semidirect
302. C₂²×SD₁₆	C₂²×C₈₉	semidirect
303. C₂³×C₈	C₂²×C₈₉	semidirect
308. C₂×C₂²⋊Q₈	C₂²×C₈₉	semidirect
309. C₂×C₂²⋊Q₈	C₂²×C₈₉	semidirect
318. C₂²×C₄⋊C₄	C₂²×C₈₉	semidirect
321. #483	C₂²×C₈₉	semidirect
322. #481	C₂²×C₈₉	semidirect
323. #465	C₂²×C₈₉	semidirect
324. #450	C₂²×C₈₉	semidirect
325. #448	C₂²×C₈₉	semidirect
326. C₂×#106	C₂²×C₈₉	semidirect
327. C₂×#103	C₂²×C₈₉	semidirect
328. C₂×#102	C₂²×C₈₉	semidirect
330. C₂×C₂²⋊C₈	C₂²×C₈₉	semidirect
331. C₂×C₂²⋊C₈	C₂²×C₈₉	semidirect
338. C₂²×Q₈	C₂³×C₈₉	semidirect
361. C₂×SD₁₆	C₂³×C₈₉	semidirect
364. C₂²×C₈	C₂³×C₈₉	semidirect
370. C₂²⋊Q₈	C₂³×C₈₉	semidirect
387. C₂×C₄⋊C₄	C₂³×C₈₉	semidirect
394. #106	C₂³×C₈₉	semidirect
395. #103	C₂³×C₈₉	semidirect
396. #102	C₂³×C₈₉	semidirect
398. C₂²⋊C₈	C₂³×C₈₉	semidirect
406. C₂⁴	SD₁₆×C₈₉	semidirect
496. C₂³	C₂×SD₁₆×C₈₉	semidirect
571. C₂	C₈₉×#3187	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
C₂²×#121×C₈₉	2
C₈₉×#3187	2
C₂×#481×C₈₉	2
C₂×#483×C₈₉	2
C₂×#465×C₈₉	2
C₂×#448×C₈₉	2
C₂×#450×C₈₉	2
C₂²×C₂²⋊C₈×C₈₉	2
C₂×#3187	89

Subgroups — C2×C89×#3187 #36438577

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈₉×#3187 #36438577