Subgroups — C₂×C₈₉×#3094 #36438357

Order 22784.

2670 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#121×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×#117×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#3094	C₂	1	16	yes	maximal, direct product factor
4	C₂×#481×C₈₉	C₂	1	2	yes	maximal, complemented
5	C₂×#482×C₈₉	C₂	1	4	yes	maximal, complemented
6	C₂×#480×C₈₉	C₂	1	2	yes	maximal, complemented
7	C₂×#428×C₈₉	C₂	1	2	yes	maximal, complemented
8	C₂×#418×C₈₉	C₂	1	2	yes	maximal, complemented
9	C₂×#407×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×D₄×C₈₉	—	2	1	no	—
11	C₂³×D₄×C₈₉	C₂²	1	1	yes	characteristic
12	C₂⁴×C₄×C₈₉	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₈₉	C₂²	1	2	yes	—
14	C₂×#121×C₈₉	—	2	1	no	—
15	C₂×#121×C₈₉	—	2	4	no	—
16	C₂×#121×C₈₉	—	2	1	no	—
17	C₂×#121×C₈₉	C₂²	1	2	yes	—
18	C₂×#121×C₈₉	C₂²	1	2	yes	—
19	C₂×#121×C₈₉	C₂²	1	8	yes	—
20	C₂×#117×C₈₉	—	2	1	no	—
21	C₂×#117×C₈₉	—	2	1	no	—
22	C₂×#117×C₈₉	—	2	4	no	—
23	C₂×#117×C₈₉	C₂²	1	2	yes	—
24	C₂×#117×C₈₉	C₂²	1	2	yes	complemented
25	C₂×#117×C₈₉	C₂²	1	8	yes	complemented
26	C₂²×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
27	C₂²×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic, complemented
28	C₂²×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic, complemented
29	C₂²×C₂²⋊C₄×C₈₉	—	2	1	no	—
30	C₂²×C₂²⋊C₄×C₈₉	—	2	1	no	—
31	C₂²×C₄²×C₈₉	—	2	1	no	—
32	#482×C₈₉	C₂²	1	32	yes	complemented
33	#481×C₈₉	C₂²	1	16	yes	complemented
34	#480×C₈₉	C₂²	1	16	yes	complemented
35	#428×C₈₉	C₂²	1	16	yes	complemented
36	C₂×#107×C₈₉	C₂²	1	4	yes	complemented
37	C₂×#106×C₈₉	C₂²	1	4	yes	complemented
38	#418×C₈₉	C₂²	1	16	yes	complemented
39	C₂×#102×C₈₉	C₂²	1	4	yes	complemented
40	C₂×#102×C₈₉	C₂²	1	4	yes	complemented
41	#407×C₈₉	C₂²	1	8	yes	complemented
42	C₂×C₂²⋊C₈×C₈₉	C₂²	1	2	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×D₄×C₈₉	—	2	8	no	—
47	C₂²×D₄×C₈₉	—	2	2	no	—
48	C₂²×D₄×C₈₉	—	2	2	no	—
49	C₂²×D₄×C₈₉	—	2	2	no	—
50	C₂²×D₄×C₈₉	—	2	1	no	—
51	C₂²×D₄×C₈₉	—	2	8	no	—
52	C₂²×D₄×C₈₉	—	4	8	no	—
53	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
54	C₂²×D₄×C₈₉	C₂³	1	4	yes	—
55	C₂³×C₄×C₈₉	—	2	8	no	—
56	C₂³×C₄×C₈₉	—	2	1	no	—
57	C₂³×C₄×C₈₉	—	2	1	no	—
58	C₂³×C₄×C₈₉	—	2	1	no	—
59	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
60	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
61	C₂³×C₄×C₈₉	D₄	1	4	yes	—
62	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
63	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
64	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
65	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
66	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
67	C₂×M₁₆×C₈₉	—	2	8	no	—
68	C₂×M₁₆×C₈₉	C₂³	1	8	yes	—
69	C₂²×C₈×C₈₉	C₂³	1	4	yes	—
70	#121×C₈₉	—	2	8	no	—
71	#121×C₈₉	—	2	4	no	—
72	#121×C₈₉	—	2	4	no	—
73	#121×C₈₉	—	2	8	no	—
74	#121×C₈₉	C₂³	1	8	yes	—
75	#121×C₈₉	C₂³	1	8	yes	—
76	#117×C₈₉	—	2	4	no	—
77	#117×C₈₉	—	2	8	no	—
78	#117×C₈₉	—	2	8	no	—
79	#117×C₈₉	—	2	4	no	—
80	#117×C₈₉	C₂³	1	8	yes	—
81	#117×C₈₉	C₂³	1	8	yes	complemented
82	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
83	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
84	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
85	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
86	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
87	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
88	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
89	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
90	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
91	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
92	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	complemented
93	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	complemented
94	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
95	C₂×C₂²⋊C₄×C₈₉	—	2	8	no	—
96	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
97	C₂×C₂²⋊C₄×C₈₉	—	2	8	no	—
98	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
99	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
100	C₂×C₄²×C₈₉	—	2	2	no	—
101	C₂×C₄²×C₈₉	—	2	8	no	—
102	C₂×C₄²×C₈₉	—	2	2	no	—
103	#107×C₈₉	C₂³	1	16	yes	complemented
104	#106×C₈₉	C₂³	1	16	yes	complemented
105	#102×C₈₉	C₂³	1	16	yes	complemented
106	#102×C₈₉	C₂³	1	16	yes	complemented
107	C₂²⋊C₈×C₈₉	C₂³	1	8	yes	complemented
108	C₂²⋊C₈×C₈₉	C₂³	1	8	yes	complemented
109	C₂⁴×C₈₉	—	2	4	no	—
110	C₂⁴×C₈₉	—	2	4	no	—
111	C₂⁴×C₈₉	—	2	4	no	—
112	C₂⁴×C₈₉	—	2	2	no	—
113	C₂⁴×C₈₉	—	4	2	no	—
114	C₂⁴×C₈₉	—	4	4	no	—
115	C₂⁴×C₈₉	—	4	4	no	—
116	C₂⁴×C₈₉	—	4	4	no	—
117	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
118	C₂⁴×C₈₉	C₂×D₄	1	4	yes	—
119	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
120	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
121	C₂×D₄×C₈₉	—	2	8	no	—
122	C₂×D₄×C₈₉	—	2	8	no	—
123	C₂×D₄×C₈₉	—	2	4	no	—
124	C₂×D₄×C₈₉	—	2	2	no	—
125	C₂×D₄×C₈₉	—	2	8	no	—
126	C₂×D₄×C₈₉	—	2	16	no	—
127	C₂×D₄×C₈₉	—	4	32	no	—
128	C₂×D₄×C₈₉	—	4	8	no	—
129	C₂×D₄×C₈₉	—	4	8	no	—
130	C₂×D₄×C₈₉	—	4	8	no	—
131	C₂×D₄×C₈₉	—	4	4	no	—
132	C₂×D₄×C₈₉	C₂⁴	1	4	yes	—
133	C₂²×C₄×C₈₉	—	2	4	no	—
134	C₂²×C₄×C₈₉	—	2	4	no	—
135	C₂²×C₄×C₈₉	—	2	16	no	—
136	C₂²×C₄×C₈₉	—	2	4	no	—
137	C₂²×C₄×C₈₉	—	2	4	no	—
138	C₂²×C₄×C₈₉	—	2	4	no	—
139	C₂²×C₄×C₈₉	—	2	16	no	—
140	C₂²×C₄×C₈₉	—	2	4	no	—
141	C₂²×C₄×C₈₉	—	2	4	no	—
142	C₂²×C₄×C₈₉	—	2	4	no	—
143	C₂²×C₄×C₈₉	—	2	2	no	—
144	C₂²×C₄×C₈₉	—	2	2	no	—
145	C₂²×C₄×C₈₉	—	2	2	no	—
146	C₂²×C₄×C₈₉	—	4	4	no	—
147	C₂²×C₄×C₈₉	—	4	4	no	—
148	C₂²×C₄×C₈₉	—	4	4	no	—
149	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
150	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
151	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
152	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
153	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
154	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
157	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₈₉	#41	1	8	yes	—
159	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
160	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
161	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
162	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
163	M₁₆×C₈₉	—	2	16	no	—
164	C₂×C₈×C₈₉	—	2	8	no	—
165	C₂×C₈×C₈₉	C₂⁴	1	8	yes	—
166	C₄⋊C₄×C₈₉	—	2	2	no	—
167	C₄⋊C₄×C₈₉	—	2	2	no	—
168	C₄⋊C₄×C₈₉	—	2	4	no	—
169	C₄⋊C₄×C₈₉	—	2	2	no	—
170	C₄⋊C₄×C₈₉	—	2	4	no	—
171	C₄⋊C₄×C₈₉	—	2	4	no	—
172	C₄⋊C₄×C₈₉	C₂⁴	1	4	yes	—
173	C₄⋊C₄×C₈₉	C₂⁴	1	4	yes	—
174	C₄⋊C₄×C₈₉	C₂⁴	1	4	yes	—
175	C₂²⋊C₄×C₈₉	—	2	8	no	—
176	C₂²⋊C₄×C₈₉	—	2	8	no	—
177	C₂²⋊C₄×C₈₉	—	2	8	no	—
178	C₂²⋊C₄×C₈₉	—	2	8	no	—
179	C₄²×C₈₉	—	2	8	no	—
180	C₄²×C₈₉	—	2	8	no	—
181	C₂³×C₈₉	—	2	2	no	—
182	C₂³×C₈₉	—	2	2	no	—
183	C₂³×C₈₉	—	2	2	no	—
184	C₂³×C₈₉	—	2	2	no	—
185	C₂³×C₈₉	—	2	8	no	—
186	C₂³×C₈₉	—	2	2	no	—
187	C₂³×C₈₉	—	2	8	no	—
188	C₂³×C₈₉	—	2	2	no	—
189	C₂³×C₈₉	—	2	2	no	—
190	C₂³×C₈₉	—	2	4	no	—
191	C₂³×C₈₉	—	4	4	no	—
192	C₂³×C₈₉	—	4	16	no	—
193	C₂³×C₈₉	—	4	8	no	—
194	C₂³×C₈₉	—	4	8	no	—
195	C₂³×C₈₉	—	4	8	no	—
196	C₂³×C₈₉	—	4	4	no	—
197	C₂³×C₈₉	—	4	4	no	—
198	C₂³×C₈₉	—	4	4	no	—
199	C₂³×C₈₉	—	8	16	no	—
200	C₂³×C₈₉	#123	1	4	yes	—
201	C₂³×C₈₉	AGL(1,ℤ/8)	1	4	yes	—
202	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
203	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
204	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
205	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
206	D₄×C₈₉	—	2	8	no	—
207	D₄×C₈₉	—	4	16	no	—
208	D₄×C₈₉	—	4	16	no	—
209	D₄×C₈₉	—	4	8	no	—
210	D₄×C₈₉	—	4	4	no	—
211	C₂×C₄×C₈₉	—	2	8	no	—
212	C₂×C₄×C₈₉	—	2	2	no	—
213	C₂×C₄×C₈₉	—	2	2	no	—
214	C₂×C₄×C₈₉	—	2	8	no	—
215	C₂×C₄×C₈₉	—	2	4	no	—
216	C₂×C₄×C₈₉	—	2	4	no	—
217	C₂×C₄×C₈₉	—	2	8	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	—	4	8	no	—
224	C₂×C₄×C₈₉	—	4	8	no	—
225	C₂×C₄×C₈₉	—	4	8	no	—
226	C₂×C₄×C₈₉	—	4	4	no	—
227	C₂×C₄×C₈₉	—	4	4	no	—
228	C₂×C₄×C₈₉	—	4	4	no	—
229	C₂×C₄×C₈₉	#123	1	8	yes	—
230	C₂×C₄×C₈₉	#123	1	4	yes	—
231	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
232	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
233	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
234	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
235	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
236	C₂×C₄×C₈₉	C₂×#41	1	8	yes	—
237	C₂×C₄×C₈₉	C₂⁵	1	1	yes	characteristic
238	C₈×C₈₉	—	2	8	no	—
239	C₂²×C₈₉	—	2	8	no	—
240	C₂²×C₈₉	—	2	8	no	—
241	C₂²×C₈₉	—	2	2	no	—
242	C₂²×C₈₉	—	2	2	no	—
243	C₂²×C₈₉	—	2	2	no	—
244	C₂²×C₈₉	—	2	2	no	—
245	C₂²×C₈₉	—	2	2	no	—
246	C₂²×C₈₉	—	2	2	no	—
247	C₂²×C₈₉	—	2	2	no	—
248	C₂²×C₈₉	—	4	16	no	—
249	C₂²×C₈₉	—	4	4	no	—
250	C₂²×C₈₉	—	4	4	no	—
251	C₂²×C₈₉	—	4	4	no	—
252	C₂²×C₈₉	—	4	4	no	—
253	C₂²×C₈₉	—	8	16	no	—
254	C₂²×C₈₉	—	8	8	no	—
255	C₂²×C₈₉	C₂×#123	1	4	yes	—
256	C₂²×C₈₉	C₂×#123	1	2	yes	—
257	C₂²×C₈₉	C₂×AGL(1,ℤ/8)	1	4	yes	—
258	C₂²×C₈₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
259	C₂²×C₈₉	#575	1	2	yes	—
260	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
261	C₄×C₈₉	—	2	2	no	—
262	C₄×C₈₉	—	2	2	no	—
263	C₄×C₈₉	—	4	4	no	—
264	C₄×C₈₉	—	4	4	no	—
265	C₄×C₈₉	—	4	4	no	—
266	C₄×C₈₉	C₂×#123	1	4	yes	—
267	C₄×C₈₉	C₂×#123	1	2	yes	—
268	C₄×C₈₉	C₂²×#41	1	2	yes	—
269	C₂×#3094	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
270	C₂×C₈₉	—	2	4	no	—
271	C₂×C₈₉	—	2	4	no	—
272	C₂×C₈₉	—	4	2	no	—
273	C₂×C₈₉	—	8	4	no	—
274	C₂×C₈₉	#3094	1	4	yes	direct product factor
275	C₂×C₈₉	C₂²×#123	1	1	yes	characteristic
276	C₂×C₈₉	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
277	C₂×C₈₉	C₂×#575	1	1	yes	characteristic
278	C₂²×#121	C₂×C₈₉	1	1	yes	characteristic
279	C₂²×#117	C₂×C₈₉	1	1	yes	characteristic, complemented
280	#3094	C₂×C₈₉	1	16	yes	direct product factor
281	C₂×#481	C₂×C₈₉	1	2	yes	complemented
282	C₂×#482	C₂×C₈₉	1	4	yes	complemented
283	C₂×#480	C₂×C₈₉	1	2	yes	complemented
284	C₂×#428	C₂×C₈₉	1	2	yes	complemented
285	C₂×#418	C₂×C₈₉	1	2	yes	complemented
286	C₂×#407	C₂×C₈₉	1	1	yes	characteristic, complemented
287	C₈₉	C₂×#3094	1	1	yes	characteristic, direct product factor, Sylow
288	C₂³×D₄	—	2	1	no	—
289	C₂³×D₄	C₂²×C₈₉	1	1	yes	characteristic
290	C₂⁴×C₄	C₂²×C₈₉	1	1	yes	characteristic
291	C₂²×M₁₆	C₂²×C₈₉	1	2	yes	—
292	C₂×#121	—	2	1	no	—
293	C₂×#121	—	2	4	no	—
294	C₂×#121	—	2	1	no	—
295	C₂×#121	C₂²×C₈₉	1	2	yes	—
296	C₂×#121	C₂²×C₈₉	1	2	yes	—
297	C₂×#121	C₂²×C₈₉	1	8	yes	—
298	C₂×#117	—	2	1	no	—
299	C₂×#117	—	2	1	no	—
300	C₂×#117	—	2	4	no	—
301	C₂×#117	C₂²×C₈₉	1	2	yes	—
302	C₂×#117	C₂²×C₈₉	1	2	yes	complemented
303	C₂×#117	C₂²×C₈₉	1	8	yes	complemented
304	C₂²×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
305	C₂²×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic, complemented
306	C₂²×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic, complemented
307	C₂²×C₂²⋊C₄	—	2	1	no	—
308	C₂²×C₂²⋊C₄	—	2	1	no	—
309	C₂²×C₄²	—	2	1	no	—
310	#482	C₂²×C₈₉	1	32	yes	complemented
311	#481	C₂²×C₈₉	1	16	yes	complemented
312	#480	C₂²×C₈₉	1	16	yes	complemented
313	#428	C₂²×C₈₉	1	16	yes	complemented
314	C₂×#107	C₂²×C₈₉	1	4	yes	complemented
315	C₂×#106	C₂²×C₈₉	1	4	yes	complemented
316	#418	C₂²×C₈₉	1	16	yes	complemented
317	C₂×#102	C₂²×C₈₉	1	4	yes	complemented
318	C₂×#102	C₂²×C₈₉	1	4	yes	complemented
319	#407	C₂²×C₈₉	1	8	yes	complemented
320	C₂×C₂²⋊C₈	C₂²×C₈₉	1	2	yes	complemented
321	C₂×C₂²⋊C₈	C₂²×C₈₉	1	2	yes	complemented
322	C₂⁵	—	2	1	no	—
323	C₂⁵	D₄×C₈₉	1	1	yes	characteristic
324	C₂²×D₄	—	2	2	no	—
325	C₂²×D₄	—	2	8	no	—
326	C₂²×D₄	—	2	2	no	—
327	C₂²×D₄	—	2	1	no	—
328	C₂²×D₄	—	2	2	no	—
329	C₂²×D₄	—	2	8	no	—
330	C₂²×D₄	—	4	8	no	—
331	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
332	C₂²×D₄	C₂³×C₈₉	1	4	yes	—
333	C₂³×C₄	—	2	8	no	—
334	C₂³×C₄	—	2	1	no	—
335	C₂³×C₄	—	2	1	no	—
336	C₂³×C₄	—	2	1	no	—
337	C₂³×C₄	D₄×C₈₉	1	1	yes	characteristic
338	C₂³×C₄	D₄×C₈₉	1	4	yes	—
339	C₂³×C₄	D₄×C₈₉	1	1	yes	characteristic
340	C₂³×C₄	D₄×C₈₉	1	1	yes	characteristic
341	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
342	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
343	C₂³×C₄	C₂³×C₈₉	1	4	yes	—
344	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
345	C₂×M₁₆	—	2	8	no	—
346	C₂×M₁₆	C₂³×C₈₉	1	8	yes	—
347	C₂²×C₈	C₂³×C₈₉	1	4	yes	—
348	#121	—	2	8	no	—
349	#121	—	2	4	no	—
350	#121	—	2	4	no	—
351	#121	—	2	8	no	—
352	#121	C₂³×C₈₉	1	8	yes	—
353	#121	C₂³×C₈₉	1	8	yes	—
354	#117	—	2	4	no	—
355	#117	—	2	8	no	—
356	#117	—	2	8	no	—
357	#117	—	2	4	no	—
358	#117	C₂³×C₈₉	1	8	yes	—
359	#117	C₂³×C₈₉	1	8	yes	complemented
360	C₂×C₄⋊C₄	—	2	1	no	—
361	C₂×C₄⋊C₄	—	2	2	no	—
362	C₂×C₄⋊C₄	—	2	2	no	—
363	C₂×C₄⋊C₄	—	2	2	no	—
364	C₂×C₄⋊C₄	—	2	1	no	—
365	C₂×C₄⋊C₄	—	2	1	no	—
366	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
367	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
368	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
369	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
370	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	complemented
371	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	complemented
372	C₂×C₂²⋊C₄	—	2	2	no	—
373	C₂×C₂²⋊C₄	—	2	8	no	—
374	C₂×C₂²⋊C₄	—	2	2	no	—
375	C₂×C₂²⋊C₄	—	2	2	no	—
376	C₂×C₂²⋊C₄	—	2	2	no	—
377	C₂×C₂²⋊C₄	—	2	8	no	—
378	C₂×C₄²	—	2	2	no	—
379	C₂×C₄²	—	2	8	no	—
380	C₂×C₄²	—	2	2	no	—
381	#107	C₂³×C₈₉	1	16	yes	complemented
382	#106	C₂³×C₈₉	1	16	yes	complemented
383	#102	C₂³×C₈₉	1	16	yes	complemented
384	#102	C₂³×C₈₉	1	16	yes	complemented
385	C₂²⋊C₈	C₂³×C₈₉	1	8	yes	complemented
386	C₂²⋊C₈	C₂³×C₈₉	1	8	yes	complemented
387	C₂⁴	—	2	4	no	—
388	C₂⁴	—	2	4	no	—
389	C₂⁴	—	2	4	no	—
390	C₂⁴	—	2	2	no	—
391	C₂⁴	—	4	2	no	—
392	C₂⁴	—	4	4	no	—
393	C₂⁴	—	4	4	no	—
394	C₂⁴	—	4	4	no	—
395	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
396	C₂⁴	C₂×D₄×C₈₉	1	4	yes	—
397	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
398	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
399	C₂×D₄	—	2	8	no	—
400	C₂×D₄	—	2	8	no	—
401	C₂×D₄	—	2	4	no	—
402	C₂×D₄	—	2	2	no	—
403	C₂×D₄	—	2	8	no	—
404	C₂×D₄	—	2	16	no	—
405	C₂×D₄	—	4	32	no	—
406	C₂×D₄	—	4	8	no	—
407	C₂×D₄	—	4	8	no	—
408	C₂×D₄	—	4	8	no	—
409	C₂×D₄	—	4	4	no	—
410	C₂×D₄	C₂⁴×C₈₉	1	4	yes	—
411	C₂²×C₄	—	2	4	no	—
412	C₂²×C₄	—	2	4	no	—
413	C₂²×C₄	—	2	16	no	—
414	C₂²×C₄	—	2	4	no	—
415	C₂²×C₄	—	2	4	no	—
416	C₂²×C₄	—	2	4	no	—
417	C₂²×C₄	—	2	16	no	—
418	C₂²×C₄	—	2	2	no	—
419	C₂²×C₄	—	2	2	no	—
420	C₂²×C₄	—	2	4	no	—
421	C₂²×C₄	—	2	4	no	—
422	C₂²×C₄	—	2	4	no	—
423	C₂²×C₄	—	2	2	no	—
424	C₂²×C₄	—	4	4	no	—
425	C₂²×C₄	—	4	4	no	—
426	C₂²×C₄	—	4	4	no	—
427	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
428	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
429	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
430	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
431	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
432	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
433	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
434	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
435	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
436	C₂²×C₄	#41×C₈₉	1	8	yes	—
437	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
438	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
439	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
440	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
441	M₁₆	—	2	16	no	—
442	C₂×C₈	—	2	8	no	—
443	C₂×C₈	C₂⁴×C₈₉	1	8	yes	—
444	C₄⋊C₄	—	2	2	no	—
445	C₄⋊C₄	—	2	2	no	—
446	C₄⋊C₄	—	2	4	no	—
447	C₄⋊C₄	—	2	2	no	—
448	C₄⋊C₄	—	2	4	no	—
449	C₄⋊C₄	—	2	4	no	—
450	C₄⋊C₄	C₂⁴×C₈₉	1	4	yes	—
451	C₄⋊C₄	C₂⁴×C₈₉	1	4	yes	—
452	C₄⋊C₄	C₂⁴×C₈₉	1	4	yes	—
453	C₂²⋊C₄	—	2	8	no	—
454	C₂²⋊C₄	—	2	8	no	—
455	C₂²⋊C₄	—	2	8	no	—
456	C₂²⋊C₄	—	2	8	no	—
457	C₄²	—	2	8	no	—
458	C₄²	—	2	8	no	—
459	C₂³	—	2	2	no	—
460	C₂³	—	2	2	no	—
461	C₂³	—	2	2	no	—
462	C₂³	—	2	2	no	—
463	C₂³	—	2	8	no	—
464	C₂³	—	2	2	no	—
465	C₂³	—	2	8	no	—
466	C₂³	—	2	2	no	—
467	C₂³	—	2	2	no	—
468	C₂³	—	2	4	no	—
469	C₂³	—	4	4	no	—
470	C₂³	—	4	16	no	—
471	C₂³	—	4	8	no	—
472	C₂³	—	4	8	no	—
473	C₂³	—	4	8	no	—
474	C₂³	—	4	4	no	—
475	C₂³	—	4	4	no	—
476	C₂³	—	4	4	no	—
477	C₂³	—	8	16	no	—
478	C₂³	#123×C₈₉	1	4	yes	—
479	C₂³	AGL(1,ℤ/8)×C₈₉	1	4	yes	—
480	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
481	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
482	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
483	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
484	D₄	—	2	8	no	—
485	D₄	—	4	16	no	—
486	D₄	—	4	16	no	—
487	D₄	—	4	8	no	—
488	D₄	—	4	4	no	—
489	C₂×C₄	—	2	8	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	8	no	—
493	C₂×C₄	—	2	4	no	—
494	C₂×C₄	—	2	4	no	—
495	C₂×C₄	—	2	8	no	—
496	C₂×C₄	—	2	4	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	—	4	8	no	—
502	C₂×C₄	—	4	8	no	—
503	C₂×C₄	—	4	8	no	—
504	C₂×C₄	—	4	4	no	—
505	C₂×C₄	—	4	4	no	—
506	C₂×C₄	—	4	4	no	—
507	C₂×C₄	#123×C₈₉	1	8	yes	—
508	C₂×C₄	#123×C₈₉	1	4	yes	—
509	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
510	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
511	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
512	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
513	C₂×C₄	C₂×#41×C₈₉	1	4	yes	—
514	C₂×C₄	C₂×#41×C₈₉	1	8	yes	—
515	C₂×C₄	C₂⁵×C₈₉	1	1	yes	characteristic
516	C₈	—	2	8	no	—
517	C₂²	—	2	2	no	—
518	C₂²	—	2	2	no	—
519	C₂²	—	2	2	no	—
520	C₂²	—	2	8	no	—
521	C₂²	—	2	8	no	—
522	C₂²	—	2	2	no	—
523	C₂²	—	2	2	no	—
524	C₂²	—	2	2	no	—
525	C₂²	—	2	2	no	—
526	C₂²	—	4	16	no	—
527	C₂²	—	4	4	no	—
528	C₂²	—	4	4	no	—
529	C₂²	—	4	4	no	—
530	C₂²	—	4	4	no	—
531	C₂²	—	8	16	no	—
532	C₂²	—	8	8	no	—
533	C₂²	C₂×#123×C₈₉	1	4	yes	—
534	C₂²	C₂×#123×C₈₉	1	2	yes	—
535	C₂²	C₂×AGL(1,ℤ/8)×C₈₉	1	4	yes	—
536	C₂²	C₂×AGL(1,ℤ/8)×C₈₉	1	2	yes	—
537	C₂²	#575×C₈₉	1	2	yes	—
538	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
539	C₄	—	2	2	no	—
540	C₄	—	2	2	no	—
541	C₄	—	4	4	no	—
542	C₄	—	4	4	no	—
543	C₄	—	4	4	no	—
544	C₄	C₂×#123×C₈₉	1	4	yes	—
545	C₄	C₂×#123×C₈₉	1	2	yes	—
546	C₄	C₂²×#41×C₈₉	1	2	yes	—
547	C₂	—	2	4	no	—
548	C₂	—	2	4	no	—
549	C₂	—	4	2	no	—
550	C₂	—	8	4	no	—
551	C₂	C₈₉×#3094	1	4	yes	direct product factor
552	C₂	C₂²×#123×C₈₉	1	1	yes	characteristic
553	C₂	C₂²×AGL(1,ℤ/8)×C₈₉	1	1	yes	characteristic
554	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
2. C₂²×#117×C₈₉	C₂	semidirect
3. C₈₉×#3094	C₂	direct (N × H)
4. C₂×#481×C₈₉	C₂	semidirect
5. C₂×#482×C₈₉	C₂	semidirect
6. C₂×#480×C₈₉	C₂	semidirect
7. C₂×#428×C₈₉	C₂	semidirect
8. C₂×#418×C₈₉	C₂	semidirect
9. C₂×#407×C₈₉	C₂	semidirect
24. C₂×#117×C₈₉	C₂²	semidirect
25. C₂×#117×C₈₉	C₂²	semidirect
27. C₂²×C₄⋊C₄×C₈₉	C₂²	semidirect
28. C₂²×C₄⋊C₄×C₈₉	C₂²	semidirect
32. #482×C₈₉	C₂²	semidirect
33. #481×C₈₉	C₂²	semidirect
34. #480×C₈₉	C₂²	semidirect
35. #428×C₈₉	C₂²	semidirect
36. C₂×#107×C₈₉	C₂²	semidirect
37. C₂×#106×C₈₉	C₂²	semidirect
38. #418×C₈₉	C₂²	semidirect
39. C₂×#102×C₈₉	C₂²	semidirect
40. C₂×#102×C₈₉	C₂²	semidirect
41. #407×C₈₉	C₂²	semidirect
42. C₂×C₂²⋊C₈×C₈₉	C₂²	semidirect
43. C₂×C₂²⋊C₈×C₈₉	C₂²	semidirect
81. #117×C₈₉	C₂³	semidirect
92. C₂×C₄⋊C₄×C₈₉	C₂³	semidirect
93. C₂×C₄⋊C₄×C₈₉	C₂³	semidirect
103. #107×C₈₉	C₂³	semidirect
104. #106×C₈₉	C₂³	semidirect
105. #102×C₈₉	C₂³	semidirect
106. #102×C₈₉	C₂³	semidirect
107. C₂²⋊C₈×C₈₉	C₂³	semidirect
108. C₂²⋊C₈×C₈₉	C₂³	semidirect
269. C₂×#3094	C₈₉	direct (N × H)
274. C₂×C₈₉	#3094	direct (N × H)
279. C₂²×#117	C₂×C₈₉	semidirect
280. #3094	C₂×C₈₉	direct (N × H)
281. C₂×#481	C₂×C₈₉	semidirect
282. C₂×#482	C₂×C₈₉	semidirect
283. C₂×#480	C₂×C₈₉	semidirect
284. C₂×#428	C₂×C₈₉	semidirect
285. C₂×#418	C₂×C₈₉	semidirect
286. C₂×#407	C₂×C₈₉	semidirect
287. C₈₉	C₂×#3094	direct (N × H)
302. C₂×#117	C₂²×C₈₉	semidirect
303. C₂×#117	C₂²×C₈₉	semidirect
305. C₂²×C₄⋊C₄	C₂²×C₈₉	semidirect
306. C₂²×C₄⋊C₄	C₂²×C₈₉	semidirect
310. #482	C₂²×C₈₉	semidirect
311. #481	C₂²×C₈₉	semidirect
312. #480	C₂²×C₈₉	semidirect
313. #428	C₂²×C₈₉	semidirect
314. C₂×#107	C₂²×C₈₉	semidirect
315. C₂×#106	C₂²×C₈₉	semidirect
316. #418	C₂²×C₈₉	semidirect
317. C₂×#102	C₂²×C₈₉	semidirect
318. C₂×#102	C₂²×C₈₉	semidirect
319. #407	C₂²×C₈₉	semidirect
320. C₂×C₂²⋊C₈	C₂²×C₈₉	semidirect
321. C₂×C₂²⋊C₈	C₂²×C₈₉	semidirect
359. #117	C₂³×C₈₉	semidirect
370. C₂×C₄⋊C₄	C₂³×C₈₉	semidirect
371. C₂×C₄⋊C₄	C₂³×C₈₉	semidirect
381. #107	C₂³×C₈₉	semidirect
382. #106	C₂³×C₈₉	semidirect
383. #102	C₂³×C₈₉	semidirect
384. #102	C₂³×C₈₉	semidirect
385. C₂²⋊C₈	C₂³×C₈₉	semidirect
386. C₂²⋊C₈	C₂³×C₈₉	semidirect
551. C₂	C₈₉×#3094	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×#121×C₈₉	2
C₂²×#117×C₈₉	2
C₈₉×#3094	2
C₂×#481×C₈₉	2
C₂×#482×C₈₉	2
C₂×#480×C₈₉	2
C₂×#428×C₈₉	2
C₂×#418×C₈₉	2
C₂×#407×C₈₉	2
C₂×#3094	89

Subgroups — C2×C89×#3094 #36438357

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈₉×#3094 #36438357