Subgroups — C₂×C₈₉×#2076 #36401485

Order 22784.

1270 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#107×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₂²⋊C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#380×C₈₉	C₂	1	2	yes	maximal
4	C₈₉×#2076	C₂	1	8	yes	maximal, direct product factor
5	C₂×#340×C₈₉	C₂	1	2	yes	maximal, complemented
6	C₂×#340×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁴×C₄×C₈₉	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
11	C₂²×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
12	C₂²×C₂²⋊C₄×C₈₉	—	2	2	no	—
13	C₂×#107×C₈₉	—	2	4	no	—
14	C₂×#107×C₈₉	C₂²	1	4	yes	complemented
15	C₂×C₂²⋊C₈×C₈₉	—	2	4	no	—
16	C₂×C₂²⋊C₈×C₈₉	C₂²	1	4	yes	—
17	#380×C₈₉	—	2	4	no	—
18	#380×C₈₉	C₂²	1	8	yes	—
19	C₂×#95×C₈₉	—	2	2	no	—
20	#340×C₈₉	C₂²	1	8	yes	complemented
21	#340×C₈₉	C₂²	1	4	yes	complemented
22	C₂⁵×C₈₉	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₈₉	—	2	2	no	—
24	C₂³×C₄×C₈₉	—	2	2	no	—
25	C₂³×C₄×C₈₉	—	2	4	no	—
26	C₂³×C₄×C₈₉	—	2	4	no	—
27	C₂³×C₄×C₈₉	—	2	2	no	—
28	C₂³×C₄×C₈₉	—	2	2	no	—
29	C₂³×C₄×C₈₉	D₄	1	2	yes	—
30	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
31	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
32	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₈₉	—	2	4	no	—
34	C₂²×C₈×C₈₉	—	2	2	no	—
35	C₂²×C₈×C₈₉	—	2	4	no	—
36	C₂²×C₈×C₈₉	D₄	1	4	yes	—
37	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
38	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
39	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
40	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
41	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
42	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
43	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
44	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
45	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
46	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
47	C₂×C₂²⋊C₄×C₈₉	—	2	8	no	—
48	C₂×C₂²⋊C₄×C₈₉	—	4	8	no	—
49	#107×C₈₉	—	2	8	no	—
50	C₂²⋊C₈×C₈₉	—	2	8	no	—
51	#95×C₈₉	—	2	8	no	—
52	C₂⁴×C₈₉	—	2	2	no	—
53	C₂⁴×C₈₉	—	2	4	no	—
54	C₂⁴×C₈₉	—	2	2	no	—
55	C₂⁴×C₈₉	—	2	4	no	—
56	C₂⁴×C₈₉	Q₁₆	1	4	yes	—
57	C₂⁴×C₈₉	C₂×D₄	1	2	yes	—
58	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₈₉	—	2	2	no	—
60	C₂²×C₄×C₈₉	—	2	4	no	—
61	C₂²×C₄×C₈₉	—	2	4	no	—
62	C₂²×C₄×C₈₉	—	2	8	no	—
63	C₂²×C₄×C₈₉	—	2	2	no	—
64	C₂²×C₄×C₈₉	—	2	2	no	—
65	C₂²×C₄×C₈₉	—	2	4	no	—
66	C₂²×C₄×C₈₉	—	2	4	no	—
67	C₂²×C₄×C₈₉	—	2	4	no	—
68	C₂²×C₄×C₈₉	—	2	4	no	—
69	C₂²×C₄×C₈₉	—	4	4	no	—
70	C₂²×C₄×C₈₉	—	4	4	no	—
71	C₂²×C₄×C₈₉	—	4	8	no	—
72	C₂²×C₄×C₈₉	—	4	4	no	—
73	C₂²×C₄×C₈₉	—	4	4	no	—
74	C₂²×C₄×C₈₉	—	4	4	no	—
75	C₂²×C₄×C₈₉	—	4	4	no	—
76	C₂²×C₄×C₈₉	—	4	4	no	—
77	C₂²×C₄×C₈₉	—	4	4	no	—
78	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₈₉	#41	1	2	yes	—
82	C₂²×C₄×C₈₉	#41	1	4	yes	—
83	C₂²×C₄×C₈₉	#41	1	4	yes	—
84	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
85	C₂×C₈×C₈₉	—	2	8	no	—
86	C₂×C₈×C₈₉	—	2	4	no	—
87	C₂×C₈×C₈₉	—	2	2	no	—
88	C₂×C₈×C₈₉	—	2	4	no	—
89	C₂×C₈×C₈₉	—	4	4	no	—
90	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
91	C₄⋊C₄×C₈₉	—	2	4	no	—
92	C₄⋊C₄×C₈₉	—	2	4	no	—
93	C₄⋊C₄×C₈₉	—	2	4	no	—
94	C₄⋊C₄×C₈₉	—	2	4	no	—
95	C₄⋊C₄×C₈₉	—	4	4	no	—
96	C₄⋊C₄×C₈₉	—	4	4	no	—
97	C₂²⋊C₄×C₈₉	—	4	16	no	—
98	C₂³×C₈₉	—	2	2	no	—
99	C₂³×C₈₉	—	2	4	no	—
100	C₂³×C₈₉	—	2	2	no	—
101	C₂³×C₈₉	—	2	4	no	—
102	C₂³×C₈₉	—	2	8	no	—
103	C₂³×C₈₉	—	2	16	no	—
104	C₂³×C₈₉	—	4	4	no	—
105	C₂³×C₈₉	—	4	4	no	—
106	C₂³×C₈₉	—	4	8	no	—
107	C₂³×C₈₉	#121	1	2	yes	—
108	C₂³×C₈₉	#123	1	4	yes	—
109	C₂³×C₈₉	C₂×Q₁₆	1	4	yes	—
110	C₂³×C₈₉	C₂×Q₁₆	1	2	yes	—
111	C₂³×C₈₉	#135	1	2	yes	—
112	C₂³×C₈₉	AGL(1,ℤ/8)	1	4	yes	—
113	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₈₉	—	2	2	no	—
115	C₂×C₄×C₈₉	—	2	4	no	—
116	C₂×C₄×C₈₉	—	4	4	no	—
117	C₂×C₄×C₈₉	—	4	2	no	—
118	C₂×C₄×C₈₉	—	4	2	no	—
119	C₂×C₄×C₈₉	—	4	4	no	—
120	C₂×C₄×C₈₉	—	4	8	no	—
121	C₂×C₄×C₈₉	—	4	4	no	—
122	C₂×C₄×C₈₉	—	4	4	no	—
123	C₂×C₄×C₈₉	—	4	4	no	—
124	C₂×C₄×C₈₉	—	4	4	no	—
125	C₂×C₄×C₈₉	—	4	4	no	—
126	C₂×C₄×C₈₉	—	4	4	no	—
127	C₂×C₄×C₈₉	—	8	8	no	—
128	C₂×C₄×C₈₉	—	8	8	no	—
129	C₂×C₄×C₈₉	#121	1	2	yes	—
130	C₂×C₄×C₈₉	#123	1	4	yes	—
131	C₂×C₄×C₈₉	#123	1	4	yes	—
132	C₂×C₄×C₈₉	#124	1	4	yes	—
133	C₂×C₄×C₈₉	#126	1	8	yes	—
134	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
136	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
137	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
138	C₈×C₈₉	—	2	4	no	—
139	C₈×C₈₉	—	4	2	no	—
140	C₂²×C₈₉	—	2	8	no	—
141	C₂²×C₈₉	—	2	4	no	—
142	C₂²×C₈₉	—	4	4	no	—
143	C₂²×C₈₉	—	4	16	no	—
144	C₂²×C₈₉	—	4	2	no	—
145	C₂²×C₈₉	—	4	2	no	—
146	C₂²×C₈₉	#397	1	4	yes	—
147	C₂²×C₈₉	#467	1	4	yes	—
148	C₂²×C₈₉	#469	1	4	yes	—
149	C₂²×C₈₉	#482	1	8	yes	—
150	C₂²×C₈₉	#484	1	8	yes	—
151	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
152	C₂²×C₈₉	C₂×#123	1	2	yes	—
153	C₂²×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
154	C₂²×C₈₉	C₂×#135	1	1	yes	characteristic
155	C₂²×C₈₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
156	C₄×C₈₉	—	4	2	no	—
157	C₄×C₈₉	—	8	4	no	—
158	C₄×C₈₉	—	8	4	no	—
159	C₄×C₈₉	#397	1	4	yes	—
160	C₄×C₈₉	C₂×#123	1	1	yes	characteristic
161	C₄×C₈₉	C₂×#124	1	1	yes	characteristic
162	C₄×C₈₉	C₂×#126	1	2	yes	—
163	C₂×#2076	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
164	C₂×C₈₉	—	4	4	no	—
165	C₂×C₈₉	#2076	1	8	yes	direct product factor
166	C₂×C₈₉	C₂×#397	1	1	yes	characteristic
167	C₂×C₈₉	C₂×#467	1	1	yes	characteristic
168	C₂×C₈₉	C₂×#469	1	1	yes	characteristic
169	C₂×C₈₉	C₂×#482	1	2	yes	—
170	C₂×C₈₉	C₂×#484	1	2	yes	—
171	C₂²×#107	C₂×C₈₉	1	1	yes	characteristic, complemented
172	C₂²×C₂²⋊C₈	C₂×C₈₉	1	1	yes	characteristic
173	C₂×#380	C₂×C₈₉	1	2	yes	—
174	#2076	C₂×C₈₉	1	8	yes	direct product factor
175	C₂×#340	C₂×C₈₉	1	2	yes	complemented
176	C₂×#340	C₂×C₈₉	1	1	yes	characteristic, complemented
177	C₈₉	C₂×#2076	1	1	yes	characteristic, direct product factor, Sylow
178	C₂⁴×C₄	C₂²×C₈₉	1	1	yes	characteristic
179	C₂³×C₈	C₂²×C₈₉	1	1	yes	characteristic
180	C₂³×C₈	C₂²×C₈₉	1	1	yes	characteristic
181	C₂²×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
182	C₂²×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
183	C₂²×C₂²⋊C₄	—	2	2	no	—
184	C₂×#107	—	2	4	no	—
185	C₂×#107	C₂²×C₈₉	1	4	yes	complemented
186	C₂×C₂²⋊C₈	—	2	4	no	—
187	C₂×C₂²⋊C₈	C₂²×C₈₉	1	4	yes	—
188	#380	—	2	4	no	—
189	#380	C₂²×C₈₉	1	8	yes	—
190	C₂×#95	—	2	2	no	—
191	#340	C₂²×C₈₉	1	4	yes	complemented
192	#340	C₂²×C₈₉	1	8	yes	complemented
193	C₂⁵	D₄×C₈₉	1	1	yes	characteristic
194	C₂³×C₄	—	2	2	no	—
195	C₂³×C₄	—	2	2	no	—
196	C₂³×C₄	—	2	4	no	—
197	C₂³×C₄	—	2	4	no	—
198	C₂³×C₄	—	2	2	no	—
199	C₂³×C₄	—	2	2	no	—
200	C₂³×C₄	D₄×C₈₉	1	1	yes	characteristic
201	C₂³×C₄	D₄×C₈₉	1	2	yes	—
202	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
203	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
204	C₂²×C₈	—	2	4	no	—
205	C₂²×C₈	—	2	2	no	—
206	C₂²×C₈	—	2	4	no	—
207	C₂²×C₈	D₄×C₈₉	1	4	yes	—
208	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
209	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
210	C₂×C₄⋊C₄	—	2	4	no	—
211	C₂×C₄⋊C₄	—	2	4	no	—
212	C₂×C₄⋊C₄	—	2	4	no	—
213	C₂×C₄⋊C₄	—	2	4	no	—
214	C₂×C₄⋊C₄	—	2	2	no	—
215	C₂×C₄⋊C₄	—	2	2	no	—
216	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
217	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
218	C₂×C₂²⋊C₄	—	2	8	no	—
219	C₂×C₂²⋊C₄	—	4	8	no	—
220	#107	—	2	8	no	—
221	C₂²⋊C₈	—	2	8	no	—
222	#95	—	2	8	no	—
223	C₂⁴	—	2	2	no	—
224	C₂⁴	—	2	4	no	—
225	C₂⁴	—	2	2	no	—
226	C₂⁴	—	2	4	no	—
227	C₂⁴	Q₁₆×C₈₉	1	4	yes	—
228	C₂⁴	C₂×D₄×C₈₉	1	2	yes	—
229	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	2	8	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	—	2	4	no	—
239	C₂²×C₄	—	2	4	no	—
240	C₂²×C₄	—	4	4	no	—
241	C₂²×C₄	—	4	4	no	—
242	C₂²×C₄	—	4	8	no	—
243	C₂²×C₄	—	4	4	no	—
244	C₂²×C₄	—	4	4	no	—
245	C₂²×C₄	—	4	4	no	—
246	C₂²×C₄	—	4	4	no	—
247	C₂²×C₄	—	4	4	no	—
248	C₂²×C₄	—	4	4	no	—
249	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
250	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
251	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
252	C₂²×C₄	#41×C₈₉	1	2	yes	—
253	C₂²×C₄	#41×C₈₉	1	4	yes	—
254	C₂²×C₄	#41×C₈₉	1	4	yes	—
255	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
256	C₂×C₈	—	2	8	no	—
257	C₂×C₈	—	2	4	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	4	no	—
260	C₂×C₈	—	4	4	no	—
261	C₂×C₈	C₂×D₄×C₈₉	1	4	yes	—
262	C₄⋊C₄	—	2	4	no	—
263	C₄⋊C₄	—	2	4	no	—
264	C₄⋊C₄	—	2	4	no	—
265	C₄⋊C₄	—	2	4	no	—
266	C₄⋊C₄	—	4	4	no	—
267	C₄⋊C₄	—	4	4	no	—
268	C₂²⋊C₄	—	4	16	no	—
269	C₂³	—	2	8	no	—
270	C₂³	—	2	16	no	—
271	C₂³	—	2	2	no	—
272	C₂³	—	2	4	no	—
273	C₂³	—	2	2	no	—
274	C₂³	—	2	4	no	—
275	C₂³	—	4	4	no	—
276	C₂³	—	4	4	no	—
277	C₂³	—	4	8	no	—
278	C₂³	#121×C₈₉	1	2	yes	—
279	C₂³	#123×C₈₉	1	4	yes	—
280	C₂³	C₂×Q₁₆×C₈₉	1	4	yes	—
281	C₂³	C₂×Q₁₆×C₈₉	1	2	yes	—
282	C₂³	#135×C₈₉	1	2	yes	—
283	C₂³	AGL(1,ℤ/8)×C₈₉	1	4	yes	—
284	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	—	4	8	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	4	no	—
292	C₂×C₄	—	4	4	no	—
293	C₂×C₄	—	4	4	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	—	4	4	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	8	8	no	—
299	C₂×C₄	—	8	8	no	—
300	C₂×C₄	#121×C₈₉	1	2	yes	—
301	C₂×C₄	#123×C₈₉	1	4	yes	—
302	C₂×C₄	#123×C₈₉	1	4	yes	—
303	C₂×C₄	#124×C₈₉	1	4	yes	—
304	C₂×C₄	#126×C₈₉	1	8	yes	—
305	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
306	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
307	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
308	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
309	C₈	—	2	4	no	—
310	C₈	—	4	2	no	—
311	C₂²	—	2	8	no	—
312	C₂²	—	2	4	no	—
313	C₂²	—	4	4	no	—
314	C₂²	—	4	16	no	—
315	C₂²	—	4	2	no	—
316	C₂²	—	4	2	no	—
317	C₂²	#397×C₈₉	1	4	yes	—
318	C₂²	#467×C₈₉	1	4	yes	—
319	C₂²	#469×C₈₉	1	4	yes	—
320	C₂²	#482×C₈₉	1	8	yes	—
321	C₂²	#484×C₈₉	1	8	yes	—
322	C₂²	C₂×#121×C₈₉	1	1	yes	characteristic
323	C₂²	C₂×#123×C₈₉	1	2	yes	—
324	C₂²	C₂²×Q₁₆×C₈₉	1	1	yes	characteristic
325	C₂²	C₂×#135×C₈₉	1	1	yes	characteristic
326	C₂²	C₂×AGL(1,ℤ/8)×C₈₉	1	2	yes	—
327	C₄	—	4	2	no	—
328	C₄	—	8	4	no	—
329	C₄	—	8	4	no	—
330	C₄	#397×C₈₉	1	4	yes	—
331	C₄	C₂×#123×C₈₉	1	1	yes	characteristic
332	C₄	C₂×#124×C₈₉	1	1	yes	characteristic
333	C₄	C₂×#126×C₈₉	1	2	yes	—
334	C₂	—	4	4	no	—
335	C₂	C₈₉×#2076	1	8	yes	direct product factor
336	C₂	C₂×#397×C₈₉	1	1	yes	characteristic
337	C₂	C₂×#467×C₈₉	1	1	yes	characteristic
338	C₂	C₂×#469×C₈₉	1	1	yes	characteristic
339	C₂	C₂×#482×C₈₉	1	2	yes	—
340	C₂	C₂×#484×C₈₉	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₈₉	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
prime radical	C₂⁵×C₈₉	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₈₉ ◁ C₂⁴×C₄×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#107×C₈₉	C₂	semidirect
4. C₈₉×#2076	C₂	direct (N × H)
5. C₂×#340×C₈₉	C₂	semidirect
6. C₂×#340×C₈₉	C₂	semidirect
14. C₂×#107×C₈₉	C₂²	semidirect
20. #340×C₈₉	C₂²	semidirect
21. #340×C₈₉	C₂²	semidirect
163. C₂×#2076	C₈₉	direct (N × H)
165. C₂×C₈₉	#2076	direct (N × H)
171. C₂²×#107	C₂×C₈₉	semidirect
174. #2076	C₂×C₈₉	direct (N × H)
175. C₂×#340	C₂×C₈₉	semidirect
176. C₂×#340	C₂×C₈₉	semidirect
177. C₈₉	C₂×#2076	direct (N × H)
185. C₂×#107	C₂²×C₈₉	semidirect
191. #340	C₂²×C₈₉	semidirect
192. #340	C₂²×C₈₉	semidirect
335. C₂	C₈₉×#2076	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×#107×C₈₉	2
C₂²×C₂²⋊C₈×C₈₉	2
C₂×#380×C₈₉	2
C₈₉×#2076	2
C₂×#340×C₈₉	2
C₂×#2076	89

Subgroups — C2×C89×#2076 #36401485

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈₉×#2076 #36401485