Subgroups — C₂×#26018997 #36523399

Order 22784.

830 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151650	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×D₈₉	C₂	1	1	yes	maximal, characteristic
3	#26018997	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19151350	C₂	1	2	yes	maximal, complemented
5	C₂×#19151349	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#334×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₈×D₈₉	C₄	1	1	yes	characteristic
8	C₂²×C₈×D₈₉	C₄	1	1	yes	characteristic
9	C₂×C₄²×D₈₉	C₂²	1	1	yes	characteristic
10	#19151650	—	2	4	no	—
11	#19151650	C₂²	1	8	yes	—
12	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
13	C₂×#15667924	—	2	2	no	—
14	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
15	C₂×C₈×Dic₈₉	—	2	2	no	—
16	C₂×C₈×Dic₈₉	C₄	1	1	yes	characteristic
17	C₂×C₈×Dic₈₉	C₄	1	1	yes	characteristic
18	C₄×C₈×D₈₉	C₄	1	4	yes	—
19	C₄×C₈×D₈₉	C₂²	1	4	yes	—
20	#19151350	C₂²	1	8	yes	complemented
21	#19151349	C₂²	1	4	yes	complemented
22	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
24	#334×C₈₉	C₂²	1	4	yes	complemented
25	C₂²×C₄×D₈₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×D₈₉	D₄	1	1	yes	characteristic
27	C₂²×C₄×D₈₉	Q₈	1	1	yes	characteristic
28	C₂²×#10623494	—	2	2	no	—
29	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
30	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×D₈₉	—	2	4	no	—
32	C₂×C₈×D₈₉	C₈	1	8	yes	complemented
33	C₂×C₈×D₈₉	C₂×C₄	1	4	yes	—
34	C₂×C₈×D₈₉	C₂×C₄	1	4	yes	—
35	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×Dic₈₉	D₄	1	1	yes	characteristic
37	C₂×C₄×Dic₈₉	Q₈	1	1	yes	characteristic
38	C₄²×D₈₉	C₂×C₄	1	2	yes	—
39	C₄²×D₈₉	D₄	1	2	yes	—
40	C₄²×D₈₉	Q₈	1	2	yes	—
41	C₄²×D₈₉	C₂³	1	2	yes	—
42	C₄⋊#10623494	—	2	2	no	—
43	C₄⋊#10623494	C₂³	1	4	yes	—
44	#15667924	—	2	8	no	—
45	C₄×#10623494	C₂×C₄	1	2	yes	—
46	C₄×#10623494	C₂³	1	2	yes	—
47	C₈×Dic₈₉	—	2	8	no	—
48	C₈×Dic₈₉	C₂×C₄	1	4	yes	—
49	C₈×Dic₈₉	C₂×C₄	1	4	yes	—
50	C₂²×C₈×C₈₉	—	2	2	no	—
51	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
54	C₄⋊C₈×C₈₉	—	2	2	no	—
55	C₄⋊C₈×C₈₉	C₂³	1	4	yes	—
56	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
57	C₄×C₈×C₈₉	C₂³	1	2	yes	—
58	C₂³×D₈₉	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×Dic₈₉	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×Dic₈₉	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×Dic₈₉	C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×D₈₉	—	2	4	no	—
63	C₂×C₄×D₈₉	—	2	4	no	—
64	C₂×C₄×D₈₉	C₄⋊C₄	1	2	yes	—
65	C₂×C₄×D₈₉	C₄⋊C₄	1	2	yes	—
66	C₂×C₄×D₈₉	C₄⋊C₄	1	2	yes	—
67	C₂×C₄×D₈₉	C₂×C₈	1	4	yes	—
68	C₂×C₄×D₈₉	M₁₆	1	4	yes	—
69	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
70	C₂×C₄×D₈₉	C₂×D₄	1	2	yes	—
71	C₂×C₄×D₈₉	C₂×Q₈	1	2	yes	—
72	C₂×#10623494	—	2	2	no	—
73	C₂×#10623494	—	2	4	no	—
74	C₂×#10623494	—	4	4	no	—
75	C₂×#10623494	C₂×C₈	1	4	yes	—
76	C₂×#10623494	C₂²×C₄	1	2	yes	—
77	C₂×#10623494	C₂²×C₄	1	2	yes	—
78	C₈×D₈₉	—	2	8	no	—
79	C₈×D₈₉	C₂×C₈	1	16	yes	complemented
80	C₄×Dic₈₉	C₄⋊C₄	1	2	yes	—
81	C₄×Dic₈₉	C₄⋊C₄	1	2	yes	—
82	C₄×Dic₈₉	C₄⋊C₄	1	2	yes	—
83	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
84	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
85	C₄×Dic₈₉	C₂×Q₈	1	2	yes	—
86	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
89	C₂×C₈×C₈₉	—	2	2	no	—
90	C₂×C₈×C₈₉	—	2	4	no	—
91	C₂×C₈×C₈₉	—	4	4	no	—
92	C₂×C₈×C₈₉	C₂×C₈	1	4	yes	—
93	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
94	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
95	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
98	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
99	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂²×D₈₉	—	2	2	no	—
101	C₂²×D₈₉	—	2	2	no	—
102	C₂²×D₈₉	C₄⋊C₈	1	4	yes	—
103	C₂²×D₈₉	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₄×C₈₉	—	2	2	no	—
105	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
109	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
110	C₂×C₄×C₈₉	#135	1	4	yes	—
111	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
114	C₂×Dic₈₉	—	2	2	no	—
115	C₂×Dic₈₉	C₄⋊C₈	1	4	yes	—
116	C₂×Dic₈₉	#108	1	4	yes	—
117	C₂×Dic₈₉	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×Dic₈₉	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×Dic₈₉	C₂×C₄⋊C₄	1	2	yes	—
120	C₄×D₈₉	—	2	4	no	—
121	C₄×D₈₉	—	2	4	no	—
122	C₄×D₈₉	—	2	4	no	—
123	C₄×D₈₉	—	2	4	no	—
124	C₄×D₈₉	C₄⋊C₈	1	8	yes	—
125	C₄×D₈₉	C₂²×C₈	1	4	yes	—
126	C₄×D₈₉	C₂×M₁₆	1	4	yes	—
127	C₈×C₈₉	—	2	2	no	—
128	C₈×C₈₉	—	4	4	no	—
129	C₈×C₈₉	C₂²×C₈	1	4	yes	—
130	#10623494	—	2	2	no	—
131	#10623494	—	4	4	no	—
132	#10623494	C₂²×C₈	1	4	yes	—
133	C₂²×C₈₉	C₂×C₄⋊C₈	1	2	yes	—
134	C₂²×C₈₉	#427	1	2	yes	—
135	C₂²×C₈₉	C₂×#108	1	2	yes	—
136	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
137	C₂×D₈₉	—	2	8	no	—
138	C₂×D₈₉	—	2	2	no	—
139	C₂×D₈₉	—	2	2	no	—
140	C₂×D₈₉	C₂×C₄⋊C₈	1	4	yes	—
141	C₄×C₈₉	—	2	1	no	—
142	C₄×C₈₉	—	2	1	no	—
143	C₄×C₈₉	C₂×C₄⋊C₈	1	2	yes	—
144	C₄×C₈₉	#427	1	2	yes	—
145	C₄×C₈₉	C₂³×C₈	1	1	yes	characteristic
146	C₄×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
147	C₄×C₈₉	C₂×#135	1	2	yes	—
148	Dic₈₉	—	2	2	no	—
149	Dic₈₉	C₂×C₄⋊C₈	1	4	yes	—
150	Dic₈₉	C₂×#108	1	4	yes	—
151	C₂×#1564	—	89	1	no	maximal, Sylow
152	C₂×C₈₉	#1564	1	4	yes	complemented
153	C₂×C₈₉	C₂²×C₄⋊C₈	1	1	yes	characteristic
154	C₂×C₈₉	C₂×#427	1	1	yes	characteristic
155	C₂×C₈₉	C₂²×#108	1	1	yes	characteristic
156	D₈₉	—	2	4	no	—
157	C₂×#431	—	89	2	no	—
158	C₂²×C₄×C₈	—	89	1	no	—
159	#1564	—	89	8	no	—
160	C₂×#335	—	89	2	no	—
161	C₂×#334	—	89	1	no	—
162	C₂×#334	D₈₉	1	1	yes	characteristic, complemented
163	C₈₉	C₂×#1564	1	1	yes	characteristic, complemented, Sylow
164	C₂³×C₈	—	89	1	no	—
165	C₂³×C₈	—	89	1	no	—
166	C₂²×C₄²	—	89	1	no	—
167	#431	—	89	8	no	—
168	#431	—	178	4	no	—
169	C₂×C₄⋊C₈	—	89	2	no	—
170	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
171	C₂×C₂²⋊C₈	—	178	2	no	—
172	C₂×C₄×C₈	—	89	1	no	—
173	C₂×C₄×C₈	—	89	1	no	—
174	C₂×C₄×C₈	—	89	1	no	—
175	C₂×C₄×C₈	—	89	4	no	—
176	C₂×C₄×C₈	—	89	4	no	—
177	C₂×C₄×C₈	—	178	2	no	—
178	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
179	#335	—	89	8	no	—
180	#334	—	89	4	no	—
181	#334	C₂×D₈₉	1	4	yes	complemented
182	C₂³×C₄	—	89	1	no	—
183	C₂³×C₄	—	89	1	no	—
184	C₂³×C₄	—	89	1	no	—
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	—	89	1	no	—
187	C₂²×C₈	—	89	8	no	—
188	C₂²×C₈	—	89	4	no	—
189	C₂²×C₈	—	89	1	no	—
190	C₂²×C₈	—	89	4	no	—
191	C₂²×C₈	—	178	4	no	—
192	C₂²×C₈	—	178	2	no	—
193	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
194	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
195	C₂×C₄²	—	89	1	no	—
196	C₂×C₄²	—	89	1	no	—
197	C₂×C₄²	—	89	1	no	—
198	C₂×C₄²	—	89	2	no	—
199	C₂×C₄²	—	89	2	no	—
200	C₂×C₄²	—	89	2	no	—
201	C₂×C₄²	—	89	2	no	—
202	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
203	C₄⋊C₈	—	2	2	no	—
204	C₄⋊C₈	—	89	4	no	—
205	C₄⋊C₈	—	178	2	no	—
206	C₄⋊C₈	C₂²×D₈₉	1	4	yes	—
207	C₂²⋊C₈	—	178	8	no	—
208	C₄×C₈	—	89	4	no	—
209	C₄×C₈	—	89	4	no	—
210	C₄×C₈	—	89	2	no	—
211	C₄×C₈	—	89	2	no	—
212	C₄×C₈	—	178	8	no	—
213	C₄×C₈	C₄×D₈₉	1	2	yes	—
214	C₄×C₈	C₂²×D₈₉	1	2	yes	—
215	C₂⁴	—	89	1	no	—
216	C₂²×C₄	—	89	1	no	—
217	C₂²×C₄	—	89	1	no	—
218	C₂²×C₄	—	89	4	no	—
219	C₂²×C₄	—	89	4	no	—
220	C₂²×C₄	—	89	2	no	—
221	C₂²×C₄	—	89	2	no	—
222	C₂²×C₄	—	89	1	no	—
223	C₂²×C₄	—	89	2	no	—
224	C₂²×C₄	—	89	2	no	—
225	C₂²×C₄	—	89	2	no	—
226	C₂²×C₄	—	89	2	no	—
227	C₂²×C₄	—	178	4	no	—
228	C₂²×C₄	—	178	4	no	—
229	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
230	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
231	C₂²×C₄	Q₈×D₈₉	1	1	yes	characteristic
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	4	4	no	—
235	C₂×C₈	—	89	4	no	—
236	C₂×C₈	—	89	2	no	—
237	C₂×C₈	—	89	16	no	—
238	C₂×C₈	—	89	2	no	—
239	C₂×C₈	—	356	4	no	—
240	C₂×C₈	—	178	8	no	—
241	C₂×C₈	—	178	4	no	—
242	C₂×C₈	—	178	2	no	—
243	C₂×C₈	C₈×D₈₉	1	4	yes	—
244	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
245	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
246	C₄²	—	89	2	no	—
247	C₄²	—	89	2	no	—
248	C₄²	—	89	2	no	—
249	C₄²	—	89	2	no	—
250	C₄²	—	89	2	no	—
251	C₄²	—	89	2	no	—
252	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
253	C₄²	D₄×D₈₉	1	1	yes	characteristic
254	C₄²	Q₈×D₈₉	1	1	yes	characteristic
255	C₄²	C₂³×D₈₉	1	1	yes	characteristic
256	C₂³	—	89	4	no	—
257	C₂³	—	89	2	no	—
258	C₂³	—	178	2	no	—
259	C₂³	—	178	2	no	—
260	C₂³	C₄⋊C₄×D₈₉	1	1	yes	characteristic
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	89	4	no	—
263	C₂×C₄	—	89	4	no	—
264	C₂×C₄	—	89	2	no	—
265	C₂×C₄	—	89	2	no	—
266	C₂×C₄	—	89	8	no	—
267	C₂×C₄	—	89	4	no	—
268	C₂×C₄	—	89	4	no	—
269	C₂×C₄	—	89	2	no	—
270	C₂×C₄	—	178	4	no	—
271	C₂×C₄	—	178	4	no	—
272	C₂×C₄	—	178	2	no	—
273	C₂×C₄	—	178	4	no	—
274	C₂×C₄	—	178	4	no	—
275	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
276	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
277	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
278	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
279	C₂×C₄	M₁₆×D₈₉	1	2	yes	—
280	C₂×C₄	#15668033	1	4	yes	—
281	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
282	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
283	C₂×C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
284	C₈	—	2	2	no	—
285	C₈	—	4	4	no	—
286	C₈	—	89	4	no	—
287	C₈	—	356	4	no	—
288	C₈	—	178	2	no	—
289	C₈	C₂×C₈×D₈₉	1	4	yes	—
290	C₂²	—	89	4	no	—
291	C₂²	—	178	2	no	—
292	C₂²	—	178	2	no	—
293	C₂²	—	178	8	no	—
294	C₂²	C₄⋊C₈×D₈₉	1	2	yes	—
295	C₂²	#19151633	1	2	yes	—
296	C₂²	#108×D₈₉	1	2	yes	—
297	C₂²	C₂×C₄⋊C₄×D₈₉	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	89	4	no	—
301	C₄	—	89	4	no	—
302	C₄	—	178	2	no	—
303	C₄	C₄⋊C₈×D₈₉	1	2	yes	—
304	C₄	#19151633	1	2	yes	—
305	C₄	C₂²×C₈×D₈₉	1	1	yes	characteristic
306	C₄	C₂×M₁₆×D₈₉	1	1	yes	characteristic
307	C₄	C₂×#15668033	1	2	yes	—
308	C₂	—	178	4	no	—
309	C₂	#26018997	1	4	yes	direct product factor
310	C₂	C₂×C₄⋊C₈×D₈₉	1	1	yes	characteristic
311	C₂	C₂×#19151633	1	1	yes	characteristic
312	C₂	C₂×#108×D₈₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₈₉
commutator	C₄×C₈₉	C₂³×C₈
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂×#334×C₈₉	C₂
socle	C₂³×C₈₉	C₂×C₄⋊C₄
prime radical	C₂³×D₈₉	C₄⋊C₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	C₂×#334	D₈₉
hypocenter	C₈₉	C₂×#1564
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
3. #26018997	C₂	direct (N × H)
4. C₂×#19151350	C₂	semidirect
5. C₂×#19151349	C₂	semidirect
6. C₂×#334×C₈₉	C₂	semidirect
20. #19151350	C₂²	semidirect
21. #19151349	C₂²	semidirect
24. #334×C₈₉	C₂²	semidirect
32. C₂×C₈×D₈₉	C₈	semidirect
79. C₈×D₈₉	C₂×C₈	semidirect
152. C₂×C₈₉	#1564	semidirect
162. C₂×#334	D₈₉	semidirect
163. C₈₉	C₂×#1564	semidirect
181. #334	C₂×D₈₉	semidirect
309. C₂	#26018997	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×#19151650	2
C₂×C₄×C₈×D₈₉	2
#26018997	2
C₂×#19151350	2
C₂×#19151349	2
C₂×#334×C₈₉	2
C₂×#1564	89

Subgroups — C2×#26018997 #36523399

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26018997 #36523399