Subgroups — C₂×#26019002 #36523405

Order 22784.

834 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151651	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#13083558	C₂	1	1	yes	maximal, characteristic
3	#26019002	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19151350	C₂	1	2	yes	maximal, complemented
5	C₂×#19151348	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄⋊C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×#13083558	C₄	1	1	yes	characteristic
8	C₂²×#13083558	C₄	1	1	yes	characteristic
9	#19151651	—	2	2	no	—
10	#19151651	C₂²	1	4	yes	—
11	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊#10623494	—	2	1	no	—
14	C₄×#13083558	C₄	1	4	yes	—
15	C₄×#13083558	C₂²	1	4	yes	—
16	C₂×#15667923	C₄	1	2	yes	—
17	C₂×C₄×#10623494	—	2	1	no	—
18	C₂×C₄×#10623494	C₂²	1	2	yes	—
19	#19151350	C₂²	1	8	yes	complemented
20	#19151348	C₂²	1	8	yes	complemented
21	C₂³×C₈×C₈₉	—	2	1	no	—
22	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₈×C₈₉	—	2	2	no	—
24	C₂×C₄⋊C₈×C₈₉	—	2	1	no	—
25	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
26	C₂×C₄⋊C₈×C₈₉	C₂²	1	4	yes	—
27	C₂×#13083558	—	2	4	no	—
28	C₂×#13083558	C₈	1	8	yes	complemented
29	C₂×#13083558	C₂×C₄	1	4	yes	—
30	C₂×#13083558	C₂×C₄	1	4	yes	—
31	C₂²×#10623494	—	2	1	no	—
32	C₂²×#10623494	—	2	1	no	—
33	C₂²×#10623494	C₂×C₄	1	2	yes	—
34	C₂²×#10623494	C₂×C₄	1	2	yes	—
35	C₄⋊#10623494	—	2	1	no	—
36	C₄⋊#10623494	—	2	1	no	—
37	C₄⋊#10623494	C₂³	1	2	yes	—
38	C₄⋊#10623494	C₂³	1	2	yes	—
39	C₂²⋊#10623494	—	2	4	no	—
40	#15667923	C₂×C₄	1	8	yes	—
41	C₄×#10623494	—	2	4	no	—
42	C₄×#10623494	C₂×C₄	1	4	yes	—
43	C₄×#10623494	C₂³	1	4	yes	—
44	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
45	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
46	C₂³×C₄×C₈₉	Q₈	1	1	yes	characteristic
47	C₂²×C₈×C₈₉	—	2	4	no	—
48	C₂²×C₈×C₈₉	—	2	2	no	—
49	C₂²×C₈×C₈₉	—	4	4	no	—
50	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₈₉	D₄	1	2	yes	—
53	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₈₉	Q₈	1	1	yes	characteristic
55	C₂×C₄²×C₈₉	Q₈	1	2	yes	—
56	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
58	C₄⋊C₈×C₈₉	—	2	4	no	—
59	C₄⋊C₈×C₈₉	—	2	2	no	—
60	C₄⋊C₈×C₈₉	C₂³	1	4	yes	—
61	#13083558	—	2	8	no	—
62	#13083558	C₂×C₈	1	16	yes	complemented
63	C₂×#10623494	—	2	4	no	—
64	C₂×#10623494	—	2	2	no	—
65	C₂×#10623494	—	2	2	no	—
66	C₂×#10623494	—	4	2	no	—
67	C₂×#10623494	—	4	2	no	—
68	C₂×#10623494	C₂×C₈	1	8	yes	complemented
69	C₂×#10623494	C₂²×C₄	1	4	yes	—
70	C₂×#10623494	C₂²×C₄	1	4	yes	—
71	C₂⁴×C₈₉	C₄⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₈₉	—	2	2	no	—
73	C₂²×C₄×C₈₉	—	2	2	no	—
74	C₂²×C₄×C₈₉	—	4	2	no	—
75	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
78	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
79	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
80	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₈₉	C₂×C₈	1	4	yes	—
82	C₂²×C₄×C₈₉	M₁₆	1	4	yes	—
83	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
88	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
89	C₂×C₈×C₈₉	—	2	4	no	—
90	C₂×C₈×C₈₉	—	4	8	no	—
91	C₂×C₈×C₈₉	—	4	4	no	—
92	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
93	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
94	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
95	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
97	C₄²×C₈₉	C₂×D₄	1	2	yes	—
98	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
100	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
101	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
102	C₂³×C₈₉	—	2	2	no	—
103	C₂³×C₈₉	—	2	2	no	—
104	C₂³×C₈₉	C₄⋊C₈	1	4	yes	—
105	C₂³×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
106	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₈₉	—	2	2	no	—
108	C₂×C₄×C₈₉	—	2	2	no	—
109	C₂×C₄×C₈₉	—	2	2	no	—
110	C₂×C₄×C₈₉	—	2	2	no	—
111	C₂×C₄×C₈₉	—	2	2	no	—
112	C₂×C₄×C₈₉	—	2	4	no	—
113	C₂×C₄×C₈₉	—	4	2	no	—
114	C₂×C₄×C₈₉	—	4	2	no	—
115	C₂×C₄×C₈₉	C₄⋊C₈	1	8	yes	—
116	C₂×C₄×C₈₉	C₄⋊C₈	1	4	yes	—
117	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
120	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
124	C₂×C₄×C₈₉	C₂²×C₈	1	4	yes	—
125	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
126	C₂×C₄×C₈₉	C₂×M₁₆	1	4	yes	—
127	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
128	C₂×C₄×C₈₉	#137	1	2	yes	—
129	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
132	C₈×C₈₉	—	4	4	no	—
133	#10623494	—	2	4	no	—
134	#10623494	—	4	2	no	—
135	#10623494	—	4	2	no	—
136	#10623494	C₂²×C₈	1	8	yes	complemented
137	C₂²×C₈₉	—	2	8	no	—
138	C₂²×C₈₉	—	2	2	no	—
139	C₂²×C₈₉	—	2	2	no	—
140	C₂²×C₈₉	C₂×C₄⋊C₈	1	4	yes	—
141	C₂²×C₈₉	C₂×C₄⋊C₈	1	2	yes	—
142	C₂²×C₈₉	#428	1	2	yes	—
143	C₂²×C₈₉	#430	1	2	yes	—
144	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
145	C₄×C₈₉	—	2	2	no	—
146	C₄×C₈₉	—	2	2	no	—
147	C₄×C₈₉	—	2	1	no	—
148	C₄×C₈₉	—	2	1	no	—
149	C₄×C₈₉	C₂×C₄⋊C₈	1	2	yes	—
150	C₄×C₈₉	C₂×C₄⋊C₈	1	4	yes	—
151	C₄×C₈₉	#428	1	2	yes	—
152	C₄×C₈₉	C₂³×C₈	1	1	yes	characteristic
153	C₄×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
154	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
155	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
156	C₂×#1565	—	89	1	no	maximal, Sylow
157	C₂×C₈₉	—	2	4	no	—
158	C₂×C₈₉	#1565	1	4	yes	complemented
159	C₂×C₈₉	C₂²×C₄⋊C₈	1	1	yes	characteristic
160	C₂×C₈₉	C₂×#428	1	1	yes	characteristic
161	C₂×C₈₉	C₂×#430	1	1	yes	characteristic
162	C₂×#431	—	89	1	no	—
163	C₂²×C₄⋊C₈	D₈₉	1	1	yes	characteristic
164	C₂×C₄×M₁₆	—	89	1	no	—
165	#1565	—	89	8	no	—
166	C₂×#335	—	89	2	no	—
167	C₂×#334	—	89	2	no	—
168	C₈₉	C₂×#1565	1	1	yes	characteristic, complemented, Sylow
169	C₂²×M₁₆	—	89	1	no	—
170	C₂²×M₁₆	—	89	1	no	—
171	C₂³×C₈	—	2	1	no	—
172	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
173	#431	—	89	4	no	—
174	#431	—	178	2	no	—
175	C₂×C₄⋊C₈	—	2	2	no	—
176	C₂×C₄⋊C₈	—	2	1	no	—
177	C₂×C₄⋊C₈	—	89	1	no	—
178	C₂×C₄⋊C₈	—	89	1	no	—
179	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
180	C₂×C₄⋊C₈	C₂×D₈₉	1	4	yes	—
181	C₂×C₂²⋊C₈	—	178	1	no	—
182	C₄×M₁₆	—	89	4	no	—
183	C₄×M₁₆	—	89	4	no	—
184	C₂×#97	—	89	2	no	—
185	C₂×C₄×C₈	—	89	2	no	—
186	C₂×C₄×C₈	—	178	1	no	—
187	#335	—	89	8	no	—
188	#334	—	89	8	no	—
189	C₂³×C₄	Dic₁₇₈	1	1	yes	characteristic
190	C₂³×C₄	C₄×D₈₉	1	1	yes	characteristic
191	C₂³×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
192	C₂×M₁₆	—	89	8	no	—
193	C₂×M₁₆	—	89	4	no	—
194	C₂×M₁₆	—	89	4	no	—
195	C₂×M₁₆	—	178	4	no	—
196	C₂²×C₈	—	2	4	no	—
197	C₂²×C₈	—	2	2	no	—
198	C₂²×C₈	—	4	4	no	—
199	C₂²×C₈	—	89	2	no	—
200	C₂²×C₈	—	89	2	no	—
201	C₂²×C₈	—	178	1	no	—
202	C₂²×C₈	—	178	1	no	—
203	C₂×C₄²	Dic₁₇₈	1	1	yes	characteristic
204	C₂×C₄²	Dic₁₇₈	1	2	yes	—
205	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
206	C₂×C₄²	C₄×D₈₉	1	2	yes	—
207	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
208	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
209	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
210	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
211	C₄⋊C₈	—	2	4	no	—
212	C₄⋊C₈	—	2	2	no	—
213	C₄⋊C₈	—	89	2	no	—
214	C₄⋊C₈	—	89	2	no	—
215	C₄⋊C₈	—	178	1	no	—
216	C₄⋊C₈	—	178	1	no	—
217	C₄⋊C₈	C₂²×D₈₉	1	4	yes	—
218	C₂²⋊C₈	—	178	4	no	—
219	#97	—	89	8	no	—
220	C₄×C₈	—	89	4	no	—
221	C₄×C₈	—	89	4	no	—
222	C₄×C₈	—	178	4	no	—
223	C₂⁴	#13083553	1	1	yes	characteristic
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	4	2	no	—
227	C₂²×C₄	#13083553	1	1	yes	characteristic
228	C₂²×C₄	#13083553	1	2	yes	—
229	C₂²×C₄	#13083553	1	1	yes	characteristic
230	C₂²×C₄	#13083553	1	2	yes	—
231	C₂²×C₄	#13083553	1	2	yes	—
232	C₂²×C₄	#13083553	1	1	yes	characteristic
233	C₂²×C₄	C₈×D₈₉	1	4	yes	—
234	C₂²×C₄	#13083557	1	4	yes	—
235	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
237	C₂²×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
238	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
239	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
240	C₂²×C₄	C₂×Dic₁₇₈	1	2	yes	—
241	M₁₆	—	89	16	no	—
242	M₁₆	—	178	8	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	4	8	no	—
245	C₂×C₈	—	4	4	no	—
246	C₂×C₈	—	89	8	no	—
247	C₂×C₈	—	89	4	no	—
248	C₂×C₈	—	89	4	no	—
249	C₂×C₈	—	356	2	no	—
250	C₂×C₈	—	356	2	no	—
251	C₂×C₈	—	178	2	no	—
252	C₂×C₈	—	178	2	no	—
253	C₂×C₈	—	178	4	no	—
254	C₄²	#13083553	1	2	yes	—
255	C₄²	#13083553	1	2	yes	—
256	C₄²	#13083553	1	2	yes	—
257	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
258	C₄²	C₂×C₄×D₈₉	1	2	yes	—
259	C₄²	C₂×C₂²⋊D₈₉	1	2	yes	—
260	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
261	C₄²	C₂×Dic₁₇₈	1	2	yes	—
262	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
263	C₄²	C₂³×D₈₉	1	1	yes	characteristic
264	C₂³	—	2	2	no	—
265	C₂³	—	2	2	no	—
266	C₂³	#15667941	1	4	yes	—
267	C₂³	C₂×#13083553	1	2	yes	—
268	C₂³	C₂×#13083553	1	1	yes	characteristic
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	#15667941	1	8	yes	—
278	C₂×C₄	#15667941	1	4	yes	—
279	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
280	C₂×C₄	C₂×#13083553	1	2	yes	—
281	C₂×C₄	C₂×#13083553	1	2	yes	—
282	C₂×C₄	C₂×#13083553	1	2	yes	—
283	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
284	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
285	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
286	C₂×C₄	C₂×C₈×D₈₉	1	4	yes	—
287	C₂×C₄	C₂×#13083557	1	2	yes	—
288	C₂×C₄	C₂×#13083557	1	4	yes	—
289	C₂×C₄	#15668038	1	2	yes	—
290	C₂×C₄	#15668043	1	2	yes	—
291	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
292	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
293	C₂×C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
294	C₈	—	4	4	no	—
295	C₈	—	89	8	no	—
296	C₈	—	356	2	no	—
297	C₈	—	356	2	no	—
298	C₈	—	178	4	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	8	no	—
302	C₂²	C₂×#15667941	1	4	yes	—
303	C₂²	C₂×#15667941	1	2	yes	—
304	C₂²	#19151638	1	2	yes	—
305	C₂²	#19151646	1	2	yes	—
306	C₂²	C₂²×#13083553	1	1	yes	characteristic
307	C₄	—	2	2	no	—
308	C₄	—	2	1	no	—
309	C₄	—	2	2	no	—
310	C₄	—	2	1	no	—
311	C₄	C₂×#15667941	1	2	yes	—
312	C₄	C₂×#15667941	1	4	yes	—
313	C₄	#19151638	1	2	yes	—
314	C₄	C₂²×C₈×D₈₉	1	1	yes	characteristic
315	C₄	C₂²×#13083557	1	1	yes	characteristic
316	C₄	C₂×#15668038	1	1	yes	characteristic
317	C₄	C₂×#15668043	1	1	yes	characteristic
318	C₂	—	2	4	no	—
319	C₂	#26019002	1	4	yes	direct product factor
320	C₂	C₂²×#15667941	1	1	yes	characteristic
321	C₂	C₂×#19151638	1	1	yes	characteristic
322	C₂	C₂×#19151646	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₈₉
commutator	C₄×C₈₉	C₂³×C₈
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂²×C₄⋊C₈×C₈₉	C₂
socle	C₂³×C₈₉	C₂×C₄⋊C₄
prime radical	C₂⁴×C₈₉	C₄⋊C₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	C₂²×C₄⋊C₈	D₈₉
hypocenter	C₈₉	C₂×#1565
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
3. #26019002	C₂	direct (N × H)
4. C₂×#19151350	C₂	semidirect
5. C₂×#19151348	C₂	semidirect
19. #19151350	C₂²	semidirect
20. #19151348	C₂²	semidirect
28. C₂×#13083558	C₈	semidirect
62. #13083558	C₂×C₈	semidirect
68. C₂×#10623494	C₂×C₈	semidirect
136. #10623494	C₂²×C₈	semidirect
158. C₂×C₈₉	#1565	semidirect
168. C₈₉	C₂×#1565	semidirect
319. C₂	#26019002	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×#19151651	2
C₂×C₄×#13083558	2
#26019002	2
C₂×#19151350	2
C₂×#19151348	2
C₂²×C₄⋊C₈×C₈₉	2
C₂×#1565	89

Subgroups — C2×#26019002 #36523405

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26019002 #36523405