Subgroups — C₂×#26022146 #36707864

Order 22784.

646 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19152039	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151719	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151705	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022146	C₂	1	8	yes	maximal, direct product factor
5	C₂×#19151422	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#19151409	C₂	1	1	yes	maximal, characteristic
7	C₂×#19151375	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#362×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#15668034	C₂²	1	1	yes	characteristic
10	C₂×#15668020	—	2	1	no	—
11	C₂×Q₈×Dic₈₉	—	2	1	no	—
12	#19152039	C₄	1	4	yes	complemented
13	#19152039	C₂²	1	4	yes	—
14	C₂×#15667958	C₄	1	2	yes	complemented
15	C₂×#15667955	C₄	1	2	yes	complemented
16	C₂×#15667952	C₂²	1	1	yes	characteristic
17	#19151719	—	2	2	no	—
18	#19151719	C₂²	1	4	yes	—
19	#19151705	—	2	2	no	—
20	#19151705	C₂²	1	4	yes	complemented
21	C₂×#15667944	C₂²	1	1	yes	characteristic
22	C₂×#15667938	—	2	1	no	—
23	C₂×#15667936	—	2	1	no	—
24	C₂×#15667922	C₂²	1	1	yes	characteristic
25	#19151422	C₂²	1	4	yes	complemented
26	#19151409	C₂²	1	4	yes	—
27	#19151375	C₂²	1	4	yes	complemented
28	C₂²×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
29	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
30	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
31	#362×C₈₉	C₂²	1	4	yes	complemented
32	C₂×#13083574	—	2	1	no	—
33	#15668034	—	2	1	no	—
34	#15668034	—	2	1	no	—
35	#15668034	C₂×C₄	1	2	yes	—
36	#15668034	C₂³	1	2	yes	—
37	C₂×Dic₃₅₆	C₂×C₄	1	1	yes	characteristic
38	C₂×#13083563	—	2	1	no	—
39	C₂×D₇₁₂	C₂×C₄	1	1	yes	characteristic
40	#15668020	—	2	4	no	—
41	#15668020	—	4	2	no	—
42	C₂×#13083557	—	4	1	no	—
43	C₂²×#10623494	D₄	1	1	yes	characteristic
44	C₂×C₈×D₈₉	—	4	1	no	—
45	Q₈×Dic₈₉	—	2	4	no	—
46	Q₈×Dic₈₉	—	4	2	no	—
47	C₂×C₄⋊Dic₈₉	—	2	1	no	—
48	C₂×C₄⋊Dic₈₉	—	4	1	no	—
49	C₂×C₄×Dic₈₉	—	4	1	no	—
50	C₂×C₄×Dic₈₉	D₄	1	1	yes	characteristic
51	#15667958	C₂×C₄	1	8	yes	complemented
52	#15667955	C₂×C₄	1	8	yes	complemented
53	#15667952	C₂×C₄	1	2	yes	—
54	#15667952	C₂³	1	2	yes	—
55	C₂×#13083549	C₂×C₄	1	2	yes	—
56	#15667944	—	2	1	no	—
57	#15667944	C₂³	1	2	yes	—
58	#15667938	—	2	4	no	—
59	#15667936	—	2	4	no	—
60	#15667922	D₄	1	2	yes	—
61	#15667922	C₂³	1	2	yes	—
62	C₂²×Q₈×C₈₉	—	2	1	no	—
63	C₂×Q₁₆×C₈₉	—	2	1	no	—
64	C₂×Q₁₆×C₈₉	—	2	1	no	—
65	C₂×Q₁₆×C₈₉	—	2	1	no	—
66	C₂×Q₁₆×C₈₉	C₂×C₄	1	2	yes	—
67	C₂×Q₁₆×C₈₉	C₂×C₄	1	2	yes	—
68	C₂×Q₁₆×C₈₉	C₂³	1	2	yes	—
69	C₂×M₁₆×C₈₉	—	2	1	no	—
70	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
71	M₃₂×C₈₉	—	2	1	no	—
72	M₃₂×C₈₉	C₂³	1	2	yes	—
73	C₂×C₁₆×C₈₉	—	2	1	no	—
74	#108×C₈₉	—	2	1	no	—
75	#108×C₈₉	C₂³	1	2	yes	—
76	#13083574	—	2	2	no	—
77	#13083574	—	2	2	no	—
78	#13083574	—	4	1	no	—
79	#13083574	—	4	1	no	—
80	C₂×Dic₁₇₈	—	2	1	no	—
81	C₂×D₃₅₆	—	2	1	no	—
82	C₂×C₂²⋊D₈₉	—	4	1	no	—
83	C₂²×Dic₈₉	—	2	1	no	—
84	C₂²×Dic₈₉	—	4	1	no	—
85	C₂×C₄×D₈₉	—	4	1	no	—
86	Dic₃₅₆	—	2	1	no	—
87	Dic₃₅₆	C₂²×C₄	1	2	yes	—
88	#13083563	—	2	2	no	—
89	#13083563	—	2	2	no	—
90	D₇₁₂	—	2	1	no	—
91	D₇₁₂	C₂²×C₄	1	2	yes	—
92	#13083557	—	4	4	no	—
93	C₂×#10623494	—	2	1	no	—
94	C₂×#10623494	—	2	1	no	—
95	C₂×#10623494	C₂×D₄	1	2	yes	—
96	C₈×D₈₉	—	4	4	no	—
97	C₄⋊Dic₈₉	—	2	2	no	—
98	C₄⋊Dic₈₉	—	4	4	no	—
99	C₄⋊Dic₈₉	—	4	1	no	—
100	C₄×Dic₈₉	—	2	1	no	—
101	C₄×Dic₈₉	—	4	4	no	—
102	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
103	#13083549	C₂²×C₄	1	4	yes	—
104	C₂×Q₈×C₈₉	—	2	2	no	—
105	C₂×Q₈×C₈₉	—	2	2	no	—
106	C₂×Q₈×C₈₉	—	2	2	no	—
107	C₂×Q₈×C₈₉	—	4	1	no	—
108	C₂×Q₈×C₈₉	—	4	1	no	—
109	C₂×Q₈×C₈₉	—	4	1	no	—
110	C₂²×C₄×C₈₉	—	4	1	no	—
111	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
112	Q₁₆×C₈₉	—	2	2	no	—
113	Q₁₆×C₈₉	—	2	2	no	—
114	Q₁₆×C₈₉	—	2	2	no	—
115	Q₁₆×C₈₉	C₂²×C₄	1	4	yes	—
116	M₁₆×C₈₉	—	2	2	no	—
117	M₁₆×C₈₉	—	4	1	no	—
118	C₂×C₈×C₈₉	—	4	1	no	—
119	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
123	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
124	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
125	C₁₆×C₈₉	—	2	2	no	—
126	C₂³×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂²×D₈₉	—	4	1	no	—
128	Q₈×C₈₉	—	2	4	no	—
129	Q₈×C₈₉	—	4	2	no	—
130	Q₈×C₈₉	—	4	2	no	—
131	Q₈×C₈₉	—	4	2	no	—
132	C₂×C₄×C₈₉	—	4	2	no	—
133	C₂×C₄×C₈₉	—	4	2	no	—
134	C₂×C₄×C₈₉	—	4	2	no	—
135	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
138	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
139	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
140	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
141	C₂²⋊D₈₉	—	4	2	no	—
142	C₂²⋊D₈₉	—	4	2	no	—
143	C₂×Dic₈₉	—	2	2	no	—
144	C₂×Dic₈₉	—	4	2	no	—
145	C₂×Dic₈₉	—	4	1	no	—
146	C₂×Dic₈₉	—	4	2	no	—
147	C₂×Dic₈₉	—	8	2	no	—
148	D₃₅₆	—	2	2	no	—
149	D₃₅₆	—	4	1	no	—
150	C₄×D₈₉	—	4	2	no	—
151	C₄×D₈₉	—	4	2	no	—
152	Dic₁₇₈	—	2	2	no	—
153	Dic₁₇₈	—	4	1	no	—
154	C₈×C₈₉	—	4	2	no	—
155	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
156	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
157	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
158	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
159	#10623494	—	2	2	no	—
160	C₂²×C₈₉	—	2	2	no	—
161	C₂²×C₈₉	#442	1	1	yes	characteristic
162	C₂²×C₈₉	#442	1	1	yes	characteristic
163	C₂²×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
164	C₂×D₈₉	—	4	2	no	—
165	C₂×D₈₉	—	8	2	no	—
166	C₄×C₈₉	—	4	4	no	—
167	C₄×C₈₉	#442	1	1	yes	characteristic
168	C₄×C₈₉	#442	1	1	yes	characteristic
169	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
170	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
171	Dic₈₉	—	4	2	no	—
172	Dic₈₉	—	4	2	no	—
173	Dic₈₉	—	8	2	no	—
174	C₂×#2180	—	89	1	no	maximal, Sylow
175	C₂×C₈₉	—	2	1	no	—
176	C₂×C₈₉	—	2	1	no	—
177	C₂×C₈₉	#2180	1	2	yes	complemented
178	C₂×C₈₉	C₂×#442	1	1	yes	characteristic
179	D₈₉	—	8	2	no	—
180	C₂×#510	—	89	1	no	—
181	C₂×#444	—	89	1	no	—
182	C₂×#441	—	89	1	no	—
183	#2180	—	89	8	no	—
184	C₂×#365	—	89	1	no	—
185	C₂×#362	D₈₉	1	1	yes	characteristic, complemented
186	C₂×#360	—	89	1	no	—
187	C₂×#347	—	89	1	no	—
188	C₈₉	C₂×#2180	1	1	yes	characteristic, complemented, Sylow
189	C₂×#135	—	89	1	no	—
190	C₂²×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
191	C₂×#131	—	178	1	no	—
192	C₂×C₄×Q₈	—	178	1	no	—
193	#510	—	89	4	no	—
194	#510	—	89	4	no	—
195	C₂×Q₃₂	—	89	2	no	—
196	C₂×SD₃₂	—	89	2	no	—
197	C₂×M₃₂	—	89	1	no	—
198	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
199	#444	—	89	4	no	—
200	#444	—	178	2	no	—
201	#441	—	89	4	no	—
202	#441	—	178	2	no	—
203	C₂×#108	C₂×D₈₉	1	1	yes	characteristic
204	C₂×#106	—	89	1	no	—
205	C₂×#104	—	178	1	no	—
206	C₂×#103	—	178	1	no	—
207	C₂×#97	—	89	1	no	—
208	#365	—	89	4	no	—
209	#362	C₂×D₈₉	1	4	yes	complemented
210	#360	—	89	4	no	—
211	#347	—	89	4	no	—
212	C₂×#41	—	178	1	no	—
213	C₂²×Q₈	—	2	1	no	—
214	#135	—	89	2	no	—
215	#135	—	89	2	no	—
216	#135	—	178	1	no	—
217	#135	—	178	1	no	—
218	C₂×Q₁₆	—	2	1	no	—
219	C₂×Q₁₆	—	2	1	no	—
220	C₂×Q₁₆	—	2	1	no	—
221	C₂×Q₁₆	—	89	1	no	—
222	C₂×Q₁₆	C₄×D₈₉	1	2	yes	—
223	C₂×Q₁₆	C₄×D₈₉	1	2	yes	—
224	C₂×Q₁₆	C₂²×D₈₉	1	2	yes	—
225	C₂×SD₁₆	—	178	1	no	—
226	C₂×D₈	—	89	1	no	—
227	#131	—	356	2	no	—
228	#131	—	178	4	no	—
229	C₂×M₁₆	—	2	1	no	—
230	C₂×M₁₆	—	356	1	no	—
231	C₂²×C₈	—	89	1	no	—
232	C₂²×C₈	—	356	1	no	—
233	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
234	C₄×Q₈	—	356	2	no	—
235	C₄×Q₈	—	178	4	no	—
236	C₂×C₄⋊C₄	—	356	1	no	—
237	C₂×C₄⋊C₄	—	178	1	no	—
238	C₂×C₄²	—	89	1	no	—
239	C₂×C₄²	—	356	1	no	—
240	Q₃₂	—	89	8	no	—
241	SD₃₂	—	89	8	no	—
242	M₃₂	—	2	1	no	—
243	M₃₂	—	89	2	no	—
244	M₃₂	—	89	2	no	—
245	M₃₂	C₂²×D₈₉	1	2	yes	—
246	C₂×C₁₆	—	2	1	no	—
247	C₂×C₁₆	—	89	2	no	—
248	#108	—	2	1	no	—
249	#108	C₂²×D₈₉	1	2	yes	—
250	#106	—	89	2	no	—
251	#106	—	178	1	no	—
252	#104	—	178	4	no	—
253	#103	—	178	4	no	—
254	#97	—	89	2	no	—
255	#97	—	89	2	no	—
256	#41	—	356	1	no	—
257	#41	—	356	1	no	—
258	#41	—	178	2	no	—
259	#41	—	178	2	no	—
260	C₂×Q₈	—	2	2	no	—
261	C₂×Q₈	—	2	2	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	4	1	no	—
264	C₂×Q₈	—	4	1	no	—
265	C₂×Q₈	—	4	1	no	—
266	C₂×Q₈	—	178	1	no	—
267	C₂×D₄	—	356	1	no	—
268	C₂×D₄	—	178	1	no	—
269	C₂²×C₄	—	4	1	no	—
270	C₂²×C₄	—	356	1	no	—
271	C₂²×C₄	—	356	1	no	—
272	C₂²×C₄	—	178	1	no	—
273	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
274	Q₁₆	—	2	2	no	—
275	Q₁₆	—	2	2	no	—
276	Q₁₆	—	2	2	no	—
277	Q₁₆	—	89	2	no	—
278	Q₁₆	—	178	1	no	—
279	Q₁₆	C₂×C₄×D₈₉	1	4	yes	—
280	SD₁₆	—	178	2	no	—
281	SD₁₆	—	178	2	no	—
282	D₈	—	89	2	no	—
283	D₈	—	178	1	no	—
284	M₁₆	—	2	2	no	—
285	M₁₆	—	4	1	no	—
286	M₁₆	—	356	4	no	—
287	C₂×C₈	—	4	1	no	—
288	C₂×C₈	—	89	2	no	—
289	C₂×C₈	—	356	4	no	—
290	C₂×C₈	—	178	1	no	—
291	C₂×C₈	—	178	1	no	—
292	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
293	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
294	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
295	C₂×C₈	#13083573	1	1	yes	characteristic
296	C₂×C₈	#13083573	1	1	yes	characteristic
297	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
298	C₄⋊C₄	—	356	4	no	—
299	C₄⋊C₄	—	356	1	no	—
300	C₄⋊C₄	—	178	2	no	—
301	C₄²	—	89	2	no	—
302	C₄²	—	356	4	no	—
303	C₄²	—	178	1	no	—
304	C₁₆	—	2	2	no	—
305	C₁₆	—	89	4	no	—
306	C₂³	—	356	1	no	—
307	C₂³	#15668039	1	1	yes	characteristic
308	Q₈	—	2	4	no	—
309	Q₈	—	4	2	no	—
310	Q₈	—	4	2	no	—
311	Q₈	—	4	2	no	—
312	Q₈	—	356	1	no	—
313	Q₈	—	178	2	no	—
314	D₄	—	356	2	no	—
315	D₄	—	356	2	no	—
316	D₄	—	356	1	no	—
317	D₄	—	178	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	356	2	no	—
322	C₂×C₄	—	356	1	no	—
323	C₂×C₄	—	356	2	no	—
324	C₂×C₄	—	356	2	no	—
325	C₂×C₄	—	356	2	no	—
326	C₂×C₄	—	178	2	no	—
327	C₂×C₄	—	712	2	no	—
328	C₂×C₄	#15667972	1	1	yes	characteristic
329	C₂×C₄	#15667972	1	1	yes	characteristic
330	C₂×C₄	#15668036	1	1	yes	characteristic
331	C₂×C₄	#15668036	1	1	yes	characteristic
332	C₂×C₄	#15668039	1	1	yes	characteristic
333	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
334	C₈	—	4	2	no	—
335	C₈	—	178	2	no	—
336	C₈	#15667972	1	1	yes	characteristic
337	C₈	#15667972	1	1	yes	characteristic
338	C₈	C₂²×C₄×D₈₉	1	1	yes	characteristic
339	C₈	C₂×#13083573	1	1	yes	characteristic
340	C₂²	—	2	2	no	—
341	C₂²	—	356	2	no	—
342	C₂²	—	712	2	no	—
343	C₂²	#19151707	1	1	yes	characteristic
344	C₂²	#19151707	1	1	yes	characteristic
345	C₂²	C₂×#15668039	1	1	yes	characteristic
346	C₄	—	4	4	no	—
347	C₄	—	356	2	no	—
348	C₄	—	356	2	no	—
349	C₄	—	712	2	no	—
350	C₄	#19151707	1	1	yes	characteristic
351	C₄	#19151707	1	1	yes	characteristic
352	C₄	C₂×#15667972	1	1	yes	characteristic
353	C₄	C₂×#15668036	1	1	yes	characteristic
354	C₂	—	2	1	no	—
355	C₂	—	2	1	no	—
356	C₂	—	712	2	no	—
357	C₂	#26022146	1	2	yes	direct product factor
358	C₂	C₂×#19151707	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19151707
commutator	C₈×C₈₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×#362×C₈₉	C₂
socle	C₂²×C₈₉	#442
prime radical	C₂×#15668034	C₂²
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₂×#362	D₈₉
hypocenter	C₈₉	C₂×#2180
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#362

Complements

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

N	H ≅ G/N	type
3. C₂×#19151705	C₂	semidirect
4. #26022146	C₂	direct (N × H)
5. C₂×#19151422	C₂	semidirect
7. C₂×#19151375	C₂	semidirect
8. C₂×#362×C₈₉	C₂	semidirect
12. #19152039	C₄	semidirect
14. C₂×#15667958	C₄	semidirect
15. C₂×#15667955	C₄	semidirect
20. #19151705	C₂²	semidirect
25. #19151422	C₂²	semidirect
27. #19151375	C₂²	semidirect
31. #362×C₈₉	C₂²	semidirect
51. #15667958	C₂×C₄	semidirect
52. #15667955	C₂×C₄	semidirect
177. C₂×C₈₉	#2180	semidirect
185. C₂×#362	D₈₉	semidirect
188. C₈₉	C₂×#2180	semidirect
209. #362	C₂×D₈₉	semidirect
357. C₂	#26022146	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×#19152039	2
C₂×#19151719	2
C₂×#19151705	2
#26022146	2
C₂×#19151422	2
C₂×#19151409	2
C₂×#19151375	2
C₂×#362×C₈₉	2
C₂×#2180	89

Subgroups — C2×#26022146 #36707864

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26022146 #36707864