Subgroups — C₂×#26022468 #36709314

Order 22784.

604 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151852	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151645	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022468	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19151420	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19151412	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#19151410	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#19151379	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#365×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₇₁₂	C₂²	1	1	yes	characteristic
10	C₂²×#13083557	C₂²	1	1	yes	characteristic
11	C₂×M₁₆×D₈₉	—	2	1	no	—
12	C₂×#15667987	—	2	1	no	—
13	C₂×#15667952	C₂²	1	1	yes	characteristic
14	#19151852	—	2	2	no	—
15	#19151852	C₂²	1	4	yes	—
16	#19151645	—	2	2	no	—
17	#19151645	C₂²	1	4	yes	complemented
18	C₂×#15667948	C₂²	1	1	yes	characteristic
19	C₂×#15667947	—	2	1	no	—
20	C₂×#15667932	—	2	1	no	—
21	#19151420	C₂²	1	4	yes	complemented
22	#19151412	C₂²	1	4	yes	complemented
23	#19151410	C₂²	1	4	yes	complemented
24	#19151379	C₂²	1	4	yes	complemented
25	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
26	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×#106×C₈₉	C₂²	1	1	yes	characteristic
28	#365×C₈₉	C₂²	1	4	yes	complemented
29	C₂²×D₃₅₆	—	2	1	no	—
30	C₂²×D₃₅₆	—	4	1	no	—
31	C₂²×C₄×D₈₉	D₄	1	1	yes	characteristic
32	C₂×D₇₁₂	—	2	1	no	—
33	C₂×D₇₁₂	—	2	1	no	—
34	C₂×D₇₁₂	—	2	1	no	—
35	C₂×D₇₁₂	—	2	1	no	—
36	C₂×D₇₁₂	—	2	1	no	—
37	C₂×D₇₁₂	C₂³	1	2	yes	—
38	C₂×#13083558	—	2	1	no	—
39	C₂×#13083557	—	2	1	no	—
40	C₂×#13083557	—	2	1	no	—
41	C₂×#13083557	—	2	1	no	—
42	C₂×#13083557	—	2	1	no	—
43	C₂×#13083557	—	4	1	no	—
44	C₂×#13083557	D₄	1	2	yes	—
45	C₂×#13083557	C₂³	1	2	yes	—
46	M₁₆×D₈₉	—	2	4	no	—
47	M₁₆×D₈₉	—	4	2	no	—
48	C₂²×#10623494	D₄	1	1	yes	characteristic
49	C₂×C₈×D₈₉	—	4	1	no	—
50	#15667987	—	2	4	no	—
51	#15667987	—	4	2	no	—
52	C₂×#13083551	—	4	1	no	—
53	#15667952	—	2	1	no	—
54	#15667952	C₂³	1	2	yes	—
55	C₂×#13083549	—	2	1	no	—
56	#15667948	—	2	1	no	—
57	#15667948	C₂³	1	2	yes	—
58	#15667947	—	2	4	no	—
59	#15667932	—	2	4	no	—
60	C₂×M₁₆×C₈₉	—	2	1	no	—
61	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
63	M₃₂×C₈₉	—	2	1	no	—
64	M₃₂×C₈₉	C₂³	1	2	yes	—
65	C₂×C₁₆×C₈₉	—	2	1	no	—
66	#108×C₈₉	—	2	1	no	—
67	#108×C₈₉	C₂³	1	2	yes	—
68	#106×C₈₉	—	2	1	no	—
69	#106×C₈₉	C₂³	1	2	yes	—
70	C₂³×D₈₉	—	4	1	no	—
71	C₂³×D₈₉	SD₁₆	1	1	yes	characteristic
72	C₂×D₃₅₆	—	2	2	no	—
73	C₂×D₃₅₆	—	4	1	no	—
74	C₂×D₃₅₆	—	4	1	no	—
75	C₂×D₃₅₆	—	4	1	no	—
76	C₂×D₃₅₆	—	4	1	no	—
77	C₂×D₃₅₆	—	4	4	no	—
78	C₂×D₃₅₆	—	4	1	no	—
79	C₂×D₃₅₆	—	8	4	no	—
80	C₂²×Dic₈₉	SD₁₆	1	1	yes	characteristic
81	C₂×C₄×D₈₉	—	2	1	no	—
82	C₂×C₄×D₈₉	—	2	1	no	—
83	C₂×C₄×D₈₉	—	2	1	no	—
84	C₂×C₄×D₈₉	—	2	1	no	—
85	C₂×C₄×D₈₉	SD₁₆	1	2	yes	—
86	C₂×C₄×D₈₉	C₂×D₄	1	2	yes	—
87	D₇₁₂	—	2	2	no	—
88	D₇₁₂	—	2	2	no	—
89	D₇₁₂	—	4	1	no	—
90	D₇₁₂	—	4	1	no	—
91	#13083558	—	2	2	no	—
92	#13083558	—	4	1	no	—
93	#13083557	—	2	2	no	—
94	#13083557	—	2	2	no	—
95	#13083557	—	2	2	no	—
96	#13083557	—	2	2	no	—
97	#13083557	—	4	4	no	—
98	C₂×#10623494	—	2	1	no	—
99	C₂×#10623494	—	2	1	no	—
100	C₂×#10623494	—	4	1	no	—
101	C₂×#10623494	C₂×D₄	1	2	yes	—
102	C₈×D₈₉	—	4	4	no	—
103	#13083551	—	4	4	no	—
104	#13083549	—	2	2	no	—
105	C₂²×C₄×C₈₉	—	4	1	no	—
106	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
107	M₁₆×C₈₉	—	2	2	no	—
108	M₁₆×C₈₉	—	4	1	no	—
109	C₂×C₈×C₈₉	—	4	1	no	—
110	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
112	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
113	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
114	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
115	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
116	C₄⋊C₄×C₈₉	—	2	2	no	—
117	C₄⋊C₄×C₈₉	—	4	1	no	—
118	C₁₆×C₈₉	—	2	2	no	—
119	C₂³×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
120	C₂²×D₈₉	—	2	1	no	—
121	C₂²×D₈₉	—	2	1	no	—
122	C₂²×D₈₉	—	4	2	no	—
123	C₂²×D₈₉	—	4	2	no	—
124	C₂²×D₈₉	—	8	2	no	—
125	C₂²×D₈₉	—	8	2	no	—
126	C₂²×D₈₉	—	8	1	no	—
127	C₂²×D₈₉	—	8	1	no	—
128	C₂²×D₈₉	C₂×SD₁₆	1	2	yes	—
129	C₂×C₄×C₈₉	—	4	2	no	—
130	C₂×C₄×C₈₉	—	8	2	no	—
131	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
132	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
133	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
134	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
135	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
136	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
137	C₂×Dic₈₉	—	2	1	no	—
138	C₂×Dic₈₉	—	2	1	no	—
139	C₂×Dic₈₉	C₂×SD₁₆	1	2	yes	—
140	D₃₅₆	—	4	2	no	—
141	D₃₅₆	—	4	2	no	—
142	D₃₅₆	—	8	8	no	—
143	D₃₅₆	—	8	1	no	—
144	D₃₅₆	—	8	1	no	—
145	C₄×D₈₉	—	2	2	no	—
146	C₄×D₈₉	—	2	2	no	—
147	C₄×D₈₉	—	2	2	no	—
148	C₄×D₈₉	—	2	2	no	—
149	C₈×C₈₉	—	4	2	no	—
150	C₈×C₈₉	#123	1	1	yes	characteristic
151	C₈×C₈₉	#123	1	1	yes	characteristic
152	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
153	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
154	#10623494	—	2	2	no	—
155	#10623494	—	4	2	no	—
156	C₂²×C₈₉	—	2	2	no	—
157	C₂²×C₈₉	#481	1	1	yes	characteristic
158	C₂²×C₈₉	#481	1	1	yes	characteristic
159	C₂²×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
160	C₂×D₈₉	—	2	2	no	—
161	C₂×D₈₉	—	4	2	no	—
162	C₂×D₈₉	—	4	2	no	—
163	C₂×D₈₉	—	8	1	no	—
164	C₂×D₈₉	—	8	2	no	—
165	C₂×D₈₉	—	8	2	no	—
166	C₂×D₈₉	—	8	2	no	—
167	C₂×D₈₉	—	16	2	no	—
168	C₂×D₈₉	—	16	2	no	—
169	C₄×C₈₉	—	8	2	no	—
170	C₄×C₈₉	#481	1	1	yes	characteristic
171	C₄×C₈₉	#481	1	1	yes	characteristic
172	C₄×C₈₉	C₂×#123	1	1	yes	characteristic
173	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
174	Dic₈₉	—	2	2	no	—
175	C₂×#2238	—	89	1	no	maximal, Sylow
176	C₂×C₈₉	—	2	1	no	—
177	C₂×C₈₉	—	2	1	no	—
178	C₂×C₈₉	#2238	1	2	yes	complemented
179	C₂×C₈₉	C₂×#481	1	1	yes	characteristic
180	D₈₉	—	4	2	no	—
181	D₈₉	—	16	2	no	—
182	C₂×#469	—	89	1	no	—
183	C₂×#430	—	89	1	no	—
184	#2238	—	89	8	no	—
185	C₂×#365	—	89	1	no	—
186	C₂×#365	D₈₉	1	1	yes	characteristic, complemented
187	C₂×#361	—	89	1	no	—
188	C₂×#361	—	89	1	no	—
189	C₂×#349	—	89	1	no	—
190	C₈₉	C₂×#2238	1	1	yes	characteristic, complemented, Sylow
191	C₂²×D₈	—	89	1	no	—
192	C₂²×M₁₆	—	89	1	no	—
193	C₂²×M₁₆	—	178	1	no	—
194	C₂×#121	—	178	1	no	—
195	C₂×M₃₂	—	89	1	no	—
196	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
197	#469	—	89	4	no	—
198	#469	—	178	2	no	—
199	#430	—	89	4	no	—
200	#430	—	178	2	no	—
201	C₂×#108	—	89	1	no	—
202	C₂×#108	—	178	1	no	—
203	C₂×#108	C₂×D₈₉	1	1	yes	characteristic
204	C₂×#106	C₂×D₈₉	1	1	yes	characteristic
205	C₂×#102	—	178	1	no	—
206	#365	—	89	4	no	—
207	#365	C₂×D₈₉	1	4	yes	complemented
208	#361	—	89	4	no	—
209	#361	—	89	4	no	—
210	#349	—	89	4	no	—
211	C₂²×D₄	—	356	1	no	—
212	C₂²×D₄	—	178	1	no	—
213	C₂³×C₄	—	89	1	no	—
214	C₂×D₈	—	89	2	no	—
215	C₂×D₈	—	178	1	no	—
216	C₂×D₈	—	178	1	no	—
217	C₂×D₈	—	178	1	no	—
218	C₂×D₈	—	178	1	no	—
219	C₂×D₈	—	178	1	no	—
220	C₂×M₁₆	—	2	1	no	—
221	C₂×M₁₆	—	89	2	no	—
222	C₂×M₁₆	—	89	2	no	—
223	C₂×M₁₆	—	356	1	no	—
224	C₂×M₁₆	—	356	2	no	—
225	C₂×M₁₆	—	178	4	no	—
226	C₂×M₁₆	—	178	1	no	—
227	C₂×M₁₆	—	178	1	no	—
228	C₂×M₁₆	—	178	1	no	—
229	C₂×M₁₆	—	178	1	no	—
230	C₂×M₁₆	—	178	1	no	—
231	C₂²×C₈	—	89	1	no	—
232	C₂²×C₈	—	356	1	no	—
233	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
234	#121	—	356	2	no	—
235	#121	—	178	4	no	—
236	C₂×C₄⋊C₄	—	2	1	no	—
237	C₂×C₂²⋊C₄	—	356	1	no	—
238	M₃₂	—	2	1	no	—
239	M₃₂	—	89	2	no	—
240	M₃₂	—	178	1	no	—
241	M₃₂	C₂²×D₈₉	1	2	yes	—
242	C₂×C₁₆	—	2	1	no	—
243	C₂×C₁₆	—	178	1	no	—
244	#108	—	2	1	no	—
245	#108	—	89	2	no	—
246	#108	—	178	4	no	—
247	#108	—	178	1	no	—
248	#108	C₂²×D₈₉	1	2	yes	—
249	#106	—	2	1	no	—
250	#106	C₂²×D₈₉	1	2	yes	—
251	#102	—	178	4	no	—
252	C₂⁴	—	89	1	no	—
253	C₂⁴	—	356	1	no	—
254	C₂×D₄	—	356	4	no	—
255	C₂×D₄	—	356	1	no	—
256	C₂×D₄	—	356	1	no	—
257	C₂×D₄	—	356	1	no	—
258	C₂×D₄	—	356	1	no	—
259	C₂×D₄	—	356	1	no	—
260	C₂×D₄	—	178	2	no	—
261	C₂×D₄	—	712	4	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	—	89	1	no	—
264	C₂²×C₄	—	89	2	no	—
265	C₂²×C₄	—	89	2	no	—
266	C₂²×C₄	—	178	1	no	—
267	C₂²×C₄	—	178	1	no	—
268	C₂²×C₄	—	178	1	no	—
269	C₂²×C₄	—	178	1	no	—
270	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
271	D₈	—	356	1	no	—
272	D₈	—	356	1	no	—
273	D₈	—	178	2	no	—
274	D₈	—	178	2	no	—
275	M₁₆	—	2	2	no	—
276	M₁₆	—	4	1	no	—
277	M₁₆	—	356	4	no	—
278	M₁₆	—	356	1	no	—
279	M₁₆	—	178	2	no	—
280	M₁₆	—	178	2	no	—
281	M₁₆	—	178	2	no	—
282	M₁₆	—	178	2	no	—
283	M₁₆	—	178	2	no	—
284	C₂×C₈	—	4	1	no	—
285	C₂×C₈	—	89	2	no	—
286	C₂×C₈	—	356	4	no	—
287	C₂×C₈	—	356	1	no	—
288	C₂×C₈	—	178	1	no	—
289	C₂×C₈	—	178	1	no	—
290	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
291	C₂×C₈	#13083574	1	1	yes	characteristic
292	C₂×C₈	#13083574	1	1	yes	characteristic
293	C₂×C₈	#13083575	1	1	yes	characteristic
294	C₂×C₈	#13083575	1	1	yes	characteristic
295	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	4	1	no	—
298	C₂²⋊C₄	—	356	4	no	—
299	C₁₆	—	2	2	no	—
300	C₁₆	—	178	2	no	—
301	C₂³	—	89	2	no	—
302	C₂³	—	356	2	no	—
303	C₂³	—	356	2	no	—
304	C₂³	—	178	1	no	—
305	C₂³	—	178	1	no	—
306	C₂³	—	712	2	no	—
307	C₂³	—	712	2	no	—
308	C₂³	—	712	1	no	—
309	C₂³	—	712	1	no	—
310	C₂³	SD₁₆×D₈₉	1	1	yes	characteristic
311	D₄	—	356	2	no	—
312	D₄	—	356	2	no	—
313	D₄	—	712	8	no	—
314	D₄	—	712	1	no	—
315	D₄	—	712	1	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	8	2	no	—
318	C₂×C₄	—	89	2	no	—
319	C₂×C₄	—	178	1	no	—
320	C₂×C₄	—	178	1	no	—
321	C₂×C₄	—	178	2	no	—
322	C₂×C₄	—	178	2	no	—
323	C₂×C₄	—	178	2	no	—
324	C₂×C₄	—	178	2	no	—
325	C₂×C₄	#15667997	1	1	yes	characteristic
326	C₂×C₄	#15667997	1	1	yes	characteristic
327	C₂×C₄	SD₁₆×D₈₉	1	1	yes	characteristic
328	C₂×C₄	#15668039	1	1	yes	characteristic
329	C₂×C₄	#15668039	1	1	yes	characteristic
330	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
331	C₈	—	4	2	no	—
332	C₈	—	356	2	no	—
333	C₈	—	178	2	no	—
334	C₈	#15667997	1	1	yes	characteristic
335	C₈	#15667997	1	1	yes	characteristic
336	C₈	C₂×#13083574	1	1	yes	characteristic
337	C₈	C₂×#13083575	1	1	yes	characteristic
338	C₂²	—	2	2	no	—
339	C₂²	—	356	2	no	—
340	C₂²	—	356	2	no	—
341	C₂²	—	1424	2	no	—
342	C₂²	—	1424	2	no	—
343	C₂²	—	178	2	no	—
344	C₂²	—	712	1	no	—
345	C₂²	—	712	2	no	—
346	C₂²	—	712	2	no	—
347	C₂²	—	712	2	no	—
348	C₂²	#19151918	1	1	yes	characteristic
349	C₂²	#19151918	1	1	yes	characteristic
350	C₂²	C₂×SD₁₆×D₈₉	1	1	yes	characteristic
351	C₄	—	8	2	no	—
352	C₄	—	178	2	no	—
353	C₄	#19151918	1	1	yes	characteristic
354	C₄	#19151918	1	1	yes	characteristic
355	C₄	C₂×#15667997	1	1	yes	characteristic
356	C₄	C₂×#15668039	1	1	yes	characteristic
357	C₂	—	2	1	no	—
358	C₂	—	2	1	no	—
359	C₂	—	356	2	no	—
360	C₂	—	1424	2	no	—
361	C₂	#26022468	1	2	yes	direct product factor
362	C₂	C₂×#19151918	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19151918
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×#365×C₈₉	C₂
socle	C₂²×C₈₉	#481
prime radical	C₂×#19151852	C₂
hypercenter	C₂×#365	D₈₉
hypocenter	C₈₉	C₂×#2238
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#365

Complements

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

N	H ≅ G/N	type
2. C₂×#19151645	C₂	semidirect
3. #26022468	C₂	direct (N × H)
4. C₂×#19151420	C₂	semidirect
5. C₂×#19151412	C₂	semidirect
6. C₂×#19151410	C₂	semidirect
7. C₂×#19151379	C₂	semidirect
8. C₂×#365×C₈₉	C₂	semidirect
17. #19151645	C₂²	semidirect
21. #19151420	C₂²	semidirect
22. #19151412	C₂²	semidirect
23. #19151410	C₂²	semidirect
24. #19151379	C₂²	semidirect
28. #365×C₈₉	C₂²	semidirect
178. C₂×C₈₉	#2238	semidirect
186. C₂×#365	D₈₉	semidirect
190. C₈₉	C₂×#2238	semidirect
207. #365	C₂×D₈₉	semidirect
361. C₂	#26022468	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×#19151852	2
C₂×#19151645	2
#26022468	2
C₂×#19151420	2
C₂×#19151412	2
C₂×#19151410	2
C₂×#19151379	2
C₂×#365×C₈₉	2
C₂×#2238	89

Subgroups — C2×#26022468 #36709314

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26022468 #36709314