Subgroups — C₂×#26019829 #36537003

Order 22784.

694 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151950	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19151878	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151676	C₂	1	1	yes	maximal, characteristic
4	#26019829	C₂	1	8	yes	maximal, direct product factor
5	C₂×#19151348	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#19151347	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#19151343	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#476×C₈₉	C₂	1	1	yes	maximal, characteristic
9	C₂²×#13083558	—	2	1	no	—
10	C₂×#15668012	C₂²	1	1	yes	characteristic
11	#19151950	—	2	2	no	—
12	#19151950	C₂²	1	4	yes	complemented
13	#19151878	—	2	2	no	—
14	#19151878	C₂²	1	4	yes	—
15	#19151676	—	2	2	no	—
16	#19151676	C₂²	1	4	yes	—
17	C₂×#15667945	—	2	1	no	—
18	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
20	C₂×#15667935	—	2	1	no	—
21	C₂×#15667935	—	2	1	no	—
22	C₂×#15667933	—	2	1	no	—
23	C₂×C₂²⋊#10623494	—	2	1	no	—
24	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
25	#19151348	C₂²	1	4	yes	complemented
26	#19151347	C₂²	1	4	yes	complemented
27	#19151343	C₂²	1	4	yes	complemented
28	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
29	C₂×C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
30	#476×C₈₉	—	2	2	no	—
31	#476×C₈₉	C₂²	1	4	yes	—
32	C₂×#106×C₈₉	—	2	1	no	—
33	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
34	C₂×#102×C₈₉	—	2	1	no	—
35	C₂²×Dic₁₇₈	—	2	1	no	—
36	C₂×#13083558	—	2	4	no	—
37	C₂×#13083558	—	4	4	no	—
38	C₂²×#10623494	—	2	1	no	—
39	C₂²×#10623494	—	2	1	no	—
40	C₂²×#10623494	D₄	1	1	yes	characteristic
41	C₂²×#10623494	D₄	1	1	yes	characteristic
42	#15668012	—	2	1	no	—
43	#15668012	—	2	1	no	—
44	#15668012	—	2	1	no	—
45	#15668012	C₂³	1	2	yes	—
46	C₂×C₄⋊Dic₈₉	—	2	1	no	—
47	C₂×C₄⋊Dic₈₉	—	2	1	no	—
48	#15667945	—	2	4	no	—
49	C₄⋊#10623494	—	2	1	no	—
50	C₄⋊#10623494	—	2	1	no	—
51	C₄⋊#10623494	C₂³	1	2	yes	—
52	C₄⋊#10623494	C₂³	1	2	yes	—
53	#15667935	—	2	4	no	—
54	#15667935	—	2	4	no	—
55	#15667933	—	2	4	no	—
56	C₂²⋊#10623494	—	2	4	no	—
57	C₄×#10623494	D₄	1	2	yes	—
58	C₄×#10623494	C₂³	1	2	yes	—
59	C₂²×Q₈×C₈₉	—	2	1	no	—
60	C₂²×D₄×C₈₉	D₄	1	1	yes	characteristic
61	C₂³×C₄×C₈₉	—	2	1	no	—
62	C₂²×C₈×C₈₉	—	2	1	no	—
63	C₄⋊Q₈×C₈₉	—	2	1	no	—
64	C₄⋊Q₈×C₈₉	—	2	1	no	—
65	C₄⋊Q₈×C₈₉	—	2	1	no	—
66	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
67	C₄×D₄×C₈₉	—	2	1	no	—
68	C₄×D₄×C₈₉	—	2	1	no	—
69	C₄×D₄×C₈₉	D₄	1	2	yes	—
70	C₄×D₄×C₈₉	C₂³	1	2	yes	—
71	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
72	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
73	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
74	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
75	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
76	#106×C₈₉	—	2	4	no	—
77	C₄⋊C₈×C₈₉	—	2	1	no	—
78	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
79	#102×C₈₉	—	2	4	no	—
80	C₂×Dic₁₇₈	—	2	2	no	—
81	C₂×Dic₁₇₈	—	4	2	no	—
82	C₂×Dic₁₇₈	—	4	2	no	—
83	C₂×Dic₁₇₈	—	4	1	no	—
84	C₂²×Dic₈₉	—	4	1	no	—
85	#13083558	—	4	8	no	—
86	C₂×#10623494	—	2	2	no	—
87	C₂×#10623494	—	2	2	no	—
88	C₂×#10623494	—	2	2	no	—
89	C₂×#10623494	—	2	2	no	—
90	C₂×#10623494	—	4	2	no	—
91	C₂×#10623494	—	4	2	no	—
92	C₂×#10623494	C₂×D₄	1	2	yes	—
93	C₂×#10623494	C₂×D₄	1	2	yes	—
94	C₄⋊Dic₈₉	—	2	2	no	—
95	C₄⋊Dic₈₉	—	2	2	no	—
96	C₄⋊Dic₈₉	—	4	1	no	—
97	C₄⋊Dic₈₉	—	4	1	no	—
98	C₂⁴×C₈₉	—	2	1	no	—
99	C₂×Q₈×C₈₉	—	2	2	no	—
100	C₂×Q₈×C₈₉	—	4	1	no	—
101	C₂×Q₈×C₈₉	—	8	2	no	—
102	C₂×D₄×C₈₉	—	2	1	no	—
103	C₂×D₄×C₈₉	—	2	2	no	—
104	C₂×D₄×C₈₉	—	2	2	no	—
105	C₂×D₄×C₈₉	C₂×D₄	1	2	yes	—
106	C₂²×C₄×C₈₉	—	2	1	no	—
107	C₂²×C₄×C₈₉	—	2	2	no	—
108	C₂²×C₄×C₈₉	—	2	2	no	—
109	C₂²×C₄×C₈₉	—	4	2	no	—
110	C₂²×C₄×C₈₉	—	4	2	no	—
111	C₂²×C₄×C₈₉	—	4	1	no	—
112	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
115	C₂×C₈×C₈₉	—	2	2	no	—
116	C₂×C₈×C₈₉	—	4	2	no	—
117	C₄⋊C₄×C₈₉	—	2	1	no	—
118	C₄⋊C₄×C₈₉	—	2	2	no	—
119	C₄⋊C₄×C₈₉	—	2	2	no	—
120	C₄⋊C₄×C₈₉	—	4	1	no	—
121	C₄⋊C₄×C₈₉	—	4	1	no	—
122	C₄⋊C₄×C₈₉	C₂×D₄	1	2	yes	—
123	C₂²⋊C₄×C₈₉	—	2	2	no	—
124	C₂²⋊C₄×C₈₉	—	2	2	no	—
125	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
126	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₈₉	#41	1	1	yes	characteristic
128	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
129	C₂³×C₈₉	—	2	2	no	—
130	C₂³×C₈₉	—	4	2	no	—
131	C₂³×C₈₉	—	4	2	no	—
132	C₂³×C₈₉	—	4	2	no	—
133	C₂³×C₈₉	#121	1	1	yes	characteristic
134	Q₈×C₈₉	—	8	2	no	—
135	Q₈×C₈₉	—	8	2	no	—
136	D₄×C₈₉	—	2	2	no	—
137	D₄×C₈₉	—	2	2	no	—
138	D₄×C₈₉	—	4	2	no	—
139	C₂×C₄×C₈₉	—	2	2	no	—
140	C₂×C₄×C₈₉	—	2	2	no	—
141	C₂×C₄×C₈₉	—	4	2	no	—
142	C₂×C₄×C₈₉	—	4	2	no	—
143	C₂×C₄×C₈₉	—	4	2	no	—
144	C₂×C₄×C₈₉	—	4	4	no	—
145	C₂×C₄×C₈₉	—	4	2	no	—
146	C₂×C₄×C₈₉	—	8	2	no	—
147	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
148	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
149	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
150	C₂×C₄×C₈₉	#135	1	2	yes	—
151	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
152	C₂×C₄×C₈₉	#137	1	2	yes	—
153	C₂×C₄×C₈₉	#137	1	2	yes	—
154	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
156	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
157	C₂×Dic₈₉	—	4	2	no	—
158	C₂×Dic₈₉	—	8	2	no	—
159	Dic₁₇₈	—	4	2	no	—
160	Dic₁₇₈	—	4	2	no	—
161	Dic₁₇₈	—	8	2	no	—
162	C₈×C₈₉	—	4	2	no	—
163	#10623494	—	2	2	no	—
164	#10623494	—	2	2	no	—
165	#10623494	—	4	2	no	—
166	#10623494	—	4	2	no	—
167	C₂²×C₈₉	—	4	2	no	—
168	C₂²×C₈₉	—	4	2	no	—
169	C₂²×C₈₉	—	4	2	no	—
170	C₂²×C₈₉	—	8	4	no	—
171	C₂²×C₈₉	#463	1	2	yes	—
172	C₂²×C₈₉	#470	1	2	yes	—
173	C₂²×C₈₉	#471	1	2	yes	—
174	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
175	C₄×C₈₉	—	2	1	no	—
176	C₄×C₈₉	—	2	1	no	—
177	C₄×C₈₉	—	4	2	no	—
178	C₄×C₈₉	—	8	2	no	—
179	C₄×C₈₉	#463	1	2	yes	—
180	C₄×C₈₉	#470	1	2	yes	—
181	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
182	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
183	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
184	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
185	Dic₈₉	—	8	2	no	—
186	C₂×#1703	—	89	1	no	maximal, Sylow
187	C₂×C₈₉	—	8	2	no	—
188	C₂×C₈₉	#1703	1	4	yes	complemented
189	C₂×C₈₉	C₂×#463	1	1	yes	characteristic
190	C₂×C₈₉	C₂×#470	1	1	yes	characteristic
191	C₂×C₈₉	C₂×#471	1	1	yes	characteristic
192	C₂×#487	—	89	1	no	—
193	C₂×#476	D₈₉	1	1	yes	characteristic
194	C₂×#474	—	89	1	no	—
195	C₂×#436	—	89	1	no	—
196	#1703	—	89	8	no	—
197	C₂×#334	—	89	1	no	—
198	C₂×#333	—	89	1	no	—
199	C₂×#332	—	89	1	no	—
200	C₈₉	C₂×#1703	1	1	yes	characteristic, complemented, Sylow
201	C₂²×M₁₆	—	178	1	no	—
202	C₂×C₄⋊Q₈	—	89	1	no	—
203	C₂×C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
204	C₂×C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
205	#487	—	89	4	no	—
206	#487	—	178	2	no	—
207	#476	—	2	2	no	—
208	#476	C₂×D₈₉	1	4	yes	—
209	#474	—	89	4	no	—
210	#474	—	178	2	no	—
211	#436	—	89	4	no	—
212	#436	—	178	2	no	—
213	C₂×#107	—	178	1	no	—
214	C₂×#106	—	2	1	no	—
215	C₂×C₄⋊C₈	—	89	1	no	—
216	C₂×C₄⋊C₈	—	89	1	no	—
217	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
218	C₂×#103	—	178	1	no	—
219	C₂×#103	—	178	1	no	—
220	C₂×#102	—	2	1	no	—
221	C₂×#102	—	178	1	no	—
222	C₂×C₂²⋊C₈	—	178	1	no	—
223	C₂×C₄×C₈	—	89	1	no	—
224	#334	—	89	4	no	—
225	#333	—	89	4	no	—
226	#332	—	89	4	no	—
227	C₂²×Q₈	—	2	1	no	—
228	C₂²×Q₈	—	178	1	no	—
229	C₂²×D₄	C₂²⋊D₈₉	1	1	yes	characteristic
230	C₂³×C₄	—	2	1	no	—
231	C₂×M₁₆	—	356	4	no	—
232	C₂×M₁₆	—	178	4	no	—
233	C₂²×C₈	—	2	1	no	—
234	C₂²×C₈	—	89	1	no	—
235	C₂²×C₈	—	89	1	no	—
236	C₂²×C₈	—	178	1	no	—
237	C₂²×C₈	—	178	1	no	—
238	C₄⋊Q₈	—	2	1	no	—
239	C₄⋊Q₈	—	2	1	no	—
240	C₄⋊Q₈	—	2	1	no	—
241	C₄⋊Q₈	—	89	2	no	—
242	C₄⋊Q₈	—	178	1	no	—
243	C₄⋊Q₈	—	178	1	no	—
244	C₄⋊Q₈	—	178	1	no	—
245	C₄⋊Q₈	C₂²×D₈₉	1	2	yes	—
246	C₄×D₄	—	2	1	no	—
247	C₄×D₄	—	2	1	no	—
248	C₄×D₄	C₂²⋊D₈₉	1	2	yes	—
249	C₄×D₄	C₂²×D₈₉	1	2	yes	—
250	C₂×C₄⋊C₄	—	2	1	no	—
251	C₂×C₄⋊C₄	—	2	1	no	—
252	C₂×C₄⋊C₄	—	178	1	no	—
253	C₂×C₄⋊C₄	—	178	1	no	—
254	C₂×C₄⋊C₄	C₂²⋊D₈₉	1	1	yes	characteristic
255	C₂×C₂²⋊C₄	—	2	1	no	—
256	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
257	#107	—	178	4	no	—
258	#106	—	2	4	no	—
259	C₄⋊C₈	—	2	1	no	—
260	C₄⋊C₈	—	89	2	no	—
261	C₄⋊C₈	—	89	2	no	—
262	C₄⋊C₈	—	178	1	no	—
263	C₄⋊C₈	—	178	1	no	—
264	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
265	#103	—	178	4	no	—
266	#103	—	178	4	no	—
267	#102	—	2	4	no	—
268	#102	—	178	4	no	—
269	C₂²⋊C₈	—	178	4	no	—
270	C₄×C₈	—	89	2	no	—
271	C₄×C₈	—	89	2	no	—
272	C₂⁴	—	2	1	no	—
273	C₂×Q₈	—	2	2	no	—
274	C₂×Q₈	—	4	1	no	—
275	C₂×Q₈	—	8	2	no	—
276	C₂×Q₈	—	356	1	no	—
277	C₂×Q₈	—	356	2	no	—
278	C₂×Q₈	—	356	2	no	—
279	C₂×Q₈	—	178	2	no	—
280	C₂×D₄	—	2	1	no	—
281	C₂×D₄	—	2	2	no	—
282	C₂×D₄	—	2	2	no	—
283	C₂×D₄	C₂×C₂²⋊D₈₉	1	2	yes	—
284	C₂²×C₄	—	2	1	no	—
285	C₂²×C₄	—	2	2	no	—
286	C₂²×C₄	—	2	2	no	—
287	C₂²×C₄	—	4	2	no	—
288	C₂²×C₄	—	4	2	no	—
289	C₂²×C₄	—	4	1	no	—
290	C₂²×C₄	—	356	1	no	—
291	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
292	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
293	C₂²×C₄	#13083573	1	1	yes	characteristic
294	M₁₆	—	356	8	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	4	2	no	—
297	C₂×C₈	—	89	2	no	—
298	C₂×C₈	—	89	2	no	—
299	C₂×C₈	—	356	2	no	—
300	C₂×C₈	—	356	2	no	—
301	C₂×C₈	—	178	2	no	—
302	C₂×C₈	—	178	2	no	—
303	C₂×C₈	—	178	2	no	—
304	C₂×C₈	—	178	2	no	—
305	C₄⋊C₄	—	2	1	no	—
306	C₄⋊C₄	—	2	2	no	—
307	C₄⋊C₄	—	2	2	no	—
308	C₄⋊C₄	—	4	1	no	—
309	C₄⋊C₄	—	4	1	no	—
310	C₄⋊C₄	—	356	1	no	—
311	C₄⋊C₄	—	356	1	no	—
312	C₄⋊C₄	—	178	2	no	—
313	C₄⋊C₄	—	178	2	no	—
314	C₄⋊C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
315	C₂²⋊C₄	—	2	2	no	—
316	C₂²⋊C₄	—	2	2	no	—
317	C₄²	D₄×D₈₉	1	1	yes	characteristic
318	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
319	C₄²	#13083573	1	1	yes	characteristic
320	C₄²	C₂³×D₈₉	1	1	yes	characteristic
321	C₂³	—	2	2	no	—
322	C₂³	—	4	2	no	—
323	C₂³	—	4	2	no	—
324	C₂³	—	4	2	no	—
325	C₂³	#15667984	1	1	yes	characteristic
326	Q₈	—	8	2	no	—
327	Q₈	—	8	2	no	—
328	Q₈	—	356	2	no	—
329	Q₈	—	356	2	no	—
330	Q₈	—	712	2	no	—
331	D₄	—	2	2	no	—
332	D₄	—	2	2	no	—
333	D₄	—	4	2	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	—	4	4	no	—
339	C₂×C₄	—	4	2	no	—
340	C₂×C₄	—	4	2	no	—
341	C₂×C₄	—	8	2	no	—
342	C₂×C₄	—	356	2	no	—
343	C₂×C₄	—	712	2	no	—
344	C₂×C₄	#15667984	1	1	yes	characteristic
345	C₂×C₄	#15667984	1	1	yes	characteristic
346	C₂×C₄	#15667984	1	1	yes	characteristic
347	C₂×C₄	#15668033	1	2	yes	—
348	C₂×C₄	#15668039	1	2	yes	—
349	C₂×C₄	#15668042	1	2	yes	—
350	C₂×C₄	#15668043	1	2	yes	—
351	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
352	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
353	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
354	C₈	—	4	2	no	—
355	C₈	—	356	2	no	—
356	C₈	—	356	2	no	—
357	C₈	—	178	2	no	—
358	C₈	—	178	2	no	—
359	C₂²	—	4	2	no	—
360	C₂²	—	4	2	no	—
361	C₂²	—	4	2	no	—
362	C₂²	—	8	4	no	—
363	C₂²	#19151815	1	2	yes	—
364	C₂²	#19151854	1	2	yes	—
365	C₂²	#19151859	1	2	yes	—
366	C₂²	C₂×#15667984	1	1	yes	characteristic
367	C₄	—	2	1	no	—
368	C₄	—	2	1	no	—
369	C₄	—	4	2	no	—
370	C₄	—	8	2	no	—
371	C₄	—	712	2	no	—
372	C₄	#19151815	1	2	yes	—
373	C₄	#19151854	1	2	yes	—
374	C₄	C₂×#15668033	1	1	yes	characteristic
375	C₄	C₂×#15668039	1	1	yes	characteristic
376	C₄	C₂×#15668042	1	1	yes	characteristic
377	C₄	C₂×#15668043	1	1	yes	characteristic
378	C₂	—	8	2	no	—
379	C₂	#26019829	1	4	yes	direct product factor
380	C₂	C₂×#19151815	1	1	yes	characteristic
381	C₂	C₂×#19151854	1	1	yes	characteristic
382	C₂	C₂×#19151859	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15667984
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂×#476×C₈₉	C₂
socle	C₂³×C₈₉	#121
prime radical	C₂²×D₄×C₈₉	D₄
hypercenter	C₂×#476	D₈₉
hypocenter	C₈₉	C₂×#1703
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#476

Complements

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

N	H ≅ G/N	type
1. C₂×#19151950	C₂	semidirect
4. #26019829	C₂	direct (N × H)
5. C₂×#19151348	C₂	semidirect
6. C₂×#19151347	C₂	semidirect
7. C₂×#19151343	C₂	semidirect
12. #19151950	C₂²	semidirect
25. #19151348	C₂²	semidirect
26. #19151347	C₂²	semidirect
27. #19151343	C₂²	semidirect
188. C₂×C₈₉	#1703	semidirect
200. C₈₉	C₂×#1703	semidirect
379. C₂	#26019829	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×#19151950	2
C₂×#19151878	2
C₂×#19151676	2
#26019829	2
C₂×#19151348	2
C₂×#19151347	2
C₂×#19151343	2
C₂×#476×C₈₉	2
C₂×#1703	89

Subgroups — C2×#26019829 #36537003

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26019829 #36537003