Subgroups — C₂×#26019824 #36536998

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×#19151885	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151875	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151677	C₂	1	1	yes	maximal, characteristic
4	#26019824	C₂	1	8	yes	maximal, direct product factor
5	C₂×#19151349	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#19151347	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#19151344	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#487×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#13083557	—	2	1	no	—
10	C₂×C₄×D₃₅₆	C₂²	1	1	yes	characteristic
11	#19151885	—	2	2	no	—
12	#19151885	C₂²	1	4	yes	—
13	#19151875	—	2	2	no	—
14	#19151875	C₂²	1	4	yes	—
15	#19151677	—	2	2	no	—
16	#19151677	C₂²	1	4	yes	—
17	C₂×#15667945	—	2	1	no	—
18	C₂×#15667943	—	2	1	no	—
19	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
22	C₂×#15667932	—	2	1	no	—
23	C₂×#15667932	—	2	1	no	—
24	C₂×#15667924	—	2	1	no	—
25	#19151349	C₂²	1	4	yes	complemented
26	#19151347	C₂²	1	4	yes	complemented
27	#19151344	C₂²	1	4	yes	complemented
28	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
30	#487×C₈₉	—	2	2	no	—
31	#487×C₈₉	C₂²	1	4	yes	complemented
32	C₂×#103×C₈₉	—	2	1	no	—
33	C₂×#103×C₈₉	—	2	1	no	—
34	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
35	C₂²×D₃₅₆	D₄	1	1	yes	characteristic
36	C₂²×C₄×D₈₉	—	2	1	no	—
37	C₂×#13083557	—	2	4	no	—
38	C₂×#13083557	—	4	4	no	—
39	C₂²×#10623494	—	2	1	no	—
40	C₂²×#10623494	—	2	1	no	—
41	C₂²×#10623494	—	2	1	no	—
42	C₄×D₃₅₆	—	2	1	no	—
43	C₄×D₃₅₆	—	2	1	no	—
44	C₄×D₃₅₆	D₄	1	2	yes	—
45	C₄×D₃₅₆	C₂³	1	2	yes	—
46	C₂×C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
47	C₂×#13083551	—	2	1	no	—
48	#15667945	—	2	4	no	—
49	#15667943	—	2	4	no	—
50	C₄⋊#10623494	—	2	1	no	—
51	C₄⋊#10623494	—	2	1	no	—
52	C₄⋊#10623494	—	2	1	no	—
53	C₄⋊#10623494	C₂³	1	2	yes	—
54	C₄⋊#10623494	C₂³	1	2	yes	—
55	C₄⋊#10623494	C₂³	1	2	yes	—
56	#15667932	—	2	4	no	—
57	#15667932	—	2	4	no	—
58	#15667924	—	2	4	no	—
59	C₂²×Q₈×C₈₉	—	2	1	no	—
60	C₂²×Q₈×C₈₉	—	2	1	no	—
61	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
62	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
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	—	2	1	no	—
67	C₄⋊Q₈×C₈₉	—	2	1	no	—
68	C₄⋊Q₈×C₈₉	—	2	1	no	—
69	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
70	C₄⋊Q₈×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	—	2	1	no	—
74	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
75	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
76	#103×C₈₉	—	2	4	no	—
77	#103×C₈₉	—	2	4	no	—
78	C₄×C₈×C₈₉	D₄	1	2	yes	—
79	C₄×C₈×C₈₉	C₂³	1	2	yes	—
80	C₂³×D₈₉	—	2	1	no	—
81	C₂×D₃₅₆	—	2	2	no	—
82	C₂×D₃₅₆	—	2	2	no	—
83	C₂×D₃₅₆	—	2	1	no	—
84	C₂×D₃₅₆	C₂×D₄	1	2	yes	—
85	C₂²×Dic₈₉	—	2	1	no	—
86	C₂×C₄×D₈₉	—	2	2	no	—
87	C₂×C₄×D₈₉	—	2	2	no	—
88	C₂×C₄×D₈₉	—	4	2	no	—
89	C₂×C₄×D₈₉	—	4	2	no	—
90	#13083557	—	4	8	no	—
91	C₂×#10623494	—	2	2	no	—
92	C₂×#10623494	—	2	2	no	—
93	C₂×#10623494	—	2	2	no	—
94	C₂×#10623494	—	4	2	no	—
95	C₂×#10623494	—	4	2	no	—
96	C₂×#10623494	—	4	2	no	—
97	C₄⋊Dic₈₉	—	2	1	no	—
98	C₄⋊Dic₈₉	C₂×D₄	1	2	yes	—
99	#13083551	—	2	2	no	—
100	#13083551	—	2	2	no	—
101	C₂×Q₈×C₈₉	—	2	2	no	—
102	C₂×Q₈×C₈₉	—	2	2	no	—
103	C₂×Q₈×C₈₉	—	4	1	no	—
104	C₂×Q₈×C₈₉	—	4	1	no	—
105	C₂×Q₈×C₈₉	—	4	2	no	—
106	C₂×Q₈×C₈₉	—	4	2	no	—
107	C₂×Q₈×C₈₉	—	8	2	no	—
108	C₂²×C₄×C₈₉	—	4	1	no	—
109	C₂²×C₄×C₈₉	—	4	1	no	—
110	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
113	C₂×C₈×C₈₉	—	2	2	no	—
114	C₂×C₈×C₈₉	—	2	2	no	—
115	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
116	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
117	C₄⋊C₄×C₈₉	—	2	2	no	—
118	C₄⋊C₄×C₈₉	—	2	2	no	—
119	C₄⋊C₄×C₈₉	—	2	2	no	—
120	C₄⋊C₄×C₈₉	—	2	2	no	—
121	C₄⋊C₄×C₈₉	—	4	1	no	—
122	C₄⋊C₄×C₈₉	—	4	1	no	—
123	C₄⋊C₄×C₈₉	—	4	1	no	—
124	C₄⋊C₄×C₈₉	—	4	1	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	#121	1	1	yes	characteristic
130	C₂²×D₈₉	—	2	2	no	—
131	C₂²×D₈₉	—	4	2	no	—
132	C₂²×D₈₉	—	4	2	no	—
133	C₂²×D₈₉	—	4	2	no	—
134	Q₈×C₈₉	—	4	2	no	—
135	Q₈×C₈₉	—	4	2	no	—
136	Q₈×C₈₉	—	8	2	no	—
137	Q₈×C₈₉	—	8	2	no	—
138	Q₈×C₈₉	—	8	2	no	—
139	C₂×C₄×C₈₉	—	2	2	no	—
140	C₂×C₄×C₈₉	—	4	2	no	—
141	C₂×C₄×C₈₉	—	4	2	no	—
142	C₂×C₄×C₈₉	—	8	2	no	—
143	C₂×C₄×C₈₉	—	8	2	no	—
144	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
145	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
146	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
147	C₂×C₄×C₈₉	#135	1	2	yes	—
148	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
149	C₂×C₄×C₈₉	#137	1	2	yes	—
150	C₂×C₄×C₈₉	#137	1	2	yes	—
151	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
154	C₂×Dic₈₉	—	2	2	no	—
155	C₂×Dic₈₉	—	4	2	no	—
156	D₃₅₆	—	2	2	no	—
157	D₃₅₆	—	2	2	no	—
158	D₃₅₆	—	4	2	no	—
159	C₄×D₈₉	—	4	2	no	—
160	C₄×D₈₉	—	4	2	no	—
161	C₄×D₈₉	—	4	4	no	—
162	C₈×C₈₉	—	2	2	no	—
163	C₈×C₈₉	—	2	2	no	—
164	#10623494	—	4	2	no	—
165	#10623494	—	4	2	no	—
166	#10623494	—	4	2	no	—
167	C₂²×C₈₉	#463	1	2	yes	—
168	C₂²×C₈₉	#470	1	2	yes	—
169	C₂²×C₈₉	#471	1	2	yes	—
170	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
171	C₂×D₈₉	—	4	2	no	—
172	C₂×D₈₉	—	4	2	no	—
173	C₂×D₈₉	—	4	2	no	—
174	C₂×D₈₉	—	8	4	no	—
175	C₄×C₈₉	—	2	1	no	—
176	C₄×C₈₉	—	2	1	no	—
177	C₄×C₈₉	—	8	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	—	4	2	no	—
186	C₂×#1703	—	89	1	no	maximal, Sylow
187	C₂×C₈₉	#1703	1	4	yes	complemented
188	C₂×C₈₉	C₂×#463	1	1	yes	characteristic
189	C₂×C₈₉	C₂×#470	1	1	yes	characteristic
190	C₂×C₈₉	C₂×#471	1	1	yes	characteristic
191	D₈₉	—	8	2	no	—
192	C₂×#487	D₈₉	1	1	yes	characteristic, complemented
193	C₂×#476	—	89	1	no	—
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	C₂×D₈₉	1	1	yes	characteristic
203	C₂×C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
204	C₂×C₄×D₄	—	89	1	no	—
205	#487	—	2	2	no	—
206	#487	C₂×D₈₉	1	4	yes	complemented
207	#476	—	89	4	no	—
208	#476	—	178	2	no	—
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	—	178	1	no	—
215	C₂×C₄⋊C₈	—	89	1	no	—
216	C₂×C₄⋊C₈	—	89	1	no	—
217	C₂×C₄⋊C₈	—	89	1	no	—
218	C₂×#103	—	2	1	no	—
219	C₂×#103	—	2	1	no	—
220	C₂×#102	—	178	1	no	—
221	C₂×#102	—	178	1	no	—
222	C₂×C₂²⋊C₈	—	178	1	no	—
223	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
224	#334	—	89	4	no	—
225	#333	—	89	4	no	—
226	#332	—	89	4	no	—
227	C₂²×Q₈	—	2	1	no	—
228	C₂²×Q₈	—	2	1	no	—
229	C₂²×D₄	—	89	1	no	—
230	C₂³×C₄	—	178	1	no	—
231	C₂×M₁₆	—	356	4	no	—
232	C₂×M₁₆	—	178	4	no	—
233	C₂²×C₈	—	178	1	no	—
234	C₂²×C₈	—	178	1	no	—
235	C₂²×C₈	—	178	1	no	—
236	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
237	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
238	C₄⋊Q₈	—	2	1	no	—
239	C₄⋊Q₈	—	2	1	no	—
240	C₄⋊Q₈	—	2	1	no	—
241	C₄⋊Q₈	—	2	1	no	—
242	C₄⋊Q₈	—	2	1	no	—
243	C₄⋊Q₈	—	2	1	no	—
244	C₄⋊Q₈	C₂²×D₈₉	1	2	yes	—
245	C₄⋊Q₈	C₂²×D₈₉	1	2	yes	—
246	C₄×D₄	—	89	2	no	—
247	C₄×D₄	—	89	2	no	—
248	C₄×D₄	—	178	1	no	—
249	C₄×D₄	—	178	1	no	—
250	C₂×C₄⋊C₄	—	2	1	no	—
251	C₂×C₄⋊C₄	—	2	1	no	—
252	C₂×C₄⋊C₄	—	2	1	no	—
253	C₂×C₄⋊C₄	—	2	1	no	—
254	C₂×C₄⋊C₄	—	89	1	no	—
255	C₂×C₂²⋊C₄	—	178	1	no	—
256	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
257	#107	—	178	4	no	—
258	#106	—	178	4	no	—
259	C₄⋊C₈	—	89	2	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	—	178	1	no	—
265	#103	—	2	4	no	—
266	#103	—	2	4	no	—
267	#102	—	178	4	no	—
268	#102	—	178	4	no	—
269	C₂²⋊C₈	—	178	4	no	—
270	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
271	C₄×C₈	C₂²×D₈₉	1	2	yes	—
272	C₂⁴	—	178	1	no	—
273	C₂×Q₈	—	2	2	no	—
274	C₂×Q₈	—	2	2	no	—
275	C₂×Q₈	—	4	1	no	—
276	C₂×Q₈	—	4	1	no	—
277	C₂×Q₈	—	4	2	no	—
278	C₂×Q₈	—	4	2	no	—
279	C₂×Q₈	—	8	2	no	—
280	C₂×D₄	—	89	2	no	—
281	C₂×D₄	—	178	1	no	—
282	C₂×D₄	—	178	2	no	—
283	C₂×D₄	—	178	2	no	—
284	C₂²×C₄	—	4	1	no	—
285	C₂²×C₄	—	4	1	no	—
286	C₂²×C₄	—	356	2	no	—
287	C₂²×C₄	—	356	2	no	—
288	C₂²×C₄	—	178	1	no	—
289	C₂²×C₄	—	178	2	no	—
290	C₂²×C₄	—	178	2	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	—	2	2	no	—
297	C₂×C₈	—	356	2	no	—
298	C₂×C₈	—	356	2	no	—
299	C₂×C₈	—	356	2	no	—
300	C₂×C₈	—	178	2	no	—
301	C₂×C₈	—	178	2	no	—
302	C₂×C₈	—	178	2	no	—
303	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
304	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
305	C₄⋊C₄	—	2	2	no	—
306	C₄⋊C₄	—	2	2	no	—
307	C₄⋊C₄	—	2	2	no	—
308	C₄⋊C₄	—	2	2	no	—
309	C₄⋊C₄	—	4	1	no	—
310	C₄⋊C₄	—	4	1	no	—
311	C₄⋊C₄	—	4	1	no	—
312	C₄⋊C₄	—	4	1	no	—
313	C₄⋊C₄	—	89	2	no	—
314	C₄⋊C₄	—	178	1	no	—
315	C₂²⋊C₄	—	178	2	no	—
316	C₂²⋊C₄	—	178	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	—	356	2	no	—
322	C₂³	—	356	2	no	—
323	C₂³	—	356	2	no	—
324	C₂³	—	178	2	no	—
325	C₂³	#15667983	1	1	yes	characteristic
326	Q₈	—	4	2	no	—
327	Q₈	—	4	2	no	—
328	Q₈	—	8	2	no	—
329	Q₈	—	8	2	no	—
330	Q₈	—	8	2	no	—
331	D₄	—	356	2	no	—
332	D₄	—	178	2	no	—
333	D₄	—	178	2	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	8	2	no	—
338	C₂×C₄	—	8	2	no	—
339	C₂×C₄	—	356	2	no	—
340	C₂×C₄	—	356	2	no	—
341	C₂×C₄	—	356	4	no	—
342	C₂×C₄	—	356	2	no	—
343	C₂×C₄	—	178	2	no	—
344	C₂×C₄	#15667983	1	1	yes	characteristic
345	C₂×C₄	#15667983	1	1	yes	characteristic
346	C₂×C₄	#15667983	1	1	yes	characteristic
347	C₂×C₄	#15668032	1	2	yes	—
348	C₂×C₄	#15668037	1	2	yes	—
349	C₂×C₄	#15668044	1	2	yes	—
350	C₂×C₄	#15668044	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	—	2	2	no	—
355	C₈	—	2	2	no	—
356	C₈	—	356	2	no	—
357	C₈	—	356	2	no	—
358	C₈	—	356	2	no	—
359	C₂²	—	356	2	no	—
360	C₂²	—	356	2	no	—
361	C₂²	—	356	2	no	—
362	C₂²	—	712	4	no	—
363	C₂²	#19151810	1	2	yes	—
364	C₂²	#19151853	1	2	yes	—
365	C₂²	#19151858	1	2	yes	—
366	C₂²	C₂×#15667983	1	1	yes	characteristic
367	C₄	—	2	1	no	—
368	C₄	—	2	1	no	—
369	C₄	—	8	2	no	—
370	C₄	—	8	2	no	—
371	C₄	—	356	2	no	—
372	C₄	#19151810	1	2	yes	—
373	C₄	#19151853	1	2	yes	—
374	C₄	C₂×#15668032	1	1	yes	characteristic
375	C₄	C₂×#15668037	1	1	yes	characteristic
376	C₄	C₂×#15668044	1	1	yes	characteristic
377	C₄	C₂×#15668044	1	1	yes	characteristic
378	C₂	—	712	2	no	—
379	C₂	#26019824	1	4	yes	direct product factor
380	C₂	C₂×#19151810	1	1	yes	characteristic
381	C₂	C₂×#19151853	1	1	yes	characteristic
382	C₂	C₂×#19151858	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15667983
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂×#487×C₈₉	C₂
socle	C₂³×C₈₉	#121
prime radical	C₂²×D₃₅₆	D₄
hypercenter	C₂×#487	D₈₉
hypocenter	C₈₉	C₂×#1703
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#487

Complements

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

N	H ≅ G/N	type
4. #26019824	C₂	direct (N × H)
5. C₂×#19151349	C₂	semidirect
6. C₂×#19151347	C₂	semidirect
7. C₂×#19151344	C₂	semidirect
8. C₂×#487×C₈₉	C₂	semidirect
25. #19151349	C₂²	semidirect
26. #19151347	C₂²	semidirect
27. #19151344	C₂²	semidirect
31. #487×C₈₉	C₂²	semidirect
187. C₂×C₈₉	#1703	semidirect
192. C₂×#487	D₈₉	semidirect
200. C₈₉	C₂×#1703	semidirect
206. #487	C₂×D₈₉	semidirect
379. C₂	#26019824	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×#19151885	2
C₂×#19151875	2
C₂×#19151677	2
#26019824	2
C₂×#19151349	2
C₂×#19151347	2
C₂×#19151344	2
C₂×#487×C₈₉	2
C₂×#1703	89

Subgroups — C2×#26019824 #36536998

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26019824 #36536998