Subgroups — C₂×#26019794 #36537078

Order 22784.

702 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151946	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151798	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151729	C₂	1	1	yes	maximal, characteristic, complemented
4	#26019794	C₂	1	8	yes	maximal, direct product factor
5	C₂×#19151350	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#19151347	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#19151341	C₂	1	1	yes	maximal, characteristic
8	C₂×#477×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#13083561	—	2	1	no	—
10	C₂×#15668009	C₂²	1	1	yes	characteristic
11	#19151946	—	2	2	no	—
12	#19151946	C₂²	1	4	yes	—
13	#19151798	—	2	2	no	—
14	#19151798	C₂²	1	4	yes	—
15	#19151729	—	2	2	no	—
16	#19151729	C₂²	1	4	yes	complemented
17	C₂×C₄⋊#10623494	—	2	1	no	—
18	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
20	C₂×#15667932	—	2	1	no	—
21	C₂×#15667932	—	2	1	no	—
22	C₂×#15667932	—	2	1	no	—
23	C₂×#15667923	—	2	1	no	—
24	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
25	#19151350	C₂²	1	4	yes	complemented
26	#19151347	C₂²	1	4	yes	complemented
27	#19151341	C₂²	1	4	yes	—
28	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
29	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
30	#477×C₈₉	—	2	2	no	—
31	#477×C₈₉	C₂²	1	4	yes	complemented
32	C₂×#107×C₈₉	—	2	1	no	—
33	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
34	C₂×#103×C₈₉	—	2	1	no	—
35	C₂²×D₃₅₆	—	2	1	no	—
36	C₂²×D₃₅₆	—	2	1	no	—
37	C₂²×D₃₅₆	—	2	1	no	—
38	C₂×#13083561	—	2	4	no	—
39	C₂×#13083561	—	4	4	no	—
40	C₂²×#10623494	—	2	1	no	—
41	C₂²×#10623494	—	2	1	no	—
42	C₂²×#10623494	D₄	1	1	yes	characteristic
43	C₂²×#10623494	D₄	1	1	yes	characteristic
44	#15668009	—	2	1	no	—
45	#15668009	—	2	1	no	—
46	#15668009	—	2	1	no	—
47	#15668009	C₂³	1	2	yes	—
48	C₄⋊#10623494	—	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	#15667932	—	2	4	no	—
54	#15667932	—	2	4	no	—
55	#15667932	—	2	4	no	—
56	#15667923	—	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×Q₈×C₈₉	D₄	1	1	yes	characteristic
61	C₂²×C₈×C₈₉	—	2	1	no	—
62	C₄⋊Q₈×C₈₉	—	2	1	no	—
63	C₄⋊Q₈×C₈₉	—	2	1	no	—
64	C₄⋊Q₈×C₈₉	—	2	1	no	—
65	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
66	C₄×Q₈×C₈₉	—	2	1	no	—
67	C₄×Q₈×C₈₉	—	2	1	no	—
68	C₄×Q₈×C₈₉	D₄	1	2	yes	—
69	C₄×Q₈×C₈₉	C₂³	1	2	yes	—
70	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
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×C₄²×C₈₉	—	2	1	no	—
75	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
76	#107×C₈₉	—	2	4	no	—
77	C₄⋊C₈×C₈₉	—	2	1	no	—
78	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
79	#103×C₈₉	—	2	4	no	—
80	C₂³×D₈₉	—	4	1	no	—
81	C₂×D₃₅₆	—	2	2	no	—
82	C₂×D₃₅₆	—	2	2	no	—
83	C₂×D₃₅₆	—	2	2	no	—
84	C₂×D₃₅₆	—	4	2	no	—
85	C₂×D₃₅₆	—	4	2	no	—
86	C₂×D₃₅₆	—	4	2	no	—
87	C₂×D₃₅₆	—	4	2	no	—
88	C₂×D₃₅₆	—	4	1	no	—
89	C₂×D₃₅₆	—	4	1	no	—
90	C₂×D₃₅₆	—	4	1	no	—
91	C₂×D₃₅₆	—	8	2	no	—
92	#13083561	—	4	8	no	—
93	C₂×#10623494	—	2	2	no	—
94	C₂×#10623494	—	2	2	no	—
95	C₂×#10623494	—	2	2	no	—
96	C₂×#10623494	—	2	2	no	—
97	C₂×#10623494	—	4	2	no	—
98	C₂×#10623494	—	4	2	no	—
99	C₂×#10623494	C₂×D₄	1	2	yes	—
100	C₂×#10623494	C₂×D₄	1	2	yes	—
101	C₂×Q₈×C₈₉	—	2	1	no	—
102	C₂×Q₈×C₈₉	—	2	2	no	—
103	C₂×Q₈×C₈₉	—	2	2	no	—
104	C₂×Q₈×C₈₉	—	2	2	no	—
105	C₂×Q₈×C₈₉	—	4	1	no	—
106	C₂×Q₈×C₈₉	—	8	2	no	—
107	C₂×Q₈×C₈₉	C₂×D₄	1	2	yes	—
108	C₂²×C₄×C₈₉	—	2	1	no	—
109	C₂²×C₄×C₈₉	—	2	1	no	—
110	C₂²×C₄×C₈₉	—	4	1	no	—
111	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
114	C₂×C₈×C₈₉	—	2	2	no	—
115	C₂×C₈×C₈₉	—	4	2	no	—
116	C₄⋊C₄×C₈₉	—	2	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	—	2	2	no	—
121	C₄⋊C₄×C₈₉	—	4	1	no	—
122	C₄⋊C₄×C₈₉	—	4	1	no	—
123	C₄⋊C₄×C₈₉	C₂×D₄	1	2	yes	—
124	C₄²×C₈₉	—	2	2	no	—
125	C₄²×C₈₉	—	2	2	no	—
126	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₈₉	#41	1	1	yes	characteristic
129	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
130	C₂³×C₈₉	#121	1	1	yes	characteristic
131	C₂²×D₈₉	—	4	2	no	—
132	C₂²×D₈₉	—	8	2	no	—
133	C₂²×D₈₉	—	8	2	no	—
134	C₂²×D₈₉	—	8	2	no	—
135	Q₈×C₈₉	—	2	2	no	—
136	Q₈×C₈₉	—	2	2	no	—
137	Q₈×C₈₉	—	4	2	no	—
138	Q₈×C₈₉	—	8	2	no	—
139	Q₈×C₈₉	—	8	2	no	—
140	C₂×C₄×C₈₉	—	2	2	no	—
141	C₂×C₄×C₈₉	—	2	2	no	—
142	C₂×C₄×C₈₉	—	2	2	no	—
143	C₂×C₄×C₈₉	—	4	2	no	—
144	C₂×C₄×C₈₉	—	4	2	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	D₃₅₆	—	4	2	no	—
158	D₃₅₆	—	4	2	no	—
159	D₃₅₆	—	4	2	no	—
160	D₃₅₆	—	4	2	no	—
161	D₃₅₆	—	8	2	no	—
162	D₃₅₆	—	8	2	no	—
163	D₃₅₆	—	8	2	no	—
164	D₃₅₆	—	8	2	no	—
165	C₈×C₈₉	—	4	2	no	—
166	#10623494	—	2	2	no	—
167	#10623494	—	2	2	no	—
168	#10623494	—	4	2	no	—
169	#10623494	—	4	2	no	—
170	C₂²×C₈₉	#464	1	2	yes	—
171	C₂²×C₈₉	#468	1	2	yes	—
172	C₂²×C₈₉	#472	1	2	yes	—
173	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
174	C₂×D₈₉	—	8	2	no	—
175	C₂×D₈₉	—	8	2	no	—
176	C₂×D₈₉	—	8	2	no	—
177	C₂×D₈₉	—	16	4	no	—
178	C₄×C₈₉	—	2	1	no	—
179	C₄×C₈₉	—	2	1	no	—
180	C₄×C₈₉	—	4	2	no	—
181	C₄×C₈₉	—	4	2	no	—
182	C₄×C₈₉	—	8	2	no	—
183	C₄×C₈₉	#464	1	2	yes	—
184	C₄×C₈₉	#468	1	2	yes	—
185	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
186	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
187	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
188	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
189	C₂×#1698	—	89	1	no	maximal, Sylow
190	C₂×C₈₉	#1698	1	4	yes	complemented
191	C₂×C₈₉	C₂×#464	1	1	yes	characteristic
192	C₂×C₈₉	C₂×#468	1	1	yes	characteristic
193	C₂×C₈₉	C₂×#472	1	1	yes	characteristic
194	D₈₉	—	16	2	no	—
195	C₂×#486	—	89	1	no	—
196	C₂×#477	D₈₉	1	1	yes	characteristic, complemented
197	C₂×#460	—	89	1	no	—
198	C₂×#446	—	89	1	no	—
199	#1698	—	89	8	no	—
200	C₂×#335	—	89	1	no	—
201	C₂×#333	—	89	1	no	—
202	C₂×#331	—	89	1	no	—
203	C₈₉	C₂×#1698	1	1	yes	characteristic, complemented, Sylow
204	C₂²×SD₁₆	—	178	1	no	—
205	C₂×C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
206	C₂×#127	—	89	1	no	—
207	C₂×C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
208	#486	—	89	4	no	—
209	#486	—	178	2	no	—
210	#477	—	2	2	no	—
211	#477	C₂×D₈₉	1	4	yes	complemented
212	#460	—	89	4	no	—
213	#460	—	178	2	no	—
214	#446	—	89	4	no	—
215	#446	—	178	2	no	—
216	C₂×#107	—	2	1	no	—
217	C₂×C₄⋊C₈	—	89	1	no	—
218	C₂×C₄⋊C₈	—	89	1	no	—
219	C₂×C₄⋊C₈	—	178	1	no	—
220	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
221	C₂×#103	—	2	1	no	—
222	C₂×#102	—	178	1	no	—
223	C₂×#102	—	178	1	no	—
224	C₂×#102	—	178	1	no	—
225	C₂×#97	—	178	1	no	—
226	C₂×C₄×C₈	—	89	1	no	—
227	#335	—	89	4	no	—
228	#333	—	89	4	no	—
229	#331	—	89	4	no	—
230	C₂²×Q₈	—	2	1	no	—
231	C₂²×Q₈	C₂²⋊D₈₉	1	1	yes	characteristic
232	C₂²×D₄	—	178	1	no	—
233	C₂²×D₄	—	178	1	no	—
234	C₂²×D₄	—	178	1	no	—
235	C₂×SD₁₆	—	356	4	no	—
236	C₂×SD₁₆	—	178	4	no	—
237	C₂²×C₈	—	2	1	no	—
238	C₂²×C₈	—	89	1	no	—
239	C₂²×C₈	—	89	1	no	—
240	C₂²×C₈	—	178	1	no	—
241	C₂²×C₈	—	178	1	no	—
242	C₄⋊Q₈	—	2	1	no	—
243	C₄⋊Q₈	—	2	1	no	—
244	C₄⋊Q₈	—	2	1	no	—
245	C₄⋊Q₈	C₂²×D₈₉	1	2	yes	—
246	#127	—	89	2	no	—
247	#127	—	178	1	no	—
248	#127	—	178	1	no	—
249	#127	—	178	1	no	—
250	C₄×Q₈	—	2	1	no	—
251	C₄×Q₈	—	2	1	no	—
252	C₄×Q₈	C₂²⋊D₈₉	1	2	yes	—
253	C₄×Q₈	C₂²×D₈₉	1	2	yes	—
254	C₂×C₄⋊C₄	—	2	1	no	—
255	C₂×C₄⋊C₄	—	2	1	no	—
256	C₂×C₄⋊C₄	—	2	1	no	—
257	C₂×C₄⋊C₄	C₂²⋊D₈₉	1	1	yes	characteristic
258	C₂×C₄²	—	2	1	no	—
259	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
260	#107	—	2	4	no	—
261	C₄⋊C₈	—	2	1	no	—
262	C₄⋊C₈	—	89	2	no	—
263	C₄⋊C₈	—	89	2	no	—
264	C₄⋊C₈	—	178	4	no	—
265	C₄⋊C₈	—	178	1	no	—
266	C₄⋊C₈	—	178	1	no	—
267	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
268	#103	—	2	4	no	—
269	#102	—	178	4	no	—
270	#102	—	178	4	no	—
271	#102	—	178	4	no	—
272	#97	—	178	4	no	—
273	C₄×C₈	—	89	2	no	—
274	C₄×C₈	—	89	2	no	—
275	C₂⁴	—	356	1	no	—
276	C₂×Q₈	—	2	1	no	—
277	C₂×Q₈	—	2	2	no	—
278	C₂×Q₈	—	2	2	no	—
279	C₂×Q₈	—	2	2	no	—
280	C₂×Q₈	—	4	1	no	—
281	C₂×Q₈	—	8	2	no	—
282	C₂×Q₈	C₂×C₂²⋊D₈₉	1	2	yes	—
283	C₂×D₄	—	356	1	no	—
284	C₂×D₄	—	356	1	no	—
285	C₂×D₄	—	356	1	no	—
286	C₂×D₄	—	356	2	no	—
287	C₂×D₄	—	356	2	no	—
288	C₂×D₄	—	356	2	no	—
289	C₂×D₄	—	356	2	no	—
290	C₂×D₄	—	178	2	no	—
291	C₂×D₄	—	178	2	no	—
292	C₂×D₄	—	178	2	no	—
293	C₂×D₄	—	712	2	no	—
294	C₂²×C₄	—	2	1	no	—
295	C₂²×C₄	—	2	1	no	—
296	C₂²×C₄	—	4	1	no	—
297	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
298	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
299	C₂²×C₄	#13083573	1	1	yes	characteristic
300	SD₁₆	—	356	8	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	4	2	no	—
303	C₂×C₈	—	89	2	no	—
304	C₂×C₈	—	89	2	no	—
305	C₂×C₈	—	356	2	no	—
306	C₂×C₈	—	356	2	no	—
307	C₂×C₈	—	178	2	no	—
308	C₂×C₈	—	178	2	no	—
309	C₂×C₈	—	178	2	no	—
310	C₂×C₈	—	178	2	no	—
311	C₄⋊C₄	—	2	2	no	—
312	C₄⋊C₄	—	2	1	no	—
313	C₄⋊C₄	—	2	2	no	—
314	C₄⋊C₄	—	2	2	no	—
315	C₄⋊C₄	—	2	2	no	—
316	C₄⋊C₄	—	4	1	no	—
317	C₄⋊C₄	—	4	1	no	—
318	C₄⋊C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
319	C₄²	—	2	2	no	—
320	C₄²	—	2	2	no	—
321	C₄²	D₄×D₈₉	1	1	yes	characteristic
322	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
323	C₄²	#13083573	1	1	yes	characteristic
324	C₄²	C₂³×D₈₉	1	1	yes	characteristic
325	C₂³	—	356	2	no	—
326	C₂³	—	712	2	no	—
327	C₂³	—	712	2	no	—
328	C₂³	—	712	2	no	—
329	C₂³	#15667984	1	1	yes	characteristic
330	Q₈	—	2	2	no	—
331	Q₈	—	2	2	no	—
332	Q₈	—	4	2	no	—
333	Q₈	—	8	2	no	—
334	Q₈	—	8	2	no	—
335	D₄	—	356	2	no	—
336	D₄	—	356	2	no	—
337	D₄	—	356	2	no	—
338	D₄	—	356	2	no	—
339	D₄	—	712	2	no	—
340	D₄	—	712	2	no	—
341	D₄	—	712	2	no	—
342	D₄	—	712	2	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	4	2	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	4	2	no	—
349	C₂×C₄	—	8	2	no	—
350	C₂×C₄	#15667984	1	1	yes	characteristic
351	C₂×C₄	#15667984	1	1	yes	characteristic
352	C₂×C₄	#15667984	1	1	yes	characteristic
353	C₂×C₄	#15668035	1	2	yes	—
354	C₂×C₄	#15668037	1	2	yes	—
355	C₂×C₄	#15668043	1	2	yes	—
356	C₂×C₄	#15668044	1	2	yes	—
357	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
358	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
359	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
360	C₈	—	4	2	no	—
361	C₈	—	356	2	no	—
362	C₈	—	356	2	no	—
363	C₈	—	178	2	no	—
364	C₈	—	178	2	no	—
365	C₂²	—	1424	4	no	—
366	C₂²	—	712	2	no	—
367	C₂²	—	712	2	no	—
368	C₂²	—	712	2	no	—
369	C₂²	#19151822	1	2	yes	—
370	C₂²	#19151846	1	2	yes	—
371	C₂²	#19151866	1	2	yes	—
372	C₂²	C₂×#15667984	1	1	yes	characteristic
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	4	2	no	—
376	C₄	—	4	2	no	—
377	C₄	—	8	2	no	—
378	C₄	#19151822	1	2	yes	—
379	C₄	#19151846	1	2	yes	—
380	C₄	C₂×#15668035	1	1	yes	characteristic
381	C₄	C₂×#15668037	1	1	yes	characteristic
382	C₄	C₂×#15668043	1	1	yes	characteristic
383	C₄	C₂×#15668044	1	1	yes	characteristic
384	C₂	—	1424	2	no	—
385	C₂	#26019794	1	4	yes	direct product factor
386	C₂	C₂×#19151822	1	1	yes	characteristic
387	C₂	C₂×#19151846	1	1	yes	characteristic
388	C₂	C₂×#19151866	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15667984
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂×#477×C₈₉	C₂
socle	C₂³×C₈₉	#121
prime radical	C₂×#15668009	C₂²
hypercenter	C₂×#477	D₈₉
hypocenter	C₈₉	C₂×#1698
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#477

Complements

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

N	H ≅ G/N	type
3. C₂×#19151729	C₂	semidirect
4. #26019794	C₂	direct (N × H)
5. C₂×#19151350	C₂	semidirect
6. C₂×#19151347	C₂	semidirect
8. C₂×#477×C₈₉	C₂	semidirect
16. #19151729	C₂²	semidirect
25. #19151350	C₂²	semidirect
26. #19151347	C₂²	semidirect
31. #477×C₈₉	C₂²	semidirect
190. C₂×C₈₉	#1698	semidirect
196. C₂×#477	D₈₉	semidirect
203. C₈₉	C₂×#1698	semidirect
211. #477	C₂×D₈₉	semidirect
385. C₂	#26019794	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×#19151946	2
C₂×#19151798	2
C₂×#19151729	2
#26019794	2
C₂×#19151350	2
C₂×#19151347	2
C₂×#19151341	2
C₂×#477×C₈₉	2
C₂×#1698	89

Subgroups — C2×#26019794 #36537078

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26019794 #36537078