Subgroups — C₂×#26022385 #36709000

Order 22784.

658 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15667955	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19152016	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19151870	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#19151869	C₂	1	1	yes	maximal, characteristic
5	#26022385	C₂	1	8	yes	maximal, direct product factor
6	C₂×#19151427	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#19151412	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#362×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#15668043	—	2	1	no	—
10	C₂×#15668040	—	2	1	no	—
11	C₂²×D₇₁₂	C₂²	1	1	yes	characteristic
12	C₂×#15668018	C₂²	1	1	yes	characteristic
13	C₂×#15667955	—	2	1	no	—
14	C₂×#15667955	—	2	1	no	—
15	C₂×#15667955	—	2	2	no	—
16	C₂×#15667955	C₂²	1	4	yes	complemented
17	#19152016	—	2	2	no	—
18	#19152016	C₂²	1	4	yes	complemented
19	C₂×#15667951	—	2	1	no	—
20	C₂²×#13083549	C₂²	1	1	yes	characteristic
21	C₂×C₁₆×D₈₉	—	2	1	no	—
22	#19151870	—	2	2	no	—
23	#19151870	C₂²	1	4	yes	complemented
24	#19151869	—	2	2	no	—
25	#19151869	C₂²	1	4	yes	—
26	C₂×#15667948	C₂²	1	1	yes	characteristic
27	C₂×#15667931	—	2	1	no	—
28	C₂×#15667928	—	2	1	no	—
29	#19151427	C₂²	1	4	yes	complemented
30	#19151412	C₂²	1	4	yes	complemented
31	C₂²×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
32	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
33	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
34	#362×C₈₉	C₂²	1	4	yes	complemented
35	C₂×#13083574	D₄	1	1	yes	characteristic
36	C₂²×D₃₅₆	—	2	1	no	—
37	#15668043	—	2	4	no	—
38	#15668043	—	4	2	no	—
39	#15668040	—	2	4	no	—
40	#15668040	—	4	2	no	—
41	C₂×#13083564	—	4	1	no	—
42	C₂×#13083563	—	4	1	no	—
43	C₂×#13083561	—	4	1	no	—
44	C₂×D₇₁₂	—	2	1	no	—
45	C₂×D₇₁₂	—	2	1	no	—
46	C₂×D₇₁₂	—	2	1	no	—
47	C₂×D₇₁₂	—	2	1	no	—
48	C₂×D₇₁₂	—	2	1	no	—
49	C₂×D₇₁₂	—	4	1	no	—
50	C₂×D₇₁₂	C₂³	1	2	yes	—
51	#15668018	—	2	1	no	—
52	#15668018	—	2	1	no	—
53	#15668018	D₄	1	2	yes	—
54	#15668018	C₂³	1	2	yes	—
55	C₂×#13083558	—	2	1	no	—
56	C₂×#13083558	D₄	1	1	yes	characteristic
57	C₂×#13083557	—	2	1	no	—
58	C₂×C₈×D₈₉	—	2	1	no	—
59	#15667955	—	2	4	no	—
60	#15667955	—	2	4	no	—
61	#15667951	—	2	4	no	—
62	C₂×#13083549	D₄	1	1	yes	characteristic
63	C₂×#13083549	D₄	1	1	yes	characteristic
64	C₂×#13083549	D₄	1	2	yes	—
65	C₂×#13083549	C₂³	1	2	yes	—
66	C₁₆×D₈₉	—	2	4	no	—
67	#15667948	—	2	1	no	—
68	#15667948	C₂³	1	2	yes	—
69	#15667931	—	2	4	no	—
70	#15667928	—	2	4	no	—
71	C₂²×Q₈×C₈₉	—	2	1	no	—
72	C₂×Q₁₆×C₈₉	—	2	1	no	—
73	C₂×Q₁₆×C₈₉	—	2	1	no	—
74	C₂×Q₁₆×C₈₉	—	2	1	no	—
75	C₂×Q₁₆×C₈₉	—	2	1	no	—
76	C₂×Q₁₆×C₈₉	—	2	1	no	—
77	C₂×Q₁₆×C₈₉	C₂³	1	2	yes	—
78	C₂×M₁₆×C₈₉	—	2	1	no	—
79	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
80	M₃₂×C₈₉	—	2	1	no	—
81	M₃₂×C₈₉	C₂³	1	2	yes	—
82	C₂×C₁₆×C₈₉	—	2	1	no	—
83	#108×C₈₉	—	2	1	no	—
84	#108×C₈₉	C₂³	1	2	yes	—
85	C₂³×D₈₉	—	4	1	no	—
86	#13083574	—	2	1	no	—
87	#13083574	—	2	1	no	—
88	#13083574	D₈	1	2	yes	—
89	#13083574	C₂×D₄	1	2	yes	—
90	C₂×Dic₁₇₈	D₈	1	1	yes	characteristic
91	C₂×D₃₅₆	—	2	2	no	—
92	C₂×D₃₅₆	—	4	1	no	—
93	C₂×D₃₅₆	—	4	1	no	—
94	C₂×D₃₅₆	—	4	1	no	—
95	C₂×D₃₅₆	—	4	1	no	—
96	C₂×D₃₅₆	—	4	1	no	—
97	C₂×D₃₅₆	D₈	1	1	yes	characteristic
98	C₂×C₂²⋊D₈₉	—	2	1	no	—
99	C₂×C₄×D₈₉	—	2	1	no	—
100	#13083564	—	4	4	no	—
101	#13083563	—	4	4	no	—
102	#13083561	—	4	4	no	—
103	D₇₁₂	—	2	2	no	—
104	D₇₁₂	—	2	2	no	—
105	D₇₁₂	—	4	4	no	—
106	D₇₁₂	—	4	1	no	—
107	D₇₁₂	—	4	1	no	—
108	#13083558	—	2	1	no	—
109	#13083558	—	2	2	no	—
110	#13083558	—	4	1	no	—
111	#13083558	C₂×D₄	1	2	yes	—
112	#13083557	—	2	2	no	—
113	#13083557	—	2	2	no	—
114	C₂×#10623494	—	2	1	no	—
115	C₂×#10623494	—	4	1	no	—
116	C₈×D₈₉	—	2	2	no	—
117	C₈×D₈₉	—	2	2	no	—
118	#13083549	C₂×D₄	1	2	yes	—
119	#13083549	C₂×D₄	1	2	yes	—
120	C₂×Q₈×C₈₉	—	2	2	no	—
121	C₂×Q₈×C₈₉	—	4	1	no	—
122	C₂×Q₈×C₈₉	—	4	1	no	—
123	C₂×Q₈×C₈₉	—	4	1	no	—
124	C₂×Q₈×C₈₉	—	4	1	no	—
125	C₂×Q₈×C₈₉	—	4	1	no	—
126	C₂²×C₄×C₈₉	—	4	1	no	—
127	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
128	Q₁₆×C₈₉	—	2	2	no	—
129	Q₁₆×C₈₉	—	2	2	no	—
130	Q₁₆×C₈₉	—	4	1	no	—
131	Q₁₆×C₈₉	—	4	1	no	—
132	M₁₆×C₈₉	—	2	2	no	—
133	M₁₆×C₈₉	—	4	1	no	—
134	C₂×C₈×C₈₉	—	4	1	no	—
135	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
136	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
137	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
138	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
139	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
140	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
141	C₁₆×C₈₉	—	2	2	no	—
142	C₂³×C₈₉	#135	1	1	yes	characteristic
143	C₂²×D₈₉	—	2	1	no	—
144	C₂²×D₈₉	—	4	2	no	—
145	C₂²×D₈₉	—	8	1	no	—
146	C₂²×D₈₉	—	8	1	no	—
147	C₂²×D₈₉	—	16	2	no	—
148	Q₈×C₈₉	—	4	2	no	—
149	Q₈×C₈₉	—	4	2	no	—
150	Q₈×C₈₉	—	8	1	no	—
151	Q₈×C₈₉	—	8	1	no	—
152	C₂×C₄×C₈₉	—	4	2	no	—
153	C₂×C₄×C₈₉	—	8	1	no	—
154	C₂×C₄×C₈₉	—	8	1	no	—
155	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
156	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
157	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
158	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
159	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
160	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
161	C₂²⋊D₈₉	—	2	2	no	—
162	C₂²⋊D₈₉	—	2	2	no	—
163	C₂×Dic₈₉	—	2	1	no	—
164	D₃₅₆	—	2	1	no	—
165	D₃₅₆	—	4	2	no	—
166	D₃₅₆	—	4	2	no	—
167	D₃₅₆	—	8	1	no	—
168	D₃₅₆	—	8	1	no	—
169	D₃₅₆	C₂×D₈	1	2	yes	—
170	C₄×D₈₉	—	2	2	no	—
171	C₄×D₈₉	—	2	2	no	—
172	Dic₁₇₈	—	2	1	no	—
173	Dic₁₇₈	C₂×D₈	1	2	yes	—
174	C₈×C₈₉	—	4	2	no	—
175	C₈×C₈₉	#121	1	1	yes	characteristic
176	C₈×C₈₉	#121	1	1	yes	characteristic
177	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
178	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
179	#10623494	—	2	2	no	—
180	#10623494	—	4	2	no	—
181	C₂²×C₈₉	—	2	2	no	—
182	C₂²×C₈₉	#466	1	1	yes	characteristic
183	C₂²×C₈₉	#466	1	1	yes	characteristic
184	C₂²×C₈₉	C₂×#135	1	1	yes	characteristic
185	C₂×D₈₉	—	2	2	no	—
186	C₂×D₈₉	—	4	2	no	—
187	C₂×D₈₉	—	8	2	no	—
188	C₂×D₈₉	—	16	2	no	—
189	C₂×D₈₉	—	16	2	no	—
190	C₂×D₈₉	—	16	2	no	—
191	C₄×C₈₉	—	8	2	no	—
192	C₄×C₈₉	#466	1	1	yes	characteristic
193	C₄×C₈₉	#466	1	1	yes	characteristic
194	C₄×C₈₉	C₂×#121	1	1	yes	characteristic
195	C₄×C₈₉	C₂²×D₈	1	1	yes	characteristic
196	Dic₈₉	—	2	2	no	—
197	C₂×#2223	—	89	1	no	maximal, Sylow
198	C₂×C₈₉	—	2	1	no	—
199	C₂×C₈₉	—	2	1	no	—
200	C₂×C₈₉	#2223	1	2	yes	complemented
201	C₂×C₈₉	C₂×#466	1	1	yes	characteristic
202	D₈₉	—	4	2	no	—
203	D₈₉	—	16	2	no	—
204	C₂²×SD₃₂	—	89	1	no	—
205	C₂×#504	—	89	1	no	—
206	C₂×#473	—	89	1	no	—
207	C₂×#472	—	89	1	no	—
208	#2223	—	89	8	no	—
209	C₂×#368	—	89	1	no	—
210	C₂×#362	D₈₉	1	1	yes	characteristic, complemented
211	C₂×#361	—	89	1	no	—
212	C₈₉	C₂×#2223	1	1	yes	characteristic, complemented, Sylow
213	C₂×#137	—	178	1	no	—
214	C₂×AGL(1,ℤ/8)	—	178	1	no	—
215	C₂²×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
216	C₂²×D₈	—	89	1	no	—
217	C₂×#131	—	89	1	no	—
218	C₂×SD₃₂	—	89	4	no	—
219	C₂×SD₃₂	—	178	1	no	—
220	C₂×SD₃₂	—	178	1	no	—
221	C₂×SD₃₂	—	178	2	no	—
222	#504	—	89	4	no	—
223	#504	—	178	2	no	—
224	C₂×M₃₂	—	178	1	no	—
225	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
226	C₂²×C₁₆	—	89	1	no	—
227	C₂²×C₁₆	—	178	1	no	—
228	#473	—	89	4	no	—
229	#473	—	178	2	no	—
230	#472	—	89	4	no	—
231	#472	—	178	2	no	—
232	C₂×#108	—	89	1	no	—
233	C₂×#108	C₂×D₈₉	1	1	yes	characteristic
234	C₂×#101	—	178	1	no	—
235	C₂×#100	—	178	1	no	—
236	#368	—	89	4	no	—
237	#362	C₂×D₈₉	1	4	yes	complemented
238	#361	—	89	4	no	—
239	C₂×#41	—	89	1	no	—
240	C₂²×Q₈	—	2	1	no	—
241	C₂²×D₄	—	178	1	no	—
242	#137	—	356	2	no	—
243	#137	—	178	4	no	—
244	AGL(1,ℤ/8)	—	356	2	no	—
245	AGL(1,ℤ/8)	—	178	4	no	—
246	C₂×Q₁₆	—	2	1	no	—
247	C₂×Q₁₆	—	2	1	no	—
248	C₂×Q₁₆	—	2	1	no	—
249	C₂×Q₁₆	—	2	1	no	—
250	C₂×Q₁₆	—	2	1	no	—
251	C₂×Q₁₆	—	356	1	no	—
252	C₂×Q₁₆	C₂²×D₈₉	1	2	yes	—
253	C₂×SD₁₆	—	356	1	no	—
254	C₂×SD₁₆	—	356	1	no	—
255	C₂×D₈	—	89	2	no	—
256	C₂×D₈	—	356	1	no	—
257	C₂×D₈	—	178	1	no	—
258	C₂×D₈	—	178	1	no	—
259	C₂×D₈	—	178	1	no	—
260	C₂×D₈	—	178	1	no	—
261	C₂×D₈	—	178	1	no	—
262	#131	—	89	2	no	—
263	#131	—	89	2	no	—
264	#131	—	178	1	no	—
265	#131	—	178	1	no	—
266	C₂×M₁₆	—	2	1	no	—
267	C₂×M₁₆	—	89	1	no	—
268	C₂×M₁₆	—	178	1	no	—
269	C₂×M₁₆	—	178	1	no	—
270	C₂²×C₈	—	178	1	no	—
271	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
272	SD₃₂	—	178	4	no	—
273	SD₃₂	—	178	4	no	—
274	M₃₂	—	2	1	no	—
275	M₃₂	—	178	4	no	—
276	M₃₂	C₂²×D₈₉	1	2	yes	—
277	C₂×C₁₆	—	2	1	no	—
278	C₂×C₁₆	—	89	1	no	—
279	C₂×C₁₆	—	89	1	no	—
280	C₂×C₁₆	—	89	2	no	—
281	C₂×C₁₆	—	89	2	no	—
282	C₂×C₁₆	—	178	4	no	—
283	#108	—	2	1	no	—
284	#108	—	89	2	no	—
285	#108	—	178	1	no	—
286	#108	C₂²×D₈₉	1	2	yes	—
287	#101	—	178	4	no	—
288	#100	—	178	4	no	—
289	C₂⁴	—	356	1	no	—
290	#41	—	89	2	no	—
291	#41	—	89	2	no	—
292	#41	—	178	1	no	—
293	#41	—	178	1	no	—
294	C₂×Q₈	—	2	2	no	—
295	C₂×Q₈	—	4	1	no	—
296	C₂×Q₈	—	4	1	no	—
297	C₂×Q₈	—	4	1	no	—
298	C₂×Q₈	—	4	1	no	—
299	C₂×Q₈	—	4	1	no	—
300	C₂×Q₈	—	89	1	no	—
301	C₂×D₄	—	89	1	no	—
302	C₂×D₄	—	356	1	no	—
303	C₂×D₄	—	356	1	no	—
304	C₂×D₄	—	356	1	no	—
305	C₂×D₄	—	356	1	no	—
306	C₂×D₄	—	356	1	no	—
307	C₂×D₄	—	178	1	no	—
308	C₂×D₄	—	178	2	no	—
309	C₂²×C₄	—	4	1	no	—
310	C₂²×C₄	—	178	1	no	—
311	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
312	Q₁₆	—	2	2	no	—
313	Q₁₆	—	2	2	no	—
314	Q₁₆	—	4	1	no	—
315	Q₁₆	—	4	1	no	—
316	Q₁₆	—	356	4	no	—
317	SD₁₆	—	356	4	no	—
318	SD₁₆	—	356	4	no	—
319	D₈	—	356	4	no	—
320	D₈	—	356	1	no	—
321	D₈	—	356	1	no	—
322	D₈	—	178	2	no	—
323	D₈	—	178	2	no	—
324	M₁₆	—	2	2	no	—
325	M₁₆	—	4	1	no	—
326	M₁₆	—	89	2	no	—
327	M₁₆	—	356	1	no	—
328	M₁₆	—	178	2	no	—
329	M₁₆	—	178	1	no	—
330	M₁₆	—	178	2	no	—
331	M₁₆	—	178	2	no	—
332	C₂×C₈	—	4	1	no	—
333	C₂×C₈	—	356	1	no	—
334	C₂×C₈	—	178	1	no	—
335	C₂×C₈	—	178	2	no	—
336	C₂×C₈	—	178	2	no	—
337	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
338	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
339	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
340	C₂×C₈	#13083574	1	1	yes	characteristic
341	C₂×C₈	#13083574	1	1	yes	characteristic
342	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
343	C₁₆	—	2	2	no	—
344	C₁₆	—	89	2	no	—
345	C₁₆	—	89	2	no	—
346	C₂³	—	356	2	no	—
347	C₂³	—	1424	2	no	—
348	C₂³	—	178	1	no	—
349	C₂³	—	712	1	no	—
350	C₂³	—	712	1	no	—
351	C₂³	#15668033	1	1	yes	characteristic
352	Q₈	—	4	2	no	—
353	Q₈	—	4	2	no	—
354	Q₈	—	8	1	no	—
355	Q₈	—	8	1	no	—
356	Q₈	—	89	2	no	—
357	Q₈	—	178	1	no	—
358	D₄	—	89	2	no	—
359	D₄	—	356	2	no	—
360	D₄	—	356	2	no	—
361	D₄	—	178	2	no	—
362	D₄	—	178	2	no	—
363	D₄	—	178	1	no	—
364	D₄	—	712	1	no	—
365	D₄	—	712	1	no	—
366	C₂×C₄	—	4	2	no	—
367	C₂×C₄	—	8	1	no	—
368	C₂×C₄	—	8	1	no	—
369	C₂×C₄	—	178	1	no	—
370	C₂×C₄	—	178	2	no	—
371	C₂×C₄	—	178	2	no	—
372	C₂×C₄	#15667985	1	1	yes	characteristic
373	C₂×C₄	#15667985	1	1	yes	characteristic
374	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
375	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
376	C₂×C₄	#15668033	1	1	yes	characteristic
377	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
378	C₈	—	4	2	no	—
379	C₈	—	356	2	no	—
380	C₈	—	178	2	no	—
381	C₈	#15667985	1	1	yes	characteristic
382	C₈	#15667985	1	1	yes	characteristic
383	C₈	C₂×D₄×D₈₉	1	1	yes	characteristic
384	C₈	C₂×#13083574	1	1	yes	characteristic
385	C₂²	—	2	2	no	—
386	C₂²	—	356	2	no	—
387	C₂²	—	1424	2	no	—
388	C₂²	—	1424	2	no	—
389	C₂²	—	1424	2	no	—
390	C₂²	—	178	2	no	—
391	C₂²	—	712	2	no	—
392	C₂²	#19151833	1	1	yes	characteristic
393	C₂²	#19151833	1	1	yes	characteristic
394	C₂²	C₂×#15668033	1	1	yes	characteristic
395	C₄	—	8	2	no	—
396	C₄	—	178	2	no	—
397	C₄	#19151833	1	1	yes	characteristic
398	C₄	#19151833	1	1	yes	characteristic
399	C₄	C₂×#15667985	1	1	yes	characteristic
400	C₄	C₂×D₈×D₈₉	1	1	yes	characteristic
401	C₂	—	2	1	no	—
402	C₂	—	2	1	no	—
403	C₂	—	356	2	no	—
404	C₂	—	1424	2	no	—
405	C₂	#26022385	1	2	yes	direct product factor
406	C₂	C₂×#19151833	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19151833
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×#362×C₈₉	C₂
socle	C₂²×C₈₉	#466
prime radical	C₂×#19151869	C₂
hypercenter	C₂×#362	D₈₉
hypocenter	C₈₉	C₂×#2223
LCS	G ▷ C₂×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
1. C₂²×#15667955	C₂	semidirect
2. C₂×#19152016	C₂	semidirect
3. C₂×#19151870	C₂	semidirect
5. #26022385	C₂	direct (N × H)
6. C₂×#19151427	C₂	semidirect
7. C₂×#19151412	C₂	semidirect
8. C₂×#362×C₈₉	C₂	semidirect
16. C₂×#15667955	C₂²	semidirect
18. #19152016	C₂²	semidirect
23. #19151870	C₂²	semidirect
29. #19151427	C₂²	semidirect
30. #19151412	C₂²	semidirect
34. #362×C₈₉	C₂²	semidirect
200. C₂×C₈₉	#2223	semidirect
210. C₂×#362	D₈₉	semidirect
212. C₈₉	C₂×#2223	semidirect
237. #362	C₂×D₈₉	semidirect
405. C₂	#26022385	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#15667955	2
C₂×#19152016	2
C₂×#19151870	2
C₂×#19151869	2
#26022385	2
C₂×#19151427	2
C₂×#19151412	2
C₂×#362×C₈₉	2
C₂×#2223	89

Subgroups — C2×#26022385 #36709000

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26022385 #36709000