Subgroups — #36471302

Order 22784.

470 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19152016	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19151713	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151381	C₂	1	1	yes	maximal, characteristic, complemented
4	#26021866	C₂	1	1	yes	maximal, characteristic, complemented
5	#26018333	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018329	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#350×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15668032	C₄	1	1	yes	characteristic
9	C₂×#15668018	C₂²	1	1	yes	characteristic
10	#19152016	—	2	2	no	—
11	#19152016	—	2	2	no	—
12	C₂×#15667952	C₂²	1	1	yes	characteristic
13	C₂×#15667951	—	2	1	no	—
14	C₂²×#13083549	C₂²	1	1	yes	characteristic
15	C₂×C₁₆×D₈₉	—	2	1	no	—
16	#19151713	—	2	2	no	—
17	#19151713	—	2	2	no	—
18	C₂×#15667947	C₄	1	1	yes	characteristic
19	C₂×#15667939	C₄	1	1	yes	characteristic
20	C₂×#15667938	C₄	1	1	yes	characteristic
21	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
22	#19151381	—	2	2	no	—
23	C₂²⋊#13083549	—	2	1	no	—
24	C₂²⋊#13083549	—	2	1	no	—
25	C₂×#131×C₈₉	C₂²	1	1	yes	characteristic
26	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
27	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
28	#350×C₈₉	—	2	2	no	—
29	C₂×#13083574	C₂×C₄	1	1	yes	characteristic
30	#15668032	—	2	2	no	—
31	#15668032	—	2	2	no	—
32	C₂×#13083564	C₈	1	1	yes	characteristic
33	C₂×#13083562	C₈	1	1	yes	characteristic
34	C₂×#13083561	C₈	1	1	yes	characteristic
35	C₂×#13083559	C₈	1	1	yes	characteristic
36	#15668018	—	2	1	no	—
37	#15668018	—	2	1	no	—
38	#15668018	D₄	1	2	yes	—
39	#15668018	D₄	1	2	yes	—
40	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
41	C₂×#13083557	—	2	1	no	—
42	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×D₈₉	—	2	1	no	—
44	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
45	#15667952	—	2	1	no	—
46	#15667952	—	2	1	no	—
47	#15667951	—	2	4	no	—
48	C₂×#13083549	—	2	1	no	—
49	C₂×#13083549	—	2	1	no	—
50	C₂×#13083549	—	2	1	no	—
51	C₂×#13083549	D₄	1	1	yes	characteristic
52	C₂×#13083549	D₄	1	1	yes	characteristic
53	C₁₆×D₈₉	—	2	4	no	—
54	#15667947	—	2	2	no	—
55	#15667939	—	2	2	no	—
56	#15667938	—	2	2	no	—
57	C₈×Dic₈₉	—	2	1	no	—
58	C₈×Dic₈₉	—	2	1	no	—
59	C₂×#41×C₈₉	C₂×C₄	1	1	yes	characteristic
60	#131×C₈₉	—	2	1	no	—
61	#131×C₈₉	—	2	1	no	—
62	#131×C₈₉	—	2	1	no	—
63	#131×C₈₉	—	2	1	no	—
64	C₂×M₁₆×C₈₉	—	2	1	no	—
65	C₂×M₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
66	C₂²×C₈×C₈₉	—	2	1	no	—
67	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
68	M₃₂×C₈₉	—	2	1	no	—
69	M₃₂×C₈₉	D₄	1	2	yes	—
70	C₂×C₁₆×C₈₉	—	2	1	no	—
71	C₂×C₁₆×C₈₉	—	2	1	no	—
72	C₂×C₁₆×C₈₉	—	2	1	no	—
73	C₂×C₁₆×C₈₉	—	2	1	no	—
74	#13083574	—	2	1	no	—
75	#13083574	—	2	1	no	—
76	#13083574	C₂²⋊C₄	1	2	yes	—
77	#13083574	M₁₆	1	2	yes	—
78	C₂×Dic₁₇₈	C₂×C₈	1	1	yes	characteristic
79	C₂×D₃₅₆	C₂×C₈	1	1	yes	characteristic
80	C₂×C₂²⋊D₈₉	—	2	1	no	—
81	C₂²×Dic₈₉	—	2	1	no	—
82	C₂×C₄×D₈₉	—	2	1	no	—
83	#13083564	—	2	2	no	—
84	#13083562	—	2	2	no	—
85	#13083561	—	2	2	no	—
86	#13083559	—	2	2	no	—
87	#13083558	—	2	1	no	—
88	#13083558	C₂²⋊C₄	1	2	yes	—
89	#13083557	—	2	2	no	—
90	#13083557	—	2	2	no	—
91	C₂×#10623494	—	2	1	no	—
92	C₂×#10623494	—	2	1	no	—
93	C₂×#10623494	—	2	1	no	—
94	C₂×#10623494	C₂×C₈	1	1	yes	characteristic
95	C₂×#10623494	C₂×C₈	1	1	yes	characteristic
96	C₈×D₈₉	—	2	2	no	—
97	C₈×D₈₉	—	2	2	no	—
98	C₄×Dic₈₉	—	2	1	no	—
99	C₄×Dic₈₉	—	2	1	no	—
100	#13083549	—	2	1	no	—
101	#13083549	—	2	1	no	—
102	#13083549	—	4	1	no	—
103	#41×C₈₉	—	2	1	no	—
104	#41×C₈₉	—	2	1	no	—
105	#41×C₈₉	—	2	1	no	—
106	#41×C₈₉	—	2	1	no	—
107	C₂×Q₈×C₈₉	C₂×C₈	1	1	yes	characteristic
108	C₂×D₄×C₈₉	—	2	1	no	—
109	C₂×D₄×C₈₉	C₂×C₈	1	1	yes	characteristic
110	C₂²×C₄×C₈₉	—	2	1	no	—
111	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
112	M₁₆×C₈₉	—	2	1	no	—
113	M₁₆×C₈₉	—	2	1	no	—
114	M₁₆×C₈₉	—	4	1	no	—
115	M₁₆×C₈₉	—	4	1	no	—
116	C₂×C₈×C₈₉	—	2	1	no	—
117	C₂×C₈×C₈₉	—	4	1	no	—
118	C₂×C₈×C₈₉	—	4	1	no	—
119	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
124	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
125	C₁₆×C₈₉	—	2	2	no	—
126	C₁₆×C₈₉	—	4	1	no	—
127	C₂³×C₈₉	—	2	1	no	—
128	C₂³×C₈₉	#131	1	1	yes	characteristic
129	C₂²×D₈₉	—	2	1	no	—
130	Q₈×C₈₉	—	2	1	no	—
131	Q₈×C₈₉	—	2	1	no	—
132	D₄×C₈₉	—	2	1	no	—
133	D₄×C₈₉	—	2	1	no	—
134	D₄×C₈₉	—	4	1	no	—
135	D₄×C₈₉	—	4	1	no	—
136	C₂×C₄×C₈₉	—	2	1	no	—
137	C₂×C₄×C₈₉	—	4	1	no	—
138	C₂×C₄×C₈₉	—	4	1	no	—
139	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
143	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
144	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
145	C₂²⋊D₈₉	—	2	2	no	—
146	C₂²⋊D₈₉	—	2	2	no	—
147	C₂×Dic₈₉	—	2	1	no	—
148	C₂×Dic₈₉	—	2	2	no	—
149	C₂×Dic₈₉	—	4	1	no	—
150	C₂×Dic₈₉	—	4	1	no	—
151	D₃₅₆	—	2	1	no	—
152	D₃₅₆	C₂²⋊C₈	1	2	yes	—
153	C₄×D₈₉	—	2	2	no	—
154	C₄×D₈₉	—	2	2	no	—
155	Dic₁₇₈	—	2	1	no	—
156	Dic₁₇₈	C₂²⋊C₈	1	2	yes	—
157	C₈×C₈₉	—	4	1	no	—
158	C₈×C₈₉	#120	1	1	yes	characteristic
159	C₈×C₈₉	#121	1	1	yes	characteristic
160	C₈×C₈₉	#121	1	1	yes	characteristic
161	C₈×C₈₉	#123	1	1	yes	characteristic
162	#10623494	—	2	2	no	—
163	#10623494	—	2	1	no	—
164	#10623494	—	2	1	no	—
165	C₂²×C₈₉	—	2	1	no	—
166	C₂²×C₈₉	—	2	1	no	—
167	C₂²×C₈₉	—	4	1	no	—
168	C₂²×C₈₉	—	4	1	no	—
169	C₂²×C₈₉	—	4	1	no	—
170	C₂²×C₈₉	#408	1	1	yes	characteristic
171	C₂²×C₈₉	C₈×D₄	1	1	yes	characteristic
172	C₂²×C₈₉	#435	1	1	yes	characteristic
173	C₂×D₈₉	—	2	2	no	—
174	C₂×D₈₉	—	4	1	no	—
175	C₂×D₈₉	—	4	1	no	—
176	C₄×C₈₉	—	4	1	no	—
177	C₄×C₈₉	#386	1	1	yes	characteristic
178	C₄×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
179	C₄×C₈₉	C₈×D₄	1	1	yes	characteristic
180	C₄×C₈₉	#435	1	1	yes	characteristic
181	Dic₈₉	—	2	2	no	—
182	Dic₈₉	—	4	2	no	—
183	#12980	—	89	1	no	maximal, Sylow
184	C₂×C₈₉	—	2	1	no	—
185	C₂×C₈₉	—	2	1	no	—
186	C₂×C₈₉	—	8	1	no	—
187	C₂×C₈₉	#1869	1	1	yes	characteristic
188	C₂×C₈₉	#2170	1	1	yes	characteristic
189	C₂×C₈₉	#2171	1	1	yes	characteristic
190	D₈₉	—	4	2	no	—
191	C₂×#504	—	89	1	no	—
192	C₂×#443	—	89	1	no	—
193	C₂×#350	—	89	1	no	—
194	C₂×#350	D₈₉	1	1	yes	characteristic, complemented
195	#2113	—	89	1	no	—
196	#1376	—	89	1	no	—
197	#1374	—	89	1	no	—
198	C₈₉	#12980	1	1	yes	characteristic, complemented, Sylow
199	C₂×#135	—	89	1	no	—
200	C₂×#131	—	89	1	no	—
201	C₂×#131	C₂×D₈₉	1	1	yes	characteristic
202	#504	—	178	2	no	—
203	#504	—	178	2	no	—
204	C₂×M₃₂	—	89	1	no	—
205	C₂×M₃₂	—	178	1	no	—
206	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
207	C₂²×C₁₆	—	89	1	no	—
208	C₂²×C₁₆	—	178	1	no	—
209	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
210	#443	—	178	2	no	—
211	#443	—	178	2	no	—
212	C₂×#108	—	89	1	no	—
213	C₂×#104	—	89	1	no	—
214	C₂×#104	—	89	1	no	—
215	C₂×C₄×C₈	—	89	1	no	—
216	#350	—	2	2	no	—
217	#350	—	178	2	no	—
218	C₂²⋊C₁₆	—	178	1	no	—
219	C₂²⋊C₁₆	—	178	1	no	—
220	C₂×#41	—	89	1	no	—
221	C₂×#41	C₄×D₈₉	1	1	yes	characteristic
222	#135	—	178	2	no	—
223	#135	—	178	2	no	—
224	C₂×Q₁₆	—	89	1	no	—
225	C₂×SD₁₆	—	89	1	no	—
226	C₂×SD₁₆	—	89	1	no	—
227	C₂×D₈	—	89	1	no	—
228	#131	—	2	1	no	—
229	#131	—	2	1	no	—
230	#131	—	2	1	no	—
231	#131	—	2	1	no	—
232	#131	—	89	2	no	—
233	#131	—	89	2	no	—
234	#131	—	178	1	no	—
235	#131	—	178	1	no	—
236	C₂×M₁₆	—	2	1	no	—
237	C₂×M₁₆	—	89	1	no	—
238	C₂×M₁₆	—	178	1	no	—
239	C₂×M₁₆	C₄×D₈₉	1	1	yes	characteristic
240	C₂²×C₈	—	2	1	no	—
241	C₂²×C₈	—	89	1	no	—
242	C₂²×C₈	—	178	1	no	—
243	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
244	C₂×C₄²	—	89	1	no	—
245	M₃₂	—	2	1	no	—
246	M₃₂	—	178	4	no	—
247	M₃₂	—	178	1	no	—
248	M₃₂	—	178	1	no	—
249	M₃₂	C₂²⋊D₈₉	1	2	yes	—
250	C₂×C₁₆	—	2	1	no	—
251	C₂×C₁₆	—	2	1	no	—
252	C₂×C₁₆	—	2	1	no	—
253	C₂×C₁₆	—	2	1	no	—
254	C₂×C₁₆	—	89	1	no	—
255	C₂×C₁₆	—	89	1	no	—
256	C₂×C₁₆	—	178	4	no	—
257	C₂×C₁₆	—	178	1	no	—
258	C₂×C₁₆	—	178	1	no	—
259	C₂×C₁₆	—	178	1	no	—
260	#108	—	178	2	no	—
261	#104	—	178	2	no	—
262	#104	—	178	2	no	—
263	C₄×C₈	—	178	1	no	—
264	C₄×C₈	—	178	1	no	—
265	#41	—	2	1	no	—
266	#41	—	2	1	no	—
267	#41	—	2	1	no	—
268	#41	—	2	1	no	—
269	#41	—	89	2	no	—
270	#41	—	89	2	no	—
271	#41	—	178	1	no	—
272	#41	—	178	1	no	—
273	C₂×Q₈	—	89	1	no	—
274	C₂×Q₈	C₈×D₈₉	1	1	yes	characteristic
275	C₂×D₄	—	2	1	no	—
276	C₂×D₄	—	89	1	no	—
277	C₂×D₄	—	178	1	no	—
278	C₂×D₄	C₈×D₈₉	1	1	yes	characteristic
279	C₂²×C₄	—	2	1	no	—
280	C₂²×C₄	—	178	1	no	—
281	C₂²×C₄	—	178	1	no	—
282	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
283	Q₁₆	—	178	2	no	—
284	SD₁₆	—	178	2	no	—
285	SD₁₆	—	178	2	no	—
286	D₈	—	178	2	no	—
287	M₁₆	—	2	1	no	—
288	M₁₆	—	2	1	no	—
289	M₁₆	—	4	1	no	—
290	M₁₆	—	4	1	no	—
291	M₁₆	—	89	2	no	—
292	M₁₆	—	178	1	no	—
293	M₁₆	—	178	2	no	—
294	M₁₆	—	178	2	no	—
295	C₂×C₈	—	2	1	no	—
296	C₂×C₈	—	4	1	no	—
297	C₂×C₈	—	4	1	no	—
298	C₂×C₈	—	89	1	no	—
299	C₂×C₈	—	89	1	no	—
300	C₂×C₈	—	178	1	no	—
301	C₂×C₈	—	178	1	no	—
302	C₂×C₈	—	178	1	no	—
303	C₂×C₈	—	178	2	no	—
304	C₂×C₈	—	178	2	no	—
305	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
306	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
307	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
308	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
309	C₂×C₈	#13083573	1	1	yes	characteristic
310	C₂×C₈	#13083574	1	1	yes	characteristic
311	C₄²	—	178	1	no	—
312	C₄²	—	178	1	no	—
313	C₁₆	—	2	2	no	—
314	C₁₆	—	4	1	no	—
315	C₁₆	—	356	1	no	—
316	C₁₆	—	178	1	no	—
317	C₁₆	—	178	1	no	—
318	C₂³	—	2	1	no	—
319	C₂³	—	178	1	no	—
320	C₂³	#15668020	1	1	yes	characteristic
321	Q₈	—	2	1	no	—
322	Q₈	—	2	1	no	—
323	Q₈	—	89	2	no	—
324	Q₈	—	178	1	no	—
325	D₄	—	2	1	no	—
326	D₄	—	2	1	no	—
327	D₄	—	4	1	no	—
328	D₄	—	4	1	no	—
329	D₄	—	89	2	no	—
330	D₄	—	178	2	no	—
331	D₄	—	178	2	no	—
332	D₄	—	178	1	no	—
333	C₂×C₄	—	2	1	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	4	1	no	—
336	C₂×C₄	—	356	1	no	—
337	C₂×C₄	—	356	1	no	—
338	C₂×C₄	—	178	2	no	—
339	C₂×C₄	—	178	1	no	—
340	C₂×C₄	—	178	2	no	—
341	C₂×C₄	—	178	2	no	—
342	C₂×C₄	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
343	C₂×C₄	#15667972	1	1	yes	characteristic
344	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
345	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
346	C₂×C₄	M₁₆×D₈₉	1	1	yes	characteristic
347	C₂×C₄	#15668018	1	1	yes	characteristic
348	C₈	—	4	1	no	—
349	C₈	—	178	2	no	—
350	C₈	—	178	1	no	—
351	C₈	—	178	1	no	—
352	C₈	#15667980	1	1	yes	characteristic
353	C₈	#15667983	1	1	yes	characteristic
354	C₈	#15667985	1	1	yes	characteristic
355	C₈	#15667994	1	1	yes	characteristic
356	C₂²	—	2	1	no	—
357	C₂²	—	2	1	no	—
358	C₂²	—	4	1	no	—
359	C₂²	—	4	1	no	—
360	C₂²	—	4	1	no	—
361	C₂²	—	356	1	no	—
362	C₂²	—	356	1	no	—
363	C₂²	—	178	2	no	—
364	C₂²	#19151563	1	1	yes	characteristic
365	C₂²	#19151661	1	1	yes	characteristic
366	C₂²	#19151667	1	1	yes	characteristic
367	C₄	—	4	1	no	—
368	C₄	—	356	2	no	—
369	C₄	—	178	2	no	—
370	C₄	#19151474	1	1	yes	characteristic
371	C₄	C₂²⋊C₈×D₈₉	1	1	yes	characteristic
372	C₄	C₈×C₂²⋊D₈₉	1	1	yes	characteristic
373	C₄	#19151672	1	1	yes	characteristic
374	C₂	—	2	1	no	—
375	C₂	—	2	1	no	—
376	C₂	—	8	1	no	—
377	C₂	—	356	2	no	—
378	C₂	#26020541	1	1	yes	characteristic
379	C₂	#26022086	1	1	yes	characteristic
380	C₂	#26022091	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×#350×C₈₉	C₂
socle	C₂²×C₈₉	C₈×D₄
prime radical	C₂×#15668032	C₄
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×#350	D₈₉
hypocenter	C₈₉	#12980
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#350

Complements

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

N	H ≅ G/N	type
1. C₂×#19152016	C₂	semidirect
3. C₂×#19151381	C₂	semidirect
4. #26021866	C₂	semidirect
5. #26018333	C₂	semidirect
6. #26018329	C₂	semidirect
7. C₂×#350×C₈₉	C₂	semidirect
194. C₂×#350	D₈₉	semidirect
198. C₈₉	#12980	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19152016	2
C₂×#19151713	2
C₂×#19151381	2
#26021866	2
#26018333	2
#26018329	2
C₂×#350×C₈₉	2
#12980	89