Subgroups — #36613316

Order 22784.

674 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030780	C₂	1	1	yes	maximal, characteristic, complemented
2	#26030107	C₂	1	2	yes	maximal
3	#26028847	C₂	1	1	yes	maximal, characteristic
4	#26021761	C₂	1	2	yes	maximal, complemented
5	#26021603	C₂	1	2	yes	maximal, complemented
6	#26021033	C₂	1	1	yes	maximal, characteristic, complemented
7	#26020998	C₂	1	1	yes	maximal, characteristic, complemented
8	#26020483	C₂	1	2	yes	maximal
9	#26020413	C₂	1	1	yes	maximal, characteristic, complemented
10	#26020087	C₂	1	1	yes	maximal, characteristic
11	C₈₉×#1815	C₂	1	1	yes	maximal, characteristic
12	#19152229	C₂²	1	2	yes	—
13	#19152081	C₂²	1	2	yes	—
14	#19152005	—	2	2	no	—
15	#19151998	—	2	1	no	—
16	#19151995	—	2	1	no	—
17	#19151903	—	2	2	no	—
18	#19151900	—	2	2	no	—
19	#19151891	—	2	2	no	—
20	#19151876	—	2	2	no	—
21	#19151640	—	2	2	no	—
22	#19151640	C₂²	1	2	yes	—
23	#19151640	C₂²	1	2	yes	—
24	#19151638	C₂²	1	2	yes	—
25	#19151636	—	2	1	no	—
26	#19151636	C₂²	1	1	yes	characteristic
27	C₂×#15667945	C₂²	1	2	yes	—
28	C₂×#15667944	—	2	1	no	—
29	C₂×#15667944	C₂²	1	1	yes	characteristic
30	#19151626	C₂²	1	2	yes	—
31	C₂×#15667939	C₂²	1	1	yes	characteristic
32	#19151594	C₂²	1	1	yes	characteristic
33	#19151553	C₂²	1	1	yes	characteristic
34	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
35	#19151463	C₂²	1	2	yes	—
36	#19151366	C₂²	1	2	yes	—
37	#19151362	C₂²	1	1	yes	characteristic
38	#19151359	C₂²	1	2	yes	—
39	#19151359	C₂²	1	1	yes	characteristic
40	C₂×#131×C₈₉	C₂²	1	1	yes	characteristic
41	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
42	C₂×#106×C₈₉	C₂²	1	1	yes	characteristic
43	#417×C₈₉	C₂²	1	2	yes	—
44	#343×C₈₉	C₂²	1	2	yes	—
45	C₂×#13083558	C₂³	1	1	yes	characteristic
46	C₂²×#10623494	C₂³	1	1	yes	characteristic
47	#15668011	—	2	2	no	—
48	#15668011	—	2	2	no	—
49	#15668003	—	2	2	no	—
50	#15668003	—	2	2	no	—
51	#15667991	—	4	2	no	—
52	Q₈×Dic₈₉	—	2	2	no	—
53	D₄×Dic₈₉	—	2	2	no	—
54	D₄×Dic₈₉	—	4	2	no	—
55	#15667971	—	2	2	no	—
56	#15667971	C₂³	1	2	yes	—
57	#15667971	C₂³	1	1	yes	characteristic
58	C₂×C₄⋊Dic₈₉	—	2	2	no	—
59	C₂×C₄⋊Dic₈₉	C₂³	1	2	yes	—
60	C₂×C₄×Dic₈₉	C₂³	1	1	yes	characteristic
61	#15667946	—	2	2	no	—
62	#15667946	—	2	2	no	—
63	#15667946	D₄	1	2	yes	—
64	#15667945	—	2	2	no	—
65	#15667945	—	2	2	no	—
66	#15667945	—	2	2	no	—
67	#15667944	—	2	2	no	—
68	#15667944	—	2	2	no	—
69	#15667944	—	2	1	no	—
70	#15667944	—	2	1	no	—
71	#15667944	D₄	1	1	yes	characteristic
72	#15667944	D₄	1	1	yes	characteristic
73	#15667943	—	2	2	no	—
74	#15667941	—	2	2	no	—
75	#15667941	—	2	2	no	—
76	#15667939	Q₈	1	4	yes	complemented
77	#15667936	D₄	1	2	yes	—
78	#15667933	D₄	1	2	yes	—
79	#15667922	—	2	1	no	—
80	C₈×Dic₈₉	—	2	1	no	—
81	C₈×Dic₈₉	—	2	2	no	—
82	#15667919	—	4	2	no	—
83	C₂×#41×C₈₉	C₂³	1	1	yes	characteristic
84	#131×C₈₉	—	2	2	no	—
85	#131×C₈₉	Q₈	1	4	yes	complemented
86	C₂×M₁₆×C₈₉	—	2	1	no	—
87	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
88	C₂×M₁₆×C₈₉	C₂³	1	1	yes	characteristic
89	C₂²×C₈×C₈₉	—	2	1	no	—
90	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
91	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
92	#108×C₈₉	—	2	2	no	—
93	#106×C₈₉	—	2	1	no	—
94	#106×C₈₉	—	2	1	no	—
95	#103×C₈₉	—	2	2	no	—
96	#102×C₈₉	—	2	2	no	—
97	C₂×Dic₁₇₈	—	4	2	no	—
98	C₂²×Dic₈₉	—	2	1	no	—
99	C₂²×Dic₈₉	—	2	2	no	—
100	C₂²×Dic₈₉	—	4	2	no	—
101	#13083558	—	2	1	no	—
102	#13083558	C₂×Q₈	1	2	yes	—
103	C₂×#10623494	—	2	1	no	—
104	C₂×#10623494	—	2	1	no	—
105	C₂×#10623494	—	2	1	no	—
106	C₂×#10623494	C₂×D₄	1	2	yes	—
107	C₄⋊Dic₈₉	—	2	2	no	—
108	C₄⋊Dic₈₉	—	2	2	no	—
109	C₄⋊Dic₈₉	—	4	2	no	—
110	C₄⋊Dic₈₉	—	4	2	no	—
111	C₄⋊Dic₈₉	C₂×D₄	1	2	yes	—
112	C₄⋊Dic₈₉	C₂×D₄	1	2	yes	—
113	C₄⋊Dic₈₉	C₂×D₄	1	2	yes	—
114	#13083553	—	4	2	no	—
115	#13083553	—	4	2	no	—
116	C₂²⋊Dic₈₉	—	2	2	no	—
117	C₂²⋊Dic₈₉	—	2	1	no	—
118	C₂²⋊Dic₈₉	—	4	2	no	—
119	C₄×Dic₈₉	—	2	2	no	—
120	C₄×Dic₈₉	—	2	1	no	—
121	C₄×Dic₈₉	—	2	1	no	—
122	C₄×Dic₈₉	—	4	2	no	—
123	C₄×Dic₈₉	C₂×Q₈	1	2	yes	—
124	#41×C₈₉	—	2	1	no	—
125	#41×C₈₉	—	2	1	no	—
126	#41×C₈₉	C₂×Q₈	1	2	yes	—
127	#41×C₈₉	C₂×Q₈	1	2	yes	—
128	C₂×Q₈×C₈₉	C₂×D₄	1	1	yes	characteristic
129	C₂×D₄×C₈₉	—	2	1	no	—
130	C₂×D₄×C₈₉	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₈₉	—	2	1	no	—
132	C₂²×C₄×C₈₉	—	2	2	no	—
133	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
134	M₁₆×C₈₉	—	2	1	no	—
135	M₁₆×C₈₉	—	2	4	no	—
136	M₁₆×C₈₉	—	2	2	no	—
137	M₁₆×C₈₉	—	2	2	no	—
138	M₁₆×C₈₉	C₂×Q₈	1	2	yes	—
139	C₂×C₈×C₈₉	—	2	1	no	—
140	C₂×C₈×C₈₉	—	2	4	no	—
141	C₂×C₈×C₈₉	—	2	2	no	—
142	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
143	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
144	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
145	C₂×C₈×C₈₉	#41	1	2	yes	—
146	C₄⋊C₄×C₈₉	—	2	2	no	—
147	C₄⋊C₄×C₈₉	—	2	2	no	—
148	C₂³×C₈₉	—	2	1	no	—
149	C₂³×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
150	Q₈×C₈₉	—	2	1	no	—
151	Q₈×C₈₉	C₄⋊Q₈	1	2	yes	—
152	D₄×C₈₉	—	2	2	no	—
153	D₄×C₈₉	—	2	2	no	—
154	D₄×C₈₉	—	2	1	no	—
155	D₄×C₈₉	C₄⋊Q₈	1	2	yes	—
156	C₂×C₄×C₈₉	—	2	1	no	—
157	C₂×C₄×C₈₉	—	2	2	no	—
158	C₂×C₄×C₈₉	—	2	2	no	—
159	C₂×C₄×C₈₉	—	4	2	no	—
160	C₂×C₄×C₈₉	—	8	2	no	—
161	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
163	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
164	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
165	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
166	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
167	C₂×Dic₈₉	—	2	2	no	—
168	C₂×Dic₈₉	—	2	2	no	—
169	C₂×Dic₈₉	—	2	2	no	—
170	C₂×Dic₈₉	—	2	2	no	—
171	C₂×Dic₈₉	—	4	2	no	—
172	C₂×Dic₈₉	—	4	2	no	—
173	C₂×Dic₈₉	—	8	2	no	—
174	C₂×Dic₈₉	—	16	2	no	—
175	Dic₁₇₈	—	16	2	no	—
176	C₈×C₈₉	—	2	2	no	—
177	C₈×C₈₉	—	4	2	no	—
178	C₈×C₈₉	C₂²⋊Q₈	1	2	yes	—
179	C₈×C₈₉	C₂²⋊Q₈	1	2	yes	—
180	#10623494	—	2	2	no	—
181	#10623494	—	2	2	no	—
182	C₂²×C₈₉	—	2	2	no	—
183	C₂²×C₈₉	—	2	2	no	—
184	C₂²×C₈₉	—	4	2	no	—
185	C₂²×C₈₉	#544	1	1	yes	characteristic
186	C₂²×C₈₉	D₄²	1	1	yes	characteristic
187	C₂²×C₈₉	#554	1	1	yes	characteristic
188	C₄×C₈₉	—	2	2	no	—
189	C₄×C₈₉	—	8	2	no	—
190	C₄×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
191	C₄×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
192	C₄×C₈₉	#547	1	1	yes	characteristic
193	C₄×C₈₉	D₄×Q₈	1	1	yes	characteristic
194	Dic₈₉	—	4	2	no	—
195	Dic₈₉	—	4	2	no	—
196	Dic₈₉	—	4	2	no	—
197	Dic₈₉	—	4	2	no	—
198	Dic₈₉	—	8	2	no	—
199	#26080	—	89	1	no	maximal, Sylow
200	C₂×C₈₉	—	2	1	no	—
201	C₂×C₈₉	—	2	1	no	—
202	C₂×C₈₉	—	4	2	no	—
203	C₂×C₈₉	#2440	1	1	yes	characteristic
204	C₂×C₈₉	#3290	1	2	yes	—
205	#3163	—	89	1	no	—
206	#3083	—	89	2	no	—
207	#2912	—	89	1	no	—
208	#2089	—	89	2	no	—
209	#2061	—	89	2	no	—
210	#1951	—	89	1	no	—
211	#1945	—	89	1	no	—
212	#1860	—	89	2	no	—
213	#1846	—	89	1	no	—
214	#1815	D₈₉	1	1	yes	characteristic
215	#1764	—	89	1	no	—
216	C₈₉	#26080	1	1	yes	characteristic, complemented, Sylow
217	C₂×#131	C₂×D₈₉	1	1	yes	characteristic
218	#544	—	89	2	no	—
219	#520	—	89	2	no	—
220	#501	—	178	2	no	—
221	#499	—	178	1	no	—
222	#498	—	178	1	no	—
223	#479	—	178	2	no	—
224	#478	—	178	2	no	—
225	#477	—	178	2	no	—
226	#474	—	178	2	no	—
227	C₂×#108	C₂×D₈₉	1	1	yes	characteristic
228	#428	—	89	2	no	—
229	#428	—	89	2	no	—
230	#428	—	89	2	no	—
231	#428	—	178	2	no	—
232	#427	—	89	1	no	—
233	#427	—	178	1	no	—
234	C₂×#107	—	89	2	no	—
235	C₂×#106	—	89	1	no	—
236	C₂×#106	—	178	1	no	—
237	C₂×#106	C₂×D₈₉	1	1	yes	characteristic
238	#424	—	89	2	no	—
239	C₂×#104	—	89	1	no	—
240	#417	C₂×D₈₉	1	2	yes	—
241	#416	—	89	1	no	—
242	#405	—	89	1	no	—
243	C₂×C₄×C₈	—	89	1	no	—
244	#382	—	89	2	no	—
245	#343	—	89	2	no	—
246	#343	C₂×D₈₉	1	2	yes	—
247	#341	—	89	1	no	—
248	#339	—	89	1	no	—
249	#339	—	89	2	no	—
250	C₂×#41	C₂²×D₈₉	1	1	yes	characteristic
251	#131	—	2	2	no	—
252	#131	Dic₁₇₈	1	4	yes	complemented
253	C₂×M₁₆	—	2	1	no	—
254	C₂×M₁₆	—	89	1	no	—
255	C₂×M₁₆	C₂²×D₈₉	1	2	yes	—
256	C₂×M₁₆	C₂²×D₈₉	1	1	yes	characteristic
257	C₂²×C₈	—	2	1	no	—
258	C₂²×C₈	—	89	1	no	—
259	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
260	C₄⋊Q₈	—	178	2	no	—
261	C₄⋊Q₈	—	178	2	no	—
262	#125	—	178	2	no	—
263	#125	—	178	2	no	—
264	C₂²⋊Q₈	—	356	2	no	—
265	C₄×Q₈	—	178	2	no	—
266	C₄×D₄	—	356	2	no	—
267	C₄×D₄	—	178	2	no	—
268	#117	—	89	2	no	—
269	#117	—	89	1	no	—
270	#117	—	178	2	no	—
271	C₂×C₄⋊C₄	—	89	2	no	—
272	C₂×C₄⋊C₄	—	178	2	no	—
273	C₂×C₄⋊C₄	C₂²×D₈₉	1	2	yes	—
274	C₂×C₄²	—	89	1	no	—
275	#108	—	2	2	no	—
276	#107	—	89	2	no	—
277	#107	—	178	2	no	—
278	#107	—	178	2	no	—
279	#107	—	178	2	no	—
280	#107	—	178	2	no	—
281	#107	—	178	2	no	—
282	#106	—	2	1	no	—
283	#106	—	2	1	no	—
284	#106	—	89	1	no	—
285	#106	—	89	1	no	—
286	#106	—	178	2	no	—
287	#106	—	178	2	no	—
288	#106	—	178	2	no	—
289	#106	—	178	1	no	—
290	#106	—	178	1	no	—
291	C₄⋊C₈	—	178	2	no	—
292	C₄⋊C₈	—	178	2	no	—
293	#104	—	89	4	no	—
294	#103	—	2	2	no	—
295	#103	—	89	2	no	—
296	#102	—	2	2	no	—
297	#102	—	89	2	no	—
298	#97	—	178	1	no	—
299	C₄×C₈	—	178	1	no	—
300	C₄×C₈	—	178	2	no	—
301	#95	—	356	2	no	—
302	#41	—	2	1	no	—
303	#41	—	2	1	no	—
304	#41	Q₈×D₈₉	1	2	yes	—
305	#41	C₂×Dic₁₇₈	1	2	yes	—
306	C₂×Q₈	—	356	2	no	—
307	C₂×Q₈	D₄×D₈₉	1	1	yes	characteristic
308	C₂×D₄	—	2	1	no	—
309	C₂×D₄	D₄×D₈₉	1	1	yes	characteristic
310	C₂²×C₄	—	2	1	no	—
311	C₂²×C₄	—	2	2	no	—
312	C₂²×C₄	—	356	2	no	—
313	C₂²×C₄	—	178	1	no	—
314	C₂²×C₄	—	178	2	no	—
315	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
316	M₁₆	—	2	1	no	—
317	M₁₆	—	2	4	no	—
318	M₁₆	—	2	2	no	—
319	M₁₆	—	2	2	no	—
320	M₁₆	—	89	2	no	—
321	M₁₆	—	178	1	no	—
322	M₁₆	C₂×Dic₁₇₈	1	2	yes	—
323	C₂×C₈	—	2	1	no	—
324	C₂×C₈	—	2	4	no	—
325	C₂×C₈	—	2	2	no	—
326	C₂×C₈	—	89	2	no	—
327	C₂×C₈	—	178	1	no	—
328	C₂×C₈	—	178	1	no	—
329	C₂×C₈	—	178	1	no	—
330	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
331	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
332	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
333	C₂×C₈	#13083573	1	2	yes	—
334	C₄⋊C₄	—	2	2	no	—
335	C₄⋊C₄	—	2	2	no	—
336	C₄⋊C₄	—	89	2	no	—
337	C₄⋊C₄	—	89	2	no	—
338	C₄⋊C₄	—	89	2	no	—
339	C₄⋊C₄	—	356	2	no	—
340	C₄⋊C₄	—	356	2	no	—
341	C₄⋊C₄	—	356	2	no	—
342	C₄⋊C₄	—	356	2	no	—
343	C₄⋊C₄	—	178	2	no	—
344	C₄⋊C₄	—	178	2	no	—
345	C₂²⋊C₄	—	356	2	no	—
346	C₂²⋊C₄	—	178	2	no	—
347	C₂²⋊C₄	—	178	1	no	—
348	C₄²	—	89	2	no	—
349	C₄²	—	356	2	no	—
350	C₄²	—	178	1	no	—
351	C₄²	—	178	2	no	—
352	C₄²	—	178	1	no	—
353	C₂³	—	2	1	no	—
354	C₂³	#15668058	1	1	yes	characteristic
355	Q₈	—	2	1	no	—
356	Q₈	—	1424	2	no	—
357	Q₈	#15668011	1	2	yes	—
358	D₄	—	2	2	no	—
359	D₄	—	2	2	no	—
360	D₄	—	2	1	no	—
361	D₄	#15668011	1	2	yes	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	4	2	no	—
366	C₂×C₄	—	8	2	no	—
367	C₂×C₄	—	356	2	no	—
368	C₂×C₄	—	356	2	no	—
369	C₂×C₄	—	1424	2	no	—
370	C₂×C₄	—	178	2	no	—
371	C₂×C₄	—	178	2	no	—
372	C₂×C₄	—	178	2	no	—
373	C₂×C₄	—	178	2	no	—
374	C₂×C₄	—	712	2	no	—
375	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
376	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
377	C₂×C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
378	C₂×C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
379	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
380	C₂×C₄	#15668059	1	1	yes	characteristic
381	C₈	—	2	2	no	—
382	C₈	—	4	2	no	—
383	C₈	—	178	2	no	—
384	C₈	—	178	2	no	—
385	C₈	#15667989	1	2	yes	—
386	C₈	#15667989	1	2	yes	—
387	C₂²	—	2	2	no	—
388	C₂²	—	2	2	no	—
389	C₂²	—	4	2	no	—
390	C₂²	#19152226	1	1	yes	characteristic
391	C₂²	#19152232	1	1	yes	characteristic
392	C₂²	#19152306	1	1	yes	characteristic
393	C₄	—	2	2	no	—
394	C₄	—	8	2	no	—
395	C₄	—	356	2	no	—
396	C₄	—	356	2	no	—
397	C₄	—	356	2	no	—
398	C₄	—	356	2	no	—
399	C₄	—	712	2	no	—
400	C₄	C₂×#15667989	1	1	yes	characteristic
401	C₄	C₂×#15668011	1	1	yes	characteristic
402	C₄	#19152248	1	1	yes	characteristic
403	C₄	#19152261	1	1	yes	characteristic
404	C₂	—	2	1	no	—
405	C₂	—	2	1	no	—
406	C₂	—	4	2	no	—
407	C₂	#26023955	1	1	yes	characteristic
408	C₂	#26032035	1	2	yes	—

	subgroup	G/H
center	C₂²	#19152232
commutator, cosocle	C₂²×C₄×C₈₉	C₂⁴
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₈₉×#1815	C₂
socle	C₂²×C₈₉	D₄²
prime radical	C₂×#41×C₈₉	C₂³
hypercenter	#1815	D₈₉
hypocenter	C₈₉	#26080
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1815

Complements

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

N	H ≅ G/N	type
1. #26030780	C₂	semidirect
4. #26021761	C₂	semidirect
5. #26021603	C₂	semidirect
6. #26021033	C₂	semidirect
7. #26020998	C₂	semidirect
9. #26020413	C₂	semidirect
76. #15667939	Q₈	semidirect
85. #131×C₈₉	Q₈	semidirect
216. C₈₉	#26080	semidirect
252. #131	Dic₁₇₈	semidirect

Maximal subgroups

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

subgroup	index
#26030780	2
#26030107	2
#26028847	2
#26021761	2
#26021603	2
#26021033	2
#26020998	2
#26020483	2
#26020413	2
#26020087	2
C₈₉×#1815	2
#26080	89