Subgroups — #36599553

Order 22784.

656 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151868	C₂	1	1	yes	maximal, characteristic
2	#26029105	C₂	1	1	yes	maximal, characteristic, complemented
3	#26021715	C₂	1	2	yes	maximal, complemented
4	#26021525	C₂	1	2	yes	maximal, complemented
5	#26021162	C₂	1	1	yes	maximal, characteristic, complemented
6	#26021042	C₂	1	1	yes	maximal, characteristic, complemented
7	#26020742	C₂	1	2	yes	maximal, complemented
8	#26020376	C₂	1	1	yes	maximal, characteristic, complemented
9	#26020362	C₂	1	2	yes	maximal, complemented
10	#26020306	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#1757	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15668037	C₂²	1	2	yes	—
13	C₂²×D₇₁₂	C₂²	1	1	yes	characteristic
14	#19152072	C₂²	1	2	yes	—
15	#19151868	—	2	2	no	—
16	#19151868	—	2	2	no	—
17	#19151711	—	2	2	no	—
18	C₄×D₇₁₂	—	2	2	no	—
19	#19151646	C₂²	1	2	yes	—
20	C₂×#15667948	C₂²	1	1	yes	characteristic
21	C₂×#15667948	C₂²	1	1	yes	characteristic
22	#19151636	C₂²	1	1	yes	characteristic
23	#19151615	C₂²	1	2	yes	—
24	#19151604	C₂²	1	2	yes	—
25	#19151602	C₂²	1	2	yes	—
26	C₂×#15667928	C₂²	1	2	yes	—
27	C₂×#15667928	C₂²	1	2	yes	—
28	#19151370	C₂²	1	2	yes	—
29	#19151369	C₂²	1	2	yes	—
30	#19151368	C₂²	1	2	yes	—
31	#19151354	C₂²	1	2	yes	—
32	C₂×#131×C₈₉	C₂²	1	1	yes	characteristic
33	#413×C₈₉	C₂²	1	1	yes	characteristic
34	#410×C₈₉	C₂²	1	1	yes	characteristic
35	#405×C₈₉	C₂²	1	1	yes	characteristic
36	#405×C₈₉	C₂²	1	1	yes	characteristic
37	#344×C₈₉	C₂²	1	1	yes	characteristic
38	#344×C₈₉	C₂²	1	1	yes	characteristic
39	C₂²×D₃₅₆	C₂³	1	2	yes	—
40	#15668037	—	2	4	no	—
41	#15668037	—	4	2	no	—
42	C₂×#13083561	—	2	2	no	—
43	C₂×D₇₁₂	—	2	1	no	—
44	C₂×D₇₁₂	—	2	1	no	—
45	C₂×D₇₁₂	—	2	1	no	—
46	C₂×D₇₁₂	—	2	1	no	—
47	C₂×D₇₁₂	D₄	1	2	yes	—
48	C₂×D₇₁₂	D₄	1	2	yes	—
49	C₂×#13083559	—	2	2	no	—
50	C₂×#13083558	C₂³	1	2	yes	—
51	C₂×#13083558	C₂³	1	2	yes	—
52	C₄×D₃₅₆	—	2	4	no	—
53	#15667974	—	4	2	no	—
54	#15667971	C₂³	1	2	yes	—
55	C₂²⋊C₄×D₈₉	—	2	4	no	—
56	#15667948	—	2	1	no	—
57	#15667948	—	2	1	no	—
58	#15667948	—	2	1	no	—
59	#15667948	—	2	1	no	—
60	#15667947	—	4	2	no	—
61	#15667946	—	2	1	no	—
62	#15667944	—	2	1	no	—
63	#15667939	—	4	2	no	—
64	#15667934	—	2	4	no	—
65	#15667932	—	4	2	no	—
66	#15667928	—	2	4	no	—
67	#15667928	—	4	2	no	—
68	C₂×#41×C₈₉	C₂³	1	1	yes	characteristic
69	#131×C₈₉	—	2	1	no	—
70	#131×C₈₉	—	2	1	no	—
71	#131×C₈₉	—	2	1	no	—
72	#131×C₈₉	—	2	1	no	—
73	C₂×M₁₆×C₈₉	—	2	1	no	—
74	C₂×M₁₆×C₈₉	C₂³	1	1	yes	characteristic
75	C₂×M₁₆×C₈₉	C₂³	1	1	yes	characteristic
76	C₂×M₁₆×C₈₉	C₂³	1	1	yes	characteristic
77	C₂²×C₈×C₈₉	—	2	1	no	—
78	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
79	#117×C₈₉	C₂³	1	1	yes	characteristic
80	#117×C₈₉	C₂³	1	1	yes	characteristic
81	#101×C₈₉	—	2	1	no	—
82	#100×C₈₉	—	2	1	no	—
83	#99×C₈₉	—	2	2	no	—
84	#97×C₈₉	—	2	1	no	—
85	#97×C₈₉	—	2	1	no	—
86	C₄×C₈×C₈₉	—	2	1	no	—
87	C₄×C₈×C₈₉	—	2	1	no	—
88	C₂³×D₈₉	—	2	2	no	—
89	C₂×D₃₅₆	—	2	2	no	—
90	C₂×D₃₅₆	—	2	2	no	—
91	C₂×D₃₅₆	—	2	2	no	—
92	C₂×D₃₅₆	—	2	2	no	—
93	C₂×D₃₅₆	—	2	2	no	—
94	C₂×D₃₅₆	C₂×D₄	1	4	yes	—
95	C₂×C₄×D₈₉	—	4	4	no	—
96	#13083561	—	4	4	no	—
97	D₇₁₂	—	2	2	no	—
98	D₇₁₂	—	2	2	no	—
99	D₇₁₂	—	2	2	no	—
100	D₇₁₂	—	2	2	no	—
101	#13083559	—	4	4	no	—
102	#13083558	—	2	2	no	—
103	#13083558	—	2	2	no	—
104	#13083558	—	2	2	no	—
105	#13083558	—	2	2	no	—
106	C₂×#10623494	—	2	2	no	—
107	C₂×#10623494	—	2	2	no	—
108	C₄⋊Dic₈₉	—	2	2	no	—
109	C₂²⋊Dic₈₉	—	2	2	no	—
110	#13083551	—	4	4	no	—
111	C₄×Dic₈₉	—	2	2	no	—
112	#41×C₈₉	—	2	1	no	—
113	#41×C₈₉	—	2	1	no	—
114	#41×C₈₉	—	2	1	no	—
115	#41×C₈₉	—	2	1	no	—
116	C₂×Q₈×C₈₉	#41	1	1	yes	characteristic
117	C₂×D₄×C₈₉	—	2	1	no	—
118	C₂×D₄×C₈₉	#41	1	1	yes	characteristic
119	C₂²×C₄×C₈₉	—	2	1	no	—
120	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
121	M₁₆×C₈₉	—	2	1	no	—
122	M₁₆×C₈₉	—	2	1	no	—
123	M₁₆×C₈₉	—	2	1	no	—
124	M₁₆×C₈₉	—	2	1	no	—
125	M₁₆×C₈₉	—	2	1	no	—
126	M₁₆×C₈₉	—	2	1	no	—
127	M₁₆×C₈₉	—	4	1	no	—
128	M₁₆×C₈₉	—	4	1	no	—
129	C₂×C₈×C₈₉	—	2	1	no	—
130	C₂×C₈×C₈₉	—	2	1	no	—
131	C₂×C₈×C₈₉	—	2	1	no	—
132	C₂×C₈×C₈₉	—	4	1	no	—
133	C₂×C₈×C₈₉	—	4	1	no	—
134	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
135	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
136	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
137	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
138	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
139	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
140	C₄⋊C₄×C₈₉	—	2	1	no	—
141	C₄⋊C₄×C₈₉	—	2	1	no	—
142	C₂²⋊C₄×C₈₉	—	2	1	no	—
143	C₂²⋊C₄×C₈₉	#41	1	2	yes	—
144	C₄²×C₈₉	—	2	1	no	—
145	C₄²×C₈₉	—	2	1	no	—
146	C₂³×C₈₉	—	2	1	no	—
147	C₂³×C₈₉	C₂×#41	1	1	yes	characteristic
148	C₂²×D₈₉	—	2	4	no	—
149	C₂²×D₈₉	—	4	2	no	—
150	C₂²×D₈₉	—	4	2	no	—
151	C₂²×D₈₉	—	16	2	no	—
152	Q₈×C₈₉	—	2	1	no	—
153	Q₈×C₈₉	—	2	1	no	—
154	D₄×C₈₉	—	2	1	no	—
155	D₄×C₈₉	—	2	1	no	—
156	D₄×C₈₉	—	4	1	no	—
157	D₄×C₈₉	—	4	1	no	—
158	C₂×C₄×C₈₉	—	2	1	no	—
159	C₂×C₄×C₈₉	—	2	2	no	—
160	C₂×C₄×C₈₉	—	4	1	no	—
161	C₂×C₄×C₈₉	—	4	1	no	—
162	C₂×C₄×C₈₉	—	4	1	no	—
163	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
164	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
165	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
166	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
167	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
168	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
169	C₂×Dic₈₉	—	4	2	no	—
170	D₃₅₆	—	2	4	no	—
171	D₃₅₆	—	2	4	no	—
172	D₃₅₆	—	4	2	no	—
173	D₃₅₆	—	4	2	no	—
174	C₄×D₈₉	—	16	4	no	—
175	C₈×C₈₉	—	4	1	no	—
176	C₈×C₈₉	—	4	1	no	—
177	C₈×C₈₉	—	4	1	no	—
178	C₈×C₈₉	#123	1	1	yes	characteristic
179	C₈×C₈₉	#123	1	1	yes	characteristic
180	C₈×C₈₉	#123	1	1	yes	characteristic
181	C₈×C₈₉	#123	1	1	yes	characteristic
182	#10623494	—	4	2	no	—
183	#10623494	—	4	2	no	—
184	C₂²×C₈₉	—	4	1	no	—
185	C₂²×C₈₉	—	4	1	no	—
186	C₂²×C₈₉	—	8	1	no	—
187	C₂²×C₈₉	#529	1	1	yes	characteristic
188	C₂²×C₈₉	#546	1	1	yes	characteristic
189	C₂²×C₈₉	#555	1	1	yes	characteristic
190	C₂×D₈₉	—	4	4	no	—
191	C₂×D₈₉	—	8	4	no	—
192	C₂×D₈₉	—	16	2	no	—
193	C₂×D₈₉	—	16	2	no	—
194	C₄×C₈₉	—	4	2	no	—
195	C₄×C₈₉	—	4	1	no	—
196	C₄×C₈₉	—	8	1	no	—
197	C₄×C₈₉	C₂×#123	1	1	yes	characteristic
198	C₄×C₈₉	#540	1	1	yes	characteristic
199	C₄×C₈₉	#546	1	1	yes	characteristic
200	C₄×C₈₉	#553	1	1	yes	characteristic
201	Dic₈₉	—	8	2	no	—
202	#24988	—	89	1	no	maximal, Sylow
203	C₂×C₈₉	—	2	1	no	—
204	C₂×C₈₉	—	2	1	no	—
205	C₂×C₈₉	—	2	1	no	—
206	C₂×C₈₉	—	8	1	no	—
207	C₂×C₈₉	#2420	1	1	yes	characteristic
208	D₈₉	—	8	4	no	—
209	C₂×#472	—	89	1	no	—
210	#2948	—	89	1	no	—
211	#2081	—	89	2	no	—
212	#2048	—	89	2	no	—
213	#1978	—	89	1	no	—
214	#1953	—	89	1	no	—
215	#1904	—	89	2	no	—
216	#1833	—	89	1	no	—
217	#1829	—	89	2	no	—
218	#1814	—	89	1	no	—
219	#1757	D₈₉	1	1	yes	characteristic, complemented
220	C₈₉	#24988	1	1	yes	characteristic, complemented, Sylow
221	C₂×AGL(1,ℤ/8)	—	89	2	no	—
222	C₂²×D₈	—	89	1	no	—
223	C₂×#131	C₂×D₈₉	1	1	yes	characteristic
224	#518	—	89	2	no	—
225	#472	—	178	2	no	—
226	#472	—	178	2	no	—
227	#442	—	178	2	no	—
228	C₄×D₈	—	178	2	no	—
229	#430	—	89	2	no	—
230	C₂×#108	—	89	1	no	—
231	C₂×#108	—	89	1	no	—
232	#427	—	89	1	no	—
233	#421	—	89	2	no	—
234	#418	—	89	2	no	—
235	#418	—	89	2	no	—
236	#413	C₂×D₈₉	1	1	yes	characteristic
237	C₂×#100	—	89	2	no	—
238	C₂×#100	—	89	2	no	—
239	#410	C₂×D₈₉	1	1	yes	characteristic
240	#405	C₂×D₈₉	1	1	yes	characteristic
241	#405	C₂×D₈₉	1	1	yes	characteristic
242	#344	—	89	2	no	—
243	#344	—	89	2	no	—
244	#344	—	89	2	no	—
245	#344	C₂×D₈₉	1	1	yes	characteristic
246	#344	C₂×D₈₉	1	1	yes	characteristic
247	H(3,ℤ/4)	—	89	2	no	—
248	C₂×#41	C₂²×D₈₉	1	1	yes	characteristic
249	C₂²×D₄	—	89	2	no	—
250	AGL(1,ℤ/8)	—	356	2	no	—
251	AGL(1,ℤ/8)	—	178	4	no	—
252	C₂×SD₁₆	—	178	2	no	—
253	C₂×D₈	—	89	2	no	—
254	C₂×D₈	—	89	2	no	—
255	C₂×D₈	—	178	2	no	—
256	C₂×D₈	—	178	1	no	—
257	C₂×D₈	—	178	1	no	—
258	C₂×D₈	—	178	1	no	—
259	C₂×D₈	—	178	1	no	—
260	#131	—	2	1	no	—
261	#131	—	2	1	no	—
262	#131	—	2	1	no	—
263	#131	—	2	1	no	—
264	C₂×M₁₆	—	2	1	no	—
265	C₂×M₁₆	—	89	2	no	—
266	C₂×M₁₆	—	89	2	no	—
267	C₂×M₁₆	C₂²×D₈₉	1	1	yes	characteristic
268	C₂×M₁₆	C₂²×D₈₉	1	1	yes	characteristic
269	C₂×M₁₆	C₂²×D₈₉	1	1	yes	characteristic
270	C₂²×C₈	—	2	1	no	—
271	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
272	C₄×D₄	—	356	2	no	—
273	C₄×D₄	—	178	4	no	—
274	#117	—	89	2	no	—
275	#117	C₂²×D₈₉	1	1	yes	characteristic
276	#117	C₂²×D₈₉	1	1	yes	characteristic
277	C₂×C₂²⋊C₄	—	178	4	no	—
278	#108	—	356	2	no	—
279	#108	—	178	1	no	—
280	#108	—	178	1	no	—
281	#108	—	178	1	no	—
282	#108	—	178	1	no	—
283	#107	—	178	1	no	—
284	#106	—	178	1	no	—
285	#104	—	356	2	no	—
286	#102	—	356	2	no	—
287	#102	—	178	4	no	—
288	#101	—	2	1	no	—
289	#100	—	2	1	no	—
290	#100	—	356	2	no	—
291	#100	—	178	4	no	—
292	#99	—	2	2	no	—
293	#97	—	2	1	no	—
294	#97	—	2	1	no	—
295	C₄×C₈	—	2	1	no	—
296	C₄×C₈	—	2	1	no	—
297	C₂⁴	—	178	2	no	—
298	#41	—	2	1	no	—
299	#41	—	2	1	no	—
300	#41	—	2	1	no	—
301	#41	—	2	1	no	—
302	C₂×Q₈	#13083574	1	1	yes	characteristic
303	C₂×D₄	—	2	1	no	—
304	C₂×D₄	—	89	4	no	—
305	C₂×D₄	—	178	2	no	—
306	C₂×D₄	—	178	2	no	—
307	C₂×D₄	—	178	2	no	—
308	C₂×D₄	—	178	2	no	—
309	C₂×D₄	—	178	2	no	—
310	C₂×D₄	#13083574	1	1	yes	characteristic
311	C₂²×C₄	—	2	1	no	—
312	C₂²×C₄	—	356	4	no	—
313	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
314	SD₁₆	—	356	4	no	—
315	D₈	—	356	4	no	—
316	D₈	—	178	2	no	—
317	D₈	—	178	2	no	—
318	D₈	—	178	2	no	—
319	D₈	—	178	2	no	—
320	M₁₆	—	2	1	no	—
321	M₁₆	—	2	1	no	—
322	M₁₆	—	2	1	no	—
323	M₁₆	—	2	1	no	—
324	M₁₆	—	2	1	no	—
325	M₁₆	—	2	1	no	—
326	M₁₆	—	4	1	no	—
327	M₁₆	—	4	1	no	—
328	M₁₆	—	178	2	no	—
329	M₁₆	—	178	2	no	—
330	M₁₆	—	178	2	no	—
331	M₁₆	—	178	2	no	—
332	C₂×C₈	—	2	1	no	—
333	C₂×C₈	—	2	1	no	—
334	C₂×C₈	—	2	1	no	—
335	C₂×C₈	—	4	1	no	—
336	C₂×C₈	—	4	1	no	—
337	C₂×C₈	—	178	2	no	—
338	C₂×C₈	—	178	2	no	—
339	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
340	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
341	C₂×C₈	#13083573	1	1	yes	characteristic
342	C₂×C₈	#13083573	1	1	yes	characteristic
343	C₂×C₈	#13083574	1	1	yes	characteristic
344	C₂×C₈	#13083574	1	1	yes	characteristic
345	C₄⋊C₄	—	2	1	no	—
346	C₄⋊C₄	—	2	1	no	—
347	C₄⋊C₄	—	178	2	no	—
348	C₂²⋊C₄	—	2	1	no	—
349	C₂²⋊C₄	—	356	4	no	—
350	C₂²⋊C₄	—	178	2	no	—
351	C₂²⋊C₄	#13083574	1	2	yes	—
352	C₄²	—	2	1	no	—
353	C₄²	—	2	1	no	—
354	C₄²	—	178	2	no	—
355	C₂³	—	2	1	no	—
356	C₂³	—	356	2	no	—
357	C₂³	—	356	2	no	—
358	C₂³	—	1424	2	no	—
359	C₂³	—	178	4	no	—
360	C₂³	C₂×#13083574	1	1	yes	characteristic
361	Q₈	—	2	1	no	—
362	Q₈	—	2	1	no	—
363	D₄	—	2	1	no	—
364	D₄	—	2	1	no	—
365	D₄	—	4	1	no	—
366	D₄	—	4	1	no	—
367	D₄	—	356	2	no	—
368	D₄	—	356	2	no	—
369	D₄	—	178	4	no	—
370	D₄	—	178	4	no	—
371	C₂×C₄	—	2	1	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	4	1	no	—
374	C₂×C₄	—	4	1	no	—
375	C₂×C₄	—	4	1	no	—
376	C₂×C₄	—	356	2	no	—
377	C₂×C₄	—	1424	4	no	—
378	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
379	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
380	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
381	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
382	C₂×C₄	#15668058	1	1	yes	characteristic
383	C₂×C₄	#15668058	1	1	yes	characteristic
384	C₈	—	4	1	no	—
385	C₈	—	4	1	no	—
386	C₈	—	4	1	no	—
387	C₈	—	356	2	no	—
388	C₈	—	356	2	no	—
389	C₈	#15667994	1	1	yes	characteristic
390	C₈	#15667994	1	1	yes	characteristic
391	C₈	#15667994	1	1	yes	characteristic
392	C₈	#15667994	1	1	yes	characteristic
393	C₂²	—	4	1	no	—
394	C₂²	—	4	1	no	—
395	C₂²	—	8	1	no	—
396	C₂²	—	356	4	no	—
397	C₂²	—	1424	2	no	—
398	C₂²	—	1424	2	no	—
399	C₂²	—	712	4	no	—
400	C₂²	#19152132	1	1	yes	characteristic
401	C₂²	#19152244	1	1	yes	characteristic
402	C₂²	#19152313	1	1	yes	characteristic
403	C₄	—	4	2	no	—
404	C₄	—	4	1	no	—
405	C₄	—	8	1	no	—
406	C₄	—	712	2	no	—
407	C₄	C₂×#15667994	1	1	yes	characteristic
408	C₄	#19152203	1	1	yes	characteristic
409	C₄	#19152244	1	1	yes	characteristic
410	C₄	#19152294	1	1	yes	characteristic
411	C₂	—	2	1	no	—
412	C₂	—	2	1	no	—
413	C₂	—	2	1	no	—
414	C₂	—	8	1	no	—
415	C₂	—	712	4	no	—
416	C₂	#26023744	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26023744
commutator, cosocle	C₂²×C₄×C₈₉	C₂⁴
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₈₉×#1757	C₂
socle	C₂×C₈₉	#2420
prime radical	C₂×#19151868	C₂
hypercenter	#1757	D₈₉
hypocenter	C₈₉	#24988
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1757

Complements

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

N	H ≅ G/N	type
2. #26029105	C₂	semidirect
3. #26021715	C₂	semidirect
4. #26021525	C₂	semidirect
5. #26021162	C₂	semidirect
6. #26021042	C₂	semidirect
7. #26020742	C₂	semidirect
8. #26020376	C₂	semidirect
9. #26020362	C₂	semidirect
10. #26020306	C₂	semidirect
11. C₈₉×#1757	C₂	semidirect
219. #1757	D₈₉	semidirect
220. C₈₉	#24988	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151868	2
#26029105	2
#26021715	2
#26021525	2
#26021162	2
#26021042	2
#26020742	2
#26020376	2
#26020362	2
#26020306	2
C₈₉×#1757	2
#24988	89