Subgroups — #36537517

Order 22784.

746 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26031559	C₂	1	2	yes	maximal, complemented
2	#26028965	C₂	1	1	yes	maximal, characteristic, complemented
3	#26019714	C₂	1	4	yes	maximal, complemented
4	#26019614	C₂	1	2	yes	maximal, complemented
5	#26019604	C₂	1	2	yes	maximal, complemented
6	C₂×#19151351	C₂	1	1	yes	maximal, characteristic, complemented
7	#26018929	C₂	1	2	yes	maximal, complemented
8	C₈₉×#3147	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#15668018	—	2	1	no	—
10	#19152148	C₂²	1	1	yes	characteristic
11	#19151878	—	2	2	no	—
12	#19151875	C₂²	1	4	yes	—
13	#19151832	—	2	4	no	—
14	#19151793	—	2	2	no	—
15	#19151790	C₂²	1	4	yes	—
16	#19151677	—	2	1	no	—
17	#19151677	C₂²	1	2	yes	—
18	C₈×D₃₅₆	—	2	1	no	—
19	C₈×D₃₅₆	C₂²	1	2	yes	—
20	#19151622	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
22	#19151564	—	2	1	no	—
23	#19151351	C₂²	1	2	yes	—
24	#19151351	C₂²	1	2	yes	—
25	#19151344	C₂²	1	4	yes	—
26	#19151342	C₂²	1	4	yes	—
27	C₂×#135×C₈₉	—	2	1	no	—
28	C₂×#135×C₈₉	—	2	1	no	—
29	#532×C₈₉	C₂²	1	2	yes	—
30	#495×C₈₉	—	2	1	no	—
31	#495×C₈₉	—	2	1	no	—
32	#494×C₈₉	C₂²	1	1	yes	characteristic
33	#493×C₈₉	C₂²	1	1	yes	characteristic
34	#492×C₈₉	C₂²	1	2	yes	—
35	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
36	C₂×#13083574	—	2	1	no	—
37	C₂×#13083559	—	2	4	no	—
38	#15668018	—	4	4	no	—
39	C₂×#13083558	—	2	1	no	—
40	C₂×#13083557	—	2	2	no	—
41	C₂²×#10623494	—	2	2	no	—
42	C₂×C₈×D₈₉	—	2	2	no	—
43	#15668012	D₄	1	1	yes	characteristic
44	#15668009	D₄	1	1	yes	characteristic
45	#15668000	—	2	1	no	—
46	#15667986	—	2	1	no	—
47	#15667986	D₄	1	2	yes	—
48	C₄×D₃₅₆	—	2	1	no	—
49	C₄×D₃₅₆	C₂³	1	2	yes	—
50	#15667945	—	2	4	no	—
51	C₄⋊#10623494	—	2	2	no	—
52	C₄⋊#10623494	—	2	1	no	—
53	C₄⋊#10623494	C₂³	1	4	yes	—
54	C₄⋊#10623494	C₂³	1	2	yes	—
55	#15667933	—	2	4	no	—
56	#15667932	—	2	4	no	—
57	#15667924	—	2	2	no	—
58	#15667924	—	2	2	no	—
59	C₂×#41×C₈₉	—	2	2	no	—
60	#135×C₈₉	—	4	4	no	—
61	#135×C₈₉	—	4	4	no	—
62	C₂×Q₁₆×C₈₉	—	2	1	no	—
63	C₂×Q₁₆×C₈₉	—	2	1	no	—
64	C₂×SD₁₆×C₈₉	—	2	2	no	—
65	C₂×SD₁₆×C₈₉	—	2	2	no	—
66	C₂×D₈×C₈₉	—	2	1	no	—
67	C₂×D₈×C₈₉	—	2	1	no	—
68	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
69	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
70	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
71	#127×C₈₉	C₂³	1	2	yes	—
72	#124×C₈₉	—	2	2	no	—
73	#121×C₈₉	—	2	2	no	—
74	#121×C₈₉	—	2	2	no	—
75	C₄×D₄×C₈₉	—	2	2	no	—
76	C₄×D₄×C₈₉	—	2	2	no	—
77	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
79	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
80	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
82	#13083574	—	4	2	no	—
83	#13083574	—	4	2	no	—
84	C₂×Dic₁₇₈	—	2	1	no	—
85	C₂×D₃₅₆	—	2	1	no	—
86	C₂×D₃₅₆	—	2	1	no	—
87	C₂×D₃₅₆	C₂×D₄	1	2	yes	—
88	C₂×C₂²⋊D₈₉	—	2	2	no	—
89	C₂×C₄×D₈₉	—	2	2	no	—
90	#13083559	—	8	4	no	—
91	#13083558	—	4	2	no	—
92	#13083557	—	4	4	no	—
93	C₂×#10623494	—	2	2	no	—
94	C₂×#10623494	—	2	4	no	—
95	C₂×#10623494	—	8	2	no	—
96	C₈×D₈₉	—	4	4	no	—
97	C₄⋊Dic₈₉	—	2	1	no	—
98	C₄⋊Dic₈₉	C₂×D₄	1	2	yes	—
99	#13083551	—	2	2	no	—
100	#13083551	—	2	2	no	—
101	#41×C₈₉	—	4	4	no	—
102	#41×C₈₉	—	4	4	no	—
103	C₂×Q₈×C₈₉	—	2	2	no	—
104	C₂×D₄×C₈₉	—	2	2	no	—
105	C₂×D₄×C₈₉	—	2	2	no	—
106	C₂×D₄×C₈₉	—	2	2	no	—
107	C₂×D₄×C₈₉	—	4	2	no	—
108	C₂²×C₄×C₈₉	—	4	2	no	—
109	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
112	Q₁₆×C₈₉	—	4	2	no	—
113	Q₁₆×C₈₉	—	4	2	no	—
114	SD₁₆×C₈₉	—	4	4	no	—
115	SD₁₆×C₈₉	—	4	4	no	—
116	D₈×C₈₉	—	4	2	no	—
117	D₈×C₈₉	—	4	2	no	—
118	C₂×C₈×C₈₉	—	2	2	no	—
119	C₂×C₈×C₈₉	D₈	1	4	yes	complemented
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	C₂×D₄	1	1	yes	characteristic
124	C₄⋊C₄×C₈₉	—	2	2	no	—
125	C₄⋊C₄×C₈₉	—	2	2	no	—
126	C₂²⋊C₄×C₈₉	—	4	2	no	—
127	C₂²⋊C₄×C₈₉	—	4	2	no	—
128	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
129	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
130	C₄²×C₈₉	#41	1	1	yes	characteristic
131	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
132	C₂³×C₈₉	—	4	2	no	—
133	C₂³×C₈₉	#121	1	1	yes	characteristic
134	C₂²×D₈₉	—	2	2	no	—
135	Q₈×C₈₉	—	4	2	no	—
136	Q₈×C₈₉	—	4	2	no	—
137	D₄×C₈₉	—	4	2	no	—
138	D₄×C₈₉	—	4	2	no	—
139	D₄×C₈₉	—	4	2	no	—
140	D₄×C₈₉	—	4	2	no	—
141	D₄×C₈₉	—	8	2	no	—
142	D₄×C₈₉	—	8	4	no	—
143	C₂×C₄×C₈₉	—	2	1	no	—
144	C₂×C₄×C₈₉	—	2	1	no	—
145	C₂×C₄×C₈₉	—	4	1	no	—
146	C₂×C₄×C₈₉	—	4	2	no	—
147	C₂×C₄×C₈₉	—	8	4	no	—
148	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
149	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
150	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
151	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
152	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
153	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
156	C₂²⋊D₈₉	—	4	4	no	—
157	C₂×Dic₈₉	—	2	2	no	—
158	D₃₅₆	—	2	2	no	—
159	D₃₅₆	—	2	2	no	—
160	D₃₅₆	—	4	2	no	—
161	D₃₅₆	—	4	2	no	—
162	C₄×D₈₉	—	4	2	no	—
163	C₄×D₈₉	—	4	2	no	—
164	Dic₁₇₈	—	4	2	no	—
165	C₈×C₈₉	—	2	1	no	—
166	C₈×C₈₉	—	2	1	no	—
167	C₈×C₈₉	C₂×D₈	1	2	yes	—
168	C₈×C₈₉	C₂×D₈	1	2	yes	—
169	#10623494	—	4	2	no	—
170	#10623494	—	4	4	no	—
171	C₂²×C₈₉	—	2	1	no	—
172	C₂²×C₈₉	—	2	1	no	—
173	C₂²×C₈₉	—	8	2	no	—
174	C₂²×C₈₉	—	8	2	no	—
175	C₂²×C₈₉	—	8	2	no	—
176	C₂²×C₈₉	#459	1	2	yes	—
177	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
178	C₂×D₈₉	—	4	2	no	—
179	C₂×D₈₉	—	4	2	no	—
180	C₂×D₈₉	—	4	2	no	—
181	C₄×C₈₉	—	2	1	no	—
182	C₄×C₈₉	—	2	1	no	—
183	C₄×C₈₉	—	2	1	no	—
184	C₄×C₈₉	—	8	2	no	—
185	C₄×C₈₉	#459	1	2	yes	—
186	C₄×C₈₉	C₂²×D₈	1	1	yes	characteristic
187	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
188	C₄×C₈₉	#576	1	1	yes	characteristic
189	C₄×C₈₉	#578	1	1	yes	characteristic
190	Dic₈₉	—	4	2	no	—
191	#19190	—	89	1	no	maximal, Sylow
192	C₂×C₈₉	—	2	2	no	—
193	C₂×C₈₉	—	16	2	no	—
194	C₂×C₈₉	C₂×#459	1	1	yes	characteristic
195	C₂×C₈₉	#3061	1	1	yes	characteristic
196	C₂×C₈₉	#3067	1	1	yes	characteristic
197	D₈₉	—	8	2	no	—
198	#3245	—	89	2	no	—
199	#3147	D₈₉	1	1	yes	characteristic, complemented
200	#2931	—	89	1	no	—
201	#1685	—	89	4	no	—
202	#1669	—	89	2	no	—
203	#1667	—	89	2	no	—
204	C₂×#335	—	89	1	no	—
205	#1547	—	89	2	no	—
206	C₈₉	#19190	1	1	yes	characteristic, complemented, Sylow
207	C₂×#135	—	2	1	no	—
208	C₂×#135	—	2	1	no	—
209	C₂×#131	—	178	1	no	—
210	#532	—	89	1	no	—
211	#532	C₂×D₈₉	1	2	yes	—
212	#495	—	2	1	no	—
213	#495	—	2	1	no	—
214	#494	C₂×D₈₉	1	1	yes	characteristic
215	#493	C₂×D₈₉	1	1	yes	characteristic
216	#492	C₂×D₈₉	1	2	yes	—
217	#474	—	89	4	no	—
218	#474	—	178	2	no	—
219	#466	—	178	4	no	—
220	#459	—	89	4	no	—
221	#459	—	178	2	no	—
222	#436	—	89	2	no	—
223	#436	—	178	1	no	—
224	C₈×D₄	—	89	2	no	—
225	C₈×D₄	—	178	1	no	—
226	#423	—	89	1	no	—
227	C₂×C₄⋊C₈	—	89	2	no	—
228	#408	—	178	1	no	—
229	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
230	#335	—	89	2	no	—
231	#335	—	89	2	no	—
232	#332	—	89	4	no	—
233	#331	—	89	4	no	—
234	C₂×#41	—	2	2	no	—
235	C₂×#41	—	178	1	no	—
236	#135	—	4	4	no	—
237	#135	—	4	4	no	—
238	C₂×Q₁₆	—	2	1	no	—
239	C₂×Q₁₆	—	2	1	no	—
240	C₂×SD₁₆	—	2	2	no	—
241	C₂×SD₁₆	—	2	2	no	—
242	C₂×D₈	—	2	1	no	—
243	C₂×D₈	—	2	1	no	—
244	C₂×D₈	—	178	4	no	—
245	#131	—	356	4	no	—
246	C₂×M₁₆	—	178	1	no	—
247	C₂×M₁₆	—	178	2	no	—
248	C₂²×C₈	—	178	2	no	—
249	C₂²×C₈	—	178	2	no	—
250	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
251	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
252	C₄⋊Q₈	—	89	1	no	—
253	C₄⋊Q₈	C₂²×D₈₉	1	2	yes	—
254	#127	—	89	1	no	—
255	#127	C₂²×D₈₉	1	2	yes	—
256	#124	—	2	2	no	—
257	#124	—	178	1	no	—
258	#121	—	2	2	no	—
259	#121	—	2	2	no	—
260	#121	—	89	2	no	—
261	#121	—	178	1	no	—
262	C₄×D₄	—	2	2	no	—
263	C₄×D₄	—	2	2	no	—
264	C₄×D₄	—	89	2	no	—
265	C₄×D₄	—	178	1	no	—
266	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
267	#107	—	178	4	no	—
268	C₄⋊C₈	—	89	2	no	—
269	C₄⋊C₈	—	89	4	no	—
270	C₄⋊C₈	—	178	2	no	—
271	C₄⋊C₈	—	178	1	no	—
272	#102	—	178	4	no	—
273	#102	—	178	4	no	—
274	C₂²⋊C₈	—	178	2	no	—
275	C₂²⋊C₈	—	178	2	no	—
276	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
277	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
278	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
279	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
280	#41	—	4	4	no	—
281	#41	—	4	4	no	—
282	#41	—	356	2	no	—
283	#41	—	356	2	no	—
284	C₂×Q₈	—	2	2	no	—
285	C₂×Q₈	—	178	1	no	—
286	C₂×D₄	—	2	2	no	—
287	C₂×D₄	—	2	2	no	—
288	C₂×D₄	—	2	2	no	—
289	C₂×D₄	—	4	2	no	—
290	C₂×D₄	—	89	2	no	—
291	C₂×D₄	—	178	2	no	—
292	C₂×D₄	—	178	1	no	—
293	C₂×D₄	—	178	1	no	—
294	C₂²×C₄	—	4	2	no	—
295	C₂²×C₄	—	178	2	no	—
296	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
297	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
298	C₂²×C₄	#13083573	1	1	yes	characteristic
299	Q₁₆	—	4	2	no	—
300	Q₁₆	—	4	2	no	—
301	SD₁₆	—	4	4	no	—
302	SD₁₆	—	4	4	no	—
303	D₈	—	4	2	no	—
304	D₈	—	4	2	no	—
305	D₈	—	712	4	no	—
306	M₁₆	—	356	2	no	—
307	M₁₆	—	356	4	no	—
308	C₂×C₈	—	2	2	no	—
309	C₂×C₈	—	356	4	no	—
310	C₂×C₈	—	178	4	no	—
311	C₂×C₈	—	178	2	no	—
312	C₂×C₈	—	712	2	no	—
313	C₂×C₈	#13083559	1	4	yes	complemented
314	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
315	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
316	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
317	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
318	C₄⋊C₄	—	2	2	no	—
319	C₄⋊C₄	—	2	2	no	—
320	C₄⋊C₄	—	89	2	no	—
321	C₄⋊C₄	—	178	1	no	—
322	C₂²⋊C₄	—	4	2	no	—
323	C₂²⋊C₄	—	4	2	no	—
324	C₂²⋊C₄	—	178	2	no	—
325	C₂²⋊C₄	—	178	2	no	—
326	C₄²	D₄×D₈₉	1	1	yes	characteristic
327	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
328	C₄²	#13083573	1	1	yes	characteristic
329	C₄²	C₂³×D₈₉	1	1	yes	characteristic
330	C₂³	—	4	2	no	—
331	C₂³	—	178	2	no	—
332	C₂³	#15667983	1	1	yes	characteristic
333	Q₈	—	4	2	no	—
334	Q₈	—	4	2	no	—
335	Q₈	—	356	2	no	—
336	D₄	—	4	2	no	—
337	D₄	—	4	2	no	—
338	D₄	—	4	2	no	—
339	D₄	—	4	2	no	—
340	D₄	—	8	2	no	—
341	D₄	—	8	4	no	—
342	D₄	—	356	2	no	—
343	D₄	—	356	4	no	—
344	D₄	—	356	2	no	—
345	D₄	—	178	2	no	—
346	D₄	—	178	2	no	—
347	C₂×C₄	—	2	1	no	—
348	C₂×C₄	—	2	1	no	—
349	C₂×C₄	—	4	1	no	—
350	C₂×C₄	—	4	2	no	—
351	C₂×C₄	—	8	4	no	—
352	C₂×C₄	—	356	2	no	—
353	C₂×C₄	—	356	2	no	—
354	C₂×C₄	—	178	2	no	—
355	C₂×C₄	#15667983	1	1	yes	characteristic
356	C₂×C₄	#15667983	1	1	yes	characteristic
357	C₂×C₄	#15667983	1	1	yes	characteristic
358	C₂×C₄	C₂×#13083559	1	2	yes	—
359	C₂×C₄	#15668038	1	2	yes	—
360	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
361	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
362	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
363	C₈	—	2	1	no	—
364	C₈	—	2	1	no	—
365	C₈	—	356	4	no	—
366	C₈	—	356	2	no	—
367	C₈	C₂×#13083559	1	2	yes	—
368	C₈	C₂×#13083559	1	2	yes	—
369	C₂²	—	2	1	no	—
370	C₂²	—	2	1	no	—
371	C₂²	—	8	2	no	—
372	C₂²	—	8	2	no	—
373	C₂²	—	8	2	no	—
374	C₂²	—	356	2	no	—
375	C₂²	—	356	2	no	—
376	C₂²	—	356	2	no	—
377	C₂²	#19151790	1	2	yes	—
378	C₂²	C₂×#15667983	1	1	yes	characteristic
379	C₄	—	2	1	no	—
380	C₄	—	2	1	no	—
381	C₄	—	2	1	no	—
382	C₄	—	8	2	no	—
383	C₄	—	356	2	no	—
384	C₄	#19151790	1	2	yes	—
385	C₄	C₂²×#13083559	1	1	yes	characteristic
386	C₄	C₂×#15668038	1	1	yes	characteristic
387	C₄	#19152400	1	1	yes	characteristic
388	C₄	#19152412	1	1	yes	characteristic
389	C₂	—	2	2	no	—
390	C₂	—	16	2	no	—
391	C₂	—	712	2	no	—
392	C₂	C₂×#19151790	1	1	yes	characteristic
393	C₂	#26029918	1	1	yes	characteristic
394	C₂	#26029968	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#15667983
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#3147	C₂
socle	C₂²×C₈₉	C₂×#121
hypercenter	#3147	D₈₉
hypocenter	C₈₉	#19190
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3147

Complements

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

N	H ≅ G/N	type
1. #26031559	C₂	semidirect
2. #26028965	C₂	semidirect
3. #26019714	C₂	semidirect
4. #26019614	C₂	semidirect
5. #26019604	C₂	semidirect
6. C₂×#19151351	C₂	semidirect
7. #26018929	C₂	semidirect
8. C₈₉×#3147	C₂	semidirect
119. C₂×C₈×C₈₉	D₈	semidirect
199. #3147	D₈₉	semidirect
206. C₈₉	#19190	semidirect
313. C₂×C₈	#13083559	semidirect

Maximal subgroups

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

subgroup	index
#26031559	2
#26028965	2
#26019714	2
#26019614	2
#26019604	2
C₂×#19151351	2
#26018929	2
C₈₉×#3147	2
#19190	89