Subgroups — #36711732

Order 22784.

546 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26032675	C₂	1	1	yes	maximal, characteristic
2	#26022479	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022457	C₂	1	2	yes	maximal, complemented
4	#26022404	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022367	C₂	1	2	yes	maximal, complemented
6	#26022294	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022169	C₂	1	2	yes	maximal, complemented
8	C₄×#15667955	C₂	1	1	yes	maximal, characteristic, complemented
9	#26022084	C₂	1	1	yes	maximal, characteristic, complemented
10	#26021972	C₂	1	2	yes	maximal, complemented
11	D₄×Q₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152313	—	2	1	no	—
13	C₂×#15667955	C₂²	1	1	yes	characteristic
14	C₂×#15667955	C₂²	1	2	yes	—
15	C₂×#15667952	C₂²	1	2	yes	—
16	#19152003	C₂²	1	1	yes	characteristic
17	#19151923	—	2	2	no	—
18	#19151889	—	2	1	no	—
19	#19151834	C₂²	1	2	yes	—
20	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
21	#19151636	C₂²	1	2	yes	—
22	#19151425	C₂²	1	1	yes	characteristic
23	#19151423	C₂²	1	2	yes	—
24	C₄⋊#13083549	C₂²	1	1	yes	characteristic
25	#19151402	C₂²	1	2	yes	—
26	#19151402	C₂²	1	2	yes	—
27	#19151402	C₂²	1	2	yes	—
28	#19151402	C₂²	1	2	yes	—
29	#19151402	C₂²	1	1	yes	characteristic
30	#19151398	C₂²	1	1	yes	characteristic
31	C₂²⋊#13083549	C₂²	1	2	yes	—
32	C₄×#13083549	C₂²	1	1	yes	characteristic
33	C₂²×Q₁₆×C₈₉	C₂²	1	2	yes	—
34	D₄×Q₈×C₈₉	—	2	1	no	—
35	#494×C₈₉	C₂²	1	1	yes	characteristic
36	#467×C₈₉	C₂²	1	2	yes	—
37	#462×C₈₉	—	2	1	no	—
38	#451×C₈₉	—	2	2	no	—
39	C₄×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
40	C₈×D₄×C₈₉	C₂²	1	1	yes	characteristic
41	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
42	#15668012	—	2	1	no	—
43	#15667998	—	4	2	no	—
44	#15667986	—	2	2	no	—
45	#15667983	—	4	1	no	—
46	C₄×D₃₅₆	—	2	1	no	—
47	#15667971	—	2	2	no	—
48	#15667970	—	4	1	no	—
49	#15667955	—	2	2	no	—
50	#15667955	—	4	2	no	—
51	#15667952	—	4	2	no	—
52	C₂×#13083549	C₂³	1	1	yes	characteristic
53	C₂×#13083549	C₂³	1	1	yes	characteristic
54	C₂×#13083549	C₂³	1	2	yes	—
55	#15667946	—	2	1	no	—
56	#15667946	C₂³	1	1	yes	characteristic
57	#15667946	C₂³	1	2	yes	—
58	#15667944	—	2	2	no	—
59	#15667934	—	2	2	no	—
60	#15667934	—	2	1	no	—
61	C₂²×Q₈×C₈₉	—	2	2	no	—
62	C₂×Q₁₆×C₈₉	—	2	4	no	—
63	C₂×Q₁₆×C₈₉	—	2	2	no	—
64	C₂×Q₁₆×C₈₉	—	2	1	no	—
65	C₂×Q₁₆×C₈₉	—	4	2	no	—
66	C₂×Q₁₆×C₈₉	C₂³	1	2	yes	—
67	C₂×Q₁₆×C₈₉	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
69	C₄⋊Q₈×C₈₉	—	2	1	no	—
70	C₄⋊Q₈×C₈₉	—	4	1	no	—
71	C₂²⋊Q₈×C₈₉	—	2	2	no	—
72	C₂²⋊Q₈×C₈₉	—	4	2	no	—
73	C₄×Q₈×C₈₉	—	2	1	no	—
74	C₄×D₄×C₈₉	—	4	1	no	—
75	C₄×D₄×C₈₉	D₄	1	1	yes	characteristic
76	#107×C₈₉	C₂³	1	1	yes	characteristic
77	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
78	#103×C₈₉	—	2	1	no	—
79	#103×C₈₉	—	2	2	no	—
80	C₂²⋊C₈×C₈₉	D₄	1	2	yes	—
81	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
82	C₂×Dic₁₇₈	—	4	1	no	—
83	C₂×D₃₅₆	—	2	1	no	—
84	C₂×C₂²⋊D₈₉	—	4	2	no	—
85	C₂×C₄×D₈₉	—	4	1	no	—
86	D₇₁₂	—	2	1	no	—
87	D₇₁₂	—	2	1	no	—
88	C₄⋊Dic₈₉	—	2	1	no	—
89	C₄⋊Dic₈₉	—	2	2	no	—
90	C₄⋊Dic₈₉	—	4	1	no	—
91	C₂²⋊Dic₈₉	—	4	2	no	—
92	#13083551	—	4	2	no	—
93	#13083551	—	4	1	no	—
94	C₄×Dic₈₉	—	4	2	no	—
95	#13083549	—	2	1	no	—
96	#13083549	—	2	1	no	—
97	#13083549	—	2	2	no	—
98	C₂×Q₈×C₈₉	—	2	1	no	—
99	C₂×Q₈×C₈₉	—	2	2	no	—
100	C₂×Q₈×C₈₉	—	4	4	no	—
101	C₂×Q₈×C₈₉	—	4	2	no	—
102	C₂×Q₈×C₈₉	—	8	2	no	—
103	C₂×D₄×C₈₉	D₈	1	1	yes	characteristic
104	C₂²×C₄×C₈₉	—	4	2	no	—
105	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
106	Q₁₆×C₈₉	—	2	2	no	—
107	Q₁₆×C₈₉	—	2	1	no	—
108	Q₁₆×C₈₉	—	2	2	no	—
109	Q₁₆×C₈₉	—	4	4	no	—
110	Q₁₆×C₈₉	—	8	1	no	—
111	Q₁₆×C₈₉	#41	1	2	yes	—
112	C₂×C₈×C₈₉	—	2	2	no	—
113	C₂×C₈×C₈₉	—	2	2	no	—
114	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
116	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
117	C₄⋊C₄×C₈₉	—	2	1	no	—
118	C₄⋊C₄×C₈₉	—	4	1	no	—
119	C₄⋊C₄×C₈₉	—	4	1	no	—
120	C₄⋊C₄×C₈₉	—	4	2	no	—
121	C₄⋊C₄×C₈₉	—	4	1	no	—
122	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
123	C₂²⋊C₄×C₈₉	—	4	2	no	—
124	C₂²⋊C₄×C₈₉	D₈	1	2	yes	—
125	C₄²×C₈₉	—	4	1	no	—
126	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
127	C₂³×C₈₉	C₂×D₈	1	2	yes	—
128	C₂²×D₈₉	—	4	1	no	—
129	Q₈×C₈₉	—	2	2	no	—
130	Q₈×C₈₉	—	4	1	no	—
131	Q₈×C₈₉	—	4	2	no	—
132	Q₈×C₈₉	—	4	2	no	—
133	Q₈×C₈₉	—	8	4	no	—
134	D₄×C₈₉	—	2	2	no	—
135	C₂×C₄×C₈₉	—	2	2	no	—
136	C₂×C₄×C₈₉	—	2	2	no	—
137	C₂×C₄×C₈₉	—	4	1	no	—
138	C₂×C₄×C₈₉	—	4	1	no	—
139	C₂×C₄×C₈₉	—	4	2	no	—
140	C₂×C₄×C₈₉	—	8	4	no	—
141	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
143	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
145	C₂²⋊D₈₉	—	16	2	no	—
146	C₂×Dic₈₉	—	4	1	no	—
147	C₂×Dic₈₉	—	4	2	no	—
148	D₃₅₆	—	4	1	no	—
149	D₃₅₆	—	4	1	no	—
150	C₄×D₈₉	—	8	2	no	—
151	Dic₁₇₈	—	16	1	no	—
152	C₈×C₈₉	—	2	2	no	—
153	C₈×C₈₉	—	2	1	no	—
154	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
155	C₈×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
156	C₂²×C₈₉	—	2	2	no	—
157	C₂²×C₈₉	—	2	2	no	—
158	C₂²×C₈₉	—	2	2	no	—
159	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
160	C₂×D₈₉	—	8	1	no	—
161	C₂×D₈₉	—	8	1	no	—
162	C₂×D₈₉	—	8	1	no	—
163	C₄×C₈₉	—	2	1	no	—
164	C₄×C₈₉	—	2	2	no	—
165	C₄×C₈₉	—	4	2	no	—
166	C₄×C₈₉	—	8	1	no	—
167	C₄×C₈₉	—	8	2	no	—
168	C₄×C₈₉	#510	1	2	yes	—
169	C₄×C₈₉	#546	1	1	yes	characteristic
170	C₄×C₈₉	#577	1	1	yes	characteristic
171	Dic₈₉	—	8	1	no	—
172	Dic₈₉	—	8	2	no	—
173	#33896	—	89	1	no	maximal, Sylow
174	C₂×C₈₉	—	4	2	no	—
175	C₂×C₈₉	#3294	1	1	yes	characteristic
176	C₂×C₈₉	C₂×#510	1	1	yes	characteristic
177	C₂×C₈₉	#3416	1	1	yes	characteristic
178	D₈₉	—	16	1	no	—
179	#3341	—	89	1	no	—
180	D₄×Q₁₆	D₈₉	1	1	yes	characteristic, complemented
181	#2241	—	89	1	no	—
182	#2237	—	89	2	no	—
183	#2227	—	89	1	no	—
184	#2220	—	89	2	no	—
185	#2208	—	89	1	no	—
186	#2185	—	89	2	no	—
187	C₄×SD₃₂	—	89	1	no	—
188	#2169	—	89	1	no	—
189	#2141	—	89	2	no	—
190	C₈₉	#33896	1	1	yes	characteristic, complemented, Sylow
191	C₂²×Q₁₆	C₂×D₈₉	1	2	yes	—
192	#555	—	178	1	no	—
193	D₄×Q₈	—	2	1	no	—
194	C₂×SD₃₂	—	89	2	no	—
195	C₂×SD₃₂	—	89	1	no	—
196	C₂×M₃₂	—	89	2	no	—
197	#500	—	89	1	no	—
198	#494	C₂×D₈₉	1	1	yes	characteristic
199	#482	—	178	2	no	—
200	#476	—	178	1	no	—
201	#467	C₂×D₈₉	1	2	yes	—
202	#466	—	89	2	no	—
203	#462	—	2	1	no	—
204	#451	—	2	2	no	—
205	C₄×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
206	C₄×D₈	—	89	1	no	—
207	C₈×D₄	C₂×D₈₉	1	1	yes	characteristic
208	#427	—	89	2	no	—
209	#367	—	89	1	no	—
210	#366	—	89	2	no	—
211	C₄⋊C₁₆	—	89	1	no	—
212	#358	—	89	1	no	—
213	#358	—	89	2	no	—
214	#358	—	89	2	no	—
215	#358	—	89	2	no	—
216	#358	—	89	2	no	—
217	#357	—	89	1	no	—
218	C₂²⋊C₁₆	—	89	2	no	—
219	C₄×C₁₆	—	89	1	no	—
220	C₂²×Q₈	—	2	2	no	—
221	C₂×Q₁₆	—	2	4	no	—
222	C₂×Q₁₆	—	2	2	no	—
223	C₂×Q₁₆	—	2	1	no	—
224	C₂×Q₁₆	—	4	2	no	—
225	C₂×Q₁₆	C₂²×D₈₉	1	2	yes	—
226	C₂×Q₁₆	C₂²×D₈₉	1	1	yes	characteristic
227	C₂×D₈	—	89	1	no	—
228	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
229	C₄⋊Q₈	—	2	1	no	—
230	C₄⋊Q₈	—	4	1	no	—
231	C₄⋊Q₈	—	178	1	no	—
232	#124	—	356	2	no	—
233	C₂²⋊Q₈	—	2	2	no	—
234	C₂²⋊Q₈	—	4	2	no	—
235	#121	—	356	1	no	—
236	#121	—	178	2	no	—
237	C₄×Q₈	—	2	1	no	—
238	C₄×D₄	—	4	1	no	—
239	C₄×D₄	—	178	1	no	—
240	C₄×D₄	C₂²⋊D₈₉	1	1	yes	characteristic
241	#117	—	356	1	no	—
242	#117	—	178	2	no	—
243	SD₃₂	—	356	2	no	—
244	SD₃₂	—	178	2	no	—
245	M₃₂	—	356	2	no	—
246	C₂×C₁₆	—	89	1	no	—
247	C₂×C₁₆	—	89	1	no	—
248	C₂×C₁₆	—	89	2	no	—
249	#107	—	89	2	no	—
250	#107	—	89	1	no	—
251	#107	—	178	1	no	—
252	#107	C₂²×D₈₉	1	1	yes	characteristic
253	#106	—	178	2	no	—
254	C₄⋊C₈	C₂²⋊D₈₉	1	1	yes	characteristic
255	#103	—	2	1	no	—
256	#103	—	2	2	no	—
257	#102	—	178	1	no	—
258	#102	—	178	2	no	—
259	C₂²⋊C₈	C₂²⋊D₈₉	1	2	yes	—
260	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
261	C₂×Q₈	—	2	1	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	4	4	no	—
264	C₂×Q₈	—	4	2	no	—
265	C₂×Q₈	—	8	2	no	—
266	C₂×Q₈	—	356	1	no	—
267	C₂×D₄	—	356	2	no	—
268	C₂×D₄	—	178	1	no	—
269	C₂×D₄	#13083559	1	1	yes	characteristic
270	C₂²×C₄	—	4	2	no	—
271	C₂²×C₄	—	356	1	no	—
272	C₂²×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
273	Q₁₆	—	2	2	no	—
274	Q₁₆	—	2	1	no	—
275	Q₁₆	—	2	2	no	—
276	Q₁₆	—	4	4	no	—
277	Q₁₆	—	8	1	no	—
278	Q₁₆	#13083573	1	2	yes	—
279	D₈	—	178	1	no	—
280	D₈	—	178	1	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
284	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
285	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
286	C₄⋊C₄	—	2	1	no	—
287	C₄⋊C₄	—	4	1	no	—
288	C₄⋊C₄	—	4	1	no	—
289	C₄⋊C₄	—	4	2	no	—
290	C₄⋊C₄	—	4	1	no	—
291	C₄⋊C₄	—	356	1	no	—
292	C₄⋊C₄	—	178	1	no	—
293	C₄⋊C₄	—	178	2	no	—
294	C₄⋊C₄	#13083559	1	1	yes	characteristic
295	C₂²⋊C₄	—	4	2	no	—
296	C₂²⋊C₄	—	356	2	no	—
297	C₂²⋊C₄	—	356	2	no	—
298	C₂²⋊C₄	—	356	1	no	—
299	C₂²⋊C₄	#13083559	1	2	yes	—
300	C₄²	—	4	1	no	—
301	C₄²	—	356	2	no	—
302	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
303	C₁₆	—	178	1	no	—
304	C₁₆	—	178	1	no	—
305	C₁₆	—	178	2	no	—
306	C₂³	—	356	1	no	—
307	C₂³	C₂×#13083559	1	2	yes	—
308	Q₈	—	2	2	no	—
309	Q₈	—	4	1	no	—
310	Q₈	—	4	2	no	—
311	Q₈	—	4	2	no	—
312	Q₈	—	8	4	no	—
313	Q₈	—	1424	1	no	—
314	D₄	—	2	2	no	—
315	D₄	—	356	1	no	—
316	D₄	—	356	1	no	—
317	D₄	—	1424	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	8	4	no	—
324	C₂×C₄	—	356	1	no	—
325	C₂×C₄	—	356	2	no	—
326	C₂×C₄	—	712	2	no	—
327	C₂×C₄	C₂×#13083559	1	2	yes	—
328	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
329	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
330	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
331	C₈	—	2	1	no	—
332	C₈	—	2	2	no	—
333	C₈	C₂×#13083573	1	1	yes	characteristic
334	C₈	#15668057	1	1	yes	characteristic
335	C₂²	—	2	2	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	2	2	no	—
338	C₂²	—	712	1	no	—
339	C₂²	—	712	1	no	—
340	C₂²	—	712	1	no	—
341	C₂²	C₂²×#13083559	1	1	yes	characteristic
342	C₄	—	2	2	no	—
343	C₄	—	2	1	no	—
344	C₄	—	4	2	no	—
345	C₄	—	8	2	no	—
346	C₄	—	8	1	no	—
347	C₄	—	712	1	no	—
348	C₄	—	712	2	no	—
349	C₄	#19152039	1	2	yes	—
350	C₄	#19152242	1	1	yes	characteristic
351	C₄	#19152409	1	1	yes	characteristic
352	C₂	—	4	2	no	—
353	C₂	—	1424	1	no	—
354	C₂	#26032065	1	1	yes	characteristic
355	C₂	C₂×#19152039	1	1	yes	characteristic
356	C₂	#26033396	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#13083559
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	D₄×Q₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	#26032675	C₂
hypercenter	D₄×Q₁₆	D₈₉
hypocenter	C₈₉	#33896
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #26022479	C₂	semidirect
3. #26022457	C₂	semidirect
4. #26022404	C₂	semidirect
5. #26022367	C₂	semidirect
6. #26022294	C₂	semidirect
7. #26022169	C₂	semidirect
8. C₄×#15667955	C₂	semidirect
9. #26022084	C₂	semidirect
10. #26021972	C₂	semidirect
11. D₄×Q₁₆×C₈₉	C₂	semidirect
180. D₄×Q₁₆	D₈₉	semidirect
190. C₈₉	#33896	semidirect

Maximal subgroups

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

subgroup	index
#26032675	2
#26022479	2
#26022457	2
#26022404	2
#26022367	2
#26022294	2
#26022169	2
C₄×#15667955	2
#26022084	2
#26021972	2
D₄×Q₁₆×C₈₉	2
#33896	89