Subgroups — #36524375

Order 22784.

678 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×#10623494	C₂	1	1	yes	maximal, characteristic, complemented
2	#26028964	C₂	1	1	yes	maximal, characteristic
3	#26019131	C₂	1	4	yes	maximal, complemented
4	SD₁₆⋊#10623494	C₂	1	4	yes	maximal, complemented
5	#26019027	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018681	C₂	1	1	yes	maximal, characteristic, complemented
7	#26018651	C₂	1	1	yes	maximal, characteristic
8	#26018545	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#3103	C₂	1	1	yes	maximal, characteristic
10	C₂×#15668019	—	2	1	no	—
11	Q₈×#10623494	—	2	1	no	—
12	Q₈×#10623494	—	2	1	no	—
13	Q₈×#10623494	C₂²	1	2	yes	—
14	Q₈×#10623494	C₂²	1	2	yes	—
15	#19151676	—	2	1	no	—
16	#19151676	C₂²	1	2	yes	—
17	D₄×#10623494	—	2	1	no	—
18	D₄×#10623494	C₂²	1	2	yes	—
19	#19151651	C₂²	1	1	yes	characteristic
20	#19151622	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊#10623494	—	2	1	no	—
22	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
23	#19151565	—	2	1	no	—
24	C₂×C₄×#10623494	—	2	1	no	—
25	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
26	#19151351	C₂²	1	4	yes	—
27	Q₈⋊#10623494	C₂²	1	4	yes	—
28	D₄⋊#10623494	C₂²	1	4	yes	—
29	#19151320	C₂²	1	2	yes	—
30	#19151319	C₂²	1	2	yes	—
31	#532×C₈₉	C₂²	1	1	yes	characteristic
32	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
33	#481×C₈₉	C₄	1	2	yes	—
34	#475×C₈₉	C₄	1	2	yes	—
35	#460×C₈₉	C₄	1	2	yes	—
36	#448×C₈₉	C₄	1	2	yes	—
37	C₄×SD₁₆×C₈₉	C₂²	1	4	yes	—
38	#431×C₈₉	C₂²	1	1	yes	characteristic
39	#15668019	—	4	4	no	—
40	C₂×#13083558	—	2	2	no	—
41	C₂×#13083558	—	2	1	no	—
42	C₂²×#10623494	—	2	2	no	—
43	C₂²×#10623494	—	2	1	no	—
44	C₂²×#10623494	D₄	1	1	yes	characteristic
45	C₂²×#10623494	D₄	1	1	yes	characteristic
46	C₄⋊#10623494	—	2	2	no	—
47	C₄⋊#10623494	—	2	2	no	—
48	C₄⋊#10623494	—	2	1	no	—
49	C₄⋊#10623494	—	2	1	no	—
50	C₄⋊#10623494	—	2	1	no	—
51	C₄⋊#10623494	C₂³	1	2	yes	—
52	C₄⋊#10623494	C₂³	1	2	yes	—
53	C₂²⋊#10623494	—	2	2	no	—
54	C₂²⋊#10623494	—	2	2	no	—
55	C₂²⋊#10623494	—	2	1	no	—
56	C₄×#10623494	—	2	2	no	—
57	C₄×#10623494	—	2	2	no	—
58	C₄×#10623494	D₄	1	1	yes	characteristic
59	C₄×#10623494	D₄	1	1	yes	characteristic
60	C₄×#10623494	C₂³	1	2	yes	—
61	C₄×#10623494	C₂³	1	1	yes	characteristic
62	C₄×#10623494	C₂³	1	1	yes	characteristic
63	C₂×#41×C₈₉	—	2	1	no	—
64	C₂²×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
65	C₂×SD₁₆×C₈₉	C₂×C₄	1	4	yes	—
66	C₄⋊Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
67	#127×C₈₉	C₂×C₄	1	1	yes	characteristic
68	#124×C₈₉	—	2	1	no	—
69	#121×C₈₉	—	2	1	no	—
70	#121×C₈₉	C₂×C₄	1	2	yes	—
71	C₄×Q₈×C₈₉	—	2	1	no	—
72	C₄×Q₈×C₈₉	—	2	1	no	—
73	C₄×Q₈×C₈₉	C₂×C₄	1	2	yes	—
74	C₄×Q₈×C₈₉	C₂³	1	2	yes	—
75	C₄×D₄×C₈₉	—	2	1	no	—
76	C₄×D₄×C₈₉	C₂³	1	2	yes	—
77	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
78	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
79	C₂×C₄²×C₈₉	—	2	1	no	—
80	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
81	#106×C₈₉	C₂×C₄	1	4	yes	—
82	C₄⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
83	#103×C₈₉	C₂×C₄	1	4	yes	—
84	#102×C₈₉	C₂×C₄	1	4	yes	—
85	C₂²⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
86	C₄×C₈×C₈₉	C₂³	1	2	yes	—
87	#13083558	—	4	4	no	—
88	#13083558	—	4	2	no	—
89	C₂×#10623494	—	2	2	no	—
90	C₂×#10623494	—	2	2	no	—
91	C₂×#10623494	—	2	2	no	—
92	C₂×#10623494	—	2	2	no	—
93	C₂×#10623494	—	2	2	no	—
94	C₂×#10623494	—	4	4	no	—
95	C₂×#10623494	—	4	2	no	—
96	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
97	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
98	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
99	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
100	#41×C₈₉	—	4	2	no	—
101	#41×C₈₉	—	4	2	no	—
102	C₂×Q₈×C₈₉	—	2	4	no	—
103	C₂×Q₈×C₈₉	—	2	1	no	—
104	C₂×Q₈×C₈₉	—	2	1	no	—
105	C₂×Q₈×C₈₉	C₂²×C₄	1	2	yes	—
106	C₂×D₄×C₈₉	—	2	2	no	—
107	C₂×D₄×C₈₉	—	2	1	no	—
108	C₂×D₄×C₈₉	—	2	1	no	—
109	C₂×D₄×C₈₉	C₂²×C₄	1	2	yes	—
110	C₂²×C₄×C₈₉	—	2	1	no	—
111	C₂²×C₄×C₈₉	—	2	1	no	—
112	C₂²×C₄×C₈₉	—	2	2	no	—
113	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
114	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
116	SD₁₆×C₈₉	—	2	8	no	—
117	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
118	C₄⋊C₄×C₈₉	—	2	2	no	—
119	C₄⋊C₄×C₈₉	—	2	1	no	—
120	C₄⋊C₄×C₈₉	—	2	1	no	—
121	C₄⋊C₄×C₈₉	—	2	2	no	—
122	C₄⋊C₄×C₈₉	C₂²×C₄	1	2	yes	—
123	C₄⋊C₄×C₈₉	C₂²×C₄	1	2	yes	—
124	C₂²⋊C₄×C₈₉	—	2	2	no	—
125	C₂²⋊C₄×C₈₉	—	2	2	no	—
126	C₄²×C₈₉	—	2	2	no	—
127	C₄²×C₈₉	—	2	2	no	—
128	C₄²×C₈₉	C₂²×C₄	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	—	2	2	no	—
133	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
134	Q₈×C₈₉	—	2	4	no	—
135	Q₈×C₈₉	—	2	2	no	—
136	Q₈×C₈₉	—	4	2	no	—
137	Q₈×C₈₉	#131	1	4	yes	complemented
138	D₄×C₈₉	—	2	4	no	—
139	D₄×C₈₉	—	4	2	no	—
140	D₄×C₈₉	—	4	4	no	—
141	D₄×C₈₉	—	4	2	no	—
142	C₂×C₄×C₈₉	—	2	4	no	—
143	C₂×C₄×C₈₉	—	2	2	no	—
144	C₂×C₄×C₈₉	—	2	2	no	—
145	C₂×C₄×C₈₉	—	2	2	no	—
146	C₂×C₄×C₈₉	—	2	2	no	—
147	C₂×C₄×C₈₉	—	4	1	no	—
148	C₂×C₄×C₈₉	—	4	2	no	—
149	C₂×C₄×C₈₉	—	4	2	no	—
150	C₂×C₄×C₈₉	—	4	2	no	—
151	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
152	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
153	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
154	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
155	C₂×C₄×C₈₉	#137	1	2	yes	—
156	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
157	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
159	C₈×C₈₉	—	2	4	no	—
160	#10623494	—	2	1	no	—
161	#10623494	—	2	1	no	—
162	#10623494	—	2	1	no	—
163	#10623494	—	2	1	no	—
164	#10623494	—	4	2	no	—
165	#10623494	—	4	2	no	—
166	#10623494	—	4	2	no	—
167	C₂²×C₈₉	—	2	1	no	—
168	C₂²×C₈₉	—	2	1	no	—
169	C₂²×C₈₉	—	4	2	no	—
170	C₂²×C₈₉	—	4	2	no	—
171	C₂²×C₈₉	—	4	2	no	—
172	C₂²×C₈₉	#440	1	2	yes	—
173	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
174	C₄×C₈₉	—	2	4	no	—
175	C₄×C₈₉	—	2	1	no	—
176	C₄×C₈₉	—	2	1	no	—
177	C₄×C₈₉	—	2	1	no	—
178	C₄×C₈₉	—	4	2	no	—
179	C₄×C₈₉	—	4	2	no	—
180	C₄×C₈₉	#440	1	2	yes	—
181	C₄×C₈₉	C₂×#131	1	2	yes	—
182	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
183	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
184	#17908	—	89	1	no	maximal, Sylow
185	C₂×C₈₉	—	2	2	no	—
186	C₂×C₈₉	—	8	2	no	—
187	C₂×C₈₉	#2929	1	1	yes	characteristic
188	C₂×C₈₉	C₂×#440	1	1	yes	characteristic
189	C₂×C₈₉	#2955	1	1	yes	characteristic
190	#3103	D₈₉	1	1	yes	characteristic
191	C₂×C₈×Q₈	—	89	1	no	—
192	#2931	—	89	1	no	—
193	#1588	—	89	4	no	—
194	SD₁₆⋊C₈	—	89	4	no	—
195	#1571	—	89	1	no	—
196	#1487	—	89	1	no	—
197	#1480	—	89	1	no	—
198	#1452	—	89	1	no	—
199	C₈₉	#17908	1	1	yes	characteristic, complemented, Sylow
200	C₂×#131	—	178	1	no	—
201	#532	C₂×D₈₉	1	1	yes	characteristic
202	C₂×C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
203	#481	Dic₈₉	1	2	yes	—
204	#475	Dic₈₉	1	2	yes	—
205	#460	Dic₈₉	1	2	yes	—
206	#448	Dic₈₉	1	2	yes	—
207	C₈×Q₈	—	89	2	no	—
208	C₈×Q₈	—	89	2	no	—
209	C₈×Q₈	—	178	1	no	—
210	C₈×Q₈	—	178	1	no	—
211	C₄×SD₁₆	C₂×D₈₉	1	4	yes	—
212	#436	—	89	2	no	—
213	#436	—	178	1	no	—
214	C₈×D₄	—	89	2	no	—
215	C₈×D₄	—	178	1	no	—
216	#431	—	89	1	no	—
217	#431	C₂×D₈₉	1	1	yes	characteristic
218	#423	—	89	1	no	—
219	C₂×C₄⋊C₈	—	89	1	no	—
220	C₂×C₄⋊C₈	—	178	1	no	—
221	#408	—	178	1	no	—
222	C₂×C₄×C₈	—	89	1	no	—
223	C₂×C₄×C₈	—	178	1	no	—
224	#335	—	89	4	no	—
225	Q₈⋊C₈	—	89	4	no	—
226	D₄⋊C₈	—	89	4	no	—
227	#322	—	89	2	no	—
228	#322	—	89	2	no	—
229	C₂×#41	—	2	1	no	—
230	C₂²×Q₈	C₂×Dic₈₉	1	1	yes	characteristic
231	C₂×SD₁₆	C₂×Dic₈₉	1	4	yes	—
232	#131	—	356	4	no	—
233	C₂×M₁₆	—	178	2	no	—
234	C₂×M₁₆	—	178	1	no	—
235	C₂²×C₈	—	89	1	no	—
236	C₂²×C₈	—	89	1	no	—
237	C₂²×C₈	—	178	2	no	—
238	C₂²×C₈	—	178	1	no	—
239	C₄⋊Q₈	C₂×Dic₈₉	1	1	yes	characteristic
240	#127	C₂×Dic₈₉	1	1	yes	characteristic
241	#124	—	2	1	no	—
242	#121	—	2	1	no	—
243	#121	C₂×Dic₈₉	1	2	yes	—
244	C₄×Q₈	—	2	1	no	—
245	C₄×Q₈	—	2	1	no	—
246	C₄×Q₈	C₂×Dic₈₉	1	2	yes	—
247	C₄×Q₈	C₂²×D₈₉	1	2	yes	—
248	C₄×D₄	—	2	1	no	—
249	C₄×D₄	C₂²×D₈₉	1	2	yes	—
250	C₂×C₄⋊C₄	—	2	1	no	—
251	C₂×C₄⋊C₄	C₂×Dic₈₉	1	1	yes	characteristic
252	C₂×C₄²	—	2	1	no	—
253	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
254	#106	C₂×Dic₈₉	1	4	yes	—
255	C₄⋊C₈	—	89	2	no	—
256	C₄⋊C₈	—	89	2	no	—
257	C₄⋊C₈	—	178	2	no	—
258	C₄⋊C₈	—	178	2	no	—
259	C₄⋊C₈	—	178	1	no	—
260	C₄⋊C₈	—	178	1	no	—
261	C₄⋊C₈	—	178	1	no	—
262	C₄⋊C₈	C₂×Dic₈₉	1	2	yes	—
263	#103	C₂×Dic₈₉	1	4	yes	—
264	#102	C₂×Dic₈₉	1	4	yes	—
265	C₂²⋊C₈	—	178	2	no	—
266	C₂²⋊C₈	—	178	2	no	—
267	C₂²⋊C₈	—	178	1	no	—
268	C₂²⋊C₈	C₂×Dic₈₉	1	2	yes	—
269	C₄×C₈	—	89	1	no	—
270	C₄×C₈	—	89	1	no	—
271	C₄×C₈	—	89	1	no	—
272	C₄×C₈	—	89	1	no	—
273	C₄×C₈	—	89	2	no	—
274	C₄×C₈	—	178	2	no	—
275	C₄×C₈	—	178	2	no	—
276	C₄×C₈	C₂²×D₈₉	1	2	yes	—
277	#41	—	4	2	no	—
278	#41	—	4	2	no	—
279	C₂×Q₈	—	2	4	no	—
280	C₂×Q₈	—	2	1	no	—
281	C₂×Q₈	—	2	1	no	—
282	C₂×Q₈	C₂²×Dic₈₉	1	2	yes	—
283	C₂×D₄	—	2	2	no	—
284	C₂×D₄	—	2	1	no	—
285	C₂×D₄	—	2	1	no	—
286	C₂×D₄	C₂²×Dic₈₉	1	2	yes	—
287	C₂²×C₄	—	2	1	no	—
288	C₂²×C₄	—	2	1	no	—
289	C₂²×C₄	—	2	2	no	—
290	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
291	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
292	C₂²×C₄	#13083573	1	1	yes	characteristic
293	SD₁₆	—	2	8	no	—
294	M₁₆	—	356	2	no	—
295	M₁₆	—	356	4	no	—
296	C₂×C₈	—	89	1	no	—
297	C₂×C₈	—	89	1	no	—
298	C₂×C₈	—	89	1	no	—
299	C₂×C₈	—	89	1	no	—
300	C₂×C₈	—	356	2	no	—
301	C₂×C₈	—	356	4	no	—
302	C₂×C₈	—	178	2	no	—
303	C₂×C₈	—	178	2	no	—
304	C₂×C₈	—	178	2	no	—
305	C₂×C₈	—	178	2	no	—
306	C₂×C₈	—	178	2	no	—
307	C₂×C₈	C₂²×Dic₈₉	1	4	yes	—
308	C₄⋊C₄	—	2	2	no	—
309	C₄⋊C₄	—	2	1	no	—
310	C₄⋊C₄	—	2	1	no	—
311	C₄⋊C₄	—	2	2	no	—
312	C₄⋊C₄	C₂²×Dic₈₉	1	2	yes	—
313	C₄⋊C₄	C₂²×Dic₈₉	1	2	yes	—
314	C₂²⋊C₄	—	2	2	no	—
315	C₂²⋊C₄	—	2	2	no	—
316	C₄²	—	2	2	no	—
317	C₄²	—	2	2	no	—
318	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
319	C₄²	D₄×D₈₉	1	1	yes	characteristic
320	C₄²	#13083573	1	1	yes	characteristic
321	C₄²	C₂³×D₈₉	1	1	yes	characteristic
322	C₂³	—	2	2	no	—
323	C₂³	D₄×Dic₈₉	1	1	yes	characteristic
324	Q₈	—	2	4	no	—
325	Q₈	—	2	2	no	—
326	Q₈	—	4	2	no	—
327	Q₈	#15668019	1	4	yes	complemented
328	D₄	—	2	4	no	—
329	D₄	—	4	2	no	—
330	D₄	—	4	4	no	—
331	D₄	—	4	2	no	—
332	C₂×C₄	—	2	4	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	4	1	no	—
338	C₂×C₄	—	4	2	no	—
339	C₂×C₄	—	4	2	no	—
340	C₂×C₄	—	4	2	no	—
341	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
342	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
343	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
344	C₂×C₄	#15668039	1	2	yes	—
345	C₂×C₄	#15668041	1	2	yes	—
346	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
347	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
348	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
349	C₈	—	2	4	no	—
350	C₈	—	356	2	no	—
351	C₈	—	356	2	no	—
352	C₈	—	356	2	no	—
353	C₈	—	178	1	no	—
354	C₈	—	178	1	no	—
355	C₈	—	178	1	no	—
356	C₈	—	178	1	no	—
357	C₂²	—	2	1	no	—
358	C₂²	—	2	1	no	—
359	C₂²	—	4	2	no	—
360	C₂²	—	4	2	no	—
361	C₂²	—	4	2	no	—
362	C₂²	#19151699	1	2	yes	—
363	C₂²	C₂×D₄×Dic₈₉	1	1	yes	characteristic
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	2	4	no	—
367	C₄	—	2	1	no	—
368	C₄	—	4	2	no	—
369	C₄	—	4	2	no	—
370	C₄	#19151699	1	2	yes	—
371	C₄	C₂×#15668019	1	2	yes	—
372	C₄	C₂×#15668039	1	1	yes	characteristic
373	C₄	C₂×#15668041	1	1	yes	characteristic
374	C₂	—	2	2	no	—
375	C₂	—	8	2	no	—
376	C₂	#26028947	1	1	yes	characteristic
377	C₂	C₂×#19151699	1	1	yes	characteristic
378	C₂	#26029155	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#3103	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	#532×C₈₉	C₂²
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	#3103	D₈₉
hypocenter	C₈₉	#17908
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3103

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×#10623494	C₂	semidirect
3. #26019131	C₂	semidirect
4. SD₁₆⋊#10623494	C₂	semidirect
5. #26019027	C₂	semidirect
6. #26018681	C₂	semidirect
8. #26018545	C₂	semidirect
137. Q₈×C₈₉	#131	semidirect
199. C₈₉	#17908	semidirect
327. Q₈	#15668019	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×#10623494	2
#26028964	2
#26019131	2
SD₁₆⋊#10623494	2
#26019027	2
#26018681	2
#26018651	2
#26018545	2
C₈₉×#3103	2
#17908	89