Subgroups — #36707994

Order 22784.

516 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029229	C₂	1	1	yes	maximal, characteristic
2	Q₈×#13083549	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022083	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022083	C₂	1	2	yes	maximal, complemented
5	#26022079	C₂	1	2	yes	maximal, complemented
6	C₁₆×D₃₅₆	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022058	C₂	1	2	yes	maximal, complemented
8	#26022050	C₂	1	1	yes	maximal, characteristic, complemented
9	#26021850	C₂	1	2	yes	maximal, complemented
10	C₄×#15667951	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#2183	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152135	C₄	1	1	yes	characteristic
13	C₂×#15667951	C₂²	1	1	yes	characteristic
14	C₂×#15667951	C₂²	1	1	yes	characteristic
15	C₂×#15667951	C₂²	1	2	yes	—
16	C₂×C₁₆×D₈₉	C₂²	1	2	yes	—
17	#19151729	C₄	1	1	yes	characteristic
18	#19151675	C₄	1	1	yes	characteristic
19	#19151668	C₄	1	2	yes	—
20	C₈×D₃₅₆	C₂²	1	1	yes	characteristic
21	C₈×D₃₅₆	C₂²	1	2	yes	—
22	#19151658	C₄	1	2	yes	—
23	#19151654	C₄	1	1	yes	characteristic
24	#19151552	C₂²	1	2	yes	—
25	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
26	C₄⋊#13083549	C₂²	1	1	yes	characteristic
27	C₄⋊#13083549	C₂²	1	2	yes	—
28	#19151377	C₂²	1	2	yes	—
29	#19151377	C₂²	1	2	yes	—
30	#19151377	C₂²	1	2	yes	—
31	#19151373	C₂²	1	1	yes	characteristic
32	#19151373	C₂²	1	1	yes	characteristic
33	C₄×#13083549	C₂²	1	1	yes	characteristic
34	C₄×#13083549	C₂²	1	2	yes	—
35	C₁₆×Dic₈₉	C₂²	1	2	yes	—
36	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
37	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
38	C₄⋊C₁₆×C₈₉	C₂²	1	2	yes	—
39	#346×C₈₉	C₂²	1	2	yes	—
40	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
41	C₂×#13083557	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×D₈₉	C₂³	1	2	yes	—
44	C₂×C₈×D₈₉	C₂³	1	1	yes	characteristic
45	#15668007	C₈	1	2	yes	—
46	#15668003	C₈	1	1	yes	characteristic
47	#15668001	C₈	1	1	yes	characteristic
48	#15667997	C₈	1	2	yes	—
49	#15667988	C₈	1	1	yes	characteristic
50	#15667987	C₈	1	1	yes	characteristic
51	C₄×D₃₅₆	C₂×C₄	1	2	yes	—
52	C₄×D₃₅₆	C₂×C₄	1	1	yes	characteristic
53	#15667969	C₂×C₄	1	2	yes	—
54	C₄²×D₈₉	C₂×C₄	1	1	yes	characteristic
55	#15667951	—	2	2	no	—
56	#15667951	—	2	4	no	—
57	#15667951	—	2	2	no	—
58	C₂×#13083549	C₂³	1	2	yes	—
59	C₂×#13083549	C₂³	1	1	yes	characteristic
60	C₂×#13083549	C₂³	1	1	yes	characteristic
61	C₁₆×D₈₉	—	2	4	no	—
62	C₄⋊#10623494	C₂×C₄	1	1	yes	characteristic
63	C₄⋊#10623494	C₂×C₄	1	2	yes	—
64	#15667924	C₂×C₄	1	2	yes	—
65	#15667924	C₂×C₄	1	2	yes	—
66	#15667924	C₂×C₄	1	2	yes	—
67	#15667923	C₂×C₄	1	2	yes	—
68	#15667922	C₂×C₄	1	2	yes	—
69	C₄×#10623494	C₂×C₄	1	1	yes	characteristic
70	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
71	C₈×Dic₈₉	C₂³	1	1	yes	characteristic
72	C₄×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
73	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
74	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
75	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
76	C₄⋊C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
77	C₄⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
78	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₈₉	C₂³	1	2	yes	—
80	C₂×D₃₅₆	C₂×C₈	1	1	yes	characteristic
81	C₂×D₃₅₆	C₂×C₈	1	2	yes	—
82	C₂×C₄×D₈₉	C₂×C₈	1	1	yes	characteristic
83	C₂×C₄×D₈₉	C₂×C₈	1	1	yes	characteristic
84	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
85	C₂×C₄×D₈₉	C₂²×C₄	1	1	yes	characteristic
86	#13083557	—	2	4	no	—
87	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
88	C₂×#10623494	C₂²×C₄	1	2	yes	—
89	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
90	C₈×D₈₉	—	2	2	no	—
91	C₈×D₈₉	—	2	2	no	—
92	C₈×D₈₉	—	2	1	no	—
93	C₈×D₈₉	—	2	1	no	—
94	C₈×D₈₉	—	2	2	no	—
95	C₄⋊Dic₈₉	C₂×C₈	1	1	yes	characteristic
96	C₄⋊Dic₈₉	C₂×C₈	1	2	yes	—
97	#13083553	C₂×C₈	1	2	yes	—
98	#13083553	C₂×C₈	1	2	yes	—
99	#13083551	C₂×C₈	1	2	yes	—
100	#13083551	C₂×C₈	1	2	yes	—
101	#13083551	C₂×C₈	1	2	yes	—
102	#13083551	C₂×C₈	1	2	yes	—
103	#13083551	C₂×C₈	1	2	yes	—
104	C₄×Dic₈₉	C₂×C₈	1	1	yes	characteristic
105	C₄×Dic₈₉	C₂×C₈	1	1	yes	characteristic
106	C₄×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
107	#13083549	—	2	2	no	—
108	#13083549	—	2	1	no	—
109	#13083549	#41	1	2	yes	—
110	C₂×Q₈×C₈₉	C₂×C₈	1	1	yes	characteristic
111	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
113	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
114	C₄⋊C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
115	C₄⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
116	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
117	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
118	C₁₆×C₈₉	—	2	2	no	—
119	C₁₆×C₈₉	—	2	1	no	—
120	C₁₆×C₈₉	#41	1	2	yes	—
121	C₂²×D₈₉	C₂²×C₈	1	2	yes	—
122	C₂²×D₈₉	C₂²×C₈	1	1	yes	characteristic
123	Q₈×C₈₉	—	2	2	no	—
124	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
126	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
127	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
129	C₂×Dic₈₉	C₂²×C₈	1	1	yes	characteristic
130	C₂×Dic₈₉	C₂²×C₈	1	1	yes	characteristic
131	C₂×Dic₈₉	C₂²×C₈	1	1	yes	characteristic
132	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
133	D₃₅₆	—	2	2	no	—
134	D₃₅₆	—	4	2	no	—
135	C₄×D₈₉	—	2	2	no	—
136	C₄×D₈₉	—	2	1	no	—
137	C₄×D₈₉	—	2	1	no	—
138	C₄×D₈₉	—	2	2	no	—
139	C₄×D₈₉	—	2	2	no	—
140	C₄×D₈₉	—	4	1	no	—
141	C₈×C₈₉	—	2	1	no	—
142	C₈×C₈₉	—	2	2	no	—
143	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
144	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
145	#10623494	—	2	2	no	—
146	#10623494	—	2	1	no	—
147	#10623494	#131	1	2	yes	—
148	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
149	C₂×D₈₉	—	2	1	no	—
150	C₂×D₈₉	—	2	2	no	—
151	C₂×D₈₉	—	2	1	no	—
152	C₂×D₈₉	—	2	2	no	—
153	C₂×D₈₉	—	2	1	no	—
154	C₂×D₈₉	—	2	2	no	—
155	C₄×C₈₉	—	2	2	no	—
156	C₄×C₈₉	—	2	1	no	—
157	C₄×C₈₉	—	2	2	no	—
158	C₄×C₈₉	#504	1	2	yes	—
159	C₄×C₈₉	C₄×#41	1	1	yes	characteristic
160	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
161	Dic₈₉	—	2	1	no	—
162	Dic₈₉	—	2	1	no	—
163	Dic₈₉	—	2	1	no	—
164	Dic₈₉	—	2	2	no	—
165	#33483	—	89	1	no	maximal, Sylow
166	C₂×C₈₉	C₈×#41	1	1	yes	characteristic
167	C₂×C₈₉	C₂×#504	1	1	yes	characteristic
168	C₂×C₈₉	#3407	1	1	yes	characteristic
169	D₈₉	—	4	1	no	—
170	D₈₉	—	4	2	no	—
171	#2966	—	89	1	no	—
172	#2183	D₈₉	1	1	yes	characteristic, complemented
173	Q₈×C₁₆	—	89	1	no	—
174	#2169	—	89	1	no	—
175	#2169	—	89	2	no	—
176	#2168	—	89	2	no	—
177	D₄×C₁₆	—	89	1	no	—
178	#2164	—	89	2	no	—
179	#2162	—	89	1	no	—
180	#2108	—	89	2	no	—
181	C₄×M₃₂	—	89	1	no	—
182	C₈₉	#33483	1	1	yes	characteristic, complemented, Sylow
183	#529	—	89	1	no	—
184	C₂×M₃₂	—	89	1	no	—
185	C₂×M₃₂	—	89	1	no	—
186	C₂×M₃₂	—	89	2	no	—
187	C₂²×C₁₆	—	89	2	no	—
188	#446	—	89	1	no	—
189	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
190	#436	—	89	1	no	—
191	#435	—	89	2	no	—
192	C₈×D₄	—	89	1	no	—
193	C₈×D₄	—	89	2	no	—
194	#433	—	89	2	no	—
195	#432	—	89	1	no	—
196	#405	—	89	2	no	—
197	C₂×C₄×C₈	—	89	1	no	—
198	C₄⋊C₁₆	—	89	1	no	—
199	C₄⋊C₁₆	—	89	2	no	—
200	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
201	C₄⋊C₁₆	C₂×D₈₉	1	2	yes	—
202	C₂²⋊C₁₆	—	89	2	no	—
203	C₂²⋊C₁₆	—	89	2	no	—
204	C₂²⋊C₁₆	—	89	2	no	—
205	#346	—	89	1	no	—
206	#346	—	89	1	no	—
207	#346	C₂×D₈₉	1	2	yes	—
208	C₄×C₁₆	—	89	2	no	—
209	C₄×C₁₆	—	89	1	no	—
210	C₄×C₁₆	—	89	2	no	—
211	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
212	C₂×M₁₆	—	89	2	no	—
213	C₂²×C₈	—	89	1	no	—
214	C₂²×C₈	—	89	1	no	—
215	C₂²×C₈	—	89	2	no	—
216	#126	—	89	2	no	—
217	#125	—	89	1	no	—
218	#124	—	89	1	no	—
219	#123	—	89	2	no	—
220	C₂²⋊Q₈	—	89	1	no	—
221	#121	—	89	1	no	—
222	C₄×Q₈	C₄×D₈₉	1	1	yes	characteristic
223	C₄×D₄	—	89	2	no	—
224	C₄×D₄	—	89	1	no	—
225	#117	—	89	2	no	—
226	C₂×C₄²	—	89	1	no	—
227	M₃₂	—	178	2	no	—
228	M₃₂	—	178	4	no	—
229	M₃₂	—	178	2	no	—
230	C₂×C₁₆	—	89	1	no	—
231	C₂×C₁₆	—	89	2	no	—
232	C₂×C₁₆	—	89	1	no	—
233	C₂×C₁₆	—	178	4	no	—
234	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
235	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
236	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
237	C₄⋊C₈	—	89	1	no	—
238	C₄⋊C₈	—	89	2	no	—
239	C₄⋊C₈	C₄×D₈₉	1	1	yes	characteristic
240	C₄⋊C₈	C₄×D₈₉	1	2	yes	—
241	C₂²⋊C₈	—	89	2	no	—
242	C₂²⋊C₈	—	89	2	no	—
243	C₂²⋊C₈	—	89	2	no	—
244	#97	—	89	2	no	—
245	#97	—	89	2	no	—
246	C₄×C₈	—	89	1	no	—
247	C₄×C₈	—	89	1	no	—
248	C₄×C₈	—	89	1	no	—
249	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
250	C₄×C₈	C₂²×D₈₉	1	2	yes	—
251	C₂×Q₈	C₈×D₈₉	1	1	yes	characteristic
252	C₂×D₄	—	89	2	no	—
253	C₂×D₄	—	89	1	no	—
254	C₂²×C₄	—	89	1	no	—
255	C₂²×C₄	—	89	1	no	—
256	C₂²×C₄	—	89	1	no	—
257	C₂²×C₄	—	89	2	no	—
258	M₁₆	—	178	4	no	—
259	C₂×C₈	—	89	1	no	—
260	C₂×C₈	—	89	1	no	—
261	C₂×C₈	—	89	2	no	—
262	C₂×C₈	—	178	2	no	—
263	C₂×C₈	—	178	1	no	—
264	C₂×C₈	—	178	2	no	—
265	C₂×C₈	—	178	2	no	—
266	C₂×C₈	—	178	1	no	—
267	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
268	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
269	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
270	C₄⋊C₄	—	89	2	no	—
271	C₄⋊C₄	—	89	2	no	—
272	C₄⋊C₄	—	89	1	no	—
273	C₄⋊C₄	—	89	2	no	—
274	C₄⋊C₄	C₈×D₈₉	1	1	yes	characteristic
275	C₄⋊C₄	C₈×D₈₉	1	2	yes	—
276	C₂²⋊C₄	—	89	2	no	—
277	C₂²⋊C₄	—	89	2	no	—
278	C₂²⋊C₄	—	89	2	no	—
279	C₂²⋊C₄	—	89	2	no	—
280	C₂²⋊C₄	—	89	2	no	—
281	C₄²	—	89	1	no	—
282	C₄²	—	89	1	no	—
283	C₄²	—	89	1	no	—
284	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
285	C₄²	C₂×C₄×D₈₉	1	2	yes	—
286	C₁₆	—	2	2	no	—
287	C₁₆	—	2	1	no	—
288	C₁₆	—	89	2	no	—
289	C₁₆	—	178	2	no	—
290	C₁₆	—	178	1	no	—
291	C₁₆	#13083575	1	2	yes	—
292	C₂³	—	89	2	no	—
293	C₂³	—	89	1	no	—
294	Q₈	—	2	2	no	—
295	D₄	—	356	2	no	—
296	D₄	—	178	2	no	—
297	C₂×C₄	—	89	1	no	—
298	C₂×C₄	—	89	1	no	—
299	C₂×C₄	—	89	1	no	—
300	C₂×C₄	—	89	2	no	—
301	C₂×C₄	—	356	1	no	—
302	C₂×C₄	—	178	2	no	—
303	C₂×C₄	—	178	2	no	—
304	C₂×C₄	—	178	1	no	—
305	C₂×C₄	—	178	1	no	—
306	C₂×C₄	—	178	2	no	—
307	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
308	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
309	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
310	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
311	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
312	C₈	—	2	1	no	—
313	C₈	—	2	2	no	—
314	C₈	—	89	2	no	—
315	C₈	—	178	1	no	—
316	C₈	—	178	2	no	—
317	C₈	#41×D₈₉	1	1	yes	characteristic
318	C₈	C₂×#13083575	1	1	yes	characteristic
319	C₂²	—	178	1	no	—
320	C₂²	—	178	2	no	—
321	C₂²	—	178	1	no	—
322	C₂²	—	178	2	no	—
323	C₂²	—	178	1	no	—
324	C₂²	—	178	2	no	—
325	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
326	C₄	—	2	2	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	2	no	—
329	C₄	—	178	1	no	—
330	C₄	—	178	2	no	—
331	C₄	—	178	1	no	—
332	C₄	—	178	1	no	—
333	C₄	#19152016	1	2	yes	—
334	C₄	C₄×#13083575	1	1	yes	characteristic
335	C₄	#131×D₈₉	1	1	yes	characteristic
336	C₂	—	356	1	no	—
337	C₂	—	356	2	no	—
338	C₂	C₈×#13083575	1	1	yes	characteristic
339	C₂	C₂×#19152016	1	1	yes	characteristic
340	C₂	#26033353	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₈
Fitting	C₈₉×#2183	C₂
prime radical	#15667987	C₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2183	D₈₉
hypocenter	C₈₉	#33483
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ #2183

Complements

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

N	H ≅ G/N	type
2. Q₈×#13083549	C₂	semidirect
3. #26022083	C₂	semidirect
4. #26022083	C₂	semidirect
5. #26022079	C₂	semidirect
6. C₁₆×D₃₅₆	C₂	semidirect
7. #26022058	C₂	semidirect
8. #26022050	C₂	semidirect
9. #26021850	C₂	semidirect
10. C₄×#15667951	C₂	semidirect
11. C₈₉×#2183	C₂	semidirect
172. #2183	D₈₉	semidirect
182. C₈₉	#33483	semidirect

Maximal subgroups

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

subgroup	index
#26029229	2
Q₈×#13083549	2
#26022083	2
#26022079	2
C₁₆×D₃₅₆	2
#26022058	2
#26022050	2
#26021850	2
C₄×#15667951	2
C₈₉×#2183	2
#33483	89