Subgroups — #36707986

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	#26029227	C₂	1	1	yes	maximal, characteristic
2	#26022157	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022154	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022083	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022081	C₂	1	2	yes	maximal, complemented
6	#26022080	C₂	1	2	yes	maximal, complemented
7	#26022071	C₂	1	1	yes	maximal, characteristic, complemented
8	#26022057	C₂	1	2	yes	maximal, complemented
9	#26021847	C₂	1	2	yes	maximal, complemented
10	C₄×#15667952	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#2162	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152132	C₄	1	1	yes	characteristic
13	C₂×#15667952	C₂²	1	1	yes	characteristic
14	C₂×#15667952	C₂²	1	1	yes	characteristic
15	C₂×#15667951	C₂²	1	2	yes	—
16	C₂×C₁₆×D₈₉	C₂²	1	2	yes	—
17	#19151728	C₄	1	1	yes	characteristic
18	C₈×Dic₁₇₈	C₂²	1	1	yes	characteristic
19	#19151677	C₄	1	1	yes	characteristic
20	#19151669	C₄	1	2	yes	—
21	C₈×D₃₅₆	C₂²	1	1	yes	characteristic
22	C₈×C₂²⋊D₈₉	C₂²	1	2	yes	—
23	#19151657	C₄	1	2	yes	—
24	#19151655	C₄	1	1	yes	characteristic
25	#19151552	C₂²	1	2	yes	—
26	#19151416	C₂²	1	2	yes	—
27	#19151416	C₂²	1	2	yes	—
28	#19151377	C₂²	1	2	yes	—
29	#19151377	C₂²	1	2	yes	—
30	#19151374	C₂²	1	1	yes	characteristic
31	#19151374	C₂²	1	1	yes	characteristic
32	#19151373	C₂²	1	2	yes	—
33	C₄×#13083549	C₂²	1	1	yes	characteristic
34	C₄×#13083549	C₂²	1	1	yes	characteristic
35	C₁₆×Dic₈₉	C₂²	1	2	yes	—
36	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
37	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
38	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
39	C₂²⋊C₁₆×C₈₉	C₂²	1	2	yes	—
40	C₄×C₁₆×C₈₉	C₂²	1	2	yes	—
41	C₂×#13083557	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₈₉	C₂³	1	2	yes	—
43	#15668006	C₈	1	2	yes	—
44	#15668004	C₈	1	1	yes	characteristic
45	#15668000	C₈	1	1	yes	characteristic
46	#15667995	C₈	1	2	yes	—
47	#15667991	C₈	1	1	yes	characteristic
48	#15667986	C₈	1	1	yes	characteristic
49	C₄×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
50	C₄×D₃₅₆	C₂×C₄	1	1	yes	characteristic
51	C₄×C₂²⋊D₈₉	C₂×C₄	1	2	yes	—
52	#15667969	C₂×C₄	1	2	yes	—
53	#15667952	—	2	2	no	—
54	#15667952	—	2	2	no	—
55	#15667951	—	2	4	no	—
56	C₂×#13083549	C₂³	1	1	yes	characteristic
57	C₂×#13083549	C₂³	1	1	yes	characteristic
58	C₂×#13083549	C₂³	1	1	yes	characteristic
59	C₂×#13083549	C₂³	1	1	yes	characteristic
60	C₁₆×D₈₉	—	2	4	no	—
61	C₄⋊#10623494	C₂×C₄	1	1	yes	characteristic
62	C₄⋊#10623494	C₂×C₄	1	1	yes	characteristic
63	#15667941	C₂×C₄	1	2	yes	—
64	#15667941	C₂×C₄	1	2	yes	—
65	C₂²⋊#10623494	C₂×C₄	1	2	yes	—
66	#15667924	C₂×C₄	1	2	yes	—
67	#15667924	C₂×C₄	1	2	yes	—
68	#15667923	C₂×C₄	1	2	yes	—
69	#15667922	C₂×C₄	1	2	yes	—
70	C₈×Dic₈₉	C₂³	1	2	yes	—
71	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
72	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
73	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
74	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
75	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
76	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
77	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
78	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
80	C₂×Dic₁₇₈	C₂×C₈	1	1	yes	characteristic
81	C₂×D₃₅₆	C₂×C₈	1	1	yes	characteristic
82	C₂×C₂²⋊D₈₉	C₂×C₈	1	2	yes	—
83	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
84	#13083557	—	2	4	no	—
85	C₂×#10623494	C₂²×C₄	1	2	yes	—
86	C₂×#10623494	C₂²×C₄	1	2	yes	—
87	C₈×D₈₉	—	2	2	no	—
88	C₈×D₈₉	—	2	2	no	—
89	C₄⋊Dic₈₉	C₂×C₈	1	1	yes	characteristic
90	C₄⋊Dic₈₉	C₂×C₈	1	1	yes	characteristic
91	#13083553	C₂×C₈	1	2	yes	—
92	#13083553	C₂×C₈	1	2	yes	—
93	#13083553	C₂×C₈	1	2	yes	—
94	#13083553	C₂×C₈	1	2	yes	—
95	C₂²⋊Dic₈₉	C₂×C₈	1	2	yes	—
96	#13083551	C₂×C₈	1	2	yes	—
97	#13083551	C₂×C₈	1	2	yes	—
98	#13083551	C₂×C₈	1	2	yes	—
99	#13083551	C₂×C₈	1	2	yes	—
100	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
101	#13083549	—	2	1	no	—
102	#13083549	—	2	1	no	—
103	#13083549	#41	1	2	yes	—
104	#13083549	#41	1	2	yes	—
105	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
106	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
107	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₈₉	—	2	2	no	—
109	C₂×C₈×C₈₉	—	2	1	no	—
110	C₂×C₈×C₈₉	—	2	1	no	—
111	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
115	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
116	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
117	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
118	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
119	C₁₆×C₈₉	—	2	2	no	—
120	C₁₆×C₈₉	—	2	2	no	—
121	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
122	C₂²×D₈₉	C₂²×C₈	1	2	yes	—
123	C₂×C₄×C₈₉	—	2	1	no	—
124	C₂×C₄×C₈₉	—	2	1	no	—
125	C₂×C₄×C₈₉	—	2	2	no	—
126	C₂×C₄×C₈₉	—	4	1	no	—
127	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
129	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
130	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
131	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
132	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
133	C₂²⋊D₈₉	—	4	2	no	—
134	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
135	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
136	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
137	D₃₅₆	—	2	2	no	—
138	C₄×D₈₉	—	2	2	no	—
139	C₄×D₈₉	—	2	2	no	—
140	Dic₁₇₈	—	2	2	no	—
141	C₈×C₈₉	—	2	1	no	—
142	C₈×C₈₉	—	2	1	no	—
143	C₈×C₈₉	#131	1	2	yes	—
144	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
145	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
146	#10623494	—	2	2	no	—
147	#10623494	—	2	2	no	—
148	C₂²×C₈₉	—	2	1	no	—
149	C₂²×C₈₉	—	2	1	no	—
150	C₂²×C₈₉	—	2	1	no	—
151	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
152	C₂×D₈₉	—	2	2	no	—
153	C₂×D₈₉	—	2	2	no	—
154	C₂×D₈₉	—	2	2	no	—
155	C₄×C₈₉	—	2	1	no	—
156	C₄×C₈₉	—	2	1	no	—
157	C₄×C₈₉	—	2	1	no	—
158	C₄×C₈₉	—	2	1	no	—
159	C₄×C₈₉	#504	1	2	yes	—
160	C₄×C₈₉	C₄×#41	1	1	yes	characteristic
161	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
162	Dic₈₉	—	2	2	no	—
163	Dic₈₉	—	2	2	no	—
164	Dic₈₉	—	2	2	no	—
165	#33483	—	89	1	no	maximal, Sylow
166	C₂×C₈₉	—	4	1	no	—
167	C₂×C₈₉	C₈×#41	1	1	yes	characteristic
168	C₂×C₈₉	C₂×#504	1	1	yes	characteristic
169	C₂×C₈₉	#3407	1	1	yes	characteristic
170	D₈₉	—	4	2	no	—
171	#2966	—	89	1	no	—
172	#2183	—	89	1	no	—
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	D₈₉	1	1	yes	characteristic, complemented
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	—	89	1	no	—
190	#436	—	89	1	no	—
191	#435	—	89	2	no	—
192	C₈×D₄	—	89	2	no	—
193	C₈×D₄	—	89	1	no	—
194	#433	—	89	2	no	—
195	#432	—	89	1	no	—
196	#405	—	89	2	no	—
197	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
198	C₄⋊C₁₆	—	89	2	no	—
199	C₄⋊C₁₆	—	89	2	no	—
200	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
201	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
202	C₂²⋊C₁₆	—	89	2	no	—
203	C₂²⋊C₁₆	—	89	2	no	—
204	C₂²⋊C₁₆	C₂×D₈₉	1	2	yes	—
205	#346	—	89	2	no	—
206	#346	—	89	1	no	—
207	#346	—	89	1	no	—
208	C₄×C₁₆	—	89	2	no	—
209	C₄×C₁₆	—	89	1	no	—
210	C₄×C₁₆	—	89	1	no	—
211	C₄×C₁₆	C₂×D₈₉	1	2	yes	—
212	C₂×M₁₆	—	89	2	no	—
213	C₂²×C₈	—	89	2	no	—
214	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
215	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
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	—	89	1	no	—
223	C₄×D₄	—	89	2	no	—
224	C₄×D₄	—	89	1	no	—
225	#117	—	89	2	no	—
226	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
227	M₃₂	—	178	4	no	—
228	M₃₂	—	178	2	no	—
229	M₃₂	—	178	2	no	—
230	C₂×C₁₆	—	89	1	no	—
231	C₂×C₁₆	—	89	1	no	—
232	C₂×C₁₆	—	89	1	no	—
233	C₂×C₁₆	—	89	1	no	—
234	C₂×C₁₆	—	178	4	no	—
235	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
236	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
237	C₄⋊C₈	—	89	2	no	—
238	C₄⋊C₈	—	89	2	no	—
239	C₄⋊C₈	—	89	1	no	—
240	C₄⋊C₈	—	89	1	no	—
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	2	no	—
247	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
248	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
249	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
250	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
251	C₂×Q₈	—	89	1	no	—
252	C₂×D₄	—	89	2	no	—
253	C₂×D₄	—	89	1	no	—
254	C₂²×C₄	—	89	2	no	—
255	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
256	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
257	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
258	M₁₆	—	178	4	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	89	2	no	—
263	C₂×C₈	—	89	2	no	—
264	C₂×C₈	—	178	2	no	—
265	C₂×C₈	—	178	2	no	—
266	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
267	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
268	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
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	—	89	1	no	—
275	C₄⋊C₄	—	89	2	no	—
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	2	no	—
282	C₄²	C₈×D₈₉	1	1	yes	characteristic
283	C₄²	C₈×D₈₉	1	1	yes	characteristic
284	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
285	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
286	C₁₆	—	2	2	no	—
287	C₁₆	—	2	2	no	—
288	C₁₆	—	89	2	no	—
289	C₁₆	—	89	2	no	—
290	C₁₆	—	178	1	no	—
291	C₁₆	—	178	1	no	—
292	C₂³	—	89	2	no	—
293	C₂³	C₂×C₈×D₈₉	1	1	yes	characteristic
294	Q₈	—	178	2	no	—
295	D₄	—	356	2	no	—
296	D₄	—	178	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	89	2	no	—
302	C₂×C₄	—	89	2	no	—
303	C₂×C₄	—	89	2	no	—
304	C₂×C₄	—	178	2	no	—
305	C₂×C₄	—	178	2	no	—
306	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
307	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
308	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
309	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
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	1	no	—
314	C₈	—	178	2	no	—
315	C₈	—	178	2	no	—
316	C₈	#15668020	1	2	yes	—
317	C₈	#41×D₈₉	1	1	yes	characteristic
318	C₈	#41×D₈₉	1	1	yes	characteristic
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	178	2	no	—
323	C₂²	—	178	2	no	—
324	C₂²	—	178	2	no	—
325	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	178	2	no	—
331	C₄	—	178	2	no	—
332	C₄	—	178	2	no	—
333	C₄	#19152014	1	2	yes	—
334	C₄	#19152066	1	1	yes	characteristic
335	C₄	C₂×#15668020	1	1	yes	characteristic
336	C₂	—	4	1	no	—
337	C₂	—	356	2	no	—
338	C₂	#26029208	1	1	yes	characteristic
339	C₂	C₂×#19152014	1	1	yes	characteristic
340	C₂	#26033354	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₈
Fitting	C₈₉×#2162	C₂
prime radical	#15667986	C₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2162	D₈₉
hypocenter	C₈₉	#33483
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ #2162

Complements

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

N	H ≅ G/N	type
2. #26022157	C₂	semidirect
3. #26022154	C₂	semidirect
4. #26022083	C₂	semidirect
5. #26022081	C₂	semidirect
6. #26022080	C₂	semidirect
7. #26022071	C₂	semidirect
8. #26022057	C₂	semidirect
9. #26021847	C₂	semidirect
10. C₄×#15667952	C₂	semidirect
11. C₈₉×#2162	C₂	semidirect
179. #2162	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
#26029227	2
#26022157	2
#26022154	2
#26022083	2
#26022081	2
#26022080	2
#26022071	2
#26022057	2
#26021847	2
C₄×#15667952	2
C₈₉×#2162	2
#33483	89