Subgroups — #36707989

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	#26029224	C₂	1	1	yes	maximal, characteristic
2	#26022160	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022154	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022084	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022081	C₂	1	2	yes	maximal, complemented
6	#26022077	C₂	1	2	yes	maximal, complemented
7	#26022056	C₂	1	2	yes	maximal, complemented
8	#26022050	C₂	1	1	yes	maximal, characteristic, complemented
9	#26021848	C₂	1	2	yes	maximal, complemented
10	C₄×#15667951	C₂	1	1	yes	maximal, characteristic, complemented
11	D₄×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152130	C₄	1	1	yes	characteristic
13	C₂×#15667952	C₂²	1	2	yes	—
14	C₂×#15667951	C₂²	1	1	yes	characteristic
15	C₂×#15667951	C₂²	1	1	yes	characteristic
16	#19151725	C₄	1	1	yes	characteristic
17	C₈×Dic₁₇₈	C₂²	1	1	yes	characteristic
18	#19151676	C₄	1	1	yes	characteristic
19	#19151672	C₄	1	2	yes	—
20	C₈×C₂²⋊D₈₉	C₂²	1	2	yes	—
21	#19151656	C₄	1	2	yes	—
22	#19151654	C₄	1	1	yes	characteristic
23	#19151553	C₂²	1	2	yes	—
24	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
25	C₄⋊#13083549	C₂²	1	1	yes	characteristic
26	#19151416	C₂²	1	2	yes	—
27	#19151416	C₂²	1	2	yes	—
28	C₂²⋊#13083549	C₂²	1	2	yes	—
29	#19151377	C₂²	1	2	yes	—
30	#19151373	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	C₁₆×Dic₈₉	C₂²	1	2	yes	—
35	C₁₆×Dic₈₉	C₂²	1	2	yes	—
36	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	—
37	C₈×D₄×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	1	yes	characteristic
41	C₂×#13083558	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×D₈₉	C₂³	1	1	yes	characteristic
44	#15668005	C₈	1	2	yes	—
45	#15668003	C₈	1	1	yes	characteristic
46	#15667999	C₈	1	1	yes	characteristic
47	#15667994	C₈	1	2	yes	—
48	#15667992	C₈	1	1	yes	characteristic
49	#15667983	C₈	1	1	yes	characteristic
50	C₄×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
51	C₄×C₂²⋊D₈₉	C₂×C₄	1	2	yes	—
52	#15667971	C₂×C₄	1	2	yes	—
53	C₄²×D₈₉	C₂×C₄	1	1	yes	characteristic
54	#15667952	—	2	4	no	—
55	#15667951	—	2	2	no	—
56	#15667951	—	2	2	no	—
57	C₂×#13083549	C₂³	1	2	yes	—
58	C₂×#13083549	C₂³	1	1	yes	characteristic
59	C₂×#13083549	C₂³	1	1	yes	characteristic
60	C₄⋊#10623494	C₂×C₄	1	1	yes	characteristic
61	#15667941	C₂×C₄	1	2	yes	—
62	#15667941	C₂×C₄	1	2	yes	—
63	C₂²⋊#10623494	C₂×C₄	1	2	yes	—
64	#15667924	C₂×C₄	1	2	yes	—
65	#15667922	C₂×C₄	1	2	yes	—
66	#15667922	C₂×C₄	1	2	yes	—
67	C₄×#10623494	C₂×C₄	1	1	yes	characteristic
68	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
69	C₈×Dic₈₉	C₂³	1	1	yes	characteristic
70	C₈×Dic₈₉	C₂³	1	2	yes	—
71	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
72	C₄×D₄×C₈₉	C₂×C₄	1	1	yes	characteristic
73	C₂×C₁₆×C₈₉	—	2	4	no	—
74	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
75	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
76	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
77	C₄⋊C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
78	C₂²⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
79	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
80	C₂×Dic₁₇₈	C₂×C₈	1	1	yes	characteristic
81	C₂×C₂²⋊D₈₉	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	1	yes	characteristic
85	#13083558	—	2	4	no	—
86	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
87	C₂×#10623494	C₂²×C₄	1	2	yes	—
88	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
89	C₈×D₈₉	—	2	1	no	—
90	C₈×D₈₉	—	2	1	no	—
91	C₈×D₈₉	—	2	2	no	—
92	C₄⋊Dic₈₉	C₂×C₈	1	2	yes	—
93	C₄⋊Dic₈₉	C₂×C₈	1	2	yes	—
94	C₄⋊Dic₈₉	C₂×C₈	1	1	yes	characteristic
95	#13083553	C₂×C₈	1	2	yes	—
96	#13083553	C₂×C₈	1	2	yes	—
97	C₂²⋊Dic₈₉	C₂×C₈	1	2	yes	—
98	C₂²⋊Dic₈₉	C₂×C₈	1	2	yes	—
99	C₂²⋊Dic₈₉	C₂×C₈	1	2	yes	—
100	#13083551	C₂×C₈	1	2	yes	—
101	C₄×Dic₈₉	C₂×C₈	1	1	yes	characteristic
102	C₄×Dic₈₉	C₂×C₈	1	1	yes	characteristic
103	C₄×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
104	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
105	#13083549	—	2	1	no	—
106	#13083549	—	2	2	no	—
107	#13083549	#41	1	2	yes	—
108	C₂×D₄×C₈₉	C₂×C₈	1	1	yes	characteristic
109	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
110	C₂×C₈×C₈₉	—	2	2	no	—
111	C₂×C₈×C₈₉	—	2	2	no	—
112	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
114	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
115	C₄⋊C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
116	C₂²⋊C₄×C₈₉	C₂×C₈	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	C₂³×C₈₉	C₂²×C₈	1	2	yes	—
122	C₂²×D₈₉	C₂²×C₈	1	1	yes	characteristic
123	D₄×C₈₉	—	2	2	no	—
124	C₂×C₄×C₈₉	—	2	2	no	—
125	C₂×C₄×C₈₉	—	2	2	no	—
126	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
128	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
129	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
130	C₂²⋊D₈₉	—	4	2	no	—
131	C₂×Dic₈₉	C₂²×C₈	1	1	yes	characteristic
132	C₂×Dic₈₉	C₂²×C₈	1	1	yes	characteristic
133	C₂×Dic₈₉	C₂²×C₈	1	1	yes	characteristic
134	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
135	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
136	C₄×D₈₉	—	2	1	no	—
137	C₄×D₈₉	—	2	1	no	—
138	C₄×D₈₉	—	2	2	no	—
139	C₄×D₈₉	—	4	1	no	—
140	Dic₁₇₈	—	2	2	no	—
141	C₈×C₈₉	—	2	2	no	—
142	C₈×C₈₉	—	2	1	no	—
143	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
144	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
145	#10623494	—	2	1	no	—
146	#10623494	—	2	2	no	—
147	#10623494	#131	1	2	yes	—
148	C₂²×C₈₉	—	2	2	no	—
149	C₂²×C₈₉	—	2	2	no	—
150	C₂²×C₈₉	—	2	2	no	—
151	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
152	C₂×D₈₉	—	2	1	no	—
153	C₂×D₈₉	—	2	1	no	—
154	C₂×D₈₉	—	2	1	no	—
155	C₄×C₈₉	—	2	1	no	—
156	C₄×C₈₉	—	2	2	no	—
157	C₄×C₈₉	#504	1	2	yes	—
158	C₄×C₈₉	C₄×#41	1	1	yes	characteristic
159	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
160	Dic₈₉	—	2	2	no	—
161	Dic₈₉	—	2	1	no	—
162	Dic₈₉	—	2	1	no	—
163	Dic₈₉	—	2	2	no	—
164	Dic₈₉	—	2	1	no	—
165	#33483	—	89	1	no	maximal, Sylow
166	C₂×C₈₉	—	4	2	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	1	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	D₈₉	1	1	yes	characteristic, complemented
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	2	no	—
185	C₂×M₃₂	—	89	1	no	—
186	C₂×M₃₂	—	89	1	no	—
187	C₂²×C₁₆	C₂×D₈₉	1	2	yes	—
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	C₂×D₈₉	1	1	yes	characteristic
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	2	no	—
199	C₄⋊C₁₆	—	89	2	no	—
200	C₄⋊C₁₆	—	89	1	no	—
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	1	no	—
206	#346	—	89	1	no	—
207	#346	—	89	2	no	—
208	C₄×C₁₆	—	89	2	no	—
209	C₄×C₁₆	—	89	2	no	—
210	C₄×C₁₆	—	89	1	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	C₂²×D₈₉	1	2	yes	—
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	C₄×D₈₉	1	1	yes	characteristic
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	—	2	4	no	—
231	C₂×C₁₆	—	89	1	no	—
232	C₂×C₁₆	—	89	2	no	—
233	C₂×C₁₆	—	89	1	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	2	no	—
238	C₄⋊C₈	—	89	2	no	—
239	C₄⋊C₈	—	89	1	no	—
240	C₄⋊C₈	C₄×D₈₉	1	1	yes	characteristic
241	C₂²⋊C₈	—	89	2	no	—
242	C₂²⋊C₈	—	89	2	no	—
243	C₂²⋊C₈	C₄×D₈₉	1	2	yes	—
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	2	no	—
249	C₄×C₈	—	89	1	no	—
250	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
251	C₂×Q₈	—	89	1	no	—
252	C₂×D₄	—	89	2	no	—
253	C₂×D₄	C₈×D₈₉	1	1	yes	characteristic
254	C₂²×C₄	—	89	1	no	—
255	C₂²×C₄	—	89	1	no	—
256	C₂²×C₄	—	89	1	no	—
257	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
258	M₁₆	—	178	4	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	89	1	no	—
262	C₂×C₈	—	89	1	no	—
263	C₂×C₈	—	89	2	no	—
264	C₂×C₈	—	178	2	no	—
265	C₂×C₈	—	178	1	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	2	no	—
273	C₄⋊C₄	—	89	1	no	—
274	C₄⋊C₄	—	89	2	no	—
275	C₄⋊C₄	C₈×D₈₉	1	1	yes	characteristic
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	C₈×D₈₉	1	2	yes	—
281	C₄²	—	89	1	no	—
282	C₄²	—	89	1	no	—
283	C₄²	—	89	1	no	—
284	C₄²	—	89	2	no	—
285	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
286	C₁₆	—	2	1	no	—
287	C₁₆	—	2	2	no	—
288	C₁₆	—	89	2	no	—
289	C₁₆	—	178	2	no	—
290	C₁₆	—	178	1	no	—
291	C₁₆	#13083573	1	2	yes	—
292	C₂³	—	89	1	no	—
293	C₂³	C₂×C₈×D₈₉	1	2	yes	—
294	Q₈	—	178	2	no	—
295	D₄	—	2	2	no	—
296	D₄	—	356	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	89	1	no	—
300	C₂×C₄	—	89	1	no	—
301	C₂×C₄	—	89	1	no	—
302	C₂×C₄	—	89	2	no	—
303	C₂×C₄	—	89	2	no	—
304	C₂×C₄	—	356	1	no	—
305	C₂×C₄	—	178	2	no	—
306	C₂×C₄	—	178	1	no	—
307	C₂×C₄	—	178	1	no	—
308	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
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	2	no	—
313	C₈	—	2	1	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×#13083573	1	1	yes	characteristic
319	C₂²	—	2	2	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	178	1	no	—
323	C₂²	—	178	1	no	—
324	C₂²	—	178	1	no	—
325	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
326	C₄	—	2	2	no	—
327	C₄	—	2	1	no	—
328	C₄	—	178	1	no	—
329	C₄	—	178	1	no	—
330	C₄	—	178	2	no	—
331	C₄	—	178	1	no	—
332	C₄	—	178	2	no	—
333	C₄	#19152014	1	2	yes	—
334	C₄	C₄×#13083573	1	1	yes	characteristic
335	C₄	#131×D₈₉	1	1	yes	characteristic
336	C₂	—	4	2	no	—
337	C₂	—	356	1	no	—
338	C₂	C₈×#13083573	1	1	yes	characteristic
339	C₂	C₂×#19152014	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	D₄×C₁₆×C₈₉	C₂
prime radical	#15667983	C₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	D₄×C₁₆	D₈₉
hypocenter	C₈₉	#33483
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ D₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #26022160	C₂	semidirect
3. #26022154	C₂	semidirect
4. #26022084	C₂	semidirect
5. #26022081	C₂	semidirect
6. #26022077	C₂	semidirect
7. #26022056	C₂	semidirect
8. #26022050	C₂	semidirect
9. #26021848	C₂	semidirect
10. C₄×#15667951	C₂	semidirect
11. D₄×C₁₆×C₈₉	C₂	semidirect
177. D₄×C₁₆	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
#26029224	2
#26022160	2
#26022154	2
#26022084	2
#26022081	2
#26022077	2
#26022056	2
#26022050	2
#26021848	2
C₄×#15667951	2
D₄×C₁₆×C₈₉	2
#33483	89