Subgroups — #36708149

Order 22784.

502 conjugacy classes of subgroups.

The subgroup lattice has 340 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	#26022157	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₆×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022075	C₂	1	2	yes	maximal, complemented
5	D₄×#13083549	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022069	C₂	1	2	yes	maximal, complemented
7	#26022056	C₂	1	2	yes	maximal, complemented
8	#26022049	C₂	1	1	yes	maximal, characteristic, complemented
9	#26022012	C₂	1	2	yes	maximal, complemented
10	C₄⋊C₁₆×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#2169	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152130	C₄	1	1	yes	characteristic
13	C₂²×#13083549	C₂²	1	2	yes	—
14	C₂×C₁₆×D₈₉	C₂²	1	2	yes	—
15	#19151725	C₄	1	1	yes	characteristic
16	C₈×Dic₁₇₈	C₂²	1	1	yes	characteristic
17	#19151676	C₄	1	1	yes	characteristic
18	#19151672	C₄	1	2	yes	—
19	C₈×C₂²⋊D₈₉	C₂²	1	2	yes	—
20	#19151656	C₄	1	2	yes	—
21	#19151654	C₄	1	1	yes	characteristic
22	#19151553	C₂²	1	2	yes	—
23	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
24	C₄⋊#13083549	C₂²	1	1	yes	characteristic
25	#19151416	C₂²	1	2	yes	—
26	#19151416	C₂²	1	2	yes	—
27	#19151416	C₂²	1	2	yes	—
28	#19151416	C₂²	1	1	yes	characteristic
29	#19151416	C₂²	1	1	yes	characteristic
30	C₂²⋊#13083549	C₂²	1	2	yes	—
31	#19151377	C₂²	1	1	yes	characteristic
32	#19151377	C₂²	1	1	yes	characteristic
33	#19151373	C₂²	1	2	yes	—
34	C₄×#13083549	C₂²	1	1	yes	characteristic
35	C₁₆×Dic₈₉	C₂²	1	2	yes	—
36	C₂×M₃₂×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	C₂×#13083549	—	4	2	no	—
55	C₂×#13083549	C₂³	1	2	yes	—
56	C₂×#13083549	C₂³	1	1	yes	characteristic
57	C₂×#13083549	C₂³	1	1	yes	characteristic
58	C₁₆×D₈₉	—	4	2	no	—
59	C₄⋊#10623494	C₂×C₄	1	1	yes	characteristic
60	#15667941	C₂×C₄	1	2	yes	—
61	#15667941	C₂×C₄	1	2	yes	—
62	C₂²⋊#10623494	C₂×C₄	1	2	yes	—
63	#15667924	C₂×C₄	1	2	yes	—
64	#15667922	C₂×C₄	1	2	yes	—
65	#15667922	C₂×C₄	1	2	yes	—
66	C₄×#10623494	C₂×C₄	1	1	yes	characteristic
67	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
68	C₈×Dic₈₉	C₂³	1	2	yes	—
69	C₈×Dic₈₉	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
71	C₄×D₄×C₈₉	C₂×C₄	1	1	yes	characteristic
72	M₃₂×C₈₉	—	4	2	no	—
73	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
75	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
76	C₄⋊C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
77	C₂²⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
78	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
79	C₂×Dic₁₇₈	C₂×C₈	1	1	yes	characteristic
80	C₂×C₂²⋊D₈₉	C₂×C₈	1	2	yes	—
81	C₂×C₄×D₈₉	C₂×C₈	1	1	yes	characteristic
82	C₂×C₄×D₈₉	C₂×C₈	1	1	yes	characteristic
83	C₂×C₄×D₈₉	C₂²×C₄	1	1	yes	characteristic
84	#13083558	—	4	2	no	—
85	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
86	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
87	C₂×#10623494	C₂²×C₄	1	2	yes	—
88	C₈×D₈₉	—	2	1	no	—
89	C₈×D₈₉	—	2	1	no	—
90	C₈×D₈₉	—	4	1	no	—
91	C₄⋊Dic₈₉	C₂×C₈	1	2	yes	—
92	C₄⋊Dic₈₉	C₂×C₈	1	2	yes	—
93	C₄⋊Dic₈₉	C₂×C₈	1	1	yes	characteristic
94	#13083553	C₂×C₈	1	2	yes	—
95	#13083553	C₂×C₈	1	2	yes	—
96	C₂²⋊Dic₈₉	C₂×C₈	1	2	yes	—
97	C₂²⋊Dic₈₉	C₂×C₈	1	2	yes	—
98	C₂²⋊Dic₈₉	C₂×C₈	1	2	yes	—
99	#13083551	C₂×C₈	1	2	yes	—
100	C₄×Dic₈₉	C₂×C₈	1	1	yes	characteristic
101	C₄×Dic₈₉	C₂×C₈	1	1	yes	characteristic
102	C₄×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
103	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
104	#13083549	—	2	2	no	—
105	#13083549	—	2	1	no	—
106	#13083549	—	2	1	no	—
107	C₂×D₄×C₈₉	C₂×C₈	1	1	yes	characteristic
108	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
109	C₂×C₈×C₈₉	—	2	2	no	—
110	C₂×C₈×C₈₉	—	2	2	no	—
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	C₂²⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
116	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
117	C₁₆×C₈₉	—	2	1	no	—
118	C₁₆×C₈₉	—	2	2	no	—
119	C₁₆×C₈₉	—	2	1	no	—
120	C₂³×C₈₉	C₂²×C₈	1	2	yes	—
121	C₂²×D₈₉	C₂²×C₈	1	1	yes	characteristic
122	D₄×C₈₉	—	2	2	no	—
123	C₂×C₄×C₈₉	—	2	2	no	—
124	C₂×C₄×C₈₉	—	2	2	no	—
125	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
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	C₂²⋊D₈₉	—	2	4	no	—
130	C₂×Dic₈₉	C₂²×C₈	1	1	yes	characteristic
131	C₂×Dic₈₉	C₂²×C₈	1	1	yes	characteristic
132	C₂×Dic₈₉	C₂²×C₈	1	1	yes	characteristic
133	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
134	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
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	Dic₁₇₈	—	2	2	no	—
140	C₈×C₈₉	—	2	2	no	—
141	C₈×C₈₉	—	2	1	no	—
142	C₈×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
143	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
144	#10623494	—	2	1	no	—
145	#10623494	—	2	1	no	—
146	#10623494	—	2	2	no	—
147	C₂²×C₈₉	—	2	2	no	—
148	C₂²×C₈₉	—	2	2	no	—
149	C₂²×C₈₉	—	2	2	no	—
150	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
151	C₂×D₈₉	—	2	1	no	—
152	C₂×D₈₉	—	2	1	no	—
153	C₂×D₈₉	#504	1	2	yes	—
154	C₄×C₈₉	—	2	1	no	—
155	C₄×C₈₉	—	2	2	no	—
156	C₄×C₈₉	#504	1	2	yes	—
157	C₄×C₈₉	#520	1	1	yes	characteristic
158	C₄×C₈₉	#568	1	1	yes	characteristic
159	Dic₈₉	—	2	2	no	—
160	Dic₈₉	—	2	1	no	—
161	Dic₈₉	—	2	1	no	—
162	Dic₈₉	—	2	2	no	—
163	Dic₈₉	—	2	1	no	—
164	#33508	—	89	1	no	maximal, Sylow
165	C₂×C₈₉	—	4	2	no	—
166	C₂×C₈₉	#2988	1	1	yes	characteristic
167	C₂×C₈₉	C₂×#504	1	1	yes	characteristic
168	C₂×C₈₉	C₂×#504	1	1	yes	characteristic
169	D₈₉	—	2	2	no	—
170	#2966	—	89	1	no	—
171	#2183	—	89	1	no	—
172	Q₈×C₁₆	—	89	1	no	—
173	#2169	D₈₉	1	1	yes	characteristic, complemented
174	#2168	—	89	2	no	—
175	D₄×C₁₆	—	89	1	no	—
176	D₄×C₁₆	—	89	2	no	—
177	#2164	—	89	2	no	—
178	#2162	—	89	1	no	—
179	#2151	—	89	2	no	—
180	C₂×C₄⋊C₁₆	—	89	1	no	—
181	C₈₉	#33508	1	1	yes	characteristic, complemented, Sylow
182	#529	—	89	1	no	—
183	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
184	C₂²×C₁₆	—	89	2	no	—
185	C₂²×C₁₆	—	89	2	no	—
186	#446	—	89	1	no	—
187	C₈×Q₈	—	89	1	no	—
188	#436	—	89	1	no	—
189	#435	—	89	2	no	—
190	C₈×D₄	—	89	2	no	—
191	C₈×D₄	C₂×D₈₉	1	1	yes	characteristic
192	#433	—	89	2	no	—
193	#432	—	89	1	no	—
194	#405	—	89	2	no	—
195	C₂×C₄×C₈	—	89	1	no	—
196	C₄⋊C₁₆	—	89	2	no	—
197	C₄⋊C₁₆	—	89	1	no	—
198	C₄⋊C₁₆	—	89	2	no	—
199	C₄⋊C₁₆	—	89	2	no	—
200	C₄⋊C₁₆	—	89	1	no	—
201	C₄⋊C₁₆	—	89	1	no	—
202	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
203	C₂²⋊C₁₆	—	89	2	no	—
204	C₂²⋊C₁₆	—	89	1	no	—
205	C₂²⋊C₁₆	—	89	1	no	—
206	C₂²⋊C₁₆	C₂×D₈₉	1	2	yes	—
207	#346	—	89	2	no	—
208	C₄×C₁₆	—	89	2	no	—
209	C₄×C₁₆	—	89	1	no	—
210	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
211	C₂×M₁₆	—	89	2	no	—
212	C₂²×C₈	—	89	1	no	—
213	C₂²×C₈	—	89	1	no	—
214	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
215	#126	—	89	2	no	—
216	#125	—	89	1	no	—
217	#124	—	89	1	no	—
218	#123	—	89	2	no	—
219	C₂²⋊Q₈	—	89	1	no	—
220	#121	—	89	1	no	—
221	C₄×Q₈	—	89	1	no	—
222	C₄×D₄	—	89	2	no	—
223	C₄×D₄	C₄×D₈₉	1	1	yes	characteristic
224	#117	—	89	2	no	—
225	C₂×C₄²	—	89	1	no	—
226	M₃₂	—	4	2	no	—
227	C₂×C₁₆	—	89	1	no	—
228	C₂×C₁₆	—	89	2	no	—
229	C₂×C₁₆	—	89	1	no	—
230	C₂×C₁₆	—	356	2	no	—
231	C₂×C₁₆	—	356	2	no	—
232	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
233	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
234	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
235	C₄⋊C₈	—	89	2	no	—
236	C₄⋊C₈	—	89	2	no	—
237	C₄⋊C₈	—	89	1	no	—
238	C₄⋊C₈	C₄×D₈₉	1	1	yes	characteristic
239	C₂²⋊C₈	—	89	2	no	—
240	C₂²⋊C₈	—	89	2	no	—
241	C₂²⋊C₈	C₄×D₈₉	1	2	yes	—
242	#97	—	89	2	no	—
243	#97	—	89	2	no	—
244	C₄×C₈	—	89	1	no	—
245	C₄×C₈	—	89	2	no	—
246	C₄×C₈	—	89	1	no	—
247	C₄×C₈	—	89	1	no	—
248	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
249	C₂×Q₈	—	89	1	no	—
250	C₂×D₄	—	89	2	no	—
251	C₂×D₄	C₈×D₈₉	1	1	yes	characteristic
252	C₂²×C₄	—	89	1	no	—
253	C₂²×C₄	—	89	1	no	—
254	C₂²×C₄	—	89	1	no	—
255	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
256	M₁₆	—	356	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	89	1	no	—
260	C₂×C₈	—	89	1	no	—
261	C₂×C₈	—	89	2	no	—
262	C₂×C₈	—	356	1	no	—
263	C₂×C₈	—	178	1	no	—
264	C₂×C₈	—	178	1	no	—
265	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
266	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
267	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
268	C₄⋊C₄	—	89	2	no	—
269	C₄⋊C₄	—	89	2	no	—
270	C₄⋊C₄	—	89	2	no	—
271	C₄⋊C₄	—	89	1	no	—
272	C₄⋊C₄	—	89	2	no	—
273	C₄⋊C₄	C₈×D₈₉	1	1	yes	characteristic
274	C₂²⋊C₄	—	89	2	no	—
275	C₂²⋊C₄	—	89	2	no	—
276	C₂²⋊C₄	—	89	2	no	—
277	C₂²⋊C₄	—	89	2	no	—
278	C₂²⋊C₄	C₈×D₈₉	1	2	yes	—
279	C₄²	—	89	1	no	—
280	C₄²	—	89	1	no	—
281	C₄²	—	89	1	no	—
282	C₄²	—	89	2	no	—
283	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
284	C₁₆	—	2	2	no	—
285	C₁₆	—	2	1	no	—
286	C₁₆	—	2	1	no	—
287	C₁₆	—	178	2	no	—
288	C₁₆	—	178	1	no	—
289	C₁₆	—	178	1	no	—
290	C₂³	—	89	1	no	—
291	C₂³	C₂×C₈×D₈₉	1	2	yes	—
292	Q₈	—	178	2	no	—
293	D₄	—	2	2	no	—
294	D₄	—	178	4	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	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	—	89	2	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	C₂×C₈×D₈₉	1	1	yes	characteristic
307	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
308	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
309	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
310	C₈	—	2	2	no	—
311	C₈	—	2	1	no	—
312	C₈	—	178	1	no	—
313	C₈	—	178	1	no	—
314	C₈	—	178	2	no	—
315	C₈	#15668057	1	1	yes	characteristic
316	C₈	#15668059	1	1	yes	characteristic
317	C₂²	—	2	2	no	—
318	C₂²	—	2	2	no	—
319	C₂²	—	2	2	no	—
320	C₂²	—	89	2	no	—
321	C₂²	—	178	1	no	—
322	C₂²	—	178	1	no	—
323	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
324	C₄	—	2	2	no	—
325	C₄	—	2	1	no	—
326	C₄	—	178	1	no	—
327	C₄	—	178	1	no	—
328	C₄	—	178	2	no	—
329	C₄	—	178	1	no	—
330	C₄	—	178	2	no	—
331	C₄	#19152016	1	2	yes	—
332	C₄	#19152077	1	1	yes	characteristic
333	C₄	#19152363	1	1	yes	characteristic
334	C₂	—	4	2	no	—
335	C₂	—	178	2	no	—
336	C₂	#26029359	1	1	yes	characteristic
337	C₂	#504×D₈₉	1	1	yes	characteristic
338	C₂	C₂×#19152016	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₈
Fitting	C₈₉×#2169	C₂
prime radical	#15667983	C₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2169	D₈₉
hypocenter	C₈₉	#33508
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ #2169

Complements

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

N	H ≅ G/N	type
2. #26022157	C₂	semidirect
3. C₁₆×Dic₁₇₈	C₂	semidirect
4. #26022075	C₂	semidirect
5. D₄×#13083549	C₂	semidirect
6. #26022069	C₂	semidirect
7. #26022056	C₂	semidirect
8. #26022049	C₂	semidirect
9. #26022012	C₂	semidirect
10. C₄⋊C₁₆×D₈₉	C₂	semidirect
11. C₈₉×#2169	C₂	semidirect
173. #2169	D₈₉	semidirect
181. C₈₉	#33508	semidirect

Maximal subgroups

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

subgroup	index
#26029224	2
#26022157	2
C₁₆×Dic₁₇₈	2
#26022075	2
D₄×#13083549	2
#26022069	2
#26022056	2
#26022049	2
#26022012	2
C₄⋊C₁₆×D₈₉	2
C₈₉×#2169	2
#33508	89