Subgroups — #36523419

Order 22784.

734 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029360	C₂	1	2	yes	maximal
2	C₄×#15668018	C₂	1	1	yes	maximal, characteristic
3	#26019005	C₂	1	2	yes	maximal, complemented
4	#26019001	C₂	1	4	yes	maximal, complemented
5	#26018999	C₂	1	2	yes	maximal, complemented
6	C₂×#19151351	C₂	1	1	yes	maximal, characteristic, complemented
7	#335×D₈₉	C₂	1	2	yes	maximal, complemented
8	C₈₉×#1564	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#15668018	C₄	1	1	yes	characteristic
10	C₂×#15668018	C₄	1	1	yes	characteristic
11	C₄×#13083574	C₂²	1	1	yes	characteristic
12	C₈×Dic₁₇₈	—	2	2	no	—
13	C₈×D₃₅₆	—	2	2	no	—
14	#19151661	—	2	4	no	—
15	#19151650	C₂²	1	4	yes	—
16	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
17	C₄⋊C₈×D₈₉	C₂²	1	4	yes	—
18	#19151553	C₄	1	1	yes	characteristic
19	#19151553	C₄	1	1	yes	characteristic
20	#19151552	C₄	1	2	yes	—
21	#19151552	C₄	1	2	yes	—
22	C₄×#13083558	C₂²	1	1	yes	characteristic
23	C₄×#13083557	C₂²	1	2	yes	—
24	C₄×C₈×D₈₉	C₂²	1	2	yes	—
25	#19151351	C₂²	1	2	yes	—
26	#19151351	C₂²	1	2	yes	—
27	#19151350	C₂²	1	4	yes	—
28	#19151348	C₂²	1	4	yes	—
29	#431×C₈₉	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
31	#335×C₈₉	C₂²	1	1	yes	characteristic
32	#335×C₈₉	C₂²	1	1	yes	characteristic
33	#334×C₈₉	C₂²	1	2	yes	—
34	C₂×#13083574	C₂×C₄	1	1	yes	characteristic
35	#15668018	—	2	4	no	—
36	#15668018	C₈	1	8	yes	complemented
37	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
38	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
39	C₂×#13083557	C₂×C₄	1	2	yes	—
40	C₂×#13083557	C₂×C₄	1	2	yes	—
41	C₂²×#10623494	—	2	2	no	—
42	C₂×C₈×D₈₉	—	2	4	no	—
43	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
44	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
45	C₄×Dic₁₇₈	D₄	1	1	yes	characteristic
46	C₄×Dic₁₇₈	Q₈	1	1	yes	characteristic
47	C₄×D₃₅₆	D₄	1	1	yes	characteristic
48	C₄×D₃₅₆	Q₈	1	1	yes	characteristic
49	C₄×C₂²⋊D₈₉	D₄	1	2	yes	—
50	C₄×C₂²⋊D₈₉	Q₈	1	2	yes	—
51	#15667971	C₂×C₄	1	1	yes	characteristic
52	#15667969	C₂×C₄	1	2	yes	—
53	C₄²×D₈₉	C₂³	1	2	yes	—
54	C₄⋊#10623494	—	2	2	no	—
55	C₄⋊#10623494	C₂³	1	4	yes	—
56	#15667941	—	2	4	no	—
57	#15667924	—	2	4	no	—
58	#15667923	C₂×C₄	1	2	yes	—
59	#15667922	C₂×C₄	1	2	yes	—
60	#15667922	C₂×C₄	1	2	yes	—
61	C₄×#10623494	C₂³	1	2	yes	—
62	C₈×Dic₈₉	—	2	4	no	—
63	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
64	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
65	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
66	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
68	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
69	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
70	C₂²⋊C₈×C₈₉	—	2	2	no	—
71	C₄×C₈×C₈₉	—	2	2	no	—
72	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
73	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
74	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
76	#13083574	C₂×C₈	1	4	yes	—
77	#13083574	M₁₆	1	4	yes	—
78	C₂×Dic₁₇₈	C₄⋊C₄	1	1	yes	characteristic
79	C₂×D₃₅₆	C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₂²⋊D₈₉	C₄⋊C₄	1	2	yes	—
81	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
82	C₂×C₄×D₈₉	C₂×D₄	1	2	yes	—
83	C₂×C₄×D₈₉	C₂×Q₈	1	2	yes	—
84	#13083558	—	2	2	no	—
85	#13083558	C₂×C₈	1	4	yes	complemented
86	#13083557	—	2	4	no	—
87	#13083557	C₂×C₈	1	8	yes	complemented
88	C₂×#10623494	—	2	4	no	—
89	C₂×#10623494	—	8	2	no	—
90	C₂×#10623494	C₂²×C₄	1	2	yes	—
91	C₂×#10623494	C₂²×C₄	1	2	yes	—
92	C₈×D₈₉	—	2	4	no	—
93	C₈×D₈₉	—	8	4	no	—
94	C₈×D₈₉	C₂×C₈	1	8	yes	complemented
95	C₄⋊Dic₈₉	C₄⋊C₄	1	1	yes	characteristic
96	C₄⋊Dic₈₉	C₄⋊C₄	1	1	yes	characteristic
97	#13083553	C₄⋊C₄	1	2	yes	—
98	#13083553	C₄⋊C₄	1	2	yes	—
99	C₂²⋊Dic₈₉	C₄⋊C₄	1	2	yes	—
100	#13083551	C₄⋊C₄	1	2	yes	—
101	#13083551	C₄⋊C₄	1	2	yes	—
102	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
103	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
104	C₄×Dic₈₉	C₂×Q₈	1	2	yes	—
105	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
108	C₂×C₈×C₈₉	—	2	2	no	—
109	C₂×C₈×C₈₉	—	2	2	no	—
110	C₂×C₈×C₈₉	—	2	2	no	—
111	C₂×C₈×C₈₉	C₂×C₈	1	4	yes	complemented
112	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
116	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
117	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
119	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
120	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂²×D₈₉	C₂×C₄⋊C₄	1	2	yes	—
122	C₂×C₄×C₈₉	—	2	1	no	—
123	C₂×C₄×C₈₉	—	2	1	no	—
124	C₂×C₄×C₈₉	—	4	1	no	—
125	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
129	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
130	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
131	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
133	C₂²⋊D₈₉	C₄⋊C₈	1	8	yes	—
134	C₂×Dic₈₉	C₂×C₄⋊C₄	1	2	yes	—
135	C₂×Dic₈₉	C₂×C₄⋊C₄	1	2	yes	—
136	C₂×Dic₈₉	C₂×C₄⋊C₄	1	2	yes	—
137	D₃₅₆	C₄⋊C₈	1	4	yes	—
138	C₄×D₈₉	—	2	2	no	—
139	C₄×D₈₉	—	2	2	no	—
140	C₄×D₈₉	—	4	2	no	—
141	C₄×D₈₉	C₂²×C₈	1	4	yes	—
142	C₄×D₈₉	C₂×M₁₆	1	4	yes	—
143	Dic₁₇₈	C₄⋊C₈	1	4	yes	—
144	C₈×C₈₉	—	2	1	no	—
145	C₈×C₈₉	—	2	1	no	—
146	C₈×C₈₉	—	4	2	no	—
147	C₈×C₈₉	—	4	2	no	—
148	C₈×C₈₉	C₂²×C₈	1	2	yes	—
149	C₈×C₈₉	C₂²×C₈	1	2	yes	—
150	#10623494	—	2	2	no	—
151	#10623494	—	4	4	no	—
152	#10623494	C₂²×C₈	1	4	yes	—
153	C₂²×C₈₉	—	2	1	no	—
154	C₂²×C₈₉	—	2	1	no	—
155	C₂²×C₈₉	C₂×C₄⋊C₈	1	2	yes	—
156	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
157	C₂×D₈₉	—	2	2	no	—
158	C₂×D₈₉	—	2	2	no	—
159	C₂×D₈₉	C₂×C₄⋊C₈	1	4	yes	—
160	C₄×C₈₉	—	2	1	no	—
161	C₄×C₈₉	—	2	1	no	—
162	C₄×C₈₉	—	2	1	no	—
163	C₄×C₈₉	C₂×C₄⋊C₈	1	2	yes	—
164	C₄×C₈₉	C₂³×C₈	1	1	yes	characteristic
165	C₄×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
166	C₄×C₈₉	#576	1	1	yes	characteristic
167	C₄×C₈₉	#578	1	1	yes	characteristic
168	Dic₈₉	—	2	2	no	—
169	Dic₈₉	—	2	2	no	—
170	Dic₈₉	C₂×C₄⋊C₈	1	4	yes	—
171	#17789	—	89	1	no	maximal, Sylow
172	C₂×C₈₉	—	2	2	no	—
173	C₂×C₈₉	C₂²×C₄⋊C₈	1	1	yes	characteristic
174	C₂×C₈₉	#2913	1	1	yes	characteristic
175	C₂×C₈₉	#2916	1	1	yes	characteristic
176	D₈₉	—	2	4	no	—
177	#2988	—	89	2	no	—
178	C₄×#131	—	89	1	no	—
179	#1565	—	89	2	no	—
180	#1565	—	89	4	no	—
181	#1564	—	89	2	no	—
182	#1564	D₈₉	1	1	yes	characteristic, complemented
183	C₂×#335	—	89	1	no	—
184	C₂×#335	—	89	2	no	—
185	C₈₉	#17789	1	1	yes	characteristic, complemented, Sylow
186	C₂×#131	—	89	1	no	—
187	C₂×#131	—	89	1	no	—
188	C₄×#41	—	89	1	no	—
189	C₈×Q₈	—	178	2	no	—
190	C₈×D₄	—	178	2	no	—
191	C₈×D₄	—	178	4	no	—
192	#431	—	89	4	no	—
193	#431	C₂×D₈₉	1	2	yes	—
194	C₂×C₄⋊C₈	—	89	2	no	—
195	C₂×C₄⋊C₈	—	89	4	no	—
196	#405	—	89	1	no	—
197	#405	—	89	1	no	—
198	#405	—	89	2	no	—
199	#405	—	89	2	no	—
200	C₄×M₁₆	—	89	1	no	—
201	C₄×M₁₆	—	89	2	no	—
202	C₂×C₄×C₈	—	89	2	no	—
203	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
204	#335	—	89	4	no	—
205	#335	—	89	2	no	—
206	#335	—	89	2	no	—
207	#335	C₂×D₈₉	1	1	yes	characteristic
208	#335	C₂×D₈₉	1	1	yes	characteristic
209	#334	—	89	4	no	—
210	#334	C₂×D₈₉	1	2	yes	—
211	C₂×#41	—	89	1	no	—
212	#131	—	89	8	no	—
213	#131	—	178	4	no	—
214	C₂×M₁₆	—	89	1	no	—
215	C₂×M₁₆	—	89	1	no	—
216	C₂×M₁₆	—	89	2	no	—
217	C₂×M₁₆	—	89	2	no	—
218	C₂²×C₈	—	89	2	no	—
219	C₂²×C₈	—	89	2	no	—
220	C₂²×C₈	—	178	4	no	—
221	C₂²×C₈	—	178	2	no	—
222	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
223	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
224	C₄×Q₈	—	89	1	no	—
225	C₄×Q₈	—	89	1	no	—
226	C₄×D₄	—	89	2	no	—
227	C₄×D₄	—	89	2	no	—
228	C₄×D₄	—	89	1	no	—
229	C₄×D₄	—	89	1	no	—
230	#117	—	89	1	no	—
231	#117	—	89	2	no	—
232	C₂×C₄²	—	89	2	no	—
233	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
234	C₄⋊C₈	—	89	4	no	—
235	C₄⋊C₈	—	178	4	no	—
236	C₄⋊C₈	—	178	2	no	—
237	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
238	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
239	C₂²⋊C₈	—	2	2	no	—
240	C₂²⋊C₈	—	178	4	no	—
241	#97	—	89	2	no	—
242	#97	—	89	2	no	—
243	#97	—	89	2	no	—
244	C₄×C₈	—	2	2	no	—
245	C₄×C₈	—	89	2	no	—
246	C₄×C₈	—	89	2	no	—
247	C₄×C₈	—	89	2	no	—
248	C₄×C₈	—	178	4	no	—
249	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
250	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
251	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
252	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
253	#41	—	89	4	no	—
254	#41	—	89	4	no	—
255	C₂×Q₈	—	89	1	no	—
256	C₂×D₄	—	89	2	no	—
257	C₂×D₄	—	89	1	no	—
258	C₂²×C₄	—	89	2	no	—
259	C₂²×C₄	—	89	2	no	—
260	C₂²×C₄	—	89	2	no	—
261	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
262	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
263	C₂²×C₄	Q₈×D₈₉	1	1	yes	characteristic
264	M₁₆	—	89	4	no	—
265	M₁₆	—	89	8	no	—
266	M₁₆	—	178	4	no	—
267	M₁₆	—	178	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	89	2	no	—
272	C₂×C₈	—	89	8	no	—
273	C₂×C₈	—	89	2	no	—
274	C₂×C₈	—	178	4	no	—
275	C₂×C₈	—	178	4	no	—
276	C₂×C₈	—	712	4	no	—
277	C₂×C₈	—	712	2	no	—
278	C₂×C₈	C₈×D₈₉	1	4	yes	complemented
279	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
280	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
281	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
282	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
283	C₄⋊C₄	—	89	2	no	—
284	C₄⋊C₄	—	89	2	no	—
285	C₄⋊C₄	—	89	1	no	—
286	C₄⋊C₄	—	89	1	no	—
287	C₂²⋊C₄	—	89	2	no	—
288	C₂²⋊C₄	—	89	2	no	—
289	C₂²⋊C₄	—	89	2	no	—
290	C₄²	—	89	2	no	—
291	C₄²	—	89	2	no	—
292	C₄²	—	89	2	no	—
293	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
294	C₄²	D₄×D₈₉	1	1	yes	characteristic
295	C₄²	Q₈×D₈₉	1	1	yes	characteristic
296	C₄²	C₂³×D₈₉	1	1	yes	characteristic
297	C₂³	—	89	2	no	—
298	C₂³	C₄⋊C₄×D₈₉	1	1	yes	characteristic
299	Q₈	—	89	4	no	—
300	D₄	—	89	8	no	—
301	D₄	—	89	4	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	89	2	no	—
306	C₂×C₄	—	89	4	no	—
307	C₂×C₄	—	89	4	no	—
308	C₂×C₄	—	89	2	no	—
309	C₂×C₄	—	89	2	no	—
310	C₂×C₄	—	356	2	no	—
311	C₂×C₄	—	178	2	no	—
312	C₂×C₄	—	178	2	no	—
313	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
314	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
315	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
316	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
317	C₂×C₄	M₁₆×D₈₉	1	2	yes	—
318	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
319	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
320	C₂×C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
321	C₈	—	2	1	no	—
322	C₈	—	2	1	no	—
323	C₈	—	4	2	no	—
324	C₈	—	4	2	no	—
325	C₈	—	89	4	no	—
326	C₈	—	356	4	no	—
327	C₈	—	178	2	no	—
328	C₈	C₂×C₈×D₈₉	1	2	yes	—
329	C₈	C₂×C₈×D₈₉	1	2	yes	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	89	4	no	—
333	C₂²	—	178	2	no	—
334	C₂²	—	178	2	no	—
335	C₂²	C₄⋊C₈×D₈₉	1	2	yes	—
336	C₂²	C₂×C₄⋊C₄×D₈₉	1	1	yes	characteristic
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	89	4	no	—
341	C₄	—	178	2	no	—
342	C₄	—	178	2	no	—
343	C₄	C₄⋊C₈×D₈₉	1	2	yes	—
344	C₄	C₂²×C₈×D₈₉	1	1	yes	characteristic
345	C₄	C₂×M₁₆×D₈₉	1	1	yes	characteristic
346	C₄	#19152400	1	1	yes	characteristic
347	C₄	#19152412	1	1	yes	characteristic
348	C₂	—	2	2	no	—
349	C₂	—	178	4	no	—
350	C₂	C₂×C₄⋊C₈×D₈₉	1	1	yes	characteristic
351	C₂	#26028849	1	1	yes	characteristic
352	C₂	#26028860	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₈₉
commutator	C₄×C₈₉	C₂³×C₈
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1564	C₂
socle	C₂²×C₈₉	C₂²×C₄⋊C₄
prime radical	C₂×#13083574	C₂×C₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	#1564	D₈₉
hypocenter	C₈₉	#17789
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
3. #26019005	C₂	semidirect
4. #26019001	C₂	semidirect
5. #26018999	C₂	semidirect
6. C₂×#19151351	C₂	semidirect
7. #335×D₈₉	C₂	semidirect
8. C₈₉×#1564	C₂	semidirect
36. #15668018	C₈	semidirect
85. #13083558	C₂×C₈	semidirect
87. #13083557	C₂×C₈	semidirect
94. C₈×D₈₉	C₂×C₈	semidirect
111. C₂×C₈×C₈₉	C₂×C₈	semidirect
182. #1564	D₈₉	semidirect
185. C₈₉	#17789	semidirect
278. C₂×C₈	C₈×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26029360	2
C₄×#15668018	2
#26019005	2
#26019001	2
#26018999	2
C₂×#19151351	2
#335×D₈₉	2
C₈₉×#1564	2
#17789	89