Subgroups — #36523661

Order 22784.

610 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#13083574	C₂	1	1	yes	maximal, characteristic
2	#26028920	C₂	1	2	yes	maximal
3	#26019029	C₂	1	4	yes	maximal, complemented
4	#26019028	C₂	1	2	yes	maximal, complemented
5	#26019012	C₂	1	2	yes	maximal, complemented
6	#26018997	C₂	1	2	yes	maximal, complemented
7	C₂×#19151349	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈₉×#1566	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#13083574	C₂²	1	1	yes	characteristic
10	C₈×Dic₁₇₈	C₄	1	1	yes	characteristic
11	C₈×Dic₁₇₈	C₄	1	1	yes	characteristic
12	C₈×D₃₅₆	C₄	1	1	yes	characteristic
13	C₈×D₃₅₆	C₄	1	1	yes	characteristic
14	C₈×C₂²⋊D₈₉	C₄	1	2	yes	—
15	C₈×C₂²⋊D₈₉	C₄	1	2	yes	—
16	#19151658	—	2	4	no	—
17	#19151654	C₂²	1	4	yes	—
18	#19151651	C₂²	1	1	yes	characteristic
19	#19151650	C₂²	1	4	yes	—
20	#19151649	C₂²	1	2	yes	—
21	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
22	#19151566	—	2	2	no	—
23	#19151556	—	2	2	no	—
24	C₄×C₈×D₈₉	C₂²	1	2	yes	—
25	#19151350	C₂²	1	4	yes	—
26	#19151350	C₂²	1	4	yes	—
27	#19151349	C₂²	1	1	yes	characteristic
28	#19151349	C₂²	1	1	yes	characteristic
29	#19151349	C₂²	1	2	yes	—
30	#423×C₈₉	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
32	#334×C₈₉	C₂²	1	2	yes	—
33	#334×C₈₉	C₂²	1	2	yes	—
34	C₂×#13083574	D₄	1	1	yes	characteristic
35	C₂²×#10623494	—	2	2	no	—
36	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
37	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
38	C₄×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
39	C₄×Dic₁₇₈	Q₈	1	1	yes	characteristic
40	C₄×D₃₅₆	C₂×C₄	1	1	yes	characteristic
41	C₄×D₃₅₆	Q₈	1	1	yes	characteristic
42	C₄×C₂²⋊D₈₉	C₂×C₄	1	2	yes	—
43	C₄×C₂²⋊D₈₉	Q₈	1	2	yes	—
44	#15667971	D₄	1	1	yes	characteristic
45	#15667969	D₄	1	2	yes	—
46	C₄²×D₈₉	C₂³	1	2	yes	—
47	C₄⋊#10623494	—	2	2	no	—
48	C₄⋊#10623494	C₂×C₄	1	1	yes	characteristic
49	C₄⋊#10623494	C₂×C₄	1	1	yes	characteristic
50	C₄⋊#10623494	C₂³	1	2	yes	—
51	C₄⋊#10623494	C₂³	1	2	yes	—
52	#15667941	C₂×C₄	1	2	yes	—
53	#15667941	C₂×C₄	1	2	yes	—
54	C₂²⋊#10623494	C₂×C₄	1	2	yes	—
55	#15667924	—	2	4	no	—
56	#15667924	—	2	4	no	—
57	#15667924	C₂×C₄	1	2	yes	—
58	#15667924	C₂×C₄	1	2	yes	—
59	#15667922	—	2	4	no	—
60	C₄×#10623494	C₂³	1	2	yes	—
61	C₈×Dic₈₉	—	2	4	no	—
62	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
63	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
64	C₂×M₁₆×C₈₉	—	2	2	no	—
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	—	2	2	no	—
69	C₄⋊C₈×C₈₉	C₂³	1	4	yes	—
70	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
74	#13083574	—	2	4	no	—
75	C₂×Dic₁₇₈	C₄⋊C₄	1	1	yes	characteristic
76	C₂×D₃₅₆	C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₂²⋊D₈₉	C₄⋊C₄	1	2	yes	—
78	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
79	C₂×C₄×D₈₉	C₂×D₄	1	2	yes	—
80	C₂×C₄×D₈₉	C₂×Q₈	1	2	yes	—
81	C₂×#10623494	—	2	2	no	—
82	C₂×#10623494	—	2	2	no	—
83	C₂×#10623494	—	4	4	no	—
84	C₂×#10623494	C₂²×C₄	1	2	yes	—
85	C₂×#10623494	C₂²×C₄	1	2	yes	—
86	C₈×D₈₉	—	2	4	no	—
87	C₈×D₈₉	—	4	2	no	—
88	C₄⋊Dic₈₉	C₄⋊C₄	1	1	yes	characteristic
89	C₄⋊Dic₈₉	C₄⋊C₄	1	1	yes	characteristic
90	#13083553	C₄⋊C₄	1	2	yes	—
91	#13083553	C₄⋊C₄	1	2	yes	—
92	C₂²⋊Dic₈₉	C₄⋊C₄	1	2	yes	—
93	#13083551	C₄⋊C₄	1	2	yes	—
94	#13083551	C₄⋊C₄	1	2	yes	—
95	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
96	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
97	C₄×Dic₈₉	C₂×Q₈	1	2	yes	—
98	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
101	M₁₆×C₈₉	—	4	4	no	—
102	C₂×C₈×C₈₉	—	2	2	no	—
103	C₂×C₈×C₈₉	—	2	4	no	—
104	C₂×C₈×C₈₉	—	4	1	no	—
105	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
112	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
113	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂²×D₈₉	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₈₉	—	2	1	no	—
116	C₂×C₄×C₈₉	—	2	1	no	—
117	C₂×C₄×C₈₉	—	4	1	no	—
118	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₈₉	#135	1	4	yes	—
122	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
125	C₂²⋊D₈₉	—	2	4	no	—
126	C₂×Dic₈₉	C₂×C₄⋊C₄	1	2	yes	—
127	C₂×Dic₈₉	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×Dic₈₉	C₂×C₄⋊C₄	1	2	yes	—
129	D₃₅₆	—	2	2	no	—
130	C₄×D₈₉	—	2	2	no	—
131	C₄×D₈₉	—	2	2	no	—
132	C₄×D₈₉	—	2	4	no	—
133	C₄×D₈₉	—	4	2	no	—
134	Dic₁₇₈	—	2	2	no	—
135	C₈×C₈₉	—	2	1	no	—
136	C₈×C₈₉	—	2	1	no	—
137	C₈×C₈₉	—	2	1	no	—
138	C₈×C₈₉	—	4	4	no	—
139	C₈×C₈₉	#131	1	2	yes	—
140	#10623494	—	2	2	no	—
141	#10623494	—	2	2	no	—
142	#10623494	—	4	2	no	—
143	#10623494	—	4	2	no	—
144	C₂²×C₈₉	—	2	1	no	—
145	C₂²×C₈₉	—	2	1	no	—
146	C₂²×C₈₉	#427	1	2	yes	—
147	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
148	C₂×D₈₉	—	2	2	no	—
149	C₂×D₈₉	—	2	2	no	—
150	C₂×D₈₉	—	2	2	no	—
151	C₄×C₈₉	—	2	1	no	—
152	C₄×C₈₉	—	2	1	no	—
153	C₄×C₈₉	—	2	1	no	—
154	C₄×C₈₉	#427	1	2	yes	—
155	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
156	C₄×C₈₉	#568	1	1	yes	characteristic
157	C₄×C₈₉	C₂×#135	1	2	yes	—
158	Dic₈₉	—	2	2	no	—
159	Dic₈₉	—	2	2	no	—
160	Dic₈₉	—	2	2	no	—
161	#17827	—	89	1	no	maximal, Sylow
162	C₂×C₈₉	—	2	2	no	—
163	C₂×C₈₉	#2906	1	1	yes	characteristic
164	C₂×C₈₉	C₂×#427	1	1	yes	characteristic
165	C₂×C₈₉	#2916	1	1	yes	characteristic
166	D₈₉	—	4	2	no	—
167	C₈×#41	—	89	1	no	—
168	#2925	—	89	2	no	—
169	#1572	—	89	4	no	—
170	#1571	—	89	2	no	—
171	#1568	—	89	2	no	—
172	#1566	D₈₉	1	1	yes	characteristic, complemented
173	#1564	—	89	2	no	—
174	C₂×#334	—	89	1	no	—
175	C₈₉	#17827	1	1	yes	characteristic, complemented, Sylow
176	C₄×#41	—	89	1	no	—
177	C₈×Q₈	—	89	1	no	—
178	C₈×Q₈	—	89	1	no	—
179	C₈×D₄	—	89	2	no	—
180	C₈×D₄	—	89	2	no	—
181	C₈×D₄	—	89	1	no	—
182	C₈×D₄	—	89	1	no	—
183	#433	—	178	4	no	—
184	#432	—	89	4	no	—
185	#431	—	89	1	no	—
186	#431	—	89	2	no	—
187	#431	—	89	4	no	—
188	#423	C₂×D₈₉	1	2	yes	—
189	C₂×C₄⋊C₈	—	89	2	no	—
190	#408	—	178	2	no	—
191	#405	—	178	2	no	—
192	C₂×C₄×C₈	—	89	2	no	—
193	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
194	#335	—	89	4	no	—
195	#335	—	89	4	no	—
196	#334	—	89	1	no	—
197	#334	—	89	1	no	—
198	#334	—	89	2	no	—
199	#334	C₂×D₈₉	1	2	yes	—
200	#334	C₂×D₈₉	1	2	yes	—
201	C₂×#41	—	89	1	no	—
202	C₂×M₁₆	—	2	2	no	—
203	C₂²×C₈	—	89	2	no	—
204	C₂²×C₈	—	89	2	no	—
205	C₂²×C₈	—	178	2	no	—
206	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
207	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
208	C₄×Q₈	—	89	1	no	—
209	C₄×Q₈	—	89	1	no	—
210	C₄×D₄	—	89	2	no	—
211	C₄×D₄	—	89	2	no	—
212	C₄×D₄	—	89	1	no	—
213	C₄×D₄	—	89	1	no	—
214	#117	—	89	1	no	—
215	#117	—	89	2	no	—
216	C₂×C₄²	—	89	2	no	—
217	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
218	C₄⋊C₈	—	2	2	no	—
219	C₄⋊C₈	—	89	2	no	—
220	C₄⋊C₈	—	89	2	no	—
221	C₄⋊C₈	—	89	1	no	—
222	C₄⋊C₈	—	89	1	no	—
223	C₄⋊C₈	—	89	2	no	—
224	C₄⋊C₈	—	89	2	no	—
225	C₄⋊C₈	—	178	2	no	—
226	C₄⋊C₈	C₂²×D₈₉	1	4	yes	—
227	C₂²⋊C₈	—	89	2	no	—
228	C₂²⋊C₈	—	89	2	no	—
229	C₂²⋊C₈	—	89	2	no	—
230	C₂²⋊C₈	—	178	4	no	—
231	C₂²⋊C₈	—	178	4	no	—
232	#97	—	178	4	no	—
233	C₄×C₈	—	89	2	no	—
234	C₄×C₈	—	89	2	no	—
235	C₄×C₈	—	89	2	no	—
236	C₄×C₈	—	178	4	no	—
237	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
238	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
239	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
240	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
241	#41	—	178	4	no	—
242	C₂×Q₈	—	89	1	no	—
243	C₂×D₄	—	89	2	no	—
244	C₂×D₄	—	89	1	no	—
245	C₂²×C₄	—	89	2	no	—
246	C₂²×C₄	—	89	2	no	—
247	C₂²×C₄	—	89	2	no	—
248	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
249	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
250	C₂²×C₄	Q₈×D₈₉	1	1	yes	characteristic
251	M₁₆	—	4	4	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	4	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	—	89	2	no	—
256	C₂×C₈	—	89	2	no	—
257	C₂×C₈	—	356	2	no	—
258	C₂×C₈	—	356	4	no	—
259	C₂×C₈	—	178	2	no	—
260	C₂×C₈	—	178	2	no	—
261	C₂×C₈	—	178	4	no	—
262	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
263	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
264	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
265	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
266	C₄⋊C₄	—	89	2	no	—
267	C₄⋊C₄	—	89	2	no	—
268	C₄⋊C₄	—	89	1	no	—
269	C₄⋊C₄	—	89	1	no	—
270	C₂²⋊C₄	—	89	2	no	—
271	C₂²⋊C₄	—	89	2	no	—
272	C₂²⋊C₄	—	89	2	no	—
273	C₄²	—	89	2	no	—
274	C₄²	—	89	2	no	—
275	C₄²	—	89	2	no	—
276	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
277	C₄²	D₄×D₈₉	1	1	yes	characteristic
278	C₄²	Q₈×D₈₉	1	1	yes	characteristic
279	C₄²	C₂³×D₈₉	1	1	yes	characteristic
280	C₂³	—	89	2	no	—
281	C₂³	C₄⋊C₄×D₈₉	1	1	yes	characteristic
282	Q₈	—	178	2	no	—
283	D₄	—	178	4	no	—
284	D₄	—	178	2	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	89	2	no	—
289	C₂×C₄	—	89	2	no	—
290	C₂×C₄	—	89	2	no	—
291	C₂×C₄	—	356	2	no	—
292	C₂×C₄	—	178	2	no	—
293	C₂×C₄	—	178	2	no	—
294	C₂×C₄	—	178	4	no	—
295	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
296	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
297	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
298	C₂×C₄	#15668033	1	4	yes	—
299	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
300	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
301	C₂×C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
302	C₈	—	2	1	no	—
303	C₈	—	2	1	no	—
304	C₈	—	2	1	no	—
305	C₈	—	4	4	no	—
306	C₈	—	356	2	no	—
307	C₈	—	356	2	no	—
308	C₈	—	178	2	no	—
309	C₈	—	178	2	no	—
310	C₈	#15668020	1	2	yes	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	178	2	no	—
314	C₂²	—	178	2	no	—
315	C₂²	—	178	2	no	—
316	C₂²	#19151633	1	2	yes	—
317	C₂²	C₂×C₄⋊C₄×D₈₉	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	178	2	no	—
322	C₄	—	178	2	no	—
323	C₄	—	178	2	no	—
324	C₄	#19151633	1	2	yes	—
325	C₄	C₂×#15668020	1	1	yes	characteristic
326	C₄	#19152364	1	1	yes	characteristic
327	C₄	C₂×#15668033	1	2	yes	—
328	C₂	—	2	2	no	—
329	C₂	—	356	2	no	—
330	C₂	#26028821	1	1	yes	characteristic
331	C₂	C₂×#19151633	1	1	yes	characteristic
332	C₂	#26028860	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1566	C₂
socle	C₂²×C₈₉	C₂²×C₄⋊C₄
prime radical	C₂×#13083574	D₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	#1566	D₈₉
hypocenter	C₈₉	#17827
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

Complements

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

N	H ≅ G/N	type
3. #26019029	C₂	semidirect
4. #26019028	C₂	semidirect
5. #26019012	C₂	semidirect
6. #26018997	C₂	semidirect
7. C₂×#19151349	C₂	semidirect
8. C₈₉×#1566	C₂	semidirect
172. #1566	D₈₉	semidirect
175. C₈₉	#17827	semidirect

Maximal subgroups

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

subgroup	index
C₈×#13083574	2
#26028920	2
#26019029	2
#26019028	2
#26019012	2
#26018997	2
C₂×#19151349	2
C₈₉×#1566	2
#17827	89