Subgroups — #36460302

Order 22784.

810 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151622	C₂	1	1	yes	maximal, characteristic
2	C₄×#15667971	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151369	C₂	1	4	yes	maximal
4	C₂×#423×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#13083558	C₂²	1	2	yes	—
6	C₂×#15667971	C₂²	1	2	yes	—
7	#19151622	—	2	2	no	—
8	#19151622	C₄	1	2	yes	complemented
9	#19151622	C₄	1	2	yes	complemented
10	C₂×C₄⋊#10623494	—	2	2	no	—
11	C₄×C₄⋊Dic₈₉	—	2	2	no	—
12	C₄×C₂²⋊Dic₈₉	C₄	1	4	yes	—
13	#19151444	—	2	1	no	—
14	C₄²×Dic₈₉	—	2	1	no	—
15	#19151369	—	2	8	no	—
16	C₂²×M₁₆×C₈₉	C₂²	1	2	yes	—
17	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
18	#423×C₈₉	—	2	2	no	—
19	#423×C₈₉	—	2	1	no	—
20	#423×C₈₉	—	2	1	no	—
21	C₂×C₄⋊C₈×C₈₉	C₄	1	4	yes	complemented
22	C₂×#13083558	—	2	2	no	—
23	C₂×#13083558	—	2	4	no	—
24	C₂×#13083558	—	2	4	no	—
25	C₂×#13083558	C₂×C₄	1	4	yes	—
26	C₂²×#10623494	—	2	2	no	—
27	#15667971	—	2	2	no	—
28	#15667971	—	2	2	no	—
29	#15667971	D₄	1	4	yes	—
30	#15667971	Q₈	1	4	yes	—
31	C₂×C₄⋊Dic₈₉	—	2	2	no	—
32	C₂×C₂²⋊Dic₈₉	C₂×C₄	1	4	yes	—
33	C₂×C₄×Dic₈₉	—	2	4	no	—
34	C₂×C₄×Dic₈₉	—	2	2	no	—
35	C₄⋊#10623494	—	2	4	no	—
36	C₄⋊#10623494	—	2	4	no	—
37	#15667919	—	2	4	no	—
38	C₂³×C₄×C₈₉	C₂×C₄	1	2	yes	—
39	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
40	C₂×M₁₆×C₈₉	—	2	4	no	—
41	C₂×M₁₆×C₈₉	—	2	2	no	—
42	C₂×M₁₆×C₈₉	—	2	2	no	—
43	C₂×M₁₆×C₈₉	—	2	4	no	—
44	C₂²×C₈×C₈₉	C₂×C₄	1	4	yes	—
45	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₈₉	D₄	1	2	yes	—
49	C₂×C₄²×C₈₉	D₄	1	2	yes	—
50	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₈₉	Q₈	1	1	yes	characteristic
53	C₂×C₄²×C₈₉	Q₈	1	1	yes	characteristic
54	C₄⋊C₈×C₈₉	—	2	4	no	—
55	C₄⋊C₈×C₈₉	—	2	4	no	—
56	C₂²×Dic₈₉	—	2	4	no	—
57	#13083558	—	2	8	no	—
58	#13083558	—	4	4	no	—
59	C₂×#10623494	—	2	4	no	—
60	C₂×#10623494	—	4	2	no	—
61	C₂×#10623494	—	4	2	no	—
62	C₄⋊Dic₈₉	—	2	4	no	—
63	C₄⋊Dic₈₉	—	2	4	no	—
64	C₂²⋊Dic₈₉	—	2	4	no	—
65	C₂²⋊Dic₈₉	C₄⋊C₄	1	8	yes	—
66	C₄×Dic₈₉	—	2	4	no	—
67	C₄×Dic₈₉	—	2	4	no	—
68	C₄×Dic₈₉	—	4	8	no	—
69	C₂⁴×C₈₉	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₈₉	—	4	4	no	—
71	C₂²×C₄×C₈₉	C₄²	1	2	yes	—
72	C₂²×C₄×C₈₉	C₄²	1	2	yes	—
73	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
74	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
75	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
76	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
77	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
84	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
85	C₂²×C₄×C₈₉	#41	1	2	yes	—
86	C₂²×C₄×C₈₉	#41	1	2	yes	—
87	M₁₆×C₈₉	—	4	4	no	—
88	M₁₆×C₈₉	—	4	2	no	—
89	M₁₆×C₈₉	—	4	2	no	—
90	C₂×C₈×C₈₉	—	2	4	no	—
91	C₂×C₈×C₈₉	—	2	8	no	—
92	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
93	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
94	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
95	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
96	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
97	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
98	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
99	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
100	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
101	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
102	C₂³×C₈₉	—	4	2	no	—
103	C₂³×C₈₉	C₂×C₄²	1	1	yes	characteristic
104	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂³×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
106	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂³×C₈₉	#117	1	2	yes	—
108	C₂×C₄×C₈₉	—	2	4	no	—
109	C₂×C₄×C₈₉	—	2	4	no	—
110	C₂×C₄×C₈₉	—	2	4	no	—
111	C₂×C₄×C₈₉	—	4	2	no	—
112	C₂×C₄×C₈₉	—	4	2	no	—
113	C₂×C₄×C₈₉	—	4	2	no	—
114	C₂×C₄×C₈₉	#95	1	2	yes	—
115	C₂×C₄×C₈₉	#95	1	2	yes	—
116	C₂×C₄×C₈₉	#95	1	2	yes	—
117	C₂×C₄×C₈₉	#95	1	2	yes	—
118	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
125	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
126	C₂×C₄×C₈₉	#121	1	2	yes	—
127	C₂×C₄×C₈₉	#121	1	2	yes	—
128	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
129	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
130	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
131	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
132	C₂×C₄×C₈₉	#125	1	2	yes	—
133	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
134	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
135	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
136	C₂×Dic₈₉	—	2	8	no	—
137	C₂×Dic₈₉	—	4	8	no	—
138	C₈×C₈₉	—	4	4	no	—
139	#10623494	—	4	4	no	—
140	C₂²×C₈₉	—	2	2	no	—
141	C₂²×C₈₉	—	2	2	no	—
142	C₂²×C₈₉	—	2	2	no	—
143	C₂²×C₈₉	—	4	2	no	—
144	C₂²×C₈₉	—	4	2	no	—
145	C₂²×C₈₉	C₂×#95	1	1	yes	characteristic
146	C₂²×C₈₉	C₄×C₄⋊C₄	1	2	yes	—
147	C₂²×C₈₉	#380	1	2	yes	—
148	C₂²×C₈₉	#382	1	1	yes	characteristic
149	C₂²×C₈₉	#384	1	1	yes	characteristic
150	C₄×C₈₉	—	2	2	no	—
151	C₄×C₈₉	—	2	2	no	—
152	C₄×C₈₉	—	2	2	no	—
153	C₄×C₈₉	#344	1	4	yes	complemented
154	C₄×C₈₉	#389	1	2	yes	—
155	C₄×C₈₉	#389	1	2	yes	—
156	C₄×C₈₉	#390	1	1	yes	characteristic
157	C₄×C₈₉	#390	1	1	yes	characteristic
158	C₄×C₈₉	#391	1	1	yes	characteristic
159	C₄×C₈₉	#391	1	1	yes	characteristic
160	Dic₈₉	—	4	8	no	—
161	#10856	—	89	1	no	maximal, Sylow
162	C₂×C₈₉	—	2	2	no	—
163	C₂×C₈₉	—	2	2	no	—
164	C₂×C₈₉	—	2	1	no	—
165	C₂×C₈₉	—	2	1	no	—
166	C₂×C₈₉	C₄⋊#95	1	1	yes	characteristic
167	C₂×C₈₉	C₂×#344	1	2	yes	—
168	C₂×#423	—	89	1	no	—
169	C₂×#423	D₈₉	1	1	yes	characteristic
170	C₄×#117	—	89	1	no	—
171	C₂×#344	—	89	4	no	—
172	C₈₉	#10856	1	1	yes	characteristic, complemented, Sylow
173	C₂²×M₁₆	—	89	2	no	—
174	C₂²×M₁₆	C₂×D₈₉	1	2	yes	—
175	C₂×#117	—	89	2	no	—
176	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
177	#423	—	2	2	no	—
178	#423	—	2	1	no	—
179	#423	—	2	1	no	—
180	#423	—	89	2	no	—
181	#423	—	89	2	no	—
182	#423	—	178	2	no	—
183	C₂×C₄⋊C₈	—	178	2	no	—
184	C₂×C₄⋊C₈	Dic₈₉	1	4	yes	complemented
185	C₄×C₄⋊C₄	—	178	2	no	—
186	C₄×C₂²⋊C₄	—	89	4	no	—
187	#376	—	178	1	no	—
188	C₄³	—	178	1	no	—
189	#344	—	178	8	no	—
190	C₂³×C₄	C₄×D₈₉	1	2	yes	—
191	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
192	C₂×M₁₆	—	2	4	no	—
193	C₂×M₁₆	—	2	2	no	—
194	C₂×M₁₆	—	2	2	no	—
195	C₂×M₁₆	—	2	4	no	—
196	C₂×M₁₆	—	89	4	no	—
197	C₂×M₁₆	—	178	4	no	—
198	C₂×M₁₆	—	178	4	no	—
199	C₂×M₁₆	—	178	2	no	—
200	C₂²×C₈	—	178	2	no	—
201	C₂²×C₈	C₂×Dic₈₉	1	4	yes	—
202	#117	—	89	4	no	—
203	#117	—	89	4	no	—
204	#117	—	178	2	no	—
205	#117	—	178	2	no	—
206	C₂×C₄⋊C₄	—	178	2	no	—
207	C₂×C₂²⋊C₄	—	89	4	no	—
208	C₂×C₄²	—	178	4	no	—
209	C₂×C₄²	—	178	2	no	—
210	C₂×C₄²	Dic₁₇₈	1	1	yes	characteristic
211	C₂×C₄²	Dic₁₇₈	1	1	yes	characteristic
212	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
213	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
214	C₂×C₄²	D₃₅₆	1	2	yes	—
215	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
216	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
217	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
218	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
219	C₄⋊C₈	—	2	4	no	—
220	C₄⋊C₈	—	2	4	no	—
221	C₄⋊C₈	—	178	4	no	—
222	C₄⋊C₈	—	178	4	no	—
223	#95	—	178	4	no	—
224	C₂⁴	C₂×C₄×D₈₉	1	1	yes	characteristic
225	C₂²×C₄	—	4	4	no	—
226	C₂²×C₄	—	178	4	no	—
227	C₂²×C₄	C₄×Dic₈₉	1	2	yes	—
228	C₂²×C₄	C₄×Dic₈₉	1	2	yes	—
229	C₂²×C₄	#13083551	1	2	yes	—
230	C₂²×C₄	#13083551	1	2	yes	—
231	C₂²×C₄	#13083553	1	2	yes	—
232	C₂²×C₄	#13083553	1	2	yes	—
233	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
234	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
235	C₂²×C₄	D₄×D₈₉	1	2	yes	—
236	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
238	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
239	C₂²×C₄	Q₈×D₈₉	1	2	yes	—
240	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
241	C₂²×C₄	#13083573	1	2	yes	—
242	C₂²×C₄	#13083575	1	2	yes	—
243	M₁₆	—	4	4	no	—
244	M₁₆	—	4	2	no	—
245	M₁₆	—	4	2	no	—
246	M₁₆	—	356	4	no	—
247	M₁₆	—	178	8	no	—
248	C₂×C₈	—	2	8	no	—
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	—	356	2	no	—
251	C₂×C₈	—	356	2	no	—
252	C₂×C₈	—	178	4	no	—
253	C₄⋊C₄	—	178	4	no	—
254	C₄⋊C₄	—	178	4	no	—
255	C₂²⋊C₄	—	89	8	no	—
256	C₂²⋊C₄	—	178	4	no	—
257	C₄²	—	356	8	no	—
258	C₄²	—	178	4	no	—
259	C₄²	—	178	4	no	—
260	C₄²	#13083551	1	2	yes	—
261	C₄²	#13083551	1	2	yes	—
262	C₄²	C₂²⋊Dic₈₉	1	2	yes	—
263	C₄²	C₂²⋊Dic₈₉	1	2	yes	—
264	C₄²	#13083553	1	1	yes	characteristic
265	C₄²	#13083553	1	1	yes	characteristic
266	C₄²	#13083553	1	1	yes	characteristic
267	C₄²	#13083553	1	1	yes	characteristic
268	C₄²	C₄⋊Dic₈₉	1	2	yes	—
269	C₄²	C₄⋊Dic₈₉	1	2	yes	—
270	C₂³	—	4	2	no	—
271	C₂³	C₂×C₄×Dic₈₉	1	1	yes	characteristic
272	C₂³	C₂×#13083551	1	1	yes	characteristic
273	C₂³	C₄⋊C₄×D₈₉	1	2	yes	—
274	C₂³	C₂×#13083553	1	1	yes	characteristic
275	C₂³	#15667970	1	2	yes	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	356	8	no	—
283	C₂×C₄	—	178	8	no	—
284	C₂×C₄	#15667919	1	2	yes	—
285	C₂×C₄	#15667919	1	2	yes	—
286	C₂×C₄	#15667919	1	2	yes	—
287	C₂×C₄	#15667919	1	2	yes	—
288	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
289	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
290	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
291	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
292	C₂×C₄	#15667974	1	2	yes	—
293	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
294	C₂×C₄	#15667977	1	2	yes	—
295	C₂×C₄	Q₈×Dic₈₉	1	2	yes	—
296	C₂×C₄	#15667983	1	2	yes	—
297	C₂×C₄	#15667987	1	2	yes	—
298	C₂×C₄	#15667988	1	2	yes	—
299	C₂×C₄	#15667992	1	2	yes	—
300	C₂×C₄	#15667999	1	1	yes	characteristic
301	C₂×C₄	#15668001	1	1	yes	characteristic
302	C₂×C₄	#15668003	1	2	yes	—
303	C₂×C₄	#15668008	1	1	yes	characteristic
304	C₂×C₄	#15668010	1	1	yes	characteristic
305	C₂×C₄	#15668011	1	2	yes	—
306	C₈	—	4	4	no	—
307	C₈	—	356	4	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	4	2	no	—
312	C₂²	—	4	2	no	—
313	C₂²	C₂×#15667919	1	1	yes	characteristic
314	C₂²	C₄⋊C₄×Dic₈₉	1	2	yes	—
315	C₂²	#19151455	1	2	yes	—
316	C₂²	#19151463	1	1	yes	characteristic
317	C₂²	#19151467	1	1	yes	characteristic
318	C₄	—	2	2	no	—
319	C₄	—	2	2	no	—
320	C₄	—	2	2	no	—
321	C₄	—	356	8	no	—
322	C₄	#19151369	1	4	yes	complemented
323	C₄	#19151494	1	2	yes	—
324	C₄	#19151497	1	2	yes	—
325	C₄	#19151499	1	1	yes	characteristic
326	C₄	#19151501	1	1	yes	characteristic
327	C₄	#19151503	1	1	yes	characteristic
328	C₄	#19151505	1	1	yes	characteristic
329	C₂	—	2	2	no	—
330	C₂	—	2	2	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	#26018481	1	1	yes	characteristic
334	C₂	C₂×#19151369	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₈₉
commutator	C₂³×C₈₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₈₉
Fitting	C₂×#423×C₈₉	C₂
socle	C₂²×C₈₉	C₂×#95
prime radical	C₂⁴×C₈₉	C₂²×C₄
cosocle	C₂³×C₄×C₈₉	C₂³
hypercenter	C₂×#423	D₈₉
hypocenter	C₈₉	#10856
LCS	G ▷ C₂³×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
8. #19151622	C₄	semidirect
9. #19151622	C₄	semidirect
21. C₂×C₄⋊C₈×C₈₉	C₄	semidirect
153. C₄×C₈₉	#344	semidirect
172. C₈₉	#10856	semidirect
184. C₂×C₄⋊C₈	Dic₈₉	semidirect
322. C₄	#19151369	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151622	2
C₄×#15667971	2
C₂×#19151369	2
C₂×#423×C₈₉	2
#10856	89