Subgroups — #36460304

Order 22784.

918 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#13083558	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151468	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151366	C₂	1	4	yes	maximal
4	C₂×#423×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#13083558	C₂²	1	2	yes	—
6	C₂²×C₄⋊Dic₈₉	C₄	1	4	yes	—
7	C₂²×C₄⋊Dic₈₉	C₂²	1	2	yes	—
8	C₄×#13083558	—	2	2	no	—
9	C₄×#13083558	C₄	1	4	yes	complemented
10	C₂×#15667923	—	2	1	no	—
11	C₂×C₄×#10623494	—	2	1	no	—
12	#19151468	—	2	1	no	—
13	#19151468	—	2	1	no	—
14	#19151468	—	2	2	no	—
15	#19151366	—	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	C₄	1	4	yes	complemented
20	C₂×C₄⋊C₈×C₈₉	—	2	2	no	—
21	C₂³×Dic₈₉	C₂×C₄	1	4	yes	—
22	C₂×#13083558	—	2	2	no	—
23	C₂×#13083558	—	2	2	no	—
24	C₂×#13083558	—	2	2	no	—
25	C₂×#13083558	C₂×C₄	1	4	yes	—
26	C₂×#13083558	D₄	1	4	yes	—
27	C₂×#13083558	Q₈	1	4	yes	—
28	C₂²×#10623494	—	2	2	no	—
29	C₂×C₄⋊Dic₈₉	—	2	4	no	—
30	C₂×C₄⋊Dic₈₉	—	2	4	no	—
31	C₂×C₄⋊Dic₈₉	—	2	4	no	—
32	C₂×C₄⋊Dic₈₉	—	2	2	no	—
33	C₂×C₄⋊Dic₈₉	—	2	2	no	—
34	C₂×C₄⋊Dic₈₉	—	2	4	no	—
35	C₂×C₄⋊Dic₈₉	—	4	8	no	—
36	#15667923	—	2	4	no	—
37	C₄×#10623494	—	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	2	no	—
41	C₂×M₁₆×C₈₉	—	2	4	no	—
42	C₂×M₁₆×C₈₉	—	2	4	no	—
43	C₂×M₁₆×C₈₉	C₂×C₄	1	4	yes	—
44	C₂²×C₈×C₈₉	—	2	2	no	—
45	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₈₉	D₄	1	2	yes	—
50	C₂×C₄²×C₈₉	D₄	1	2	yes	—
51	C₂×C₄²×C₈₉	Q₈	1	2	yes	—
52	C₄⋊C₈×C₈₉	—	2	8	no	—
53	C₂²×Dic₈₉	—	2	4	no	—
54	C₂²×Dic₈₉	—	2	8	no	—
55	C₂²×Dic₈₉	—	4	8	no	—
56	#13083558	—	2	4	no	—
57	#13083558	—	2	4	no	—
58	#13083558	—	2	4	no	—
59	#13083558	C₄⋊C₄	1	8	yes	complemented
60	C₂×#10623494	—	2	4	no	—
61	C₂×#10623494	—	2	4	no	—
62	C₂×#10623494	—	2	4	no	—
63	C₄⋊Dic₈₉	—	4	8	no	—
64	C₄⋊Dic₈₉	—	4	8	no	—
65	C₄⋊Dic₈₉	—	4	4	no	—
66	C₄⋊Dic₈₉	—	4	2	no	—
67	C₄⋊Dic₈₉	—	4	2	no	—
68	C₂⁴×C₈₉	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₈₉	—	2	4	no	—
70	C₂²×C₄×C₈₉	—	2	4	no	—
71	C₂²×C₄×C₈₉	C₄²	1	4	yes	—
72	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
73	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
74	C₂²×C₄×C₈₉	C₄⋊C₄	1	4	yes	—
75	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
82	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
83	C₂²×C₄×C₈₉	#41	1	2	yes	—
84	C₂²×C₄×C₈₉	#41	1	2	yes	—
85	M₁₆×C₈₉	—	2	8	no	—
86	M₁₆×C₈₉	—	4	4	no	—
87	C₂×C₈×C₈₉	—	2	4	no	—
88	C₂×C₈×C₈₉	—	4	2	no	—
89	C₂×C₈×C₈₉	—	4	2	no	—
90	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
91	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
92	C₄²×C₈₉	C₂²⋊C₄	1	4	yes	—
93	C₄²×C₈₉	C₄⋊C₄	1	4	yes	—
94	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
95	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
96	C₂³×C₈₉	—	2	2	no	—
97	C₂³×C₈₉	—	2	2	no	—
98	C₂³×C₈₉	C₂×C₄²	1	1	yes	characteristic
99	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂³×C₈₉	C₄×D₄	1	2	yes	—
102	C₂³×C₈₉	C₄×Q₈	1	2	yes	—
103	C₂×C₄×C₈₉	—	2	2	no	—
104	C₂×C₄×C₈₉	—	2	2	no	—
105	C₂×C₄×C₈₉	—	2	8	no	—
106	C₂×C₄×C₈₉	—	2	8	no	—
107	C₂×C₄×C₈₉	#95	1	4	yes	—
108	C₂×C₄×C₈₉	#95	1	4	yes	—
109	C₂×C₄×C₈₉	#100	1	4	yes	—
110	C₂×C₄×C₈₉	#101	1	4	yes	—
111	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₈₉	#117	1	2	yes	—
117	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
119	C₂×C₄×C₈₉	#121	1	2	yes	—
120	C₂×C₄×C₈₉	#121	1	2	yes	—
121	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
122	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
123	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
124	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
125	C₂×C₄×C₈₉	#125	1	2	yes	—
126	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
129	C₂×Dic₈₉	—	4	4	no	—
130	C₂×Dic₈₉	—	4	8	no	—
131	C₂×Dic₈₉	—	8	8	no	—
132	C₈×C₈₉	—	4	4	no	—
133	#10623494	—	2	8	no	—
134	C₂²×C₈₉	—	2	2	no	—
135	C₂²×C₈₉	—	2	2	no	—
136	C₂²×C₈₉	—	2	2	no	—
137	C₂²×C₈₉	—	2	2	no	—
138	C₂²×C₈₉	—	2	4	no	—
139	C₂²×C₈₉	C₂×#95	1	1	yes	characteristic
140	C₂²×C₈₉	C₄×C₄⋊C₄	1	2	yes	—
141	C₂²×C₈₉	#389	1	2	yes	—
142	C₂²×C₈₉	#390	1	1	yes	characteristic
143	C₂²×C₈₉	#391	1	1	yes	characteristic
144	C₂²×C₈₉	C₂×#100	1	2	yes	—
145	C₂²×C₈₉	C₂×#101	1	2	yes	—
146	C₄×C₈₉	—	2	4	no	—
147	C₄×C₈₉	#343	1	8	yes	complemented
148	C₄×C₈₉	#380	1	2	yes	—
149	C₄×C₈₉	#382	1	1	yes	characteristic
150	C₄×C₈₉	#384	1	1	yes	characteristic
151	C₄×C₈₉	#389	1	2	yes	—
152	C₄×C₈₉	#390	1	1	yes	characteristic
153	C₄×C₈₉	#391	1	1	yes	characteristic
154	Dic₈₉	—	8	4	no	—
155	#10857	—	89	1	no	maximal, Sylow
156	C₂×C₈₉	—	2	2	no	—
157	C₂×C₈₉	—	2	2	no	—
158	C₂×C₈₉	C₄⋊#95	1	1	yes	characteristic
159	C₂×C₈₉	C₂×#343	1	2	yes	—
160	C₂×C₈₉	#1965	1	1	yes	characteristic
161	C₂×C₈₉	#1966	1	1	yes	characteristic
162	C₂×C₈₉	#1997	1	2	yes	—
163	C₂×#423	D₈₉	1	1	yes	characteristic
164	C₂×C₄×M₁₆	—	89	1	no	—
165	C₂×#384	—	89	1	no	—
166	C₂×#343	—	89	4	no	—
167	C₈₉	#10857	1	1	yes	characteristic, complemented, Sylow
168	C₂²×M₁₆	—	89	2	no	—
169	C₂²×M₁₆	C₂×D₈₉	1	2	yes	—
170	C₂²×C₄⋊C₄	—	89	4	no	—
171	C₂²×C₄⋊C₄	—	89	2	no	—
172	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
173	#423	—	2	2	no	—
174	#423	Dic₈₉	1	4	yes	complemented
175	C₂×C₄⋊C₈	—	2	2	no	—
176	C₄×M₁₆	—	89	4	no	—
177	C₄×M₁₆	—	178	2	no	—
178	C₂×#97	—	178	1	no	—
179	C₂×C₄×C₈	—	178	1	no	—
180	#384	—	178	1	no	—
181	#384	—	178	1	no	—
182	#384	—	178	2	no	—
183	#343	—	178	8	no	—
184	C₂³×C₄	—	89	4	no	—
185	C₂³×C₄	C₄×D₈₉	1	2	yes	—
186	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	—	2	4	no	—
189	C₂×M₁₆	—	2	4	no	—
190	C₂×M₁₆	—	89	4	no	—
191	C₂×M₁₆	—	89	4	no	—
192	C₂×M₁₆	—	89	4	no	—
193	C₂×M₁₆	—	178	2	no	—
194	C₂×M₁₆	—	178	2	no	—
195	C₂×M₁₆	—	178	2	no	—
196	C₂×M₁₆	C₂×Dic₈₉	1	4	yes	—
197	C₂²×C₈	—	2	2	no	—
198	C₂²×C₈	—	178	2	no	—
199	C₂×C₄⋊C₄	—	356	8	no	—
200	C₂×C₄⋊C₄	—	178	4	no	—
201	C₂×C₄⋊C₄	—	178	4	no	—
202	C₂×C₄⋊C₄	—	178	4	no	—
203	C₂×C₄⋊C₄	—	178	2	no	—
204	C₂×C₄⋊C₄	—	178	2	no	—
205	C₂×C₄⋊C₄	—	178	4	no	—
206	C₂×C₄²	Dic₁₇₈	1	2	yes	—
207	C₂×C₄²	C₄×D₈₉	1	2	yes	—
208	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
209	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
210	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
211	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
212	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
213	C₄⋊C₈	—	2	8	no	—
214	#97	—	178	4	no	—
215	C₄×C₈	—	178	4	no	—
216	C₂⁴	C₂×C₄×D₈₉	1	1	yes	characteristic
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	356	8	no	—
220	C₂²×C₄	—	178	4	no	—
221	C₂²×C₄	—	178	8	no	—
222	C₂²×C₄	C₄×Dic₈₉	1	4	yes	—
223	C₂²×C₄	#13083551	1	2	yes	—
224	C₂²×C₄	#13083551	1	2	yes	—
225	C₂²×C₄	#13083553	1	4	yes	—
226	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
227	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
228	C₂²×C₄	D₄×D₈₉	1	2	yes	—
229	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
232	C₂²×C₄	Q₈×D₈₉	1	2	yes	—
233	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
234	C₂²×C₄	#13083573	1	2	yes	—
235	C₂²×C₄	#13083575	1	2	yes	—
236	M₁₆	—	2	8	no	—
237	M₁₆	—	4	4	no	—
238	M₁₆	—	89	8	no	—
239	M₁₆	—	178	4	no	—
240	M₁₆	—	178	4	no	—
241	M₁₆	—	178	4	no	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	4	2	no	—
244	C₂×C₈	—	4	2	no	—
245	C₂×C₈	—	178	4	no	—
246	C₂×C₈	—	178	4	no	—
247	C₂×C₈	—	178	4	no	—
248	C₄⋊C₄	—	356	8	no	—
249	C₄⋊C₄	—	356	4	no	—
250	C₄⋊C₄	—	356	2	no	—
251	C₄⋊C₄	—	356	2	no	—
252	C₄⋊C₄	—	356	8	no	—
253	C₄²	#13083551	1	2	yes	—
254	C₄²	#13083551	1	2	yes	—
255	C₄²	C₂²⋊Dic₈₉	1	4	yes	—
256	C₄²	#13083553	1	4	yes	—
257	C₄²	C₄⋊Dic₈₉	1	2	yes	—
258	C₄²	C₄⋊Dic₈₉	1	2	yes	—
259	C₂³	—	2	2	no	—
260	C₂³	—	2	2	no	—
261	C₂³	C₂×C₄×Dic₈₉	1	1	yes	characteristic
262	C₂³	C₂×#13083551	1	1	yes	characteristic
263	C₂³	C₂×#13083553	1	1	yes	characteristic
264	C₂³	#15667974	1	2	yes	—
265	C₂³	#15667977	1	2	yes	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	8	no	—
269	C₂×C₄	—	2	8	no	—
270	C₂×C₄	—	356	4	no	—
271	C₂×C₄	—	356	8	no	—
272	C₂×C₄	—	712	8	no	—
273	C₂×C₄	#15667919	1	4	yes	—
274	C₂×C₄	#15667919	1	4	yes	—
275	C₂×C₄	#15667928	1	4	yes	—
276	C₂×C₄	#15667930	1	4	yes	—
277	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
278	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
279	C₂×C₄	C₄⋊C₄×D₈₉	1	2	yes	—
280	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
281	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
282	C₂×C₄	#15667970	1	2	yes	—
283	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
284	C₂×C₄	Q₈×Dic₈₉	1	2	yes	—
285	C₂×C₄	#15667983	1	2	yes	—
286	C₂×C₄	#15667987	1	2	yes	—
287	C₂×C₄	#15667988	1	2	yes	—
288	C₂×C₄	#15667992	1	2	yes	—
289	C₂×C₄	#15667999	1	1	yes	characteristic
290	C₂×C₄	#15668001	1	1	yes	characteristic
291	C₂×C₄	#15668003	1	2	yes	—
292	C₂×C₄	#15668008	1	1	yes	characteristic
293	C₂×C₄	#15668010	1	1	yes	characteristic
294	C₂×C₄	#15668011	1	2	yes	—
295	C₈	—	4	4	no	—
296	C₈	—	178	8	no	—
297	C₂²	—	2	4	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	2	no	—
302	C₂²	C₂×#15667919	1	1	yes	characteristic
303	C₂²	C₄⋊C₄×Dic₈₉	1	2	yes	—
304	C₂²	#19151494	1	2	yes	—
305	C₂²	#19151499	1	1	yes	characteristic
306	C₂²	#19151503	1	1	yes	characteristic
307	C₂²	C₂×#15667928	1	2	yes	—
308	C₂²	C₂×#15667930	1	2	yes	—
309	C₄	—	2	4	no	—
310	C₄	—	712	4	no	—
311	C₄	#19151366	1	8	yes	complemented
312	C₄	#19151455	1	2	yes	—
313	C₄	#19151463	1	1	yes	characteristic
314	C₄	#19151467	1	1	yes	characteristic
315	C₄	#19151497	1	2	yes	—
316	C₄	#19151501	1	1	yes	characteristic
317	C₄	#19151505	1	1	yes	characteristic
318	C₂	—	2	2	no	—
319	C₂	—	2	2	no	—
320	C₂	#26018481	1	1	yes	characteristic
321	C₂	C₂×#19151366	1	2	yes	—
322	C₂	#26021091	1	1	yes	characteristic
323	C₂	#26021095	1	1	yes	characteristic
324	C₂	#26021240	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#13083551
commutator	C₂³×C₈₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₈₉
Fitting	C₂×#423×C₈₉	C₂
socle	C₂³×C₈₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₈₉	C₂²×C₄
cosocle	C₂³×C₄×C₈₉	C₂³
hypercenter	C₂×#423	D₈₉
hypocenter	C₈₉	#10857
LCS	G ▷ C₂³×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
9. C₄×#13083558	C₄	semidirect
19. #423×C₈₉	C₄	semidirect
59. #13083558	C₄⋊C₄	semidirect
147. C₄×C₈₉	#343	semidirect
167. C₈₉	#10857	semidirect
174. #423	Dic₈₉	semidirect
311. C₄	#19151366	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#13083558	2
C₂×#19151468	2
C₂×#19151366	2
C₂×#423×C₈₉	2
#10857	89