Subgroups — #36447261

Order 22784.

598 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×#10623494	C₂	1	1	yes	maximal, characteristic
2	#26020232	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151348	C₂	1	2	yes	maximal, complemented
4	#26018086	C₂	1	2	yes	maximal, complemented
5	C₈₉×#1797	C₂	1	1	yes	maximal, characteristic
6	C₂³×#10623494	C₄	1	1	yes	characteristic
7	C₂³×#10623494	C₄	1	1	yes	characteristic
8	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
9	C₂×C₂²⋊#10623494	—	2	1	no	—
10	C₂×C₄×#10623494	—	2	4	no	—
11	C₂×C₄×#10623494	C₄	1	1	yes	characteristic
12	C₂×C₄×#10623494	C₄	1	1	yes	characteristic
13	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
14	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
15	#19151353	—	2	1	no	—
16	#19151348	—	2	4	no	—
17	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₈×C₈₉	—	2	1	no	—
20	#336×C₈₉	—	2	1	no	—
21	C₂²×#10623494	—	2	2	no	—
22	C₂²×#10623494	—	2	2	no	—
23	C₂²×#10623494	—	2	2	no	—
24	C₂²×#10623494	—	2	2	no	—
25	C₂²×#10623494	—	2	2	no	—
26	C₂²×#10623494	—	2	2	no	—
27	C₂²×#10623494	C₈	1	4	yes	complemented
28	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
29	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
30	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
31	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
32	C₄⋊#10623494	—	2	2	no	—
33	C₄⋊#10623494	—	2	2	no	—
34	C₂²⋊#10623494	—	4	2	no	—
35	C₄×#10623494	—	2	2	no	—
36	C₄×#10623494	—	2	2	no	—
37	C₄×#10623494	—	2	2	no	—
38	C₄×#10623494	D₄	1	4	yes	—
39	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
41	C₂³×C₄×C₈₉	Q₈	1	1	yes	characteristic
42	C₂²×C₈×C₈₉	—	2	2	no	—
43	C₂×C₄²×C₈₉	—	2	2	no	—
44	C₂×C₄²×C₈₉	—	2	2	no	—
45	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₈₉	Q₈	1	1	yes	characteristic
48	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
49	C₄⋊C₈×C₈₉	—	2	2	no	—
50	C₄⋊C₈×C₈₉	—	2	2	no	—
51	C₂²⋊C₈×C₈₉	—	4	2	no	—
52	C₂×#10623494	—	2	8	no	—
53	C₂×#10623494	—	2	1	no	—
54	C₂×#10623494	—	2	1	no	—
55	C₂×#10623494	—	2	1	no	—
56	C₂×#10623494	—	2	1	no	—
57	C₂×#10623494	—	2	1	no	—
58	C₂×#10623494	—	2	1	no	—
59	C₂×#10623494	—	2	8	no	—
60	C₂×#10623494	—	4	2	no	—
61	C₂×#10623494	—	4	2	no	—
62	C₂×#10623494	—	4	2	no	—
63	C₂×#10623494	C₂²⋊C₄	1	2	yes	—
64	C₂×#10623494	C₂²⋊C₄	1	2	yes	—
65	C₂×#10623494	C₂×C₈	1	2	yes	—
66	C₂×#10623494	C₂×C₈	1	2	yes	—
67	C₂×#10623494	M₁₆	1	2	yes	—
68	C₂×#10623494	M₁₆	1	2	yes	—
69	C₂⁴×C₈₉	C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₈₉	—	2	2	no	—
71	C₂²×C₄×C₈₉	—	2	2	no	—
72	C₂²×C₄×C₈₉	—	2	1	no	—
73	C₂²×C₄×C₈₉	—	2	1	no	—
74	C₂²×C₄×C₈₉	—	2	2	no	—
75	C₂²×C₄×C₈₉	—	2	2	no	—
76	C₂²×C₄×C₈₉	—	2	1	no	—
77	C₂²×C₄×C₈₉	—	2	1	no	—
78	C₂²×C₄×C₈₉	—	2	2	no	—
79	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₈₉	C₂×C₈	1	2	yes	—
83	C₂²×C₄×C₈₉	M₁₆	1	2	yes	—
84	C₂²×C₄×C₈₉	SD₁₆	1	2	yes	—
85	C₂²×C₄×C₈₉	SD₁₆	1	2	yes	—
86	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
89	C₂×C₈×C₈₉	—	4	2	no	—
90	C₂×C₈×C₈₉	—	4	2	no	—
91	C₂×C₈×C₈₉	—	4	2	no	—
92	C₄²×C₈₉	—	2	1	no	—
93	C₄²×C₈₉	—	2	1	no	—
94	C₄²×C₈₉	—	2	2	no	—
95	C₄²×C₈₉	—	2	2	no	—
96	C₄²×C₈₉	C₂×D₄	1	2	yes	—
97	C₄²×C₈₉	#41	1	2	yes	—
98	C₂³×C₈₉	—	2	1	no	—
99	C₂³×C₈₉	—	2	1	no	—
100	C₂³×C₈₉	—	2	1	no	—
101	C₂³×C₈₉	—	2	1	no	—
102	C₂³×C₈₉	C₄⋊C₈	1	2	yes	—
103	C₂³×C₈₉	#106	1	2	yes	—
104	C₂³×C₈₉	#108	1	2	yes	—
105	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₈₉	—	2	8	no	—
107	C₂×C₄×C₈₉	—	2	1	no	—
108	C₂×C₄×C₈₉	—	2	1	no	—
109	C₂×C₄×C₈₉	—	2	1	no	—
110	C₂×C₄×C₈₉	—	2	1	no	—
111	C₂×C₄×C₈₉	—	2	1	no	—
112	C₂×C₄×C₈₉	—	2	1	no	—
113	C₂×C₄×C₈₉	—	2	1	no	—
114	C₂×C₄×C₈₉	—	2	1	no	—
115	C₂×C₄×C₈₉	—	2	2	no	—
116	C₂×C₄×C₈₉	—	2	2	no	—
117	C₂×C₄×C₈₉	—	2	2	no	—
118	C₂×C₄×C₈₉	—	2	2	no	—
119	C₂×C₄×C₈₉	—	2	4	no	—
120	C₂×C₄×C₈₉	—	2	4	no	—
121	C₂×C₄×C₈₉	C₄⋊C₈	1	2	yes	—
122	C₂×C₄×C₈₉	#106	1	2	yes	—
123	C₂×C₄×C₈₉	#108	1	2	yes	—
124	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
126	C₂×C₄×C₈₉	#121	1	2	yes	—
127	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
128	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
129	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
130	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
131	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
132	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
133	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
134	C₂×C₄×C₈₉	#135	1	2	yes	—
135	C₈×C₈₉	—	8	2	no	—
136	#10623494	—	2	2	no	—
137	#10623494	—	2	2	no	—
138	#10623494	—	2	2	no	—
139	#10623494	—	8	2	no	—
140	#10623494	C₂²⋊C₈	1	4	yes	complemented
141	C₂²×C₈₉	—	2	2	no	—
142	C₂²×C₈₉	—	2	2	no	—
143	C₂²×C₈₉	—	2	2	no	—
144	C₂²×C₈₉	—	2	2	no	—
145	C₂²×C₈₉	—	2	2	no	—
146	C₂²×C₈₉	—	2	2	no	—
147	C₂²×C₈₉	#334	1	4	yes	complemented
148	C₂²×C₈₉	#380	1	1	yes	characteristic
149	C₂²×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
150	C₂²×C₈₉	#423	1	1	yes	characteristic
151	C₂²×C₈₉	C₂×#106	1	1	yes	characteristic
152	C₂²×C₈₉	#427	1	1	yes	characteristic
153	C₂²×C₈₉	C₂×#108	1	1	yes	characteristic
154	C₂²×C₈₉	C₂×#108	1	1	yes	characteristic
155	C₄×C₈₉	—	2	2	no	—
156	C₄×C₈₉	—	2	2	no	—
157	C₄×C₈₉	—	2	2	no	—
158	C₄×C₈₉	—	2	2	no	—
159	C₄×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
160	C₄×C₈₉	#407	1	1	yes	characteristic
161	C₄×C₈₉	#431	1	1	yes	characteristic
162	C₄×C₈₉	#432	1	1	yes	characteristic
163	C₄×C₈₉	#465	1	2	yes	—
164	C₄×C₈₉	#465	1	2	yes	—
165	#8595	—	89	1	no	maximal, Sylow
166	C₂×C₈₉	—	2	4	no	—
167	C₂×C₈₉	C₂×#334	1	1	yes	characteristic
168	C₂×C₈₉	#1564	1	1	yes	characteristic
169	C₂×C₈₉	#1566	1	1	yes	characteristic
170	C₂×C₈₉	#1568	1	1	yes	characteristic
171	C₂×C₈₉	#1797	1	1	yes	characteristic
172	C₂×C₈₉	#1807	1	1	yes	characteristic
173	C₂×C₈₉	#1810	1	1	yes	characteristic
174	C₂²×C₄×C₈	—	89	1	no	—
175	#1797	—	89	1	no	—
176	#1797	D₈₉	1	1	yes	characteristic
177	C₂×#334	—	89	2	no	—
178	#1277	—	89	2	no	—
179	C₈₉	#8595	1	1	yes	characteristic, complemented, Sylow
180	C₂³×C₈	—	89	1	no	—
181	C₂³×C₈	—	89	1	no	—
182	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
183	C₂×C₄⋊C₈	—	89	2	no	—
184	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
185	C₂×C₂²⋊C₈	—	2	1	no	—
186	C₂×C₂²⋊C₈	—	178	1	no	—
187	C₂×C₄×C₈	—	89	1	no	—
188	C₂×C₄×C₈	—	89	1	no	—
189	C₂×C₄×C₈	—	89	1	no	—
190	C₂×C₄×C₈	—	89	1	no	—
191	C₂×C₄×C₈	—	178	4	no	—
192	#336	—	2	1	no	—
193	#336	—	178	1	no	—
194	#334	—	178	4	no	—
195	C₂³×C₄	Dic₁₇₈	1	1	yes	characteristic
196	C₂³×C₄	C₄×D₈₉	1	1	yes	characteristic
197	C₂³×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
198	C₂²×C₈	—	2	2	no	—
199	C₂²×C₈	—	89	1	no	—
200	C₂²×C₈	—	89	4	no	—
201	C₂²×C₈	—	89	1	no	—
202	C₂²×C₈	—	89	1	no	—
203	C₂²×C₈	—	89	1	no	—
204	C₂²×C₈	—	178	2	no	—
205	C₂²×C₈	—	178	2	no	—
206	C₂²×C₈	—	178	2	no	—
207	C₂²×C₈	—	178	2	no	—
208	C₂²×C₈	—	178	2	no	—
209	C₂²×C₈	—	178	2	no	—
210	C₂×C₄²	—	2	2	no	—
211	C₂×C₄²	—	2	2	no	—
212	C₂×C₄²	Dic₁₇₈	1	1	yes	characteristic
213	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
214	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
215	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
216	C₄⋊C₈	—	2	2	no	—
217	C₄⋊C₈	—	2	2	no	—
218	C₄⋊C₈	—	178	2	no	—
219	C₄⋊C₈	—	178	2	no	—
220	C₂²⋊C₈	—	4	2	no	—
221	C₂²⋊C₈	—	356	2	no	—
222	C₄×C₈	—	89	4	no	—
223	C₄×C₈	—	178	2	no	—
224	C₄×C₈	—	178	2	no	—
225	C₄×C₈	—	178	2	no	—
226	C₂⁴	#13083553	1	1	yes	characteristic
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	1	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	#13083553	1	1	yes	characteristic
237	C₂²×C₄	#13083553	1	1	yes	characteristic
238	C₂²×C₄	#13083553	1	1	yes	characteristic
239	C₂²×C₄	C₈×D₈₉	1	2	yes	—
240	C₂²×C₄	#13083557	1	2	yes	—
241	C₂²×C₄	#13083561	1	2	yes	—
242	C₂²×C₄	#13083562	1	2	yes	—
243	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
244	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
245	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
246	C₂×C₈	—	4	2	no	—
247	C₂×C₈	—	4	2	no	—
248	C₂×C₈	—	4	2	no	—
249	C₂×C₈	—	89	2	no	—
250	C₂×C₈	—	89	2	no	—
251	C₂×C₈	—	89	2	no	—
252	C₂×C₈	—	89	2	no	—
253	C₂×C₈	—	89	2	no	—
254	C₂×C₈	—	89	2	no	—
255	C₂×C₈	—	356	2	no	—
256	C₂×C₈	—	356	2	no	—
257	C₂×C₈	—	356	2	no	—
258	C₂×C₈	—	178	8	no	—
259	C₂×C₈	—	178	8	no	—
260	C₂×C₈	—	178	1	no	—
261	C₂×C₈	—	178	1	no	—
262	C₂×C₈	—	178	1	no	—
263	C₂×C₈	—	178	1	no	—
264	C₂×C₈	—	178	1	no	—
265	C₂×C₈	—	178	1	no	—
266	C₄²	—	2	1	no	—
267	C₄²	—	2	2	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	1	no	—
270	C₄²	D₄×D₈₉	1	2	yes	—
271	C₄²	#13083573	1	2	yes	—
272	C₂³	—	2	1	no	—
273	C₂³	—	2	1	no	—
274	C₂³	—	2	1	no	—
275	C₂³	—	2	1	no	—
276	C₂³	#15667941	1	2	yes	—
277	C₂³	#15667943	1	2	yes	—
278	C₂³	#15667947	1	2	yes	—
279	C₂³	C₂×#13083553	1	1	yes	characteristic
280	C₂×C₄	—	2	8	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	#15667941	1	2	yes	—
296	C₂×C₄	#15667943	1	2	yes	—
297	C₂×C₄	#15667947	1	2	yes	—
298	C₂×C₄	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
299	C₂×C₄	#15667968	1	1	yes	characteristic
300	C₂×C₄	#15667984	1	2	yes	—
301	C₂×C₄	#15667989	1	2	yes	—
302	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
303	C₂×C₄	M₁₆×D₈₉	1	1	yes	characteristic
304	C₂×C₄	M₁₆×D₈₉	1	1	yes	characteristic
305	C₂×C₄	C₂×#13083557	1	1	yes	characteristic
306	C₂×C₄	C₂×#13083561	1	1	yes	characteristic
307	C₂×C₄	C₂×#13083562	1	1	yes	characteristic
308	C₂×C₄	#15668032	1	2	yes	—
309	C₈	—	8	2	no	—
310	C₈	—	89	4	no	—
311	C₈	—	178	2	no	—
312	C₈	—	178	2	no	—
313	C₈	—	178	2	no	—
314	C₈	—	712	2	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	2	2	no	—
318	C₂²	—	2	2	no	—
319	C₂²	—	2	2	no	—
320	C₂²	—	2	2	no	—
321	C₂²	#19151348	1	4	yes	complemented
322	C₂²	#19151456	1	1	yes	characteristic
323	C₂²	C₂×#15667941	1	1	yes	characteristic
324	C₂²	#19151621	1	1	yes	characteristic
325	C₂²	C₂×#15667943	1	1	yes	characteristic
326	C₂²	#19151634	1	1	yes	characteristic
327	C₂²	C₂×#15667947	1	1	yes	characteristic
328	C₂²	C₂×#15667947	1	1	yes	characteristic
329	C₄	—	2	2	no	—
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	2	2	no	—
333	C₄	C₂²⋊C₈×D₈₉	1	1	yes	characteristic
334	C₄	#19151560	1	1	yes	characteristic
335	C₄	#19151648	1	1	yes	characteristic
336	C₄	#19151652	1	1	yes	characteristic
337	C₄	#19151825	1	2	yes	—
338	C₄	#19151829	1	2	yes	—
339	C₂	—	2	4	no	—
340	C₂	C₂×#19151348	1	1	yes	characteristic
341	C₂	#26018998	1	1	yes	characteristic
342	C₂	#26019007	1	1	yes	characteristic
343	C₂	#26019013	1	1	yes	characteristic
344	C₂	#26020230	1	1	yes	characteristic
345	C₂	#26020275	1	1	yes	characteristic
346	C₂	#26020288	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₈₉
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1797	C₂
socle	C₂³×C₈₉	C₂×C₄⋊C₄
prime radical	C₂⁴×C₈₉	C₄⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1797	D₈₉
hypocenter	C₈₉	#8595
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #1797

Complements

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

N	H ≅ G/N	type
3. C₂×#19151348	C₂	semidirect
4. #26018086	C₂	semidirect
27. C₂²×#10623494	C₈	semidirect
140. #10623494	C₂²⋊C₈	semidirect
147. C₂²×C₈₉	#334	semidirect
179. C₈₉	#8595	semidirect
321. C₂²	#19151348	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×#10623494	2
#26020232	2
C₂×#19151348	2
#26018086	2
C₈₉×#1797	2
#8595	89