Subgroups — #36539706

Order 22784.

686 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26031190	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151798	C₂	1	1	yes	maximal, characteristic
3	#26029333	C₂	1	1	yes	maximal, characteristic, complemented
4	#26019670	C₂	1	4	yes	maximal, complemented
5	#26019602	C₂	1	4	yes	maximal, complemented
6	#26019024	C₂	1	1	yes	maximal, characteristic, complemented
7	#26018921	C₂	1	1	yes	maximal, characteristic
8	#26018909	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#3078	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#13083561	—	2	1	no	—
11	C₂×#15668009	C₂²	1	1	yes	characteristic
12	#19151972	C₂²	1	2	yes	—
13	#19151946	C₂²	1	2	yes	—
14	#19151798	—	2	2	no	—
15	#19151798	—	2	2	no	—
16	#19151798	C₂²	1	4	yes	—
17	#19151748	—	2	2	no	—
18	Q₈×#10623494	—	2	2	no	—
19	Q₈×#10623494	C₂²	1	4	yes	complemented
20	#19151651	—	2	2	no	—
21	#19151651	C₂²	1	1	yes	characteristic
22	#19151651	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
24	C₂×#15667932	—	2	1	no	—
25	#19151348	C₂²	1	4	yes	complemented
26	#19151347	C₂²	1	2	yes	complemented
27	#19151344	C₂²	1	2	yes	complemented
28	#19151341	C₂²	1	4	yes	—
29	#533×C₈₉	C₂²	1	1	yes	characteristic
30	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
31	#479×C₈₉	—	2	2	no	—
32	#477×C₈₉	C₂²	1	2	yes	complemented
33	#475×C₈₉	C₂²	1	2	yes	complemented
34	#428×C₈₉	—	2	1	no	—
35	#423×C₈₉	C₂²	1	1	yes	characteristic
36	#419×C₈₉	—	2	1	no	—
37	C₂²×D₃₅₆	—	2	1	no	—
38	C₂²×D₃₅₆	—	2	1	no	—
39	C₂²×D₃₅₆	—	2	1	no	—
40	C₂×#13083561	—	2	4	no	—
41	C₂×#13083561	—	2	4	no	—
42	C₂×#13083561	—	4	4	no	—
43	C₂²×#10623494	—	2	1	no	—
44	#15668009	—	2	1	no	—
45	#15668009	—	2	1	no	—
46	#15668009	—	2	1	no	—
47	#15668009	C₂³	1	2	yes	—
48	C₄⋊#10623494	—	2	4	no	—
49	C₄⋊#10623494	—	2	1	no	—
50	C₄⋊#10623494	D₄	1	2	yes	—
51	C₄⋊#10623494	C₂³	1	2	yes	—
52	C₄⋊#10623494	C₂³	1	2	yes	—
53	#15667932	—	2	4	no	—
54	#15667932	—	2	4	no	—
55	C₂²⋊#10623494	—	2	1	no	—
56	C₂²⋊#10623494	D₄	1	2	yes	—
57	C₄×#10623494	—	2	4	no	—
58	C₄×#10623494	—	2	1	no	—
59	C₄×#10623494	C₂³	1	2	yes	—
60	C₂²×Q₈×C₈₉	D₄	1	1	yes	characteristic
61	C₂×M₁₆×C₈₉	—	2	1	no	—
62	C₄⋊Q₈×C₈₉	C₂³	1	1	yes	characteristic
63	C₄⋊Q₈×C₈₉	C₂³	1	1	yes	characteristic
64	#125×C₈₉	—	2	1	no	—
65	C₂²⋊Q₈×C₈₉	—	2	1	no	—
66	C₂²⋊Q₈×C₈₉	—	2	1	no	—
67	C₄×Q₈×C₈₉	—	2	1	no	—
68	C₄×Q₈×C₈₉	—	2	1	no	—
69	C₄×Q₈×C₈₉	—	2	1	no	—
70	C₄×Q₈×C₈₉	—	2	1	no	—
71	C₄×Q₈×C₈₉	D₄	1	2	yes	—
72	C₄×Q₈×C₈₉	C₂³	1	2	yes	—
73	#117×C₈₉	—	2	1	no	—
74	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
75	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
76	C₂×C₄²×C₈₉	—	2	1	no	—
77	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
78	#107×C₈₉	—	2	2	no	—
79	#106×C₈₉	—	2	2	no	—
80	C₄⋊C₈×C₈₉	—	2	1	no	—
81	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
82	#103×C₈₉	—	2	2	no	—
83	#103×C₈₉	—	2	2	no	—
84	C₂³×D₈₉	—	4	1	no	—
85	C₂×D₃₅₆	—	2	2	no	—
86	C₂×D₃₅₆	—	2	2	no	—
87	C₂×D₃₅₆	—	2	2	no	—
88	C₂×D₃₅₆	—	4	2	no	—
89	C₂×D₃₅₆	—	4	2	no	—
90	C₂×D₃₅₆	—	4	1	no	—
91	C₂×D₃₅₆	—	4	1	no	—
92	C₂×D₃₅₆	—	4	1	no	—
93	C₂×D₃₅₆	—	8	2	no	—
94	C₂×D₃₅₆	—	8	2	no	—
95	#13083561	—	4	8	no	—
96	#13083561	—	8	4	no	—
97	C₂×#10623494	—	2	2	no	—
98	C₂×#10623494	—	2	2	no	—
99	C₂×#10623494	—	4	2	no	—
100	C₂×#10623494	C₂×D₄	1	4	yes	—
101	C₂×Q₈×C₈₉	—	2	1	no	—
102	C₂×Q₈×C₈₉	—	2	2	no	—
103	C₂×Q₈×C₈₉	—	2	1	no	—
104	C₂×Q₈×C₈₉	—	2	1	no	—
105	C₂×Q₈×C₈₉	SD₁₆	1	4	yes	complemented
106	C₂×Q₈×C₈₉	C₂×D₄	1	2	yes	—
107	C₂²×C₄×C₈₉	—	2	1	no	—
108	C₂²×C₄×C₈₉	—	2	1	no	—
109	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
112	M₁₆×C₈₉	—	4	2	no	—
113	C₂×C₈×C₈₉	—	2	2	no	—
114	C₄⋊C₄×C₈₉	—	2	2	no	—
115	C₄⋊C₄×C₈₉	—	2	1	no	—
116	C₄⋊C₄×C₈₉	—	2	2	no	—
117	C₄⋊C₄×C₈₉	—	2	1	no	—
118	C₄⋊C₄×C₈₉	—	2	1	no	—
119	C₄⋊C₄×C₈₉	—	2	1	no	—
120	C₄⋊C₄×C₈₉	—	2	1	no	—
121	C₄⋊C₄×C₈₉	—	4	1	no	—
122	C₄⋊C₄×C₈₉	—	4	1	no	—
123	C₄⋊C₄×C₈₉	C₂×D₄	1	2	yes	—
124	C₂²⋊C₄×C₈₉	—	4	1	no	—
125	C₄²×C₈₉	—	2	2	no	—
126	C₄²×C₈₉	—	2	2	no	—
127	C₄²×C₈₉	—	4	1	no	—
128	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
129	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
130	C₄²×C₈₉	#41	1	1	yes	characteristic
131	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
132	C₂³×C₈₉	#121	1	1	yes	characteristic
133	C₂²×D₈₉	—	4	2	no	—
134	C₂²×D₈₉	—	8	2	no	—
135	C₂²×D₈₉	—	8	2	no	—
136	C₂²×D₈₉	—	8	2	no	—
137	Q₈×C₈₉	—	2	2	no	—
138	Q₈×C₈₉	—	2	4	no	—
139	Q₈×C₈₉	—	4	2	no	—
140	Q₈×C₈₉	—	8	1	no	—
141	Q₈×C₈₉	C₂×SD₁₆	1	4	yes	—
142	C₂×C₄×C₈₉	—	2	2	no	—
143	C₂×C₄×C₈₉	—	2	4	no	—
144	C₂×C₄×C₈₉	—	2	2	no	—
145	C₂×C₄×C₈₉	—	2	2	no	—
146	C₂×C₄×C₈₉	—	4	1	no	—
147	C₂×C₄×C₈₉	—	4	1	no	—
148	C₂×C₄×C₈₉	—	4	1	no	—
149	C₂×C₄×C₈₉	—	8	1	no	—
150	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
151	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
152	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
153	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
154	C₂×C₄×C₈₉	#135	1	2	yes	—
155	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
156	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
158	D₃₅₆	—	4	2	no	—
159	D₃₅₆	—	4	2	no	—
160	D₃₅₆	—	4	4	no	—
161	D₃₅₆	—	8	2	no	—
162	D₃₅₆	—	8	2	no	—
163	D₃₅₆	—	8	2	no	—
164	D₃₅₆	—	8	2	no	—
165	C₈×C₈₉	—	4	2	no	—
166	#10623494	—	2	4	no	—
167	#10623494	—	4	2	no	—
168	#10623494	—	4	2	no	—
169	C₂²×C₈₉	—	2	1	no	—
170	C₂²×C₈₉	—	2	1	no	—
171	C₂²×C₈₉	#465	1	2	yes	—
172	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
173	C₂×D₈₉	—	8	2	no	—
174	C₂×D₈₉	—	8	2	no	—
175	C₂×D₈₉	—	8	2	no	—
176	C₂×D₈₉	—	16	4	no	—
177	C₄×C₈₉	—	2	1	no	—
178	C₄×C₈₉	—	2	4	no	—
179	C₄×C₈₉	—	2	1	no	—
180	C₄×C₈₉	—	2	1	no	—
181	C₄×C₈₉	—	4	2	no	—
182	C₄×C₈₉	—	8	1	no	—
183	C₄×C₈₉	—	8	1	no	—
184	C₄×C₈₉	#465	1	2	yes	—
185	C₄×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
186	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
187	C₄×C₈₉	#577	1	2	yes	—
188	#19396	—	89	1	no	maximal, Sylow
189	C₂×C₈₉	—	2	2	no	—
190	C₂×C₈₉	#3043	1	1	yes	characteristic
191	C₂×C₈₉	C₂×#465	1	1	yes	characteristic
192	C₂×C₈₉	#3068	1	1	yes	characteristic
193	D₈₉	—	16	2	no	—
194	#3208	—	89	1	no	—
195	#3078	D₈₉	1	1	yes	characteristic, complemented
196	C₂×#460	—	89	1	no	—
197	#2982	—	89	1	no	—
198	#1678	—	89	4	no	—
199	#1666	—	89	4	no	—
200	#1570	—	89	1	no	—
201	#1545	—	89	1	no	—
202	#1542	—	89	1	no	—
203	C₈₉	#19396	1	1	yes	characteristic, complemented, Sylow
204	C₂²×SD₁₆	—	178	1	no	—
205	#533	C₂×D₈₉	1	1	yes	characteristic
206	C₂×#127	—	89	1	no	—
207	C₂×C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
208	#492	—	89	2	no	—
209	#486	—	89	2	no	—
210	#479	—	2	2	no	—
211	#477	C₂×D₈₉	1	2	yes	complemented
212	#475	C₂×D₈₉	1	2	yes	complemented
213	#460	—	89	4	no	—
214	#460	—	178	2	no	—
215	#460	—	178	2	no	—
216	#450	—	178	2	no	—
217	C₈×Q₈	—	89	4	no	—
218	C₈×Q₈	—	178	2	no	—
219	#431	—	89	1	no	—
220	#431	—	89	1	no	—
221	#431	—	178	2	no	—
222	#428	—	2	1	no	—
223	#423	C₂×D₈₉	1	1	yes	characteristic
224	C₂×C₄⋊C₈	—	89	1	no	—
225	#419	—	2	1	no	—
226	C₂×#102	—	178	1	no	—
227	#334	—	89	4	no	—
228	#333	—	89	2	no	—
229	#332	—	89	2	no	—
230	#331	—	89	4	no	—
231	C₂²×Q₈	C₂²⋊D₈₉	1	1	yes	characteristic
232	C₂²×D₄	—	178	1	no	—
233	C₂²×D₄	—	178	1	no	—
234	C₂²×D₄	—	178	1	no	—
235	C₂×SD₁₆	—	356	4	no	—
236	C₂×SD₁₆	—	178	4	no	—
237	C₂×SD₁₆	—	178	4	no	—
238	C₂×M₁₆	—	2	1	no	—
239	C₂²×C₈	—	178	1	no	—
240	C₄⋊Q₈	C₂²×D₈₉	1	1	yes	characteristic
241	C₄⋊Q₈	C₂²×D₈₉	1	1	yes	characteristic
242	#127	—	89	2	no	—
243	#127	—	178	1	no	—
244	#127	—	178	1	no	—
245	#127	—	178	1	no	—
246	#125	—	2	1	no	—
247	C₂²⋊Q₈	—	2	1	no	—
248	C₂²⋊Q₈	—	2	1	no	—
249	C₄×Q₈	—	2	1	no	—
250	C₄×Q₈	—	2	1	no	—
251	C₄×Q₈	—	2	1	no	—
252	C₄×Q₈	—	2	1	no	—
253	C₄×Q₈	C₂²⋊D₈₉	1	2	yes	—
254	C₄×Q₈	C₂²×D₈₉	1	2	yes	—
255	#117	—	2	1	no	—
256	C₂×C₄⋊C₄	—	2	1	no	—
257	C₂×C₄⋊C₄	C₂²⋊D₈₉	1	1	yes	characteristic
258	C₂×C₄²	—	2	1	no	—
259	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
260	#107	—	2	2	no	—
261	#106	—	2	2	no	—
262	C₄⋊C₈	—	2	1	no	—
263	C₄⋊C₈	—	89	2	no	—
264	C₄⋊C₈	—	89	2	no	—
265	C₄⋊C₈	—	89	2	no	—
266	C₄⋊C₈	—	178	4	no	—
267	C₄⋊C₈	—	178	1	no	—
268	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
269	#103	—	2	2	no	—
270	#103	—	2	2	no	—
271	#102	—	178	4	no	—
272	#102	—	178	4	no	—
273	C₂²⋊C₈	—	89	2	no	—
274	C₂²⋊C₈	—	178	1	no	—
275	C₄×C₈	—	89	2	no	—
276	C₄×C₈	—	178	4	no	—
277	C₄×C₈	—	178	1	no	—
278	C₂⁴	—	356	1	no	—
279	C₂×Q₈	—	2	1	no	—
280	C₂×Q₈	—	2	2	no	—
281	C₂×Q₈	—	2	1	no	—
282	C₂×Q₈	—	2	1	no	—
283	C₂×Q₈	#13083561	1	4	yes	complemented
284	C₂×Q₈	C₂×C₂²⋊D₈₉	1	2	yes	—
285	C₂×D₄	—	356	1	no	—
286	C₂×D₄	—	356	1	no	—
287	C₂×D₄	—	356	1	no	—
288	C₂×D₄	—	356	2	no	—
289	C₂×D₄	—	356	2	no	—
290	C₂×D₄	—	178	2	no	—
291	C₂×D₄	—	178	2	no	—
292	C₂×D₄	—	178	2	no	—
293	C₂×D₄	—	712	2	no	—
294	C₂×D₄	—	712	2	no	—
295	C₂²×C₄	—	2	1	no	—
296	C₂²×C₄	—	2	1	no	—
297	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
298	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
299	C₂²×C₄	#13083573	1	1	yes	characteristic
300	SD₁₆	—	356	8	no	—
301	SD₁₆	—	712	4	no	—
302	M₁₆	—	4	2	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	89	4	no	—
305	C₂×C₈	—	356	2	no	—
306	C₂×C₈	—	178	2	no	—
307	C₂×C₈	—	178	2	no	—
308	C₄⋊C₄	—	2	2	no	—
309	C₄⋊C₄	—	2	1	no	—
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	—	2	1	no	—
312	C₄⋊C₄	—	2	1	no	—
313	C₄⋊C₄	—	2	1	no	—
314	C₄⋊C₄	—	2	1	no	—
315	C₄⋊C₄	—	4	1	no	—
316	C₄⋊C₄	—	4	1	no	—
317	C₄⋊C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
318	C₂²⋊C₄	—	4	1	no	—
319	C₄²	—	2	2	no	—
320	C₄²	—	2	2	no	—
321	C₄²	—	4	1	no	—
322	C₄²	D₄×D₈₉	1	1	yes	characteristic
323	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
324	C₄²	#13083573	1	1	yes	characteristic
325	C₄²	C₂³×D₈₉	1	1	yes	characteristic
326	C₂³	—	356	2	no	—
327	C₂³	—	712	2	no	—
328	C₂³	—	712	2	no	—
329	C₂³	—	712	2	no	—
330	C₂³	#15667984	1	1	yes	characteristic
331	Q₈	—	2	4	no	—
332	Q₈	—	2	2	no	—
333	Q₈	—	4	2	no	—
334	Q₈	—	8	1	no	—
335	Q₈	C₂×#13083561	1	4	yes	—
336	D₄	—	356	4	no	—
337	D₄	—	356	2	no	—
338	D₄	—	356	2	no	—
339	D₄	—	712	2	no	—
340	D₄	—	712	2	no	—
341	D₄	—	712	2	no	—
342	D₄	—	712	2	no	—
343	C₂×C₄	—	2	4	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	2	2	no	—
347	C₂×C₄	—	4	1	no	—
348	C₂×C₄	—	4	1	no	—
349	C₂×C₄	—	4	1	no	—
350	C₂×C₄	—	8	1	no	—
351	C₂×C₄	#15667984	1	1	yes	characteristic
352	C₂×C₄	#15667984	1	1	yes	characteristic
353	C₂×C₄	#15667984	1	1	yes	characteristic
354	C₂×C₄	C₂×#13083561	1	2	yes	—
355	C₂×C₄	#15668032	1	2	yes	—
356	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
357	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
358	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
359	C₈	—	4	2	no	—
360	C₈	—	356	2	no	—
361	C₈	—	356	2	no	—
362	C₈	—	178	4	no	—
363	C₂²	—	2	1	no	—
364	C₂²	—	2	1	no	—
365	C₂²	—	1424	4	no	—
366	C₂²	—	712	2	no	—
367	C₂²	—	712	2	no	—
368	C₂²	—	712	2	no	—
369	C₂²	#19151829	1	2	yes	—
370	C₂²	C₂×#15667984	1	1	yes	characteristic
371	C₄	—	2	1	no	—
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	2	4	no	—
375	C₄	—	4	2	no	—
376	C₄	—	8	1	no	—
377	C₄	—	8	1	no	—
378	C₄	#19151829	1	2	yes	—
379	C₄	C₂²×#13083561	1	1	yes	characteristic
380	C₄	C₂×#15668032	1	1	yes	characteristic
381	C₄	#19152407	1	2	yes	—
382	C₂	—	2	2	no	—
383	C₂	—	1424	2	no	—
384	C₂	#26029775	1	1	yes	characteristic
385	C₂	C₂×#19151829	1	1	yes	characteristic
386	C₂	#26029982	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#15667984
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#3078	C₂
socle	C₂²×C₈₉	C₂×#121
prime radical	C₂×#15668009	C₂²
hypercenter	#3078	D₈₉
hypocenter	C₈₉	#19396
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3078

Complements

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

N	H ≅ G/N	type
3. #26029333	C₂	semidirect
4. #26019670	C₂	semidirect
5. #26019602	C₂	semidirect
6. #26019024	C₂	semidirect
8. #26018909	C₂	semidirect
9. C₈₉×#3078	C₂	semidirect
19. Q₈×#10623494	C₂²	semidirect
25. #19151348	C₂²	semidirect
26. #19151347	C₂²	semidirect
27. #19151344	C₂²	semidirect
32. #477×C₈₉	C₂²	semidirect
33. #475×C₈₉	C₂²	semidirect
105. C₂×Q₈×C₈₉	SD₁₆	semidirect
195. #3078	D₈₉	semidirect
203. C₈₉	#19396	semidirect
211. #477	C₂×D₈₉	semidirect
212. #475	C₂×D₈₉	semidirect
283. C₂×Q₈	#13083561	semidirect

Maximal subgroups

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

subgroup	index
#26031190	2
C₂×#19151798	2
#26029333	2
#26019670	2
#26019602	2
#26019024	2
#26018921	2
#26018909	2
C₈₉×#3078	2
#19396	89