Subgroups — #36707781

Order 22784.

660 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19152039	C₂	1	1	yes	maximal, characteristic
2	#26029129	C₂	1	1	yes	maximal, characteristic
3	#26022121	C₂	1	2	yes	maximal, complemented
4	#26022114	C₂	1	2	yes	maximal, complemented
5	C₄×#15667958	C₂	1	2	yes	maximal, complemented
6	C₄×#15667955	C₂	1	2	yes	maximal, complemented
7	#26022041	C₂	1	1	yes	maximal, characteristic, complemented
8	#26021957	C₂	1	1	yes	maximal, characteristic
9	C₂×#19151402	C₂	1	1	yes	maximal, characteristic, complemented
10	#26021849	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#2950	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15668034	C₂²	1	1	yes	characteristic
13	#19152078	—	2	1	no	—
14	#19152039	C₄	1	8	yes	complemented
15	C₂×#15667958	C₂²	1	2	yes	—
16	C₂×#15667955	C₂²	1	2	yes	—
17	C₂×#15667952	C₂²	1	1	yes	characteristic
18	C₂²×#13083549	C₂²	1	1	yes	characteristic
19	#19151700	—	2	2	no	—
20	C₄×Dic₃₅₆	C₂²	1	2	yes	—
21	C₄×D₇₁₂	C₂²	1	2	yes	—
22	C₂×#15667946	C₂²	1	1	yes	characteristic
23	#19151600	—	2	1	no	—
24	#19151425	C₂²	1	2	yes	—
25	#19151423	C₂²	1	2	yes	—
26	#19151402	C₂²	1	2	yes	—
27	#19151402	C₂²	1	2	yes	—
28	#19151401	C₂²	1	1	yes	characteristic
29	#19151401	C₂²	1	1	yes	characteristic
30	#19151398	C₂²	1	1	yes	characteristic
31	#19151398	C₂²	1	1	yes	characteristic
32	#19151374	C₂²	1	2	yes	—
33	C₄×#13083549	C₂²	1	2	yes	—
34	C₂²×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
35	#519×C₈₉	—	2	1	no	—
36	#441×C₈₉	—	2	2	no	—
37	C₄×Q₁₆×C₈₉	C₂²	1	2	yes	—
38	C₄×Q₁₆×C₈₉	C₂²	1	2	yes	—
39	#427×C₈₉	C₂²	1	1	yes	characteristic
40	#419×C₈₉	—	2	1	no	—
41	#405×C₈₉	C₂²	1	1	yes	characteristic
42	C₂×#13083574	—	2	1	no	—
43	#15668034	—	2	2	no	—
44	#15668034	C₂×C₄	1	4	yes	—
45	C₂×Dic₃₅₆	C₂³	1	1	yes	characteristic
46	C₂×#13083563	—	2	1	no	—
47	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
48	C₄×Dic₁₇₈	—	2	2	no	—
49	C₄×D₃₅₆	—	2	2	no	—
50	#15667972	—	4	2	no	—
51	#15667970	—	4	1	no	—
52	C₂×C₄⋊Dic₈₉	—	2	1	no	—
53	C₄⋊C₄×D₈₉	—	4	1	no	—
54	#15667958	C₂×C₄	1	8	yes	complemented
55	#15667955	C₂×C₄	1	8	yes	complemented
56	#15667952	C₂×C₄	1	4	yes	complemented
57	C₂×#13083549	—	2	2	no	—
58	C₂×#13083549	C₂³	1	1	yes	characteristic
59	C₂×#13083549	C₂³	1	1	yes	characteristic
60	C₂×#13083549	C₂³	1	2	yes	—
61	#15667946	—	2	1	no	—
62	#15667946	C₂³	1	2	yes	—
63	#15667937	—	2	2	no	—
64	#15667934	—	2	2	no	—
65	C₂²×Q₈×C₈₉	—	2	1	no	—
66	C₂×Q₁₆×C₈₉	—	2	1	no	—
67	C₂×Q₁₆×C₈₉	—	2	2	no	—
68	C₂×Q₁₆×C₈₉	C₂×C₄	1	4	yes	complemented
69	C₂×Q₁₆×C₈₉	C₂³	1	2	yes	—
70	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
71	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
72	C₄×Q₈×C₈₉	—	2	2	no	—
73	C₄×Q₈×C₈₉	—	2	2	no	—
74	C₄×Q₈×C₈₉	—	4	1	no	—
75	C₄×Q₈×C₈₉	—	4	1	no	—
76	#117×C₈₉	—	2	1	no	—
77	#117×C₈₉	—	4	2	no	—
78	#117×C₈₉	D₄	1	1	yes	characteristic
79	#107×C₈₉	C₂³	1	1	yes	characteristic
80	#107×C₈₉	C₂³	1	1	yes	characteristic
81	#106×C₈₉	—	2	1	no	—
82	#103×C₈₉	—	2	2	no	—
83	#103×C₈₉	—	2	2	no	—
84	#97×C₈₉	D₄	1	2	yes	—
85	C₄×C₈×C₈₉	C₂³	1	2	yes	—
86	#13083574	—	2	4	no	—
87	#13083574	—	4	2	no	—
88	C₂×Dic₁₇₈	—	2	1	no	—
89	C₂×D₃₅₆	—	2	1	no	—
90	C₂×C₂²⋊D₈₉	—	4	1	no	—
91	C₂²×Dic₈₉	—	4	1	no	—
92	C₂×C₄×D₈₉	—	4	1	no	—
93	C₂×C₄×D₈₉	—	4	2	no	—
94	Dic₃₅₆	—	2	1	no	—
95	Dic₃₅₆	C₂²×C₄	1	2	yes	—
96	#13083563	—	2	4	no	—
97	D₇₁₂	—	2	1	no	—
98	D₇₁₂	C₂²×C₄	1	2	yes	—
99	C₄⋊Dic₈₉	—	2	2	no	—
100	C₄⋊Dic₈₉	—	4	1	no	—
101	#13083553	—	4	2	no	—
102	#13083551	—	4	2	no	—
103	C₄×Dic₈₉	—	4	2	no	—
104	#13083549	—	2	1	no	—
105	#13083549	—	2	1	no	—
106	#13083549	C₂²×C₄	1	4	yes	—
107	C₂×Q₈×C₈₉	—	2	4	no	—
108	C₂×Q₈×C₈₉	—	2	2	no	—
109	C₂×Q₈×C₈₉	—	4	1	no	—
110	C₂×Q₈×C₈₉	—	4	2	no	—
111	C₂²×C₄×C₈₉	—	4	1	no	—
112	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
113	Q₁₆×C₈₉	—	2	4	no	—
114	Q₁₆×C₈₉	—	2	2	no	—
115	Q₁₆×C₈₉	C₂²×C₄	1	4	yes	—
116	M₁₆×C₈₉	—	2	2	no	—
117	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
118	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
119	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₈₉	#41	1	2	yes	—
121	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
122	C₄⋊C₄×C₈₉	—	2	1	no	—
123	C₄⋊C₄×C₈₉	—	2	1	no	—
124	C₄⋊C₄×C₈₉	—	4	2	no	—
125	C₄⋊C₄×C₈₉	—	4	2	no	—
126	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
127	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
128	C₂²⋊C₄×C₈₉	—	4	1	no	—
129	C₂²⋊C₄×C₈₉	D₈	1	2	yes	—
130	C₄²×C₈₉	—	4	1	no	—
131	C₄²×C₈₉	—	4	2	no	—
132	C₄²×C₈₉	—	4	2	no	—
133	C₄²×C₈₉	C₂×D₄	1	2	yes	—
134	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
135	C₂²×D₈₉	—	4	1	no	—
136	Q₈×C₈₉	—	2	4	no	—
137	Q₈×C₈₉	—	4	2	no	—
138	Q₈×C₈₉	—	4	4	no	—
139	C₂×C₄×C₈₉	—	4	2	no	—
140	C₂×C₄×C₈₉	—	4	4	no	—
141	C₂×C₄×C₈₉	—	4	1	no	—
142	C₂×C₄×C₈₉	—	4	1	no	—
143	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
144	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
145	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
146	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₈₉	#135	1	2	yes	—
148	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
149	C₂²⋊D₈₉	—	4	4	no	—
150	C₂×Dic₈₉	—	4	1	no	—
151	C₂×Dic₈₉	—	4	2	no	—
152	C₂×Dic₈₉	—	16	1	no	—
153	D₃₅₆	—	2	2	no	—
154	D₃₅₆	—	4	1	no	—
155	C₄×D₈₉	—	4	4	no	—
156	C₄×D₈₉	—	16	2	no	—
157	Dic₁₇₈	—	2	2	no	—
158	Dic₁₇₈	—	4	1	no	—
159	C₈×C₈₉	—	2	2	no	—
160	C₈×C₈₉	C₄×D₄	1	2	yes	—
161	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
162	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
163	C₂²×C₈₉	—	2	1	no	—
164	C₂²×C₈₉	—	2	1	no	—
165	C₂²×C₈₉	C₄×D₈	1	2	yes	—
166	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
167	C₂×D₈₉	—	4	2	no	—
168	C₂×D₈₉	—	8	1	no	—
169	C₂×D₈₉	—	8	1	no	—
170	C₄×C₈₉	—	2	2	no	—
171	C₄×C₈₉	—	2	2	no	—
172	C₄×C₈₉	—	4	4	no	—
173	C₄×C₈₉	—	8	1	no	—
174	C₄×C₈₉	—	8	1	no	—
175	C₄×C₈₉	C₄×D₈	1	2	yes	—
176	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
177	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
178	Dic₈₉	—	4	2	no	—
179	Dic₈₉	—	8	2	no	—
180	#33455	—	89	1	no	maximal, Sylow
181	C₂×C₈₉	—	2	2	no	—
182	C₂×C₈₉	C₂×C₄×D₈	1	1	yes	characteristic
183	C₂×C₈₉	#3416	1	1	yes	characteristic
184	C₂×C₈₉	#3417	1	1	yes	characteristic
185	D₈₉	—	8	2	no	—
186	C₂×#510	—	89	1	no	—
187	#2951	—	89	1	no	—
188	#2950	D₈₉	1	1	yes	characteristic, complemented
189	#2176	—	89	2	no	—
190	#2175	—	89	2	no	—
191	C₄×Q₃₂	—	89	2	no	—
192	C₄×SD₃₂	—	89	2	no	—
193	#2159	—	89	1	no	—
194	#2138	—	89	1	no	—
195	C₂×#358	—	89	1	no	—
196	#2108	—	89	1	no	—
197	C₈₉	#33455	1	1	yes	characteristic, complemented, Sylow
198	C₂×#135	—	89	1	no	—
199	C₂²×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
200	#520	—	178	1	no	—
201	#519	—	2	1	no	—
202	#510	—	89	8	no	—
203	C₂×Q₃₂	—	89	2	no	—
204	C₂×SD₃₂	—	89	2	no	—
205	C₂×M₃₂	—	89	1	no	—
206	C₂²×C₁₆	—	89	1	no	—
207	#441	—	2	2	no	—
208	#440	—	178	2	no	—
209	C₄×Q₁₆	—	89	2	no	—
210	C₄×Q₁₆	C₂×D₈₉	1	2	yes	—
211	C₄×Q₁₆	C₂×D₈₉	1	2	yes	—
212	C₄×D₈	—	89	2	no	—
213	#427	C₂×D₈₉	1	1	yes	characteristic
214	C₂×#107	—	89	1	no	—
215	#419	—	2	1	no	—
216	#417	—	178	1	no	—
217	#405	C₂×D₈₉	1	1	yes	characteristic
218	#367	—	89	2	no	—
219	#366	—	89	2	no	—
220	#358	—	89	2	no	—
221	#358	—	89	2	no	—
222	#358	—	89	1	no	—
223	#358	—	89	1	no	—
224	#357	—	89	1	no	—
225	#357	—	89	1	no	—
226	#346	—	89	2	no	—
227	C₄×C₁₆	—	89	2	no	—
228	C₂×#41	—	178	1	no	—
229	C₂²×Q₈	—	2	1	no	—
230	#135	—	89	4	no	—
231	#135	—	178	2	no	—
232	C₂×Q₁₆	—	2	1	no	—
233	C₂×Q₁₆	—	2	2	no	—
234	C₂×Q₁₆	—	89	1	no	—
235	C₂×Q₁₆	C₄×D₈₉	1	4	yes	complemented
236	C₂×Q₁₆	C₂²×D₈₉	1	2	yes	—
237	C₂×SD₁₆	—	178	1	no	—
238	C₂×D₈	—	89	1	no	—
239	C₂×M₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
240	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
241	C₄×Q₈	—	2	2	no	—
242	C₄×Q₈	—	2	2	no	—
243	C₄×Q₈	—	4	1	no	—
244	C₄×Q₈	—	4	1	no	—
245	C₄×Q₈	—	178	2	no	—
246	C₄×D₄	—	356	2	no	—
247	C₄×D₄	—	178	2	no	—
248	#117	—	2	1	no	—
249	#117	—	4	2	no	—
250	#117	—	356	1	no	—
251	#117	C₂²⋊D₈₉	1	1	yes	characteristic
252	C₂×C₄⋊C₄	—	356	1	no	—
253	C₂×C₄⋊C₄	—	178	1	no	—
254	Q₃₂	—	89	8	no	—
255	SD₃₂	—	89	8	no	—
256	M₃₂	—	89	4	no	—
257	C₂×C₁₆	—	89	2	no	—
258	C₂×C₁₆	—	89	1	no	—
259	C₂×C₁₆	—	89	1	no	—
260	C₂×C₁₆	—	178	2	no	—
261	#107	—	89	2	no	—
262	#107	—	178	1	no	—
263	#107	C₂²×D₈₉	1	1	yes	characteristic
264	#107	C₂²×D₈₉	1	1	yes	characteristic
265	#106	—	2	1	no	—
266	#103	—	2	2	no	—
267	#103	—	2	2	no	—
268	#103	—	178	2	no	—
269	#102	—	178	2	no	—
270	#97	C₂²⋊D₈₉	1	2	yes	—
271	C₄×C₈	C₂²×D₈₉	1	2	yes	—
272	#41	—	356	2	no	—
273	#41	—	178	4	no	—
274	C₂×Q₈	—	2	4	no	—
275	C₂×Q₈	—	2	2	no	—
276	C₂×Q₈	—	4	1	no	—
277	C₂×Q₈	—	4	2	no	—
278	C₂×Q₈	—	178	1	no	—
279	C₂×D₄	—	356	1	no	—
280	C₂×D₄	—	178	1	no	—
281	C₂²×C₄	—	4	1	no	—
282	C₂²×C₄	—	356	1	no	—
283	C₂²×C₄	—	356	1	no	—
284	C₂²×C₄	—	356	2	no	—
285	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
286	Q₁₆	—	2	4	no	—
287	Q₁₆	—	2	2	no	—
288	Q₁₆	—	89	2	no	—
289	Q₁₆	—	178	1	no	—
290	Q₁₆	C₂×C₄×D₈₉	1	4	yes	—
291	SD₁₆	—	178	4	no	—
292	D₈	—	89	2	no	—
293	D₈	—	178	1	no	—
294	M₁₆	—	2	2	no	—
295	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
296	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
297	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
298	C₂×C₈	#13083574	1	2	yes	—
299	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	—	2	1	no	—
302	C₄⋊C₄	—	4	2	no	—
303	C₄⋊C₄	—	4	2	no	—
304	C₄⋊C₄	—	356	1	no	—
305	C₄⋊C₄	—	356	2	no	—
306	C₄⋊C₄	—	178	2	no	—
307	C₄⋊C₄	#13083559	1	1	yes	characteristic
308	C₄⋊C₄	#13083559	1	1	yes	characteristic
309	C₂²⋊C₄	—	4	1	no	—
310	C₂²⋊C₄	—	356	2	no	—
311	C₂²⋊C₄	#13083559	1	2	yes	—
312	C₄²	—	4	1	no	—
313	C₄²	—	4	2	no	—
314	C₄²	—	4	2	no	—
315	C₄²	—	356	2	no	—
316	C₄²	C₂×C₂²⋊D₈₉	1	2	yes	—
317	C₁₆	—	89	4	no	—
318	C₁₆	—	178	1	no	—
319	C₁₆	—	178	1	no	—
320	C₂³	—	356	1	no	—
321	C₂³	C₂×#13083559	1	1	yes	characteristic
322	Q₈	—	2	4	no	—
323	Q₈	—	4	4	no	—
324	Q₈	—	4	2	no	—
325	Q₈	—	356	1	no	—
326	Q₈	—	178	2	no	—
327	D₄	—	356	4	no	—
328	D₄	—	356	1	no	—
329	D₄	—	178	2	no	—
330	C₂×C₄	—	4	4	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	356	1	no	—
335	C₂×C₄	—	356	4	no	—
336	C₂×C₄	—	356	2	no	—
337	C₂×C₄	—	1424	2	no	—
338	C₂×C₄	—	1424	1	no	—
339	C₂×C₄	C₄×C₂²⋊D₈₉	1	2	yes	—
340	C₂×C₄	C₂×#13083559	1	2	yes	—
341	C₂×C₄	C₂×#13083559	1	2	yes	—
342	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
343	C₂×C₄	#15668032	1	2	yes	—
344	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
345	C₈	—	2	2	no	—
346	C₈	C₄×C₂²⋊D₈₉	1	2	yes	—
347	C₈	C₂²×C₄×D₈₉	1	1	yes	characteristic
348	C₈	C₂×#13083574	1	1	yes	characteristic
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	356	2	no	—
352	C₂²	—	712	1	no	—
353	C₂²	—	712	1	no	—
354	C₂²	C₄×#13083559	1	2	yes	—
355	C₂²	C₂²×#13083559	1	1	yes	characteristic
356	C₄	—	2	2	no	—
357	C₄	—	2	2	no	—
358	C₄	—	4	4	no	—
359	C₄	—	8	1	no	—
360	C₄	—	8	1	no	—
361	C₄	—	356	2	no	—
362	C₄	—	712	2	no	—
363	C₄	C₄×#13083559	1	2	yes	—
364	C₄	C₂×C₄×C₂²⋊D₈₉	1	1	yes	characteristic
365	C₄	C₂×#15668032	1	1	yes	characteristic
366	C₂	—	2	2	no	—
367	C₂	—	712	2	no	—
368	C₂	C₂×C₄×#13083559	1	1	yes	characteristic
369	C₂	#26033396	1	1	yes	characteristic
370	C₂	#26033403	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#13083559
commutator	C₈×C₈₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2950	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×#15668034	C₂²
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2950	D₈₉
hypocenter	C₈₉	#33455
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2950

Complements

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

N	H ≅ G/N	type
3. #26022121	C₂	semidirect
4. #26022114	C₂	semidirect
5. C₄×#15667958	C₂	semidirect
6. C₄×#15667955	C₂	semidirect
7. #26022041	C₂	semidirect
9. C₂×#19151402	C₂	semidirect
10. #26021849	C₂	semidirect
11. C₈₉×#2950	C₂	semidirect
14. #19152039	C₄	semidirect
54. #15667958	C₂×C₄	semidirect
55. #15667955	C₂×C₄	semidirect
56. #15667952	C₂×C₄	semidirect
68. C₂×Q₁₆×C₈₉	C₂×C₄	semidirect
188. #2950	D₈₉	semidirect
197. C₈₉	#33455	semidirect
235. C₂×Q₁₆	C₄×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19152039	2
#26029129	2
#26022121	2
#26022114	2
C₄×#15667958	2
C₄×#15667955	2
#26022041	2
#26021957	2
C₂×#19151402	2
#26021849	2
C₈₉×#2950	2
#33455	89