Subgroups — #36707844

Order 22784.

642 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	#26033386	C₂	1	1	yes	maximal, characteristic
2	#26029169	C₂	1	1	yes	maximal, characteristic, complemented
3	#26029168	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022135	C₂	1	4	yes	maximal, complemented
5	#26022130	C₂	1	4	yes	maximal, complemented
6	#26022046	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#19151406	C₂	1	1	yes	maximal, characteristic, complemented
8	#26021848	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#359×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#15668034	C₂²	1	1	yes	characteristic
11	#19152365	—	2	1	no	—
12	#19152364	—	2	1	no	—
13	#19152039	C₄	1	2	yes	complemented
14	#19152038	C₄	1	2	yes	complemented
15	#19152034	C₄	1	2	yes	complemented
16	#19152033	C₄	1	2	yes	complemented
17	#19152028	C₂²	1	4	yes	—
18	C₂×#15667952	C₂²	1	1	yes	characteristic
19	#19151720	—	2	2	no	—
20	#19151719	—	2	2	no	—
21	#19151715	C₂²	1	4	yes	complemented
22	#19151714	C₂²	1	4	yes	complemented
23	C₂×#15667948	C₂²	1	1	yes	characteristic
24	#19151615	—	2	1	no	—
25	#19151614	—	2	1	no	—
26	#19151553	C₂²	1	1	yes	characteristic
27	#19151427	C₂²	1	4	yes	complemented
28	#19151406	C₂²	1	2	yes	complemented
29	#19151406	C₂²	1	2	yes	complemented
30	#19151373	C₂²	1	2	yes	complemented
31	C₁₆×Dic₈₉	C₂²	1	2	yes	complemented
32	C₂×#135×C₈₉	C₂²	1	1	yes	characteristic
33	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
34	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
35	#359×C₈₉	C₂²	1	2	yes	complemented
36	#359×C₈₉	C₂²	1	2	yes	complemented
37	C₂×#13083574	—	2	1	no	—
38	#15668034	—	2	1	no	—
39	#15668034	—	2	1	no	—
40	#15668034	C₂×C₄	1	2	yes	—
41	#15668034	C₂³	1	2	yes	—
42	C₂×Dic₃₅₆	C₂×C₄	1	1	yes	characteristic
43	C₂×#13083563	—	2	1	no	—
44	C₂×D₇₁₂	C₂×C₄	1	1	yes	characteristic
45	#15668020	—	2	4	no	—
46	#15668019	—	2	4	no	—
47	#15668019	—	4	2	no	—
48	#15668018	—	4	2	no	—
49	C₂×#13083558	—	2	1	no	—
50	C₂×#13083558	—	4	2	no	—
51	C₂×#13083558	D₄	1	1	yes	characteristic
52	M₁₆×D₈₉	—	4	2	no	—
53	#15667971	D₄	1	1	yes	characteristic
54	#15667958	C₂×C₄	1	4	yes	—
55	#15667956	C₂×C₄	1	4	yes	—
56	#15667955	C₂×C₄	1	4	yes	—
57	#15667953	C₂×C₄	1	4	yes	—
58	#15667952	C₂×C₄	1	2	yes	—
59	#15667952	C₂×C₄	1	2	yes	—
60	C₂×#13083549	C₂³	1	2	yes	—
61	#15667948	—	2	1	no	—
62	#15667948	C₂³	1	2	yes	—
63	#15667939	—	2	4	no	—
64	#15667938	—	2	4	no	—
65	#15667922	D₄	1	2	yes	—
66	C₈×Dic₈₉	C₂³	1	2	yes	—
67	C₂×#41×C₈₉	—	2	1	no	—
68	#135×C₈₉	—	2	1	no	—
69	#135×C₈₉	—	2	1	no	—
70	#135×C₈₉	C₂×C₄	1	2	yes	—
71	#135×C₈₉	C₂³	1	2	yes	—
72	C₂×Q₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
73	C₂×SD₁₆×C₈₉	—	2	1	no	—
74	C₂×D₈×C₈₉	C₂×C₄	1	1	yes	characteristic
75	C₂×M₁₆×C₈₉	—	2	1	no	—
76	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
77	C₂×C₁₆×C₈₉	—	2	1	no	—
78	C₂×C₁₆×C₈₉	—	2	1	no	—
79	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
80	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
81	#108×C₈₉	—	2	1	no	—
82	#108×C₈₉	C₂³	1	2	yes	—
83	#13083574	—	2	2	no	—
84	#13083574	—	2	2	no	—
85	#13083574	—	4	1	no	—
86	#13083574	—	4	1	no	—
87	C₂×Dic₁₇₈	—	2	1	no	—
88	C₂×D₃₅₆	—	2	1	no	—
89	C₂×C₂²⋊D₈₉	—	4	1	no	—
90	C₂×C₄×D₈₉	—	4	1	no	—
91	Dic₃₅₆	—	2	1	no	—
92	Dic₃₅₆	C₂²×C₄	1	2	yes	—
93	#13083563	—	2	2	no	—
94	#13083563	—	2	2	no	—
95	D₇₁₂	—	2	1	no	—
96	D₇₁₂	C₂²×C₄	1	2	yes	—
97	#13083558	—	2	2	no	—
98	#13083558	—	4	1	no	—
99	#13083558	—	4	4	no	—
100	#13083558	D₈	1	2	yes	—
101	#13083558	D₈	1	2	yes	—
102	#13083557	—	4	4	no	—
103	C₂×#10623494	—	4	1	no	—
104	C₂×#10623494	—	4	4	no	—
105	C₂×#10623494	C₂×D₄	1	2	yes	—
106	C₈×D₈₉	—	4	4	no	—
107	C₄⋊Dic₈₉	—	2	1	no	—
108	C₂²⋊Dic₈₉	—	2	1	no	—
109	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
110	#13083549	C₂²×C₄	1	4	yes	—
111	#41×C₈₉	—	2	2	no	—
112	#41×C₈₉	—	2	2	no	—
113	#41×C₈₉	—	4	1	no	—
114	#41×C₈₉	—	4	1	no	—
115	C₂×Q₈×C₈₉	—	2	1	no	—
116	C₂×D₄×C₈₉	—	2	1	no	—
117	C₂×D₄×C₈₉	—	4	1	no	—
118	C₂²×C₄×C₈₉	—	4	1	no	—
119	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
120	Q₁₆×C₈₉	—	2	1	no	—
121	Q₁₆×C₈₉	C₂²×C₄	1	2	yes	—
122	SD₁₆×C₈₉	—	2	2	no	—
123	SD₁₆×C₈₉	—	2	2	no	—
124	D₈×C₈₉	—	2	1	no	—
125	D₈×C₈₉	C₂²×C₄	1	2	yes	—
126	M₁₆×C₈₉	—	2	2	no	—
127	M₁₆×C₈₉	—	4	1	no	—
128	C₂×C₈×C₈₉	—	4	1	no	—
129	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
132	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
133	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
134	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
135	C₁₆×C₈₉	—	2	1	no	—
136	C₁₆×C₈₉	—	2	1	no	—
137	C₂³×C₈₉	—	4	1	no	—
138	C₂³×C₈₉	#135	1	1	yes	characteristic
139	C₂²×D₈₉	—	4	1	no	—
140	Q₈×C₈₉	—	2	2	no	—
141	Q₈×C₈₉	—	4	1	no	—
142	D₄×C₈₉	—	2	2	no	—
143	D₄×C₈₉	—	4	2	no	—
144	D₄×C₈₉	—	4	2	no	—
145	D₄×C₈₉	—	4	1	no	—
146	C₂×C₄×C₈₉	—	4	1	no	—
147	C₂×C₄×C₈₉	—	4	2	no	—
148	C₂×C₄×C₈₉	—	4	2	no	—
149	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
150	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
151	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
152	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
153	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
154	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
155	C₂²⋊D₈₉	—	4	2	no	—
156	C₂²⋊D₈₉	—	4	2	no	—
157	C₂×Dic₈₉	—	2	2	no	—
158	C₂×Dic₈₉	—	4	1	no	—
159	D₃₅₆	—	2	2	no	—
160	D₃₅₆	—	4	1	no	—
161	C₄×D₈₉	—	4	2	no	—
162	C₄×D₈₉	—	4	2	no	—
163	Dic₁₇₈	—	2	2	no	—
164	Dic₁₇₈	—	4	1	no	—
165	C₈×C₈₉	—	4	2	no	—
166	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
167	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
168	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
169	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
170	#10623494	—	4	2	no	—
171	#10623494	C₂×D₈	1	4	yes	—
172	C₂²×C₈₉	—	4	1	no	—
173	C₂²×C₈₉	—	4	2	no	—
174	C₂²×C₈₉	—	16	1	no	—
175	C₂²×C₈₉	C₄×D₈	1	1	yes	characteristic
176	C₂²×C₈₉	C₄×D₈	1	1	yes	characteristic
177	C₂²×C₈₉	C₂×#135	1	1	yes	characteristic
178	C₂×D₈₉	—	4	2	no	—
179	C₂×D₈₉	—	16	1	no	—
180	C₄×C₈₉	—	4	2	no	—
181	C₄×C₈₉	C₄×D₈	1	1	yes	characteristic
182	C₄×C₈₉	C₄×D₈	1	1	yes	characteristic
183	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
184	C₄×C₈₉	C₂²×D₈	1	1	yes	characteristic
185	Dic₈₉	—	4	2	no	—
186	Dic₈₉	—	4	2	no	—
187	#33464	—	89	1	no	maximal, Sylow
188	C₂×C₈₉	—	2	1	no	—
189	C₂×C₈₉	—	2	1	no	—
190	C₂×C₈₉	—	2	1	no	—
191	C₂×C₈₉	—	8	2	no	—
192	C₂×C₈₉	C₂×C₄×D₈	1	1	yes	characteristic
193	D₈₉	—	8	2	no	—
194	#3414	—	89	1	no	—
195	#2957	—	89	1	no	—
196	#2957	—	89	1	no	—
197	#2179	—	89	4	no	—
198	#2178	—	89	4	no	—
199	#2161	—	89	1	no	—
200	C₂×#359	—	89	1	no	—
201	C₂×#359	D₈₉	1	1	yes	characteristic, complemented
202	#2108	—	89	1	no	—
203	C₈₉	#33464	1	1	yes	characteristic, complemented, Sylow
204	C₂×#135	—	89	1	no	—
205	C₂×#135	C₂×D₈₉	1	1	yes	characteristic
206	#568	—	178	1	no	—
207	#568	—	178	1	no	—
208	#510	—	89	2	no	—
209	#510	—	89	2	no	—
210	#509	—	89	2	no	—
211	#509	—	89	2	no	—
212	#508	—	89	4	no	—
213	C₂×M₃₂	—	89	1	no	—
214	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
215	#444	—	178	2	no	—
216	#444	—	178	2	no	—
217	#443	—	89	4	no	—
218	#443	—	89	4	no	—
219	C₂×#108	—	89	1	no	—
220	C₂×#108	C₂×D₈₉	1	1	yes	characteristic
221	#421	—	178	1	no	—
222	#421	—	178	1	no	—
223	#405	—	89	1	no	—
224	#368	—	89	4	no	—
225	#359	—	89	2	no	—
226	#359	—	89	2	no	—
227	#359	C₂×D₈₉	1	2	yes	complemented
228	#359	C₂×D₈₉	1	2	yes	complemented
229	#346	—	89	2	no	—
230	C₄×C₁₆	—	89	2	no	—
231	C₂×#41	—	2	1	no	—
232	C₂×#41	—	178	1	no	—
233	#135	—	2	1	no	—
234	#135	—	2	1	no	—
235	#135	—	89	2	no	—
236	#135	—	89	2	no	—
237	#135	—	178	1	no	—
238	#135	—	178	1	no	—
239	#135	C₄×D₈₉	1	2	yes	—
240	#135	C₂²×D₈₉	1	2	yes	—
241	C₂×Q₁₆	—	89	1	no	—
242	C₂×Q₁₆	C₄×D₈₉	1	1	yes	characteristic
243	C₂×SD₁₆	—	2	1	no	—
244	C₂×SD₁₆	—	178	1	no	—
245	C₂×D₈	—	89	1	no	—
246	C₂×D₈	C₄×D₈₉	1	1	yes	characteristic
247	#131	—	356	2	no	—
248	#131	—	356	2	no	—
249	#131	—	178	4	no	—
250	#131	—	178	4	no	—
251	C₂×M₁₆	—	2	1	no	—
252	C₂×M₁₆	—	89	1	no	—
253	C₂×M₁₆	—	356	2	no	—
254	C₂×M₁₆	—	356	2	no	—
255	C₂×M₁₆	—	178	1	no	—
256	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
257	#117	—	89	1	no	—
258	Q₃₂	—	89	4	no	—
259	SD₃₂	—	89	4	no	—
260	SD₃₂	—	89	4	no	—
261	D₁₆	—	89	4	no	—
262	M₃₂	—	89	2	no	—
263	M₃₂	—	89	2	no	—
264	C₂×C₁₆	—	2	1	no	—
265	C₂×C₁₆	—	2	1	no	—
266	C₂×C₁₆	—	89	2	no	—
267	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
268	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
269	#108	—	2	1	no	—
270	#108	—	89	2	no	—
271	#108	—	178	1	no	—
272	#108	C₂²×D₈₉	1	2	yes	—
273	#104	—	178	4	no	—
274	#104	—	178	4	no	—
275	#97	—	89	2	no	—
276	C₄×C₈	—	89	2	no	—
277	#41	—	2	2	no	—
278	#41	—	2	2	no	—
279	#41	—	4	1	no	—
280	#41	—	4	1	no	—
281	#41	—	356	1	no	—
282	#41	—	356	1	no	—
283	#41	—	178	2	no	—
284	#41	—	178	2	no	—
285	C₂×Q₈	—	2	1	no	—
286	C₂×Q₈	—	178	1	no	—
287	C₂×D₄	—	2	1	no	—
288	C₂×D₄	—	4	1	no	—
289	C₂×D₄	—	356	1	no	—
290	C₂×D₄	—	178	1	no	—
291	C₂²×C₄	—	4	1	no	—
292	C₂²×C₄	—	356	1	no	—
293	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
294	Q₁₆	—	2	1	no	—
295	Q₁₆	—	89	2	no	—
296	Q₁₆	—	178	1	no	—
297	Q₁₆	C₂×C₄×D₈₉	1	2	yes	—
298	SD₁₆	—	2	2	no	—
299	SD₁₆	—	2	2	no	—
300	SD₁₆	—	178	2	no	—
301	SD₁₆	—	178	2	no	—
302	D₈	—	2	1	no	—
303	D₈	—	89	2	no	—
304	D₈	—	178	1	no	—
305	D₈	C₂×C₄×D₈₉	1	2	yes	—
306	M₁₆	—	2	2	no	—
307	M₁₆	—	4	1	no	—
308	M₁₆	—	89	2	no	—
309	M₁₆	—	89	2	no	—
310	M₁₆	—	356	4	no	—
311	M₁₆	—	356	4	no	—
312	M₁₆	—	356	1	no	—
313	M₁₆	—	178	2	no	—
314	C₂×C₈	—	4	1	no	—
315	C₂×C₈	—	89	2	no	—
316	C₂×C₈	—	356	4	no	—
317	C₂×C₈	—	356	4	no	—
318	C₂×C₈	—	356	1	no	—
319	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
320	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
321	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
322	C₂×C₈	#13083573	1	1	yes	characteristic
323	C₂×C₈	#13083573	1	1	yes	characteristic
324	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
325	C₄⋊C₄	—	178	1	no	—
326	C₂²⋊C₄	—	178	1	no	—
327	C₄²	—	89	2	no	—
328	C₁₆	—	2	1	no	—
329	C₁₆	—	2	1	no	—
330	C₁₆	—	89	4	no	—
331	C₂³	—	4	1	no	—
332	C₂³	—	356	1	no	—
333	C₂³	#15668033	1	1	yes	characteristic
334	Q₈	—	2	2	no	—
335	Q₈	—	4	1	no	—
336	Q₈	—	356	1	no	—
337	Q₈	—	178	2	no	—
338	D₄	—	2	2	no	—
339	D₄	—	4	2	no	—
340	D₄	—	4	2	no	—
341	D₄	—	4	1	no	—
342	D₄	—	356	2	no	—
343	D₄	—	356	2	no	—
344	D₄	—	356	1	no	—
345	D₄	—	178	2	no	—
346	C₂×C₄	—	4	1	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	4	2	no	—
349	C₂×C₄	—	356	1	no	—
350	C₂×C₄	—	356	2	no	—
351	C₂×C₄	—	356	2	no	—
352	C₂×C₄	—	178	2	no	—
353	C₂×C₄	#15667972	1	1	yes	characteristic
354	C₂×C₄	#15667972	1	1	yes	characteristic
355	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
356	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
357	C₂×C₄	#15668033	1	1	yes	characteristic
358	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
359	C₈	—	4	2	no	—
360	C₈	—	89	4	no	—
361	C₈	—	356	2	no	—
362	C₈	#15667972	1	1	yes	characteristic
363	C₈	#15667972	1	1	yes	characteristic
364	C₈	C₂²×C₄×D₈₉	1	1	yes	characteristic
365	C₈	C₂×#13083573	1	1	yes	characteristic
366	C₂²	—	4	1	no	—
367	C₂²	—	4	2	no	—
368	C₂²	—	16	1	no	—
369	C₂²	—	356	2	no	—
370	C₂²	—	1424	1	no	—
371	C₂²	#19151678	1	1	yes	characteristic
372	C₂²	#19151678	1	1	yes	characteristic
373	C₂²	C₂×#15668033	1	1	yes	characteristic
374	C₄	—	4	2	no	—
375	C₄	—	356	2	no	—
376	C₄	—	356	2	no	—
377	C₄	#19151678	1	1	yes	characteristic
378	C₄	#19151678	1	1	yes	characteristic
379	C₄	C₂×#15667972	1	1	yes	characteristic
380	C₄	C₂×D₈×D₈₉	1	1	yes	characteristic
381	C₂	—	2	1	no	—
382	C₂	—	2	1	no	—
383	C₂	—	2	1	no	—
384	C₂	—	8	2	no	—
385	C₂	—	712	2	no	—
386	C₂	C₂×#19151678	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₈×D₈₉
commutator	C₈×C₈₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×#359×C₈₉	C₂
socle	C₂×C₈₉	C₂×C₄×D₈
prime radical	#26033386	C₂
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₂×#359	D₈₉
hypocenter	C₈₉	#33464
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#359

Complements

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

N	H ≅ G/N	type
2. #26029169	C₂	semidirect
3. #26029168	C₂	semidirect
4. #26022135	C₂	semidirect
5. #26022130	C₂	semidirect
6. #26022046	C₂	semidirect
7. C₂×#19151406	C₂	semidirect
8. #26021848	C₂	semidirect
9. C₂×#359×C₈₉	C₂	semidirect
13. #19152039	C₄	semidirect
14. #19152038	C₄	semidirect
15. #19152034	C₄	semidirect
16. #19152033	C₄	semidirect
21. #19151715	C₂²	semidirect
22. #19151714	C₂²	semidirect
27. #19151427	C₂²	semidirect
28. #19151406	C₂²	semidirect
29. #19151406	C₂²	semidirect
30. #19151373	C₂²	semidirect
31. C₁₆×Dic₈₉	C₂²	semidirect
35. #359×C₈₉	C₂²	semidirect
36. #359×C₈₉	C₂²	semidirect
201. C₂×#359	D₈₉	semidirect
203. C₈₉	#33464	semidirect
227. #359	C₂×D₈₉	semidirect
228. #359	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26033386	2
#26029169	2
#26029168	2
#26022135	2
#26022130	2
#26022046	2
C₂×#19151406	2
#26021848	2
C₂×#359×C₈₉	2
#33464	89