Subgroups — #36707889

Order 22784.

646 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26033388	C₂	1	1	yes	maximal, characteristic
2	#26029170	C₂	1	1	yes	maximal, characteristic, complemented
3	#26029137	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022152	C₂	1	4	yes	maximal, complemented
5	#26022145	C₂	1	4	yes	maximal, complemented
6	C₂×#19151421	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022002	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#19151409	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#2110	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#15668034	C₂²	1	1	yes	characteristic
11	C₂×#15668034	C₂²	1	1	yes	characteristic
12	#19152364	—	2	1	no	—
13	#19152078	—	2	1	no	—
14	#19152041	C₄	1	2	yes	complemented
15	#19152041	C₂²	1	2	yes	—
16	#19152036	C₄	1	2	yes	complemented
17	#19152036	C₂²	1	2	yes	—
18	#19152030	C₄	1	4	yes	complemented
19	#19151721	C₂²	1	4	yes	—
20	#19151716	—	2	2	no	—
21	#19151711	C₂²	1	2	yes	complemented
22	#19151706	C₂²	1	2	yes	complemented
23	C₄×#13083563	—	2	2	no	—
24	C₂×#15667948	C₂²	1	1	yes	characteristic
25	C₂×#15667944	C₂²	1	1	yes	characteristic
26	#19151616	—	2	1	no	—
27	#19151600	—	2	1	no	—
28	#19151421	C₂²	1	4	yes	complemented
29	#19151415	C₂²	1	2	yes	complemented
30	#19151412	C₂²	1	2	yes	complemented
31	#19151409	C₂²	1	4	yes	—
32	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
33	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
34	#405×C₈₉	C₂²	1	1	yes	characteristic
35	#347×C₈₉	C₂²	1	4	yes	complemented
36	C₂×#13083574	—	2	1	no	—
37	C₂×#13083574	—	2	1	no	—
38	#15668034	—	2	1	no	—
39	#15668034	—	2	1	no	—
40	#15668034	—	2	1	no	—
41	#15668034	—	2	1	no	—
42	#15668034	C₂×C₄	1	2	yes	—
43	#15668034	C₂×C₄	1	2	yes	—
44	#15668034	C₂×C₄	1	2	yes	complemented
45	#15668034	C₂³	1	2	yes	—
46	C₂×Dic₃₅₆	C₂×C₄	1	1	yes	characteristic, complemented
47	C₂×Dic₃₅₆	C₂³	1	1	yes	characteristic
48	C₂×#13083563	—	2	1	no	—
49	C₂×#13083563	—	2	1	no	—
50	C₂×D₇₁₂	C₂×C₄	1	1	yes	characteristic, complemented
51	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
52	#15668020	—	4	2	no	—
53	#15668018	—	2	4	no	—
54	C₂×#13083558	—	2	1	no	—
55	M₁₆×D₈₉	—	4	2	no	—
56	C₄×Dic₁₇₈	—	2	2	no	—
57	C₄×D₃₅₆	—	2	2	no	—
58	#15667972	—	4	2	no	—
59	#15667970	—	4	1	no	—
60	C₂×C₄⋊Dic₈₉	—	2	1	no	—
61	C₄⋊C₄×D₈₉	—	4	1	no	—
62	Dic₇₁₂	C₂×C₄	1	4	yes	complemented
63	#15667957	C₂×C₄	1	4	yes	complemented
64	#15667957	C₂×C₄	1	4	yes	complemented
65	D₁₄₂₄	C₂×C₄	1	4	yes	complemented
66	#15667948	—	2	1	no	—
67	#15667948	C₂³	1	2	yes	—
68	#15667944	—	2	1	no	—
69	#15667944	C₂³	1	2	yes	—
70	#15667940	—	2	4	no	—
71	#15667937	—	2	2	no	—
72	#15667934	—	2	2	no	—
73	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
74	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
75	#117×C₈₉	D₄	1	1	yes	characteristic
76	M₃₂×C₈₉	—	2	1	no	—
77	M₃₂×C₈₉	C₂×C₄	1	2	yes	complemented
78	M₃₂×C₈₉	C₂³	1	2	yes	—
79	M₃₂×C₈₉	C₂³	1	2	yes	—
80	C₂×C₁₆×C₈₉	—	2	1	no	—
81	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	complemented
82	#97×C₈₉	C₂³	1	2	yes	—
83	C₄×C₈×C₈₉	D₄	1	2	yes	—
84	#13083574	—	2	2	no	—
85	#13083574	—	2	2	no	—
86	#13083574	—	2	2	no	—
87	#13083574	—	2	2	no	—
88	#13083574	—	4	1	no	—
89	#13083574	—	4	1	no	—
90	#13083574	—	4	1	no	—
91	#13083574	—	4	1	no	—
92	C₂×Dic₁₇₈	—	2	1	no	—
93	C₂×Dic₁₇₈	—	2	1	no	—
94	C₂×D₃₅₆	—	2	1	no	—
95	C₂×D₃₅₆	—	2	1	no	—
96	C₂×C₂²⋊D₈₉	—	4	1	no	—
97	C₂×C₂²⋊D₈₉	—	4	1	no	—
98	C₂²×Dic₈₉	—	4	1	no	—
99	C₂×C₄×D₈₉	—	4	1	no	—
100	C₂×C₄×D₈₉	—	4	1	no	—
101	C₂×C₄×D₈₉	—	4	2	no	—
102	Dic₃₅₆	—	2	1	no	—
103	Dic₃₅₆	—	2	1	no	—
104	Dic₃₅₆	C₂²×C₄	1	2	yes	—
105	Dic₃₅₆	C₂²×C₄	1	2	yes	—
106	#13083563	—	2	2	no	—
107	#13083563	—	2	2	no	—
108	#13083563	—	2	2	no	—
109	#13083563	—	2	2	no	—
110	D₇₁₂	—	2	1	no	—
111	D₇₁₂	—	2	1	no	—
112	D₇₁₂	C₂²×C₄	1	2	yes	—
113	D₇₁₂	C₂²×C₄	1	2	yes	—
114	#13083558	—	2	2	no	—
115	#13083558	—	4	1	no	—
116	#13083557	—	4	4	no	—
117	C₂×#10623494	—	4	1	no	—
118	C₈×D₈₉	—	4	4	no	—
119	C₄⋊Dic₈₉	—	2	2	no	—
120	C₄⋊Dic₈₉	—	4	1	no	—
121	#13083553	—	4	2	no	—
122	#13083551	—	4	2	no	—
123	C₄×Dic₈₉	—	4	2	no	—
124	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
125	M₁₆×C₈₉	—	2	1	no	—
126	M₁₆×C₈₉	—	2	1	no	—
127	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
130	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
131	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
132	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
133	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
134	C₄⋊C₄×C₈₉	—	2	1	no	—
135	C₂²⋊C₄×C₈₉	—	2	1	no	—
136	C₄²×C₈₉	C₂×D₄	1	2	yes	—
137	C₁₆×C₈₉	—	2	2	no	—
138	C₁₆×C₈₉	C₂²×C₄	1	4	yes	—
139	C₂³×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
140	C₂²×D₈₉	—	4	1	no	—
141	C₂²×D₈₉	—	4	1	no	—
142	C₂×C₄×C₈₉	—	2	2	no	—
143	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
144	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
146	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
147	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
148	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
149	C₂²⋊D₈₉	—	4	2	no	—
150	C₂²⋊D₈₉	—	4	2	no	—
151	C₂²⋊D₈₉	—	4	2	no	—
152	C₂²⋊D₈₉	—	4	2	no	—
153	C₂×Dic₈₉	—	4	1	no	—
154	C₂×Dic₈₉	—	4	1	no	—
155	C₂×Dic₈₉	—	4	2	no	—
156	C₂×Dic₈₉	—	16	1	no	—
157	D₃₅₆	—	2	2	no	—
158	D₃₅₆	—	2	2	no	—
159	D₃₅₆	—	4	1	no	—
160	D₃₅₆	—	4	1	no	—
161	C₄×D₈₉	—	4	2	no	—
162	C₄×D₈₉	—	4	2	no	—
163	C₄×D₈₉	—	4	2	no	—
164	C₄×D₈₉	—	4	2	no	—
165	C₄×D₈₉	—	16	2	no	—
166	Dic₁₇₈	—	2	2	no	—
167	Dic₁₇₈	—	2	2	no	—
168	Dic₁₇₈	—	4	1	no	—
169	Dic₁₇₈	—	4	1	no	—
170	C₈×C₈₉	—	2	2	no	—
171	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
172	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
173	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
174	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
175	#10623494	—	4	2	no	—
176	C₂²×C₈₉	—	4	1	no	—
177	C₂²×C₈₉	#442	1	1	yes	characteristic
178	C₂²×C₈₉	#442	1	1	yes	characteristic
179	C₂²×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
180	C₂×D₈₉	—	4	2	no	—
181	C₂×D₈₉	—	4	2	no	—
182	C₂×D₈₉	—	8	1	no	—
183	C₂×D₈₉	—	8	1	no	—
184	C₂×D₈₉	—	16	1	no	—
185	C₄×C₈₉	—	4	2	no	—
186	C₄×C₈₉	#442	1	1	yes	characteristic
187	C₄×C₈₉	#442	1	1	yes	characteristic
188	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
189	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
190	Dic₈₉	—	4	2	no	—
191	Dic₈₉	—	4	2	no	—
192	Dic₈₉	—	8	2	no	—
193	#33469	—	89	1	no	maximal, Sylow
194	C₂×C₈₉	—	2	1	no	—
195	C₂×C₈₉	—	2	1	no	—
196	C₂×C₈₉	—	2	1	no	—
197	C₂×C₈₉	C₂×#442	1	1	yes	characteristic
198	D₈₉	—	8	2	no	—
199	D₈₉	—	8	2	no	—
200	#3414	—	89	1	no	—
201	#2957	—	89	1	no	—
202	#2952	—	89	1	no	—
203	AGL(1,ℤ/16)	—	89	4	no	—
204	#2180	—	89	4	no	—
205	C₂×#365	—	89	1	no	—
206	#2148	—	89	1	no	—
207	C₂×#360	—	89	1	no	—
208	#2110	D₈₉	1	1	yes	characteristic, complemented
209	C₈₉	#33469	1	1	yes	characteristic, complemented, Sylow
210	C₂×#135	—	89	1	no	—
211	C₂×#135	—	89	1	no	—
212	#568	—	178	1	no	—
213	#520	—	178	1	no	—
214	#510	—	89	2	no	—
215	#510	—	89	2	no	—
216	#509	—	89	2	no	—
217	#509	—	89	2	no	—
218	#508	—	89	4	no	—
219	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
220	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
221	#444	—	89	4	no	—
222	#443	—	178	2	no	—
223	#442	—	89	2	no	—
224	#441	—	89	2	no	—
225	C₄×SD₁₆	—	178	2	no	—
226	C₂×#108	—	89	1	no	—
227	C₂×#106	—	89	1	no	—
228	#421	—	178	1	no	—
229	#417	—	178	1	no	—
230	#405	C₂×D₈₉	1	1	yes	characteristic
231	#365	—	89	4	no	—
232	#362	—	89	2	no	—
233	#361	—	89	2	no	—
234	#360	—	89	4	no	—
235	#347	C₂×D₈₉	1	4	yes	complemented
236	C₂×#41	—	178	1	no	—
237	C₂×#41	—	178	1	no	—
238	#135	—	89	2	no	—
239	#135	—	89	2	no	—
240	#135	—	89	2	no	—
241	#135	—	89	2	no	—
242	#135	—	178	1	no	—
243	#135	—	178	1	no	—
244	#135	—	178	1	no	—
245	#135	—	178	1	no	—
246	C₂×Q₁₆	—	89	1	no	—
247	C₂×Q₁₆	—	89	1	no	—
248	C₂×SD₁₆	—	178	1	no	—
249	C₂×SD₁₆	—	178	1	no	—
250	C₂×D₈	—	89	1	no	—
251	C₂×D₈	—	89	1	no	—
252	#131	—	356	2	no	—
253	#131	—	178	4	no	—
254	C₂×M₁₆	—	356	2	no	—
255	C₂×M₁₆	—	178	1	no	—
256	C₂×M₁₆	D₃₅₆	1	1	yes	characteristic
257	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
258	C₄×Q₈	—	178	2	no	—
259	C₄×D₄	—	356	2	no	—
260	C₄×D₄	—	178	2	no	—
261	#117	—	356	1	no	—
262	#117	D₃₅₆	1	1	yes	characteristic
263	C₂×C₄⋊C₄	—	356	1	no	—
264	C₂×C₄⋊C₄	—	178	1	no	—
265	Q₃₂	—	89	4	no	—
266	SD₃₂	—	89	4	no	—
267	SD₃₂	—	89	4	no	—
268	D₁₆	—	89	4	no	—
269	M₃₂	—	2	1	no	—
270	M₃₂	C₄×D₈₉	1	2	yes	complemented
271	M₃₂	C₂²×D₈₉	1	2	yes	—
272	M₃₂	C₂²×D₈₉	1	2	yes	—
273	C₂×C₁₆	—	2	1	no	—
274	C₂×C₁₆	C₄×D₈₉	1	2	yes	complemented
275	#108	—	89	2	no	—
276	#108	—	178	1	no	—
277	#106	—	89	2	no	—
278	#106	—	178	1	no	—
279	#104	—	178	4	no	—
280	#103	—	178	2	no	—
281	#102	—	178	2	no	—
282	#97	C₂²×D₈₉	1	2	yes	—
283	C₄×C₈	D₃₅₆	1	2	yes	—
284	#41	—	356	1	no	—
285	#41	—	356	1	no	—
286	#41	—	356	1	no	—
287	#41	—	356	1	no	—
288	#41	—	178	2	no	—
289	#41	—	178	2	no	—
290	#41	—	178	2	no	—
291	#41	—	178	2	no	—
292	C₂×Q₈	—	178	1	no	—
293	C₂×Q₈	—	178	1	no	—
294	C₂×D₄	—	356	1	no	—
295	C₂×D₄	—	356	1	no	—
296	C₂×D₄	—	178	1	no	—
297	C₂×D₄	—	178	1	no	—
298	C₂²×C₄	—	356	2	no	—
299	C₂²×C₄	—	356	1	no	—
300	C₂²×C₄	—	356	1	no	—
301	C₂²×C₄	—	356	1	no	—
302	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
303	Q₁₆	—	89	2	no	—
304	Q₁₆	—	89	2	no	—
305	Q₁₆	—	178	1	no	—
306	Q₁₆	—	178	1	no	—
307	SD₁₆	—	178	2	no	—
308	SD₁₆	—	178	2	no	—
309	SD₁₆	—	178	2	no	—
310	SD₁₆	—	178	2	no	—
311	D₈	—	89	2	no	—
312	D₈	—	89	2	no	—
313	D₈	—	178	1	no	—
314	D₈	—	178	1	no	—
315	M₁₆	—	2	1	no	—
316	M₁₆	—	2	1	no	—
317	M₁₆	—	356	4	no	—
318	M₁₆	—	356	1	no	—
319	M₁₆	—	178	2	no	—
320	C₂×C₈	—	356	4	no	—
321	C₂×C₈	—	356	1	no	—
322	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
323	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
324	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
325	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
326	C₂×C₈	#13083574	1	1	yes	characteristic
327	C₂×C₈	#13083574	1	1	yes	characteristic
328	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
329	C₄⋊C₄	—	2	1	no	—
330	C₄⋊C₄	—	356	2	no	—
331	C₄⋊C₄	—	356	1	no	—
332	C₄⋊C₄	—	178	2	no	—
333	C₂²⋊C₄	—	2	1	no	—
334	C₂²⋊C₄	—	356	2	no	—
335	C₄²	—	356	2	no	—
336	C₄²	C₂×D₃₅₆	1	2	yes	—
337	C₁₆	—	2	2	no	—
338	C₁₆	C₂×C₄×D₈₉	1	4	yes	—
339	C₂³	—	356	1	no	—
340	C₂³	—	356	1	no	—
341	C₂³	#15668040	1	1	yes	characteristic
342	Q₈	—	356	1	no	—
343	Q₈	—	356	1	no	—
344	Q₈	—	178	2	no	—
345	Q₈	—	178	2	no	—
346	D₄	—	356	2	no	—
347	D₄	—	356	2	no	—
348	D₄	—	356	2	no	—
349	D₄	—	356	2	no	—
350	D₄	—	356	1	no	—
351	D₄	—	356	1	no	—
352	D₄	—	178	2	no	—
353	D₄	—	178	2	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	356	1	no	—
356	C₂×C₄	—	356	1	no	—
357	C₂×C₄	—	356	2	no	—
358	C₂×C₄	—	356	2	no	—
359	C₂×C₄	—	356	2	no	—
360	C₂×C₄	—	356	2	no	—
361	C₂×C₄	—	356	2	no	—
362	C₂×C₄	—	1424	2	no	—
363	C₂×C₄	—	1424	1	no	—
364	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
365	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
366	C₂×C₄	#15668040	1	1	yes	characteristic
367	C₂×C₄	#15668040	1	1	yes	characteristic
368	C₂×C₄	#15668040	1	1	yes	characteristic
369	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
370	C₈	—	2	2	no	—
371	C₈	—	356	2	no	—
372	C₈	C₄×D₃₅₆	1	1	yes	characteristic
373	C₈	C₄×D₃₅₆	1	1	yes	characteristic
374	C₈	C₂²×C₄×D₈₉	1	1	yes	characteristic
375	C₈	C₂×#13083574	1	1	yes	characteristic
376	C₂²	—	4	1	no	—
377	C₂²	—	356	2	no	—
378	C₂²	—	356	2	no	—
379	C₂²	—	1424	1	no	—
380	C₂²	—	712	1	no	—
381	C₂²	—	712	1	no	—
382	C₂²	#19151711	1	1	yes	characteristic
383	C₂²	#19151711	1	1	yes	characteristic
384	C₂²	C₂×#15668040	1	1	yes	characteristic
385	C₄	—	4	2	no	—
386	C₄	—	356	2	no	—
387	C₄	—	356	2	no	—
388	C₄	—	712	2	no	—
389	C₄	#19151711	1	1	yes	characteristic
390	C₄	#19151711	1	1	yes	characteristic
391	C₄	C₂×C₄×D₃₅₆	1	1	yes	characteristic
392	C₄	C₂×#15668040	1	1	yes	characteristic
393	C₂	—	2	1	no	—
394	C₂	—	2	1	no	—
395	C₂	—	2	1	no	—
396	C₂	—	712	2	no	—
397	C₂	—	712	2	no	—
398	C₂	C₂×#19151711	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#19151711
commutator	C₈×C₈₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2110	C₂
socle	C₂×C₈₉	C₂×#442
prime radical	#26033388	C₂
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2110	D₈₉
hypocenter	C₈₉	#33469
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #2110

Complements

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

N	H ≅ G/N	type
2. #26029170	C₂	semidirect
3. #26029137	C₂	semidirect
4. #26022152	C₂	semidirect
5. #26022145	C₂	semidirect
6. C₂×#19151421	C₂	semidirect
7. #26022002	C₂	semidirect
8. C₂×#19151409	C₂	semidirect
9. C₈₉×#2110	C₂	semidirect
14. #19152041	C₄	semidirect
16. #19152036	C₄	semidirect
18. #19152030	C₄	semidirect
21. #19151711	C₂²	semidirect
22. #19151706	C₂²	semidirect
28. #19151421	C₂²	semidirect
29. #19151415	C₂²	semidirect
30. #19151412	C₂²	semidirect
35. #347×C₈₉	C₂²	semidirect
44. #15668034	C₂×C₄	semidirect
46. C₂×Dic₃₅₆	C₂×C₄	semidirect
50. C₂×D₇₁₂	C₂×C₄	semidirect
62. Dic₇₁₂	C₂×C₄	semidirect
63. #15667957	C₂×C₄	semidirect
64. #15667957	C₂×C₄	semidirect
65. D₁₄₂₄	C₂×C₄	semidirect
77. M₃₂×C₈₉	C₂×C₄	semidirect
81. C₂×C₁₆×C₈₉	C₂×C₄	semidirect
208. #2110	D₈₉	semidirect
209. C₈₉	#33469	semidirect
235. #347	C₂×D₈₉	semidirect
270. M₃₂	C₄×D₈₉	semidirect
274. C₂×C₁₆	C₄×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26033388	2
#26029170	2
#26029137	2
#26022152	2
#26022145	2
C₂×#19151421	2
#26022002	2
C₂×#19151409	2
C₈₉×#2110	2
#33469	89