Subgroups — #36707906

Order 22784.

668 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19152034	C₂	1	1	yes	maximal, characteristic
2	#26029168	C₂	1	1	yes	maximal, characteristic, complemented
3	#26029104	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022153	C₂	1	8	yes	maximal, complemented
5	#26022026	C₂	1	1	yes	maximal, characteristic, complemented
6	#26021989	C₂	1	1	yes	maximal, characteristic, complemented
7	#26021856	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#361×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#15668034	C₂²	1	1	yes	characteristic
10	#19152364	—	2	1	no	—
11	#19152073	—	2	1	no	—
12	#19152034	C₄	1	4	yes	complemented
13	#19152034	C₂²	1	4	yes	—
14	C₂×#15667956	C₄	1	2	yes	complemented
15	C₂×#15667953	C₄	1	2	yes	complemented
16	C₂×#15667952	C₂²	1	1	yes	characteristic
17	#19151719	C₂²	1	4	yes	complemented
18	#19151714	—	2	2	no	—
19	#19151710	C₂²	1	4	yes	complemented
20	D₈×Dic₈₉	—	2	2	no	—
21	#19151636	C₂²	1	1	yes	characteristic
22	#19151614	—	2	1	no	—
23	#19151603	—	2	1	no	—
24	#19151553	C₂²	1	1	yes	characteristic
25	#19151422	C₂²	1	4	yes	complemented
26	#19151409	C₂²	1	4	yes	complemented
27	#19151375	C₂²	1	4	yes	complemented
28	C₂²×D₈×C₈₉	C₂²	1	1	yes	characteristic
29	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
30	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
31	#361×C₈₉	C₂²	1	4	yes	complemented
32	C₂×#13083574	—	2	1	no	—
33	#15668034	—	2	1	no	—
34	#15668034	—	2	1	no	—
35	#15668034	C₂×C₄	1	2	yes	complemented
36	#15668034	C₂³	1	2	yes	—
37	C₂×Dic₃₅₆	C₂×C₄	1	1	yes	characteristic, complemented
38	C₂×#13083563	—	2	1	no	—
39	C₂×D₇₁₂	C₂×C₄	1	1	yes	characteristic, complemented
40	#15668020	—	2	4	no	—
41	#15668018	—	4	2	no	—
42	C₂×#13083558	D₄	1	1	yes	characteristic
43	M₁₆×D₈₉	—	4	2	no	—
44	D₄×Dic₈₉	—	2	4	no	—
45	D₄×Dic₈₉	—	4	2	no	—
46	#15667971	—	2	1	no	—
47	#15667971	D₄	1	1	yes	characteristic
48	C₂×C₂²⋊Dic₈₉	—	4	2	no	—
49	#15667956	C₂×C₄	1	8	yes	complemented
50	#15667953	C₂×C₄	1	8	yes	complemented
51	#15667952	C₂×C₄	1	2	yes	complemented
52	#15667952	C₂³	1	2	yes	—
53	C₂×#13083549	C₂×C₄	1	2	yes	complemented
54	#15667946	—	2	1	no	—
55	#15667944	C₂³	1	2	yes	—
56	#15667938	—	2	4	no	—
57	#15667933	—	2	4	no	—
58	#15667922	C₂³	1	2	yes	—
59	C₈×Dic₈₉	D₄	1	2	yes	—
60	C₂²×D₄×C₈₉	—	2	1	no	—
61	C₂×D₈×C₈₉	—	2	1	no	—
62	C₂×D₈×C₈₉	—	2	1	no	—
63	C₂×D₈×C₈₉	—	2	1	no	—
64	C₂×D₈×C₈₉	C₂×C₄	1	2	yes	—
65	C₂×D₈×C₈₉	C₂×C₄	1	2	yes	—
66	C₂×D₈×C₈₉	C₂³	1	2	yes	—
67	C₂×M₁₆×C₈₉	—	2	1	no	—
68	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
69	M₃₂×C₈₉	—	2	1	no	—
70	M₃₂×C₈₉	C₂³	1	2	yes	—
71	C₂×C₁₆×C₈₉	—	2	1	no	—
72	#108×C₈₉	—	2	1	no	—
73	#108×C₈₉	C₂³	1	2	yes	—
74	#13083574	—	2	2	no	—
75	#13083574	—	2	2	no	—
76	#13083574	—	4	1	no	—
77	#13083574	—	4	1	no	—
78	C₂×Dic₁₇₈	—	2	1	no	—
79	C₂×D₃₅₆	—	2	1	no	—
80	C₂×C₂²⋊D₈₉	—	4	1	no	—
81	C₂²×Dic₈₉	—	4	4	no	—
82	C₂×C₄×D₈₉	—	4	1	no	—
83	Dic₃₅₆	—	2	1	no	—
84	Dic₃₅₆	C₂²×C₄	1	2	yes	—
85	#13083563	—	2	2	no	—
86	#13083563	—	2	2	no	—
87	D₇₁₂	—	2	1	no	—
88	D₇₁₂	C₂²×C₄	1	2	yes	—
89	#13083558	—	2	1	no	—
90	#13083558	—	2	1	no	—
91	#13083557	—	4	4	no	—
92	C₂×#10623494	C₂×D₄	1	2	yes	—
93	C₈×D₈₉	—	4	4	no	—
94	C₄⋊Dic₈₉	—	2	1	no	—
95	C₄⋊Dic₈₉	—	2	2	no	—
96	C₂²⋊Dic₈₉	—	2	1	no	—
97	C₂²⋊Dic₈₉	—	4	4	no	—
98	C₂²⋊Dic₈₉	—	4	1	no	—
99	C₄×Dic₈₉	—	4	1	no	—
100	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
101	#13083549	C₂²×C₄	1	4	yes	—
102	C₂⁴×C₈₉	—	4	1	no	—
103	C₂×D₄×C₈₉	—	2	2	no	—
104	C₂×D₄×C₈₉	—	2	2	no	—
105	C₂×D₄×C₈₉	—	2	2	no	—
106	C₂×D₄×C₈₉	—	4	1	no	—
107	C₂×D₄×C₈₉	—	4	1	no	—
108	C₂×D₄×C₈₉	—	4	1	no	—
109	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
110	D₈×C₈₉	—	2	2	no	—
111	D₈×C₈₉	—	2	2	no	—
112	D₈×C₈₉	—	2	2	no	—
113	D₈×C₈₉	C₂²×C₄	1	4	yes	—
114	M₁₆×C₈₉	—	2	2	no	—
115	M₁₆×C₈₉	—	4	1	no	—
116	C₂×C₈×C₈₉	—	4	1	no	—
117	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
121	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
122	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
123	C₁₆×C₈₉	—	2	2	no	—
124	C₂³×C₈₉	—	4	2	no	—
125	C₂³×C₈₉	—	4	2	no	—
126	C₂³×C₈₉	—	4	2	no	—
127	C₂³×C₈₉	—	16	2	no	—
128	C₂³×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
129	C₂²×D₈₉	—	4	1	no	—
130	D₄×C₈₉	—	2	4	no	—
131	D₄×C₈₉	—	4	2	no	—
132	D₄×C₈₉	—	4	2	no	—
133	D₄×C₈₉	—	4	2	no	—
134	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
137	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
138	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
139	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
140	C₂²⋊D₈₉	—	4	2	no	—
141	C₂²⋊D₈₉	—	4	2	no	—
142	C₂×Dic₈₉	—	2	2	no	—
143	C₂×Dic₈₉	—	4	1	no	—
144	C₂×Dic₈₉	—	4	2	no	—
145	C₂×Dic₈₉	—	16	4	no	—
146	D₃₅₆	—	2	2	no	—
147	D₃₅₆	—	4	1	no	—
148	C₄×D₈₉	—	4	2	no	—
149	C₄×D₈₉	—	4	2	no	—
150	Dic₁₇₈	—	2	2	no	—
151	Dic₁₇₈	—	4	1	no	—
152	C₈×C₈₉	—	4	2	no	—
153	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
154	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
155	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
156	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
157	#10623494	—	2	2	no	—
158	C₂²×C₈₉	—	4	1	no	—
159	C₂²×C₈₉	—	4	4	no	—
160	C₂²×C₈₉	—	8	4	no	—
161	C₂²×C₈₉	—	16	2	no	—
162	C₂²×C₈₉	—	16	2	no	—
163	C₂²×C₈₉	#442	1	1	yes	characteristic
164	C₂²×C₈₉	#442	1	1	yes	characteristic
165	C₂²×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
166	C₂×D₈₉	—	4	2	no	—
167	C₂×D₈₉	—	16	1	no	—
168	C₄×C₈₉	#442	1	1	yes	characteristic
169	C₄×C₈₉	#442	1	1	yes	characteristic
170	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
171	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
172	Dic₈₉	—	4	2	no	—
173	Dic₈₉	—	4	2	no	—
174	Dic₈₉	—	8	2	no	—
175	#33471	—	89	1	no	maximal, Sylow
176	C₂×C₈₉	—	2	1	no	—
177	C₂×C₈₉	—	2	1	no	—
178	C₂×C₈₉	—	2	1	no	—
179	C₂×C₈₉	—	8	4	no	—
180	C₂×C₈₉	C₂×#442	1	1	yes	characteristic
181	D₈₉	—	8	2	no	—
182	C₂×#509	—	89	1	no	—
183	#2957	—	89	1	no	—
184	#2948	—	89	1	no	—
185	AGL(1,ℤ/16)	—	89	8	no	—
186	#2155	—	89	1	no	—
187	C₂×#361	D₈₉	1	1	yes	characteristic, complemented
188	#2145	—	89	1	no	—
189	#2110	—	89	1	no	—
190	C₈₉	#33471	1	1	yes	characteristic, complemented, Sylow
191	C₂×#135	—	89	1	no	—
192	C₂²×D₈	C₂×D₈₉	1	1	yes	characteristic
193	#568	—	178	1	no	—
194	#518	—	178	1	no	—
195	#509	—	89	4	no	—
196	#509	—	89	4	no	—
197	C₂×SD₃₂	—	89	2	no	—
198	C₂×D₁₆	—	89	2	no	—
199	C₂×M₃₂	—	89	1	no	—
200	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
201	#444	—	89	4	no	—
202	#443	—	178	2	no	—
203	#442	—	89	4	no	—
204	C₄×D₈	—	178	2	no	—
205	C₂×#108	C₂×D₈₉	1	1	yes	characteristic
206	#427	—	89	1	no	—
207	#421	—	178	1	no	—
208	#418	—	178	1	no	—
209	#405	—	89	1	no	—
210	#365	—	89	4	no	—
211	#361	C₂×D₈₉	1	4	yes	complemented
212	#360	—	89	4	no	—
213	#347	—	89	4	no	—
214	C₂×#41	—	178	1	no	—
215	C₂²×D₄	—	2	1	no	—
216	#135	—	89	2	no	—
217	#135	—	89	2	no	—
218	#135	—	178	1	no	—
219	#135	—	178	1	no	—
220	C₂×Q₁₆	—	89	1	no	—
221	C₂×SD₁₆	—	178	1	no	—
222	C₂×D₈	—	2	1	no	—
223	C₂×D₈	—	2	1	no	—
224	C₂×D₈	—	2	1	no	—
225	C₂×D₈	—	89	1	no	—
226	C₂×D₈	C₄×D₈₉	1	2	yes	—
227	C₂×D₈	C₄×D₈₉	1	2	yes	—
228	C₂×D₈	C₂²×D₈₉	1	2	yes	—
229	#131	—	356	2	no	—
230	#131	—	178	4	no	—
231	C₂×M₁₆	—	2	1	no	—
232	C₂×M₁₆	—	89	1	no	—
233	C₂×M₁₆	—	356	2	no	—
234	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
235	C₄×D₄	—	356	2	no	—
236	C₄×D₄	—	178	4	no	—
237	#117	—	89	1	no	—
238	#117	—	178	1	no	—
239	C₂×C₂²⋊C₄	—	356	2	no	—
240	SD₃₂	—	89	8	no	—
241	D₁₆	—	89	8	no	—
242	M₃₂	—	2	1	no	—
243	M₃₂	—	89	2	no	—
244	M₃₂	—	89	2	no	—
245	M₃₂	C₂²×D₈₉	1	2	yes	—
246	C₂×C₁₆	—	2	1	no	—
247	C₂×C₁₆	—	89	2	no	—
248	#108	—	2	1	no	—
249	#108	C₂²×D₈₉	1	2	yes	—
250	#107	—	178	1	no	—
251	#106	—	89	2	no	—
252	#104	—	178	4	no	—
253	#102	—	178	4	no	—
254	#97	—	89	2	no	—
255	C₄×C₈	—	89	2	no	—
256	C₂⁴	—	4	1	no	—
257	#41	—	356	1	no	—
258	#41	—	356	1	no	—
259	#41	—	178	2	no	—
260	#41	—	178	2	no	—
261	C₂×Q₈	—	178	1	no	—
262	C₂×D₄	—	2	2	no	—
263	C₂×D₄	—	2	2	no	—
264	C₂×D₄	—	2	2	no	—
265	C₂×D₄	—	4	1	no	—
266	C₂×D₄	—	4	1	no	—
267	C₂×D₄	—	4	1	no	—
268	C₂×D₄	—	356	1	no	—
269	C₂×D₄	—	178	1	no	—
270	C₂²×C₄	—	356	1	no	—
271	C₂²×C₄	—	356	4	no	—
272	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
273	Q₁₆	—	89	2	no	—
274	Q₁₆	—	178	1	no	—
275	SD₁₆	—	178	2	no	—
276	SD₁₆	—	178	2	no	—
277	D₈	—	2	2	no	—
278	D₈	—	2	2	no	—
279	D₈	—	2	2	no	—
280	D₈	—	89	2	no	—
281	D₈	—	178	1	no	—
282	D₈	C₂×C₄×D₈₉	1	4	yes	—
283	M₁₆	—	2	2	no	—
284	M₁₆	—	4	1	no	—
285	M₁₆	—	356	4	no	—
286	M₁₆	—	178	1	no	—
287	M₁₆	—	178	1	no	—
288	C₂×C₈	—	4	1	no	—
289	C₂×C₈	—	89	2	no	—
290	C₂×C₈	—	356	4	no	—
291	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
292	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
293	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
294	C₂×C₈	#13083573	1	1	yes	characteristic
295	C₂×C₈	#13083573	1	1	yes	characteristic
296	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
297	C₄⋊C₄	—	178	2	no	—
298	C₄⋊C₄	—	178	1	no	—
299	C₂²⋊C₄	—	356	1	no	—
300	C₂²⋊C₄	—	356	4	no	—
301	C₂²⋊C₄	—	178	1	no	—
302	C₄²	—	89	2	no	—
303	C₄²	—	356	1	no	—
304	C₁₆	—	2	2	no	—
305	C₁₆	—	89	4	no	—
306	C₂³	—	4	2	no	—
307	C₂³	—	4	2	no	—
308	C₂³	—	4	2	no	—
309	C₂³	—	16	2	no	—
310	C₂³	—	356	1	no	—
311	C₂³	#15668039	1	1	yes	characteristic
312	Q₈	—	356	1	no	—
313	Q₈	—	178	2	no	—
314	D₄	—	2	4	no	—
315	D₄	—	4	2	no	—
316	D₄	—	4	2	no	—
317	D₄	—	4	2	no	—
318	D₄	—	356	2	no	—
319	D₄	—	356	2	no	—
320	D₄	—	356	1	no	—
321	D₄	—	178	2	no	—
322	C₂×C₄	—	356	1	no	—
323	C₂×C₄	—	356	2	no	—
324	C₂×C₄	—	356	2	no	—
325	C₂×C₄	—	356	2	no	—
326	C₂×C₄	—	1424	4	no	—
327	C₂×C₄	—	178	2	no	—
328	C₂×C₄	#15667972	1	1	yes	characteristic
329	C₂×C₄	#15667972	1	1	yes	characteristic
330	C₂×C₄	#15668036	1	1	yes	characteristic
331	C₂×C₄	#15668036	1	1	yes	characteristic
332	C₂×C₄	#15668039	1	1	yes	characteristic
333	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
334	C₈	—	4	2	no	—
335	C₈	—	178	2	no	—
336	C₈	#15667972	1	1	yes	characteristic
337	C₈	#15667972	1	1	yes	characteristic
338	C₈	C₂²×C₄×D₈₉	1	1	yes	characteristic
339	C₈	C₂×#13083573	1	1	yes	characteristic
340	C₂²	—	4	1	no	—
341	C₂²	—	4	4	no	—
342	C₂²	—	8	4	no	—
343	C₂²	—	16	2	no	—
344	C₂²	—	16	2	no	—
345	C₂²	—	356	2	no	—
346	C₂²	—	1424	1	no	—
347	C₂²	#19151707	1	1	yes	characteristic
348	C₂²	#19151707	1	1	yes	characteristic
349	C₂²	C₂×#15668039	1	1	yes	characteristic
350	C₄	—	356	2	no	—
351	C₄	—	356	2	no	—
352	C₄	—	712	2	no	—
353	C₄	#19151707	1	1	yes	characteristic
354	C₄	#19151707	1	1	yes	characteristic
355	C₄	C₂×#15667972	1	1	yes	characteristic
356	C₄	C₂×#15668036	1	1	yes	characteristic
357	C₂	—	2	1	no	—
358	C₂	—	2	1	no	—
359	C₂	—	2	1	no	—
360	C₂	—	8	4	no	—
361	C₂	—	712	2	no	—
362	C₂	C₂×#19151707	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#19151707
commutator	C₈×C₈₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×#361×C₈₉	C₂
socle	C₂×C₈₉	C₂×#442
prime radical	C₂×#19152034	C₂
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₂×#361	D₈₉
hypocenter	C₈₉	#33471
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#361

Complements

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

N	H ≅ G/N	type
2. #26029168	C₂	semidirect
3. #26029104	C₂	semidirect
4. #26022153	C₂	semidirect
5. #26022026	C₂	semidirect
6. #26021989	C₂	semidirect
7. #26021856	C₂	semidirect
8. C₂×#361×C₈₉	C₂	semidirect
12. #19152034	C₄	semidirect
14. C₂×#15667956	C₄	semidirect
15. C₂×#15667953	C₄	semidirect
17. #19151719	C₂²	semidirect
19. #19151710	C₂²	semidirect
25. #19151422	C₂²	semidirect
26. #19151409	C₂²	semidirect
27. #19151375	C₂²	semidirect
31. #361×C₈₉	C₂²	semidirect
35. #15668034	C₂×C₄	semidirect
37. C₂×Dic₃₅₆	C₂×C₄	semidirect
39. C₂×D₇₁₂	C₂×C₄	semidirect
49. #15667956	C₂×C₄	semidirect
50. #15667953	C₂×C₄	semidirect
51. #15667952	C₂×C₄	semidirect
53. C₂×#13083549	C₂×C₄	semidirect
187. C₂×#361	D₈₉	semidirect
190. C₈₉	#33471	semidirect
211. #361	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19152034	2
#26029168	2
#26029104	2
#26022153	2
#26022026	2
#26021989	2
#26021856	2
C₂×#361×C₈₉	2
#33471	89