Subgroups — #36707905

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×#19152036	C₂	1	1	yes	maximal, characteristic
2	#26029170	C₂	1	1	yes	maximal, characteristic, complemented
3	#26029105	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022152	C₂	1	8	yes	maximal, complemented
5	#26022025	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#19151412	C₂	1	1	yes	maximal, characteristic, complemented
7	#26021989	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈₉×#2110	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#15668034	C₂²	1	1	yes	characteristic
10	C₂²×D₇₁₂	C₂²	1	1	yes	characteristic
11	#19152364	—	2	1	no	—
12	#19152072	—	2	1	no	—
13	#19152036	C₄	1	4	yes	complemented
14	#19152036	C₂²	1	4	yes	—
15	C₂×#15667957	C₄	1	2	yes	complemented
16	C₂×D₁₄₂₄	C₄	1	2	yes	complemented
17	#19151721	C₂²	1	4	yes	complemented
18	#19151716	—	2	2	no	—
19	#19151711	C₂²	1	4	yes	complemented
20	C₄×D₇₁₂	—	2	2	no	—
21	C₂×#15667948	C₂²	1	1	yes	characteristic
22	#19151636	C₂²	1	1	yes	characteristic
23	#19151616	—	2	1	no	—
24	#19151604	—	2	1	no	—
25	#19151421	C₂²	1	4	yes	complemented
26	#19151412	C₂²	1	4	yes	complemented
27	#19151409	C₂²	1	4	yes	complemented
28	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
29	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
30	#405×C₈₉	C₂²	1	1	yes	characteristic
31	#347×C₈₉	C₂²	1	4	yes	complemented
32	C₂×#13083574	—	2	1	no	—
33	C₂²×D₃₅₆	—	2	1	no	—
34	#15668034	—	2	1	no	—
35	#15668034	—	2	1	no	—
36	#15668034	C₂×C₄	1	2	yes	complemented
37	#15668034	C₂³	1	2	yes	—
38	C₂×Dic₃₅₆	C₂×C₄	1	1	yes	characteristic, complemented
39	C₂×#13083563	—	2	1	no	—
40	C₂×D₇₁₂	—	2	1	no	—
41	C₂×D₇₁₂	—	2	1	no	—
42	C₂×D₇₁₂	—	2	1	no	—
43	C₂×D₇₁₂	C₂×C₄	1	2	yes	—
44	C₂×D₇₁₂	C₂×C₄	1	2	yes	—
45	C₂×D₇₁₂	C₂×C₄	1	1	yes	characteristic, complemented
46	C₂×D₇₁₂	C₂³	1	2	yes	—
47	#15668020	—	4	2	no	—
48	#15668018	—	2	4	no	—
49	C₂×#13083558	—	2	1	no	—
50	M₁₆×D₈₉	—	4	2	no	—
51	C₄×D₃₅₆	—	2	4	no	—
52	#15667974	—	4	2	no	—
53	#15667971	—	2	1	no	—
54	C₂²⋊C₄×D₈₉	—	4	2	no	—
55	#15667957	C₂×C₄	1	8	yes	complemented
56	D₁₄₂₄	C₂×C₄	1	8	yes	complemented
57	#15667948	—	2	1	no	—
58	#15667948	C₂³	1	2	yes	—
59	#15667946	—	2	1	no	—
60	#15667944	C₂³	1	2	yes	—
61	#15667940	—	2	4	no	—
62	#15667934	—	2	4	no	—
63	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
64	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
65	#117×C₈₉	D₄	1	1	yes	characteristic
66	M₃₂×C₈₉	—	2	1	no	—
67	M₃₂×C₈₉	C₂×C₄	1	2	yes	complemented
68	M₃₂×C₈₉	C₂³	1	2	yes	—
69	M₃₂×C₈₉	C₂³	1	2	yes	—
70	C₂×C₁₆×C₈₉	—	2	1	no	—
71	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	complemented
72	#97×C₈₉	C₂³	1	2	yes	—
73	C₄×C₈×C₈₉	D₄	1	2	yes	—
74	C₂³×D₈₉	—	4	1	no	—
75	#13083574	—	2	2	no	—
76	#13083574	—	2	2	no	—
77	#13083574	—	4	1	no	—
78	#13083574	—	4	1	no	—
79	C₂×Dic₁₇₈	—	2	1	no	—
80	C₂×D₃₅₆	—	2	2	no	—
81	C₂×D₃₅₆	—	2	2	no	—
82	C₂×D₃₅₆	—	2	1	no	—
83	C₂×D₃₅₆	—	2	2	no	—
84	C₂×D₃₅₆	—	4	1	no	—
85	C₂×D₃₅₆	—	4	1	no	—
86	C₂×D₃₅₆	—	4	1	no	—
87	C₂×C₂²⋊D₈₉	—	4	1	no	—
88	C₂×C₄×D₈₉	—	4	1	no	—
89	C₂×C₄×D₈₉	—	4	4	no	—
90	Dic₃₅₆	—	2	1	no	—
91	Dic₃₅₆	C₂²×C₄	1	2	yes	—
92	#13083563	—	2	2	no	—
93	#13083563	—	2	2	no	—
94	D₇₁₂	—	2	2	no	—
95	D₇₁₂	—	2	2	no	—
96	D₇₁₂	—	2	1	no	—
97	D₇₁₂	—	2	2	no	—
98	D₇₁₂	C₂²×C₄	1	4	yes	—
99	D₇₁₂	C₂²×C₄	1	2	yes	—
100	#13083558	—	2	2	no	—
101	#13083558	—	4	1	no	—
102	#13083557	—	4	4	no	—
103	C₂×#10623494	—	4	1	no	—
104	C₈×D₈₉	—	4	4	no	—
105	C₄⋊Dic₈₉	—	2	2	no	—
106	C₂²⋊Dic₈₉	—	4	1	no	—
107	#13083551	—	4	4	no	—
108	C₄×Dic₈₉	—	4	1	no	—
109	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
110	M₁₆×C₈₉	—	2	1	no	—
111	M₁₆×C₈₉	—	2	1	no	—
112	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
115	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
117	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
118	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
119	C₄⋊C₄×C₈₉	—	2	1	no	—
120	C₂²⋊C₄×C₈₉	—	2	1	no	—
121	C₄²×C₈₉	C₂×D₄	1	2	yes	—
122	C₁₆×C₈₉	—	2	2	no	—
123	C₁₆×C₈₉	C₂²×C₄	1	4	yes	—
124	C₂³×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂²×D₈₉	—	4	2	no	—
126	C₂²×D₈₉	—	4	1	no	—
127	C₂²×D₈₉	—	4	2	no	—
128	C₂²×D₈₉	—	4	2	no	—
129	C₂²×D₈₉	—	16	2	no	—
130	C₂×C₄×C₈₉	—	2	2	no	—
131	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
134	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
135	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
136	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
137	C₂²⋊D₈₉	—	4	2	no	—
138	C₂²⋊D₈₉	—	4	2	no	—
139	C₂×Dic₈₉	—	4	1	no	—
140	C₂×Dic₈₉	—	4	2	no	—
141	D₃₅₆	—	2	4	no	—
142	D₃₅₆	—	2	2	no	—
143	D₃₅₆	—	4	2	no	—
144	D₃₅₆	—	4	2	no	—
145	D₃₅₆	—	4	2	no	—
146	D₃₅₆	—	4	1	no	—
147	C₄×D₈₉	—	4	2	no	—
148	C₄×D₈₉	—	4	2	no	—
149	C₄×D₈₉	—	16	4	no	—
150	Dic₁₇₈	—	2	2	no	—
151	Dic₁₇₈	—	4	1	no	—
152	C₈×C₈₉	—	2	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	—	4	2	no	—
158	C₂²×C₈₉	—	4	1	no	—
159	C₂²×C₈₉	#442	1	1	yes	characteristic
160	C₂²×C₈₉	#442	1	1	yes	characteristic
161	C₂²×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
162	C₂×D₈₉	—	4	2	no	—
163	C₂×D₈₉	—	4	4	no	—
164	C₂×D₈₉	—	8	4	no	—
165	C₂×D₈₉	—	16	1	no	—
166	C₂×D₈₉	—	16	2	no	—
167	C₂×D₈₉	—	16	2	no	—
168	C₄×C₈₉	—	4	2	no	—
169	C₄×C₈₉	#442	1	1	yes	characteristic
170	C₄×C₈₉	#442	1	1	yes	characteristic
171	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
172	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
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	C₂×#442	1	1	yes	characteristic
180	D₈₉	—	8	4	no	—
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	—	89	1	no	—
188	#2145	—	89	1	no	—
189	#2110	D₈₉	1	1	yes	characteristic, complemented
190	C₈₉	#33471	1	1	yes	characteristic, complemented, Sylow
191	C₂×#135	—	89	1	no	—
192	C₂²×D₈	—	89	1	no	—
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	C₂×D₈₉	1	1	yes	characteristic
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	—	89	1	no	—
206	#427	—	89	1	no	—
207	#421	—	178	1	no	—
208	#418	—	178	1	no	—
209	#405	C₂×D₈₉	1	1	yes	characteristic
210	#365	—	89	4	no	—
211	#361	—	89	4	no	—
212	#360	—	89	4	no	—
213	#347	C₂×D₈₉	1	4	yes	complemented
214	C₂×#41	—	178	1	no	—
215	C₂²×D₄	—	178	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	—	89	2	no	—
223	C₂×D₈	—	89	2	no	—
224	C₂×D₈	—	89	1	no	—
225	C₂×D₈	—	89	2	no	—
226	C₂×D₈	—	178	1	no	—
227	C₂×D₈	—	178	1	no	—
228	C₂×D₈	—	178	1	no	—
229	#131	—	356	2	no	—
230	#131	—	178	4	no	—
231	C₂×M₁₆	—	356	2	no	—
232	C₂×M₁₆	—	178	1	no	—
233	C₂×M₁₆	D₃₅₆	1	1	yes	characteristic
234	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
235	C₄×D₄	—	356	2	no	—
236	C₄×D₄	—	178	4	no	—
237	#117	—	178	1	no	—
238	#117	D₃₅₆	1	1	yes	characteristic
239	C₂×C₂²⋊C₄	—	356	2	no	—
240	SD₃₂	—	89	8	no	—
241	D₁₆	—	89	8	no	—
242	M₃₂	—	2	1	no	—
243	M₃₂	C₄×D₈₉	1	2	yes	complemented
244	M₃₂	C₂²×D₈₉	1	2	yes	—
245	M₃₂	C₂²×D₈₉	1	2	yes	—
246	C₂×C₁₆	—	2	1	no	—
247	C₂×C₁₆	C₄×D₈₉	1	2	yes	complemented
248	#108	—	89	2	no	—
249	#108	—	178	1	no	—
250	#107	—	178	1	no	—
251	#106	—	89	2	no	—
252	#104	—	178	4	no	—
253	#102	—	178	4	no	—
254	#97	C₂²×D₈₉	1	2	yes	—
255	C₄×C₈	D₃₅₆	1	2	yes	—
256	C₂⁴	—	356	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	—	356	1	no	—
263	C₂×D₄	—	356	1	no	—
264	C₂×D₄	—	356	1	no	—
265	C₂×D₄	—	356	1	no	—
266	C₂×D₄	—	178	2	no	—
267	C₂×D₄	—	178	2	no	—
268	C₂×D₄	—	178	1	no	—
269	C₂×D₄	—	178	2	no	—
270	C₂²×C₄	—	356	4	no	—
271	C₂²×C₄	—	356	1	no	—
272	C₂²×C₄	C₂×D₃₅₆	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	—	89	2	no	—
278	D₈	—	89	4	no	—
279	D₈	—	178	2	no	—
280	D₈	—	178	2	no	—
281	D₈	—	178	1	no	—
282	D₈	—	178	2	no	—
283	M₁₆	—	2	1	no	—
284	M₁₆	—	2	1	no	—
285	M₁₆	—	356	4	no	—
286	M₁₆	—	356	1	no	—
287	M₁₆	—	178	2	no	—
288	C₂×C₈	—	356	4	no	—
289	C₂×C₈	—	356	1	no	—
290	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
291	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
292	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
293	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
294	C₂×C₈	#13083574	1	1	yes	characteristic
295	C₂×C₈	#13083574	1	1	yes	characteristic
296	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
297	C₄⋊C₄	—	2	1	no	—
298	C₄⋊C₄	—	178	2	no	—
299	C₂²⋊C₄	—	2	1	no	—
300	C₂²⋊C₄	—	356	4	no	—
301	C₂²⋊C₄	—	356	1	no	—
302	C₄²	—	356	1	no	—
303	C₄²	C₂×D₃₅₆	1	2	yes	—
304	C₁₆	—	2	2	no	—
305	C₁₆	C₂×C₄×D₈₉	1	4	yes	—
306	C₂³	—	356	1	no	—
307	C₂³	—	356	2	no	—
308	C₂³	—	356	2	no	—
309	C₂³	—	356	2	no	—
310	C₂³	—	1424	2	no	—
311	C₂³	#15668040	1	1	yes	characteristic
312	Q₈	—	356	1	no	—
313	Q₈	—	178	2	no	—
314	D₄	—	356	2	no	—
315	D₄	—	356	2	no	—
316	D₄	—	356	2	no	—
317	D₄	—	356	2	no	—
318	D₄	—	356	2	no	—
319	D₄	—	356	1	no	—
320	D₄	—	178	4	no	—
321	D₄	—	178	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	356	1	no	—
324	C₂×C₄	—	356	2	no	—
325	C₂×C₄	—	356	2	no	—
326	C₂×C₄	—	356	2	no	—
327	C₂×C₄	—	1424	4	no	—
328	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
329	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
330	C₂×C₄	#15668040	1	1	yes	characteristic
331	C₂×C₄	#15668040	1	1	yes	characteristic
332	C₂×C₄	#15668040	1	1	yes	characteristic
333	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
334	C₈	—	2	2	no	—
335	C₈	—	356	2	no	—
336	C₈	C₄×D₃₅₆	1	1	yes	characteristic
337	C₈	C₄×D₃₅₆	1	1	yes	characteristic
338	C₈	C₂²×C₄×D₈₉	1	1	yes	characteristic
339	C₈	C₂×#13083574	1	1	yes	characteristic
340	C₂²	—	4	1	no	—
341	C₂²	—	356	2	no	—
342	C₂²	—	356	4	no	—
343	C₂²	—	1424	2	no	—
344	C₂²	—	1424	2	no	—
345	C₂²	—	1424	1	no	—
346	C₂²	—	712	4	no	—
347	C₂²	#19151711	1	1	yes	characteristic
348	C₂²	#19151711	1	1	yes	characteristic
349	C₂²	C₂×#15668040	1	1	yes	characteristic
350	C₄	—	4	2	no	—
351	C₄	—	356	2	no	—
352	C₄	—	712	2	no	—
353	C₄	#19151711	1	1	yes	characteristic
354	C₄	#19151711	1	1	yes	characteristic
355	C₄	C₂×C₄×D₃₅₆	1	1	yes	characteristic
356	C₄	C₂×#15668040	1	1	yes	characteristic
357	C₂	—	2	1	no	—
358	C₂	—	2	1	no	—
359	C₂	—	2	1	no	—
360	C₂	—	712	2	no	—
361	C₂	—	712	4	no	—
362	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	C₂×#19152036	C₂
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2110	D₈₉
hypocenter	C₈₉	#33471
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. #26029105	C₂	semidirect
4. #26022152	C₂	semidirect
5. #26022025	C₂	semidirect
6. C₂×#19151412	C₂	semidirect
7. #26021989	C₂	semidirect
8. C₈₉×#2110	C₂	semidirect
13. #19152036	C₄	semidirect
15. C₂×#15667957	C₄	semidirect
16. C₂×D₁₄₂₄	C₄	semidirect
17. #19151721	C₂²	semidirect
19. #19151711	C₂²	semidirect
25. #19151421	C₂²	semidirect
26. #19151412	C₂²	semidirect
27. #19151409	C₂²	semidirect
31. #347×C₈₉	C₂²	semidirect
36. #15668034	C₂×C₄	semidirect
38. C₂×Dic₃₅₆	C₂×C₄	semidirect
45. C₂×D₇₁₂	C₂×C₄	semidirect
55. #15667957	C₂×C₄	semidirect
56. D₁₄₂₄	C₂×C₄	semidirect
67. M₃₂×C₈₉	C₂×C₄	semidirect
71. C₂×C₁₆×C₈₉	C₂×C₄	semidirect
189. #2110	D₈₉	semidirect
190. C₈₉	#33471	semidirect
213. #347	C₂×D₈₉	semidirect
243. M₃₂	C₄×D₈₉	semidirect
247. 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
C₂×#19152036	2
#26029170	2
#26029105	2
#26022152	2
#26022025	2
C₂×#19151412	2
#26021989	2
C₈₉×#2110	2
#33471	89