Subgroups — #36709178

Order 22784.

514 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030783	C₂	1	1	yes	maximal, characteristic, complemented
2	#26022428	C₂	1	4	yes	maximal, complemented
3	#26022423	C₂	1	4	yes	maximal, complemented
4	#26022026	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19151422	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022018	C₂	1	1	yes	maximal, characteristic, complemented
7	#26021988	C₂	1	1	yes	maximal, characteristic
8	C₂×#19151408	C₂	1	1	yes	maximal, characteristic
9	C₈₉×#2957	C₂	1	1	yes	maximal, characteristic
10	#19152226	—	2	1	no	—
11	C₂×#15667952	C₂²	1	1	yes	characteristic
12	C₂×#15667952	C₂²	1	1	yes	characteristic
13	#19152007	—	2	2	no	—
14	#19152000	C₂²	1	2	yes	complemented
15	#19151996	C₂²	1	2	yes	complemented
16	#19151640	—	2	1	no	—
17	#19151636	C₂²	1	1	yes	characteristic
18	C₂×#15667944	C₂²	1	1	yes	characteristic
19	#19151422	C₂²	1	2	yes	complemented
20	#19151422	C₂²	1	4	yes	complemented
21	#19151422	C₂²	1	2	yes	complemented
22	#19151419	C₂²	1	4	yes	complemented
23	#19151408	C₂²	1	2	yes	—
24	#19151408	C₂²	1	4	yes	—
25	#19151408	C₂²	1	2	yes	—
26	C₂×#135×C₈₉	C₂²	1	1	yes	characteristic
27	#568×C₈₉	—	2	1	no	—
28	#444×C₈₉	—	2	2	no	—
29	#443×C₈₉	C₂²	1	4	yes	—
30	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
31	#421×C₈₉	—	2	1	no	—
32	#405×C₈₉	C₂²	1	1	yes	characteristic
33	#15668012	—	2	2	no	—
34	#15668011	—	4	1	no	—
35	#15668004	—	2	2	no	—
36	#15668003	—	4	1	no	—
37	#15667989	—	4	2	no	—
38	#15667971	—	2	1	no	—
39	C₂×C₄⋊Dic₈₉	—	2	1	no	—
40	#15667952	—	2	1	no	—
41	#15667952	—	2	1	no	—
42	#15667952	C₂³	1	2	yes	—
43	#15667952	C₂³	1	2	yes	—
44	C₂×#13083549	—	2	1	no	—
45	C₂×#13083549	—	2	1	no	—
46	#15667946	—	2	1	no	—
47	#15667946	—	2	2	no	—
48	#15667944	—	2	1	no	—
49	#15667944	—	2	2	no	—
50	#15667944	C₂³	1	2	yes	—
51	#15667944	C₂³	1	1	yes	characteristic
52	#15667944	C₂³	1	1	yes	characteristic
53	C₂×#41×C₈₉	—	2	1	no	—
54	#135×C₈₉	—	2	1	no	—
55	#135×C₈₉	—	2	1	no	—
56	#135×C₈₉	Q₈	1	2	yes	complemented
57	#135×C₈₉	C₂³	1	2	yes	—
58	C₂×Q₁₆×C₈₉	Q₈	1	1	yes	characteristic, complemented
59	C₂×SD₁₆×C₈₉	—	2	1	no	—
60	C₂×D₈×C₈₉	Q₈	1	1	yes	characteristic, complemented
61	#131×C₈₉	—	2	4	no	—
62	#131×C₈₉	—	4	2	no	—
63	C₂×M₁₆×C₈₉	—	2	1	no	—
64	C₂×M₁₆×C₈₉	—	4	2	no	—
65	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
66	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
67	#117×C₈₉	D₄	1	1	yes	characteristic
68	#108×C₈₉	—	2	1	no	—
69	#108×C₈₉	C₂³	1	2	yes	—
70	#104×C₈₉	—	2	4	no	—
71	#97×C₈₉	D₄	1	2	yes	—
72	C₄×C₈×C₈₉	C₂³	1	2	yes	—
73	C₂×Dic₁₇₈	—	4	2	no	—
74	C₂²×Dic₈₉	—	4	1	no	—
75	C₄⋊Dic₈₉	—	2	1	no	—
76	C₄⋊Dic₈₉	—	2	1	no	—
77	C₄⋊Dic₈₉	—	2	2	no	—
78	C₄⋊Dic₈₉	—	4	2	no	—
79	C₄⋊Dic₈₉	—	4	1	no	—
80	#13083553	—	4	2	no	—
81	#13083553	—	4	2	no	—
82	C₂²⋊Dic₈₉	—	4	1	no	—
83	C₄×Dic₈₉	—	4	1	no	—
84	#13083549	—	2	2	no	—
85	#13083549	—	2	2	no	—
86	#41×C₈₉	—	2	2	no	—
87	#41×C₈₉	—	2	2	no	—
88	#41×C₈₉	—	4	1	no	—
89	#41×C₈₉	—	4	1	no	—
90	C₂×Q₈×C₈₉	—	2	1	no	—
91	C₂×D₄×C₈₉	—	2	1	no	—
92	C₂×D₄×C₈₉	—	4	1	no	—
93	C₂²×C₄×C₈₉	—	4	1	no	—
94	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
95	Q₁₆×C₈₉	—	2	1	no	—
96	Q₁₆×C₈₉	C₂×Q₈	1	2	yes	—
97	SD₁₆×C₈₉	—	2	2	no	—
98	SD₁₆×C₈₉	—	2	2	no	—
99	D₈×C₈₉	—	2	1	no	—
100	D₈×C₈₉	C₂×Q₈	1	2	yes	—
101	M₁₆×C₈₉	—	2	2	no	—
102	M₁₆×C₈₉	—	4	1	no	—
103	M₁₆×C₈₉	—	4	4	no	—
104	M₁₆×C₈₉	SD₁₆	1	2	yes	—
105	M₁₆×C₈₉	SD₁₆	1	2	yes	—
106	C₂×C₈×C₈₉	—	4	1	no	—
107	C₂×C₈×C₈₉	—	4	4	no	—
108	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
110	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
111	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
112	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
113	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
114	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
115	C₄⋊C₄×C₈₉	—	2	1	no	—
116	C₂²⋊C₄×C₈₉	—	2	1	no	—
117	C₄²×C₈₉	C₂×D₄	1	2	yes	—
118	C₂³×C₈₉	—	4	1	no	—
119	C₂³×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
120	Q₈×C₈₉	—	2	2	no	—
121	Q₈×C₈₉	—	4	1	no	—
122	D₄×C₈₉	—	2	2	no	—
123	D₄×C₈₉	—	4	2	no	—
124	D₄×C₈₉	—	4	2	no	—
125	D₄×C₈₉	—	4	1	no	—
126	C₂×C₄×C₈₉	—	2	2	no	—
127	C₂×C₄×C₈₉	—	4	1	no	—
128	C₂×C₄×C₈₉	—	4	2	no	—
129	C₂×C₄×C₈₉	—	4	2	no	—
130	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
131	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
132	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
133	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
134	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
135	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
136	C₂×Dic₈₉	—	4	1	no	—
137	C₂×Dic₈₉	—	4	1	no	—
138	C₂×Dic₈₉	—	4	2	no	—
139	C₂×Dic₈₉	—	16	1	no	—
140	Dic₁₇₈	—	16	2	no	—
141	C₈×C₈₉	—	4	2	no	—
142	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
143	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
144	C₈×C₈₉	C₂×SD₁₆	1	4	yes	—
145	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
146	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
147	C₂²×C₈₉	—	4	1	no	—
148	C₂²×C₈₉	—	4	2	no	—
149	C₂²×C₈₉	—	16	1	no	—
150	C₂²×C₈₉	#476	1	1	yes	characteristic
151	C₂²×C₈₉	#476	1	1	yes	characteristic
152	C₂²×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
153	C₄×C₈₉	—	4	2	no	—
154	C₄×C₈₉	—	4	2	no	—
155	C₄×C₈₉	#476	1	1	yes	characteristic
156	C₄×C₈₉	#476	1	1	yes	characteristic
157	C₄×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
158	C₄×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
159	Dic₈₉	—	8	1	no	—
160	Dic₈₉	—	8	2	no	—
161	Dic₈₉	—	8	1	no	—
162	#33638	—	89	1	no	maximal, Sylow
163	C₂×C₈₉	—	2	1	no	—
164	C₂×C₈₉	—	2	1	no	—
165	C₂×C₈₉	—	2	1	no	—
166	C₂×C₈₉	—	8	2	no	—
167	C₂×C₈₉	C₂×#476	1	1	yes	characteristic
168	#3163	—	89	1	no	—
169	#2957	D₈₉	1	1	yes	characteristic
170	#2231	—	89	4	no	—
171	#2230	—	89	4	no	—
172	#2155	—	89	1	no	—
173	C₂×#365	—	89	1	no	—
174	#2153	—	89	1	no	—
175	#2145	—	89	1	no	—
176	C₂×#360	—	89	1	no	—
177	C₈₉	#33638	1	1	yes	characteristic, complemented, Sylow
178	C₂×#135	C₂×D₈₉	1	1	yes	characteristic
179	#568	—	2	1	no	—
180	#544	—	178	1	no	—
181	C₂×M₃₂	—	89	1	no	—
182	C₂×M₃₂	—	89	1	no	—
183	#501	—	178	2	no	—
184	#499	—	89	2	no	—
185	#498	—	89	2	no	—
186	#444	—	2	2	no	—
187	#443	C₂×D₈₉	1	4	yes	—
188	C₂×#108	C₂×D₈₉	1	1	yes	characteristic
189	#428	—	178	1	no	—
190	#427	—	89	1	no	—
191	C₂×#106	—	89	1	no	—
192	#421	—	2	1	no	—
193	#405	C₂×D₈₉	1	1	yes	characteristic
194	#365	—	89	4	no	—
195	#365	—	89	2	no	—
196	#365	—	89	2	no	—
197	#364	—	89	4	no	—
198	#360	—	89	4	no	—
199	#360	—	89	2	no	—
200	#360	—	89	2	no	—
201	C₂×#41	—	2	1	no	—
202	#135	—	2	1	no	—
203	#135	—	2	1	no	—
204	#135	Dic₁₇₈	1	2	yes	complemented
205	#135	C₂²×D₈₉	1	2	yes	—
206	C₂×Q₁₆	Dic₁₇₈	1	1	yes	characteristic, complemented
207	C₂×SD₁₆	—	2	1	no	—
208	C₂×D₈	Dic₁₇₈	1	1	yes	characteristic, complemented
209	#131	—	2	4	no	—
210	#131	—	4	2	no	—
211	C₂×M₁₆	—	2	1	no	—
212	C₂×M₁₆	—	4	2	no	—
213	C₂×M₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
214	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
215	C₄⋊Q₈	—	356	1	no	—
216	C₄⋊Q₈	—	178	2	no	—
217	#125	—	356	1	no	—
218	#125	—	178	2	no	—
219	C₂²⋊Q₈	—	356	2	no	—
220	#117	—	178	1	no	—
221	#117	C₂²⋊D₈₉	1	1	yes	characteristic
222	C₂×C₄⋊C₄	—	178	1	no	—
223	M₃₂	—	89	2	no	—
224	M₃₂	—	89	2	no	—
225	M₃₂	—	178	1	no	—
226	M₃₂	—	178	1	no	—
227	C₂×C₁₆	—	178	1	no	—
228	C₂×C₁₆	—	178	1	no	—
229	#108	—	2	1	no	—
230	#108	C₂²×D₈₉	1	2	yes	—
231	#107	—	178	1	no	—
232	#107	—	178	2	no	—
233	#106	—	89	2	no	—
234	#106	—	89	1	no	—
235	#106	—	89	1	no	—
236	#106	—	178	1	no	—
237	#106	—	178	2	no	—
238	#104	—	2	4	no	—
239	#97	C₂²⋊D₈₉	1	2	yes	—
240	C₄×C₈	C₂²×D₈₉	1	2	yes	—
241	#41	—	2	2	no	—
242	#41	—	2	2	no	—
243	#41	—	4	1	no	—
244	#41	—	4	1	no	—
245	C₂×Q₈	—	2	1	no	—
246	C₂×Q₈	—	356	2	no	—
247	C₂×D₄	—	2	1	no	—
248	C₂×D₄	—	4	1	no	—
249	C₂²×C₄	—	4	1	no	—
250	C₂²×C₄	—	356	1	no	—
251	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
252	Q₁₆	—	2	1	no	—
253	Q₁₆	C₂×Dic₁₇₈	1	2	yes	—
254	SD₁₆	—	2	2	no	—
255	SD₁₆	—	2	2	no	—
256	D₈	—	2	1	no	—
257	D₈	C₂×Dic₁₇₈	1	2	yes	—
258	M₁₆	—	2	2	no	—
259	M₁₆	—	4	1	no	—
260	M₁₆	—	4	4	no	—
261	M₁₆	#13083562	1	2	yes	—
262	M₁₆	#13083562	1	2	yes	—
263	C₂×C₈	—	4	1	no	—
264	C₂×C₈	—	4	4	no	—
265	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
266	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
267	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
268	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
269	C₂×C₈	#13083574	1	1	yes	characteristic
270	C₂×C₈	#13083574	1	1	yes	characteristic
271	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	356	2	no	—
274	C₄⋊C₄	—	356	2	no	—
275	C₄⋊C₄	—	356	2	no	—
276	C₄⋊C₄	—	356	1	no	—
277	C₄⋊C₄	—	178	2	no	—
278	C₄⋊C₄	—	178	1	no	—
279	C₄⋊C₄	—	178	1	no	—
280	C₂²⋊C₄	—	2	1	no	—
281	C₂²⋊C₄	—	356	1	no	—
282	C₄²	—	356	1	no	—
283	C₄²	C₂×C₂²⋊D₈₉	1	2	yes	—
284	C₁₆	—	178	2	no	—
285	C₁₆	—	178	2	no	—
286	C₂³	—	4	1	no	—
287	C₂³	#15668037	1	1	yes	characteristic
288	Q₈	—	2	2	no	—
289	Q₈	—	4	1	no	—
290	Q₈	—	1424	2	no	—
291	D₄	—	2	2	no	—
292	D₄	—	4	2	no	—
293	D₄	—	4	2	no	—
294	D₄	—	4	1	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	356	1	no	—
300	C₂×C₄	—	356	1	no	—
301	C₂×C₄	—	356	2	no	—
302	C₂×C₄	—	1424	1	no	—
303	C₂×C₄	#15667991	1	1	yes	characteristic
304	C₂×C₄	#15667991	1	1	yes	characteristic
305	C₂×C₄	C₂×#13083562	1	1	yes	characteristic
306	C₂×C₄	C₂×#13083562	1	1	yes	characteristic
307	C₂×C₄	#15668037	1	1	yes	characteristic
308	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
309	C₈	—	4	2	no	—
310	C₈	#15667991	1	1	yes	characteristic
311	C₈	#15667991	1	1	yes	characteristic
312	C₈	C₂×#13083562	1	4	yes	—
313	C₈	C₂²×Dic₁₇₈	1	1	yes	characteristic
314	C₈	C₂×#13083574	1	1	yes	characteristic
315	C₂²	—	4	1	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	16	1	no	—
318	C₂²	#19151888	1	1	yes	characteristic
319	C₂²	#19151888	1	1	yes	characteristic
320	C₂²	C₂×#15668037	1	1	yes	characteristic
321	C₄	—	4	2	no	—
322	C₄	—	4	2	no	—
323	C₄	—	712	2	no	—
324	C₄	—	712	1	no	—
325	C₄	—	712	1	no	—
326	C₄	#19151888	1	1	yes	characteristic
327	C₄	#19151888	1	1	yes	characteristic
328	C₄	C₂×#15667991	1	1	yes	characteristic
329	C₄	C₂²×#13083562	1	1	yes	characteristic
330	C₂	—	2	1	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	8	2	no	—
334	C₂	C₂×#19151888	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#19151888
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2957	C₂
socle	C₂×C₈₉	C₂×#476
prime radical	C₂×#135×C₈₉	C₂²
hypercenter	#2957	D₈₉
hypocenter	C₈₉	#33638
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #2957

Complements

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

N	H ≅ G/N	type
1. #26030783	C₂	semidirect
2. #26022428	C₂	semidirect
3. #26022423	C₂	semidirect
4. #26022026	C₂	semidirect
5. C₂×#19151422	C₂	semidirect
6. #26022018	C₂	semidirect
14. #19152000	C₂²	semidirect
15. #19151996	C₂²	semidirect
19. #19151422	C₂²	semidirect
20. #19151422	C₂²	semidirect
21. #19151422	C₂²	semidirect
22. #19151419	C₂²	semidirect
56. #135×C₈₉	Q₈	semidirect
58. C₂×Q₁₆×C₈₉	Q₈	semidirect
60. C₂×D₈×C₈₉	Q₈	semidirect
177. C₈₉	#33638	semidirect
204. #135	Dic₁₇₈	semidirect
206. C₂×Q₁₆	Dic₁₇₈	semidirect
208. C₂×D₈	Dic₁₇₈	semidirect

Maximal subgroups

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

subgroup	index
#26030783	2
#26022428	2
#26022423	2
#26022026	2
C₂×#19151422	2
#26022018	2
#26021988	2
C₂×#19151408	2
C₈₉×#2957	2
#33638	89