Subgroups — #36445546

Order 22784.

406 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029186	C₂	1	1	yes	maximal, characteristic
2	#26018910	C₂	1	1	yes	maximal, characteristic
3	#26018750	C₂	1	1	yes	maximal, characteristic
4	#26018544	C₂	1	1	yes	maximal, characteristic
5	#26018155	C₂	1	1	yes	maximal, characteristic
6	#26018149	C₂	1	1	yes	maximal, characteristic
7	C₈×C₄⋊C₄×C₈₉	C₂	1	1	yes	maximal, characteristic
8	#19152155	C₂²	1	1	yes	characteristic
9	#19151728	—	2	1	no	—
10	#19151728	C₄	1	2	yes	complemented
11	C₈×Dic₁₇₈	—	2	1	no	—
12	C₈×Dic₁₇₈	C₄	1	2	yes	complemented
13	#19151651	C₂²	1	1	yes	characteristic
14	#19151651	C₂²	1	1	yes	characteristic
15	#19151624	—	2	1	no	—
16	#19151622	C₂²	1	1	yes	characteristic
17	#19151553	—	2	1	no	—
18	#19151346	C₄	1	2	yes	complemented
19	#19151343	C₄	1	2	yes	complemented
20	#19151334	C₄	1	4	yes	—
21	#19151320	—	2	1	no	—
22	#19151320	—	2	1	no	—
23	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₈₉	—	2	1	no	—
25	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
26	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
27	#336×C₈₉	—	2	1	no	—
28	C₂×#13083558	—	2	1	no	—
29	#15668012	C₂×C₄	1	1	yes	characteristic
30	#15668012	C₂×C₄	1	1	yes	characteristic
31	#15668004	—	2	1	no	—
32	#15667991	—	2	1	no	—
33	#15667991	C₂×C₄	1	2	yes	—
34	C₄×Dic₁₇₈	—	2	1	no	—
35	C₄×Dic₁₇₈	C₂×C₄	1	2	yes	—
36	#15667971	—	2	1	no	—
37	C₄⋊#10623494	—	2	1	no	—
38	C₄⋊#10623494	—	2	1	no	—
39	C₄⋊#10623494	C₂×C₄	1	2	yes	—
40	C₄⋊#10623494	C₂×C₄	1	2	yes	—
41	#15667941	—	2	2	no	—
42	#15667941	—	2	2	no	—
43	C₂²⋊#10623494	—	2	1	no	—
44	C₂²⋊#10623494	C₂×C₄	1	2	yes	—
45	#15667922	—	2	2	no	—
46	C₄×#10623494	—	2	1	no	—
47	C₄×#10623494	—	2	1	no	—
48	C₈×Dic₈₉	—	2	2	no	—
49	C₂²×C₈×C₈₉	—	2	1	no	—
50	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
52	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
53	C₂×C₄²×C₈₉	—	2	1	no	—
54	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₈₉	—	2	1	no	—
56	C₄⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
57	C₄×C₈×C₈₉	—	4	2	no	—
58	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
61	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
62	#95×C₈₉	C₂×C₄	1	2	yes	—
63	C₂×Dic₁₇₈	—	2	1	no	—
64	C₂×Dic₁₇₈	C₄²	1	2	yes	—
65	#13083558	—	4	1	no	—
66	#13083558	—	4	1	no	—
67	C₂×#10623494	—	2	2	no	—
68	C₂×#10623494	—	2	2	no	—
69	C₂×#10623494	—	2	2	no	—
70	C₄⋊Dic₈₉	—	2	1	no	—
71	C₄⋊Dic₈₉	—	2	1	no	—
72	C₄⋊Dic₈₉	—	2	1	no	—
73	C₄⋊Dic₈₉	C₄²	1	2	yes	—
74	#13083553	—	2	2	no	—
75	#13083553	—	2	2	no	—
76	C₂²⋊Dic₈₉	—	2	2	no	—
77	C₄×Dic₈₉	—	2	2	no	—
78	C₂²×C₄×C₈₉	—	2	2	no	—
79	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
82	C₂×C₈×C₈₉	—	2	1	no	—
83	C₂×C₈×C₈₉	—	2	1	no	—
84	C₂×C₈×C₈₉	—	2	1	no	—
85	C₂×C₈×C₈₉	—	2	1	no	—
86	C₂×C₈×C₈₉	—	2	2	no	—
87	C₂×C₈×C₈₉	—	4	1	no	—
88	C₂×C₈×C₈₉	—	4	1	no	—
89	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
93	C₄⋊C₄×C₈₉	—	2	4	no	—
94	C₄²×C₈₉	—	4	1	no	—
95	C₄²×C₈₉	—	4	1	no	—
96	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
98	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₈₉	#41	1	1	yes	characteristic
100	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
101	C₂×C₄×C₈₉	—	2	2	no	—
102	C₂×C₄×C₈₉	—	2	4	no	—
103	C₂×C₄×C₈₉	—	4	1	no	—
104	C₂×C₄×C₈₉	—	4	2	no	—
105	C₂×C₄×C₈₉	—	4	2	no	—
106	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
107	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
109	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
113	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
114	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
115	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
116	C₂×Dic₈₉	—	2	2	no	—
117	C₂×Dic₈₉	—	2	2	no	—
118	Dic₁₇₈	—	4	2	no	—
119	Dic₁₇₈	—	4	2	no	—
120	C₈×C₈₉	—	2	1	no	—
121	C₈×C₈₉	—	2	1	no	—
122	C₈×C₈₉	—	2	1	no	—
123	C₈×C₈₉	—	2	1	no	—
124	C₈×C₈₉	—	4	2	no	—
125	#10623494	—	4	2	no	—
126	#10623494	—	4	2	no	—
127	#10623494	—	4	2	no	—
128	C₂²×C₈₉	—	2	1	no	—
129	C₂²×C₈₉	—	2	1	no	—
130	C₂²×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
131	C₂²×C₈₉	#440	1	1	yes	characteristic
132	C₂²×C₈₉	#441	1	1	yes	characteristic
133	C₄×C₈₉	—	2	1	no	—
134	C₄×C₈₉	—	2	1	no	—
135	C₄×C₈₉	—	2	1	no	—
136	C₄×C₈₉	—	4	4	no	—
137	C₄×C₈₉	#417	1	1	yes	characteristic
138	C₄×C₈₉	#419	1	1	yes	characteristic
139	C₄×C₈₉	#440	1	1	yes	characteristic
140	C₄×C₈₉	#441	1	1	yes	characteristic
141	C₄×C₈₉	#443	1	2	yes	—
142	Dic₈₉	—	4	2	no	—
143	Dic₈₉	—	4	2	no	—
144	#8258	—	89	1	no	maximal, Sylow
145	C₂×C₈₉	—	2	1	no	—
146	C₂×C₈₉	—	2	1	no	—
147	C₂×C₈₉	#1757	1	1	yes	characteristic
148	C₂×C₈₉	#1763	1	1	yes	characteristic
149	C₂×C₈₉	#1764	1	1	yes	characteristic
150	#2960	—	89	1	no	—
151	C₈×C₄⋊C₄	D₈₉	1	1	yes	characteristic
152	#1542	—	89	1	no	—
153	#1503	—	89	1	no	—
154	#1452	—	89	1	no	—
155	#1303	—	89	1	no	—
156	#1301	—	89	1	no	—
157	C₈₉	#8258	1	1	yes	characteristic, complemented, Sylow
158	#533	—	89	1	no	—
159	#446	—	89	2	no	—
160	#446	—	178	1	no	—
161	C₈×Q₈	—	89	2	no	—
162	C₈×Q₈	—	178	1	no	—
163	#431	—	89	1	no	—
164	#431	—	89	1	no	—
165	#424	—	178	1	no	—
166	#423	—	89	1	no	—
167	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
168	#405	—	178	1	no	—
169	C₂×C₄×C₈	—	2	1	no	—
170	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
171	C₄×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
172	#336	—	2	1	no	—
173	#333	—	89	2	no	—
174	#332	—	89	2	no	—
175	#328	—	89	4	no	—
176	#322	—	178	1	no	—
177	#322	—	178	1	no	—
178	C₂×M₁₆	—	178	1	no	—
179	C₂²×C₈	—	2	1	no	—
180	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
181	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
182	C₄⋊Q₈	—	89	1	no	—
183	C₄⋊Q₈	—	89	1	no	—
184	#125	—	178	1	no	—
185	C₂²⋊Q₈	—	89	2	no	—
186	C₂²⋊Q₈	—	178	1	no	—
187	C₄×Q₈	—	89	2	no	—
188	C₄×Q₈	—	178	1	no	—
189	#117	—	178	1	no	—
190	C₂×C₄⋊C₄	—	2	1	no	—
191	C₂×C₄²	—	2	1	no	—
192	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
193	C₄⋊C₈	—	2	1	no	—
194	C₄⋊C₈	—	89	2	no	—
195	C₄⋊C₈	—	89	2	no	—
196	C₄⋊C₈	—	178	2	no	—
197	C₄⋊C₈	—	178	2	no	—
198	C₄⋊C₈	—	178	1	no	—
199	C₄⋊C₈	—	178	1	no	—
200	C₄⋊C₈	C₄×D₈₉	1	2	yes	—
201	C₂²⋊C₈	—	89	2	no	—
202	C₂²⋊C₈	—	178	1	no	—
203	#97	—	178	2	no	—
204	C₄×C₈	—	4	2	no	—
205	C₄×C₈	—	178	2	no	—
206	C₄×C₈	—	178	1	no	—
207	C₄×C₈	—	178	1	no	—
208	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
209	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
210	C₄×C₈	D₃₅₆	1	1	yes	characteristic
211	C₄×C₈	D₃₅₆	1	1	yes	characteristic
212	#95	C₄×D₈₉	1	2	yes	—
213	C₂×Q₈	—	89	2	no	—
214	C₂×Q₈	—	178	1	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
218	C₂²×C₄	#13083574	1	1	yes	characteristic
219	M₁₆	—	356	1	no	—
220	M₁₆	—	356	1	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	4	1	no	—
227	C₂×C₈	—	4	1	no	—
228	C₂×C₈	—	178	2	no	—
229	C₂×C₈	—	178	2	no	—
230	C₂×C₈	—	178	2	no	—
231	C₂×C₈	#13083551	1	1	yes	characteristic
232	C₂×C₈	#13083551	1	1	yes	characteristic
233	C₂×C₈	#13083551	1	1	yes	characteristic
234	C₂×C₈	#13083551	1	1	yes	characteristic
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	89	2	no	—
237	C₄⋊C₄	—	178	2	no	—
238	C₄⋊C₄	—	178	1	no	—
239	C₄⋊C₄	—	178	1	no	—
240	C₄⋊C₄	—	178	2	no	—
241	C₄⋊C₄	—	178	1	no	—
242	C₂²⋊C₄	—	178	2	no	—
243	C₄²	—	4	1	no	—
244	C₄²	—	4	1	no	—
245	C₄²	—	178	2	no	—
246	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
247	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
248	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
249	C₄²	#13083574	1	1	yes	characteristic
250	C₂³	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
251	Q₈	—	356	2	no	—
252	Q₈	—	356	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	178	2	no	—
259	C₂×C₄	—	178	2	no	—
260	C₂×C₄	C₄²×D₈₉	1	1	yes	characteristic
261	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
262	C₂×C₄	#15667969	1	1	yes	characteristic
263	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
264	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
265	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
266	C₂×C₄	#15668038	1	1	yes	characteristic
267	C₂×C₄	#15668042	1	1	yes	characteristic
268	C₂×C₄	#15668042	1	1	yes	characteristic
269	C₂×C₄	#15668043	1	1	yes	characteristic
270	C₈	—	2	1	no	—
271	C₈	—	2	1	no	—
272	C₈	—	2	1	no	—
273	C₈	—	2	1	no	—
274	C₈	—	4	2	no	—
275	C₈	—	356	2	no	—
276	C₈	—	356	2	no	—
277	C₈	—	356	2	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	2	1	no	—
280	C₂²	C₄×#13083551	1	1	yes	characteristic
281	C₂²	#19151698	1	1	yes	characteristic
282	C₂²	#19151703	1	1	yes	characteristic
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	4	4	no	—
287	C₄	—	356	2	no	—
288	C₄	—	356	2	no	—
289	C₄	#19151597	1	1	yes	characteristic
290	C₄	#19151606	1	1	yes	characteristic
291	C₄	#19151698	1	1	yes	characteristic
292	C₄	#19151703	1	1	yes	characteristic
293	C₄	#19151716	1	2	yes	—
294	C₂	—	2	1	no	—
295	C₂	—	2	1	no	—
296	C₂	#26020053	1	1	yes	characteristic
297	C₂	#26020085	1	1	yes	characteristic
298	C₂	#26020089	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#13083551
commutator	C₂×C₄×C₈₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈×C₄⋊C₄×C₈₉	C₂
socle	C₂²×C₈₉	C₄×C₂²⋊C₄
prime radical	C₂³×C₈₉	C₄×D₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₈×C₄⋊C₄	D₈₉
hypocenter	C₈₉	#8258
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
10. #19151728	C₄	semidirect
12. C₈×Dic₁₇₈	C₄	semidirect
18. #19151346	C₄	semidirect
19. #19151343	C₄	semidirect
157. C₈₉	#8258	semidirect

Maximal subgroups

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

subgroup	index
#26029186	2
#26018910	2
#26018750	2
#26018544	2
#26018155	2
#26018149	2
C₈×C₄⋊C₄×C₈₉	2
#8258	89