Subgroups — #36445683

Order 22784.

398 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#15667943	C₂	1	1	yes	maximal, characteristic
2	#26020119	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151350	C₂	1	1	yes	maximal, characteristic
4	C₂×#19151320	C₂	1	1	yes	maximal, characteristic
5	#26018093	C₂	1	1	yes	maximal, characteristic
6	#26018091	C₂	1	1	yes	maximal, characteristic
7	C₈₉×#1770	C₂	1	1	yes	maximal, characteristic
8	C₂×#15667943	C₄	1	2	yes	—
9	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
10	C₂×#15667922	—	2	1	no	—
11	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
12	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
14	#19151360	C₄	1	2	yes	—
15	#19151352	—	2	1	no	—
16	#19151350	C₄	1	4	yes	complemented
17	#19151320	C₄	1	4	yes	complemented
18	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×#97×C₈₉	—	2	1	no	—
20	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
21	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
22	#336×C₈₉	—	2	1	no	—
23	C₂²×#10623494	—	2	1	no	—
24	C₂²×#10623494	—	2	1	no	—
25	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
26	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
28	C₂×C₄×Dic₈₉	—	2	1	no	—
29	#15667943	—	2	4	no	—
30	C₄⋊#10623494	—	2	1	no	—
31	C₄⋊#10623494	C₂×C₄	1	2	yes	—
32	#15667922	—	4	2	no	—
33	C₄×#10623494	—	2	1	no	—
34	C₄×#10623494	C₂×C₄	1	2	yes	—
35	C₄×#10623494	C₂×C₄	1	2	yes	—
36	C₄×#10623494	C₂×C₄	1	2	yes	—
37	#15667919	—	2	1	no	—
38	C₂²×C₈×C₈₉	—	2	1	no	—
39	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₈₉	—	2	1	no	—
43	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₈₉	—	2	1	no	—
45	C₄⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
46	#97×C₈₉	—	4	2	no	—
47	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₈₉	Q₈	1	2	yes	—
49	#95×C₈₉	—	2	1	no	—
50	C₂²×Dic₈₉	—	2	2	no	—
51	C₂×#10623494	—	2	2	no	—
52	C₂×#10623494	—	2	2	no	—
53	C₂×#10623494	—	2	2	no	—
54	C₂×#10623494	—	4	2	no	—
55	C₂×#10623494	—	4	1	no	—
56	C₂×#10623494	—	4	1	no	—
57	C₂×#10623494	C₄²	1	2	yes	—
58	C₂×#10623494	C₄²	1	2	yes	—
59	C₂×#10623494	M₁₆	1	4	yes	—
60	C₄⋊Dic₈₉	—	2	2	no	—
61	C₄⋊Dic₈₉	—	2	2	no	—
62	C₄×Dic₈₉	—	4	1	no	—
63	C₄×Dic₈₉	—	4	1	no	—
64	C₂²×C₄×C₈₉	—	2	2	no	—
65	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
68	C₂×C₈×C₈₉	—	2	1	no	—
69	C₂×C₈×C₈₉	—	2	1	no	—
70	C₂×C₈×C₈₉	—	2	1	no	—
71	C₂×C₈×C₈₉	—	2	1	no	—
72	C₂×C₈×C₈₉	—	2	2	no	—
73	C₂×C₈×C₈₉	—	4	1	no	—
74	C₂×C₈×C₈₉	—	4	1	no	—
75	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
76	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
77	C₄⋊C₄×C₈₉	—	2	2	no	—
78	C₄⋊C₄×C₈₉	—	2	2	no	—
79	C₄²×C₈₉	—	4	1	no	—
80	C₄²×C₈₉	—	4	1	no	—
81	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
84	C₄²×C₈₉	#41	1	1	yes	characteristic
85	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₈₉	—	2	2	no	—
87	C₂×C₄×C₈₉	—	4	2	no	—
88	C₂×C₄×C₈₉	—	4	2	no	—
89	C₂×C₄×C₈₉	—	4	2	no	—
90	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
91	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
93	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
94	C₂×C₄×C₈₉	C₄×Q₈	1	1	yes	characteristic
95	C₂×C₄×C₈₉	C₄×Q₈	1	1	yes	characteristic
96	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
97	C₂×C₄×C₈₉	#131	1	2	yes	—
98	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
99	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
100	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
101	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
102	C₂×Dic₈₉	—	4	2	no	—
103	C₂×Dic₈₉	—	4	2	no	—
104	C₂×Dic₈₉	—	4	2	no	—
105	C₈×C₈₉	—	2	2	no	—
106	C₈×C₈₉	—	2	2	no	—
107	C₈×C₈₉	—	4	2	no	—
108	#10623494	—	2	2	no	—
109	#10623494	—	4	2	no	—
110	#10623494	—	4	2	no	—
111	#10623494	#97	1	4	yes	complemented
112	C₂²×C₈₉	C₄×C₄⋊C₄	1	1	yes	characteristic
113	C₂²×C₈₉	#436	1	2	yes	—
114	C₂²×C₈₉	#440	1	1	yes	characteristic
115	C₂²×C₈₉	#440	1	1	yes	characteristic
116	C₂²×C₈₉	#444	1	2	yes	—
117	C₄×C₈₉	—	2	1	no	—
118	C₄×C₈₉	—	2	1	no	—
119	C₄×C₈₉	—	8	2	no	—
120	C₄×C₈₉	C₂×#97	1	1	yes	characteristic
121	C₄×C₈₉	#405	1	1	yes	characteristic
122	C₄×C₈₉	#428	1	1	yes	characteristic
123	C₄×C₈₉	#428	1	1	yes	characteristic
124	C₄×C₈₉	#441	1	1	yes	characteristic
125	C₄×C₈₉	#442	1	1	yes	characteristic
126	C₄×C₈₉	#446	1	2	yes	—
127	Dic₈₉	—	8	2	no	—
128	#8286	—	89	1	no	maximal, Sylow
129	C₂×C₈₉	#1592	1	2	yes	—
130	C₂×C₈₉	#1592	1	2	yes	—
131	C₂×C₈₉	#1770	1	1	yes	characteristic
132	C₂×C₈₉	#1773	1	1	yes	characteristic
133	C₂×C₈₉	#1776	1	1	yes	characteristic
134	C₄×#106	—	89	1	no	—
135	#1770	—	89	1	no	—
136	#1770	D₈₉	1	1	yes	characteristic
137	C₂×#335	—	89	1	no	—
138	C₂×#322	—	89	1	no	—
139	#1279	—	89	1	no	—
140	#1279	—	89	1	no	—
141	C₈₉	#8286	1	1	yes	characteristic, complemented, Sylow
142	C₂×#106	—	89	2	no	—
143	C₂×C₄⋊C₈	—	89	1	no	—
144	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
145	C₂×#97	—	2	1	no	—
146	C₂×#97	—	178	1	no	—
147	C₂×C₄×C₈	—	89	1	no	—
148	C₂×C₄×C₈	—	89	1	no	—
149	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
150	C₄×C₄⋊C₄	—	89	1	no	—
151	C₄×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
152	#340	—	89	2	no	—
153	#336	—	2	1	no	—
154	#336	—	178	1	no	—
155	#335	—	89	4	no	—
156	#322	—	89	4	no	—
157	C₂²×C₈	—	2	1	no	—
158	C₂²×C₈	—	89	1	no	—
159	C₂²×C₈	—	89	1	no	—
160	C₂²×C₈	—	178	1	no	—
161	C₂²×C₈	—	178	1	no	—
162	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
163	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
164	C₂×C₄⋊C₄	—	89	2	no	—
165	C₂×C₄⋊C₄	C₄×D₈₉	1	2	yes	—
166	C₂×C₄²	—	2	1	no	—
167	C₂×C₄²	—	178	1	no	—
168	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
169	#106	—	178	4	no	—
170	C₄⋊C₈	—	2	1	no	—
171	C₄⋊C₈	—	89	2	no	—
172	C₄⋊C₈	—	178	1	no	—
173	C₄⋊C₈	C₄×D₈₉	1	2	yes	—
174	#97	—	4	2	no	—
175	#97	—	356	2	no	—
176	C₄×C₈	—	89	2	no	—
177	C₄×C₈	—	89	2	no	—
178	C₄×C₈	—	89	2	no	—
179	C₄×C₈	—	178	1	no	—
180	C₄×C₈	Dic₁₇₈	1	2	yes	—
181	C₄×C₈	C₄×D₈₉	1	2	yes	—
182	#95	—	2	1	no	—
183	#95	—	178	1	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	178	2	no	—
186	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
187	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
188	C₂²×C₄	#13083574	1	1	yes	characteristic
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	4	1	no	—
195	C₂×C₈	—	4	1	no	—
196	C₂×C₈	—	89	4	no	—
197	C₂×C₈	—	89	2	no	—
198	C₂×C₈	—	89	2	no	—
199	C₂×C₈	—	356	2	no	—
200	C₂×C₈	—	356	1	no	—
201	C₂×C₈	—	356	1	no	—
202	C₂×C₈	—	178	2	no	—
203	C₂×C₈	—	178	2	no	—
204	C₂×C₈	—	178	2	no	—
205	C₂×C₈	#13083553	1	2	yes	—
206	C₂×C₈	#13083553	1	2	yes	—
207	C₄⋊C₄	—	2	2	no	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	178	2	no	—
210	C₄⋊C₄	—	178	2	no	—
211	C₄²	—	4	1	no	—
212	C₄²	—	4	1	no	—
213	C₄²	—	356	1	no	—
214	C₄²	—	356	1	no	—
215	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
216	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
217	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
218	C₄²	#13083574	1	1	yes	characteristic
219	C₂³	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	356	2	no	—
225	C₂×C₄	—	356	2	no	—
226	C₂×C₄	—	356	2	no	—
227	C₂×C₄	C₄²×D₈₉	1	1	yes	characteristic
228	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
229	C₂×C₄	#15667969	1	1	yes	characteristic
230	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
231	C₂×C₄	C₄×Dic₁₇₈	1	1	yes	characteristic
232	C₂×C₄	C₄×Dic₁₇₈	1	1	yes	characteristic
233	C₂×C₄	M₁₆×D₈₉	1	2	yes	—
234	C₂×C₄	#15668020	1	2	yes	—
235	C₂×C₄	#15668037	1	1	yes	characteristic
236	C₂×C₄	#15668038	1	1	yes	characteristic
237	C₂×C₄	#15668042	1	1	yes	characteristic
238	C₂×C₄	#15668043	1	1	yes	characteristic
239	C₈	—	2	2	no	—
240	C₈	—	2	2	no	—
241	C₈	—	4	2	no	—
242	C₈	—	89	4	no	—
243	C₈	—	356	2	no	—
244	C₈	—	356	2	no	—
245	C₈	—	178	2	no	—
246	C₂²	C₄×#13083553	1	1	yes	characteristic
247	C₂²	#19151674	1	2	yes	—
248	C₂²	#19151696	1	1	yes	characteristic
249	C₂²	#19151698	1	1	yes	characteristic
250	C₂²	#19151718	1	2	yes	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	8	2	no	—
254	C₄	—	712	2	no	—
255	C₄	#97×D₈₉	1	1	yes	characteristic
256	C₄	#19151555	1	1	yes	characteristic
257	C₄	#19151638	1	1	yes	characteristic
258	C₄	#19151641	1	1	yes	characteristic
259	C₄	#19151703	1	1	yes	characteristic
260	C₄	#19151708	1	1	yes	characteristic
261	C₄	#19151726	1	2	yes	—
262	C₂	#26019152	1	2	yes	—
263	C₂	#26019154	1	2	yes	—
264	C₂	#26020120	1	1	yes	characteristic
265	C₂	#26020137	1	1	yes	characteristic
266	C₂	#26020150	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₈₉
commutator	C₂×C₄×C₈₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1770	C₂
socle, prime radical	C₂³×C₈₉	C₄×D₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1770	D₈₉
hypocenter	C₈₉	#8286
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1770

Complements

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

N	H ≅ G/N	type
16. #19151350	C₄	semidirect
17. #19151320	C₄	semidirect
111. #10623494	#97	semidirect
141. C₈₉	#8286	semidirect

Maximal subgroups

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

subgroup	index
C₄×#15667943	2
#26020119	2
C₂×#19151350	2
C₂×#19151320	2
#26018093	2
#26018091	2
C₈₉×#1770	2
#8286	89