Subgroups — #36454589

Order 22784.

438 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26020404	C₂	1	1	yes	maximal, characteristic
2	#26020105	C₂	1	1	yes	maximal, characteristic
3	#26018135	C₂	1	2	yes	maximal
4	#26018135	C₂	1	2	yes	maximal
5	C₈₉×#2859	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#10623494	—	2	1	no	—
7	C₂×C₄⋊#10623494	—	2	1	no	—
8	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
9	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
10	C₂×#15667923	—	2	1	no	—
11	#19151353	—	2	1	no	—
12	C₂×#125×C₈₉	C₄	1	2	yes	—
13	#389×C₈₉	—	2	1	no	—
14	#389×C₈₉	—	2	1	no	—
15	#389×C₈₉	—	2	1	no	—
16	#389×C₈₉	—	2	1	no	—
17	#384×C₈₉	C₂²	1	1	yes	characteristic
18	#384×C₈₉	C₂²	1	1	yes	characteristic
19	#382×C₈₉	C₄	1	2	yes	—
20	C₄³×C₈₉	C₂²	1	1	yes	characteristic
21	C₂²×#10623494	—	2	2	no	—
22	C₂²×#10623494	—	2	2	no	—
23	C₄⋊#10623494	—	2	2	no	—
24	C₄⋊#10623494	—	2	4	no	—
25	C₄⋊#10623494	—	4	2	no	—
26	C₄⋊#10623494	—	4	2	no	—
27	C₄⋊#10623494	Q₈	1	4	yes	—
28	#15667923	—	4	2	no	—
29	#125×C₈₉	—	4	4	no	—
30	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
31	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
32	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
33	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
34	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
35	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
36	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
37	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
45	#95×C₈₉	—	2	2	no	—
46	#95×C₈₉	—	2	2	no	—
47	C₂×#10623494	—	2	4	no	—
48	C₂×#10623494	—	4	2	no	—
49	C₂×#10623494	—	4	2	no	—
50	C₂×#10623494	—	4	2	no	—
51	C₂×#10623494	—	4	2	no	—
52	C₂×#10623494	—	4	2	no	—
53	C₂²×C₄×C₈₉	—	2	2	no	—
54	C₂²×C₄×C₈₉	—	2	2	no	—
55	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
62	C₄⋊C₄×C₈₉	—	4	4	no	—
63	C₄⋊C₄×C₈₉	—	4	2	no	—
64	C₄⋊C₄×C₈₉	—	4	2	no	—
65	C₄⋊C₄×C₈₉	—	4	4	no	—
66	C₄⋊C₄×C₈₉	—	4	2	no	—
67	C₄⋊C₄×C₈₉	—	4	1	no	—
68	C₄⋊C₄×C₈₉	—	4	1	no	—
69	C₄⋊C₄×C₈₉	—	4	1	no	—
70	C₄⋊C₄×C₈₉	—	4	1	no	—
71	C₄⋊C₄×C₈₉	—	4	1	no	—
72	C₄⋊C₄×C₈₉	—	4	1	no	—
73	C₄²×C₈₉	—	2	2	no	—
74	C₄²×C₈₉	—	2	2	no	—
75	C₄²×C₈₉	—	2	2	no	—
76	C₄²×C₈₉	—	2	1	no	—
77	C₄²×C₈₉	—	2	1	no	—
78	C₄²×C₈₉	—	2	1	no	—
79	C₄²×C₈₉	—	2	1	no	—
80	C₄²×C₈₉	—	2	1	no	—
81	C₄²×C₈₉	—	2	1	no	—
82	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
83	C₄²×C₈₉	#41	1	2	yes	—
84	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₈₉	—	2	1	no	—
86	C₂×C₄×C₈₉	—	2	1	no	—
87	C₂×C₄×C₈₉	—	2	1	no	—
88	C₂×C₄×C₈₉	—	2	1	no	—
89	C₂×C₄×C₈₉	—	2	1	no	—
90	C₂×C₄×C₈₉	—	2	1	no	—
91	C₂×C₄×C₈₉	—	2	1	no	—
92	C₂×C₄×C₈₉	—	2	1	no	—
93	C₂×C₄×C₈₉	—	2	1	no	—
94	C₂×C₄×C₈₉	—	2	1	no	—
95	C₂×C₄×C₈₉	—	2	1	no	—
96	C₂×C₄×C₈₉	—	2	1	no	—
97	C₂×C₄×C₈₉	—	4	2	no	—
98	C₂×C₄×C₈₉	—	4	2	no	—
99	C₂×C₄×C₈₉	—	4	2	no	—
100	C₂×C₄×C₈₉	—	4	2	no	—
101	C₂×C₄×C₈₉	—	4	2	no	—
102	C₂×C₄×C₈₉	—	4	2	no	—
103	C₂×C₄×C₈₉	#100	1	2	yes	—
104	C₂×C₄×C₈₉	#101	1	2	yes	—
105	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
106	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
107	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
108	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
109	C₂×C₄×C₈₉	#135	1	2	yes	—
110	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
111	C₂×C₄×C₈₉	AGL(1,ℤ/8)	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	#10623494	—	4	4	no	—
117	#10623494	—	8	2	no	—
118	C₂²×C₈₉	#391	1	1	yes	characteristic
119	C₂²×C₈₉	C₂×#100	1	1	yes	characteristic
120	C₂²×C₈₉	C₂×#101	1	1	yes	characteristic
121	C₂²×C₈₉	#416	1	1	yes	characteristic
122	C₂²×C₈₉	#417	1	1	yes	characteristic
123	C₂²×C₈₉	#417	1	1	yes	characteristic
124	C₂²×C₈₉	#417	1	1	yes	characteristic
125	C₄×C₈₉	—	2	2	no	—
126	C₄×C₈₉	—	2	2	no	—
127	C₄×C₈₉	—	2	2	no	—
128	C₄×C₈₉	—	4	1	no	—
129	C₄×C₈₉	—	4	1	no	—
130	C₄×C₈₉	—	8	2	no	—
131	C₄×C₈₉	—	8	2	no	—
132	C₄×C₈₉	#478	1	2	yes	—
133	C₄×C₈₉	#479	1	2	yes	—
134	C₄×C₈₉	#490	1	1	yes	characteristic
135	C₄×C₈₉	#491	1	1	yes	characteristic
136	C₄×C₈₉	#501	1	1	yes	characteristic
137	C₄×C₈₉	#501	1	1	yes	characteristic
138	#9906	—	89	1	no	maximal, Sylow
139	C₂×C₈₉	#1551	1	1	yes	characteristic
140	C₂×C₈₉	#1552	1	1	yes	characteristic
141	C₂×C₈₉	#1560	1	1	yes	characteristic
142	C₂×C₈₉	#1561	1	1	yes	characteristic
143	C₂×C₈₉	#1989	1	1	yes	characteristic
144	C₂×C₈₉	#1992	1	1	yes	characteristic
145	C₂×C₈₉	#1997	1	1	yes	characteristic
146	#2859	D₈₉	1	1	yes	characteristic
147	#1842	—	89	1	no	—
148	#1767	—	89	1	no	—
149	#1296	—	89	2	no	—
150	#1296	—	89	2	no	—
151	C₈₉	#9906	1	1	yes	characteristic, complemented, Sylow
152	C₂×#125	Dic₈₉	1	2	yes	—
153	C₂×C₄⋊C₈	—	89	2	no	—
154	C₂×C₄⋊C₈	—	89	2	no	—
155	C₂×C₄⋊C₈	—	178	1	no	—
156	C₂×C₄⋊C₈	—	178	1	no	—
157	C₂×#97	—	178	1	no	—
158	#389	—	2	1	no	—
159	#389	—	2	1	no	—
160	#389	—	2	1	no	—
161	#389	—	2	1	no	—
162	#384	C₂×D₈₉	1	1	yes	characteristic
163	#384	C₂×D₈₉	1	1	yes	characteristic
164	#382	Dic₈₉	1	2	yes	—
165	C₄³	C₂×D₈₉	1	1	yes	characteristic
166	#336	—	178	1	no	—
167	C₂²×C₈	—	178	2	no	—
168	C₂²×C₈	—	178	2	no	—
169	#125	—	4	4	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	—	2	1	no	—
173	C₂×C₄⋊C₄	—	2	1	no	—
174	C₂×C₄⋊C₄	—	2	1	no	—
175	C₂×C₄⋊C₄	—	2	1	no	—
176	C₂×C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
177	C₂×C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
178	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
179	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
180	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
181	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
182	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
183	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
184	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
185	C₄⋊C₈	—	89	4	no	—
186	C₄⋊C₈	—	356	2	no	—
187	C₄⋊C₈	—	356	2	no	—
188	C₄⋊C₈	—	178	4	no	—
189	C₄⋊C₈	—	178	2	no	—
190	#97	—	356	2	no	—
191	#95	—	2	2	no	—
192	#95	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
196	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
197	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
198	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
199	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
202	C₂×C₈	—	356	2	no	—
203	C₂×C₈	—	356	2	no	—
204	C₂×C₈	—	356	2	no	—
205	C₂×C₈	—	356	2	no	—
206	C₂×C₈	—	356	2	no	—
207	C₂×C₈	—	178	4	no	—
208	C₄⋊C₄	—	4	2	no	—
209	C₄⋊C₄	—	4	2	no	—
210	C₄⋊C₄	—	4	4	no	—
211	C₄⋊C₄	—	4	2	no	—
212	C₄⋊C₄	—	4	1	no	—
213	C₄⋊C₄	—	4	1	no	—
214	C₄⋊C₄	—	4	1	no	—
215	C₄⋊C₄	—	4	1	no	—
216	C₄⋊C₄	—	4	1	no	—
217	C₄⋊C₄	—	4	1	no	—
218	C₄⋊C₄	—	4	4	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	2	1	no	—
221	C₄²	—	2	1	no	—
222	C₄²	—	2	1	no	—
223	C₄²	—	2	1	no	—
224	C₄²	—	2	1	no	—
225	C₄²	—	2	1	no	—
226	C₄²	—	2	2	no	—
227	C₄²	—	2	2	no	—
228	C₄²	Q₈×D₈₉	1	2	yes	—
229	C₄²	#13083575	1	2	yes	—
230	C₂³	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	#15667929	1	2	yes	—
250	C₂×C₄	#15667931	1	2	yes	—
251	C₂×C₄	Q₈×Dic₈₉	1	2	yes	—
252	C₂×C₄	#15667988	1	2	yes	—
253	C₂×C₄	#15668001	1	1	yes	characteristic
254	C₂×C₄	#15668010	1	1	yes	characteristic
255	C₂×C₄	#15668032	1	2	yes	—
256	C₂×C₄	#15668037	1	1	yes	characteristic
257	C₂×C₄	#15668037	1	1	yes	characteristic
258	C₂×C₄	#15668037	1	1	yes	characteristic
259	C₂×C₄	#15668042	1	1	yes	characteristic
260	C₂×C₄	#15668042	1	1	yes	characteristic
261	C₂×C₄	#15668042	1	1	yes	characteristic
262	C₈	—	356	4	no	—
263	C₈	—	712	2	no	—
264	C₂²	#19151505	1	1	yes	characteristic
265	C₂²	C₂×#15667929	1	1	yes	characteristic
266	C₂²	C₂×#15667931	1	1	yes	characteristic
267	C₂²	#19151594	1	1	yes	characteristic
268	C₂²	#19151599	1	1	yes	characteristic
269	C₂²	#19151599	1	1	yes	characteristic
270	C₂²	#19151599	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	2	2	no	—
274	C₄	—	4	1	no	—
275	C₄	—	4	1	no	—
276	C₄	—	8	2	no	—
277	C₄	—	8	2	no	—
278	C₄	#19151895	1	2	yes	—
279	C₄	#19151902	1	2	yes	—
280	C₄	#19151964	1	1	yes	characteristic
281	C₄	#19151968	1	1	yes	characteristic
282	C₄	#19152008	1	1	yes	characteristic
283	C₄	#19152008	1	1	yes	characteristic
284	C₂	#26018947	1	1	yes	characteristic
285	C₂	#26018952	1	1	yes	characteristic
286	C₂	#26018987	1	1	yes	characteristic
287	C₂	#26018990	1	1	yes	characteristic
288	C₂	#26021210	1	1	yes	characteristic
289	C₂	#26021222	1	1	yes	characteristic
290	C₂	#26021242	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₈₉
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#2859	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#2859	D₈₉
hypocenter	C₈₉	#9906
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2859

Complements

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

N	H ≅ G/N	type
151. C₈₉	#9906	semidirect

Maximal subgroups

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

subgroup	index
#26020404	2
#26020105	2
#26018135	2
C₈₉×#2859	2
#9906	89