Subgroups — #36442894

Order 22784.

432 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26028615	C₂	1	1	yes	maximal, characteristic
2	#26019998	C₂	1	1	yes	maximal, characteristic
3	#26018074	C₂	1	4	yes	maximal
4	C₈₉×#1767	C₂	1	1	yes	maximal, characteristic
5	C₂×#15668004	—	2	1	no	—
6	C₂×C₄×#10623494	—	2	1	no	—
7	#19151496	—	2	1	no	—
8	#19151496	—	2	1	no	—
9	#19151496	C₄	1	2	yes	—
10	#19151496	C₂²	1	2	yes	—
11	#19151468	C₄	1	1	yes	characteristic
12	#19151468	C₄	1	1	yes	characteristic
13	#19151464	—	2	1	no	—
14	#19151353	—	2	1	no	—
15	#19151353	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₈₉	—	2	1	no	—
17	C₂×#97×C₈₉	—	2	1	no	—
18	C₄³×C₈₉	C₂²	1	1	yes	characteristic
19	#336×C₈₉	C₂²	1	2	yes	—
20	C₂²×#10623494	—	2	2	no	—
21	#15668004	—	4	4	no	—
22	C₂×C₄⋊Dic₈₉	—	2	1	no	—
23	C₂×C₄⋊Dic₈₉	—	2	1	no	—
24	C₂×C₄⋊Dic₈₉	—	2	1	no	—
25	C₂×C₄⋊Dic₈₉	—	2	1	no	—
26	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
27	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
28	C₂×#13083553	—	2	2	no	—
29	C₂×#13083553	—	2	2	no	—
30	C₄×#10623494	—	2	4	no	—
31	#15667919	—	2	2	no	—
32	#15667919	—	2	2	no	—
33	C₂²×C₈×C₈₉	—	2	2	no	—
34	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₈₉	—	2	4	no	—
42	#97×C₈₉	—	4	2	no	—
43	C₂²×Dic₈₉	—	2	2	no	—
44	C₂²×Dic₈₉	—	2	2	no	—
45	C₂×#10623494	—	2	4	no	—
46	C₂×#10623494	—	4	2	no	—
47	C₂×#10623494	—	4	2	no	—
48	C₄⋊Dic₈₉	—	2	4	no	—
49	C₄⋊Dic₈₉	—	4	2	no	—
50	C₄⋊Dic₈₉	—	4	2	no	—
51	C₄⋊Dic₈₉	—	4	2	no	—
52	C₄⋊Dic₈₉	—	4	2	no	—
53	C₄⋊Dic₈₉	—	4	2	no	—
54	#13083553	—	4	4	no	—
55	#13083553	—	4	4	no	—
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	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₈₉	—	2	4	no	—
64	C₂×C₈×C₈₉	—	4	2	no	—
65	C₂×C₈×C₈₉	—	4	2	no	—
66	C₄²×C₈₉	—	2	2	no	—
67	C₄²×C₈₉	—	2	2	no	—
68	C₄²×C₈₉	—	2	2	no	—
69	C₄²×C₈₉	—	2	2	no	—
70	C₄²×C₈₉	—	2	2	no	—
71	C₄²×C₈₉	—	4	1	no	—
72	C₄²×C₈₉	SD₁₆	1	2	yes	—
73	C₄²×C₈₉	Q₁₆	1	2	yes	—
74	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₈₉	—	2	1	no	—
76	C₂×C₄×C₈₉	—	2	1	no	—
77	C₂×C₄×C₈₉	—	2	1	no	—
78	C₂×C₄×C₈₉	—	2	1	no	—
79	C₂×C₄×C₈₉	—	2	1	no	—
80	C₂×C₄×C₈₉	—	2	1	no	—
81	C₂×C₄×C₈₉	—	2	1	no	—
82	C₂×C₄×C₈₉	—	2	1	no	—
83	C₂×C₄×C₈₉	—	2	1	no	—
84	C₂×C₄×C₈₉	—	2	1	no	—
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	#101	1	2	yes	—
89	C₂×C₄×C₈₉	#103	1	2	yes	—
90	C₂×C₄×C₈₉	#103	1	2	yes	—
91	C₂×C₄×C₈₉	#104	1	2	yes	—
92	C₂×C₄×C₈₉	#104	1	2	yes	—
93	C₂×C₄×C₈₉	#131	1	2	yes	—
94	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
95	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
96	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
97	C₂×Dic₈₉	—	4	2	no	—
98	C₂×Dic₈₉	—	4	2	no	—
99	C₂×Dic₈₉	—	4	2	no	—
100	C₂×Dic₈₉	—	4	2	no	—
101	C₂×Dic₈₉	—	4	2	no	—
102	C₂×Dic₈₉	—	4	2	no	—
103	C₈×C₈₉	—	4	4	no	—
104	#10623494	—	4	4	no	—
105	C₂²×C₈₉	#408	1	1	yes	characteristic
106	C₂²×C₈₉	#410	1	1	yes	characteristic
107	C₂²×C₈₉	C₂×#101	1	1	yes	characteristic
108	C₂²×C₈₉	C₂×#103	1	1	yes	characteristic
109	C₂²×C₈₉	#418	1	1	yes	characteristic
110	C₂²×C₈₉	C₂×#104	1	1	yes	characteristic
111	C₂²×C₈₉	C₂×#104	1	1	yes	characteristic
112	C₄×C₈₉	—	2	2	no	—
113	C₄×C₈₉	—	2	2	no	—
114	C₄×C₈₉	—	2	2	no	—
115	C₄×C₈₉	—	2	1	no	—
116	C₄×C₈₉	—	2	1	no	—
117	C₄×C₈₉	—	2	2	no	—
118	C₄×C₈₉	—	4	1	no	—
119	C₄×C₈₉	#330	1	4	yes	—
120	Dic₈₉	—	8	2	no	—
121	Dic₈₉	—	8	2	no	—
122	#7683	—	89	1	no	maximal, Sylow
123	C₂×C₈₉	#1470	1	1	yes	characteristic
124	C₂×C₈₉	#1484	1	1	yes	characteristic
125	C₂×C₈₉	#1493	1	1	yes	characteristic
126	C₂×C₈₉	#1503	1	1	yes	characteristic
127	C₂×C₈₉	#1505	1	1	yes	characteristic
128	C₂×C₈₉	C₂×#330	1	1	yes	characteristic
129	C₂×C₈₉	#1549	1	1	yes	characteristic
130	#2859	—	89	1	no	—
131	#1767	D₈₉	1	1	yes	characteristic
132	#1744	—	89	1	no	—
133	#1271	—	89	4	no	—
134	C₈₉	#7683	1	1	yes	characteristic, complemented, Sylow
135	C₂×#125	—	178	1	no	—
136	C₂×C₄⋊C₈	—	2	1	no	—
137	C₂×#97	—	2	1	no	—
138	C₂×C₄×C₈	—	178	1	no	—
139	#389	—	89	2	no	—
140	#389	—	89	2	no	—
141	#389	—	178	1	no	—
142	#389	—	178	1	no	—
143	#384	—	89	1	no	—
144	#384	—	89	1	no	—
145	#382	—	178	1	no	—
146	C₄³	C₂×D₈₉	1	1	yes	characteristic
147	#336	—	89	2	no	—
148	#336	—	178	1	no	—
149	#336	C₂×D₈₉	1	2	yes	—
150	C₂²×C₈	—	2	2	no	—
151	C₂²×C₈	—	178	2	no	—
152	#125	—	356	4	no	—
153	C₂×C₄⋊C₄	—	89	2	no	—
154	C₂×C₄⋊C₄	—	89	2	no	—
155	C₂×C₄⋊C₄	—	178	1	no	—
156	C₂×C₄⋊C₄	—	178	1	no	—
157	C₂×C₄⋊C₄	—	178	2	no	—
158	C₂×C₄⋊C₄	—	178	1	no	—
159	C₂×C₄⋊C₄	—	178	1	no	—
160	C₂×C₄⋊C₄	—	178	2	no	—
161	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
162	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
163	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
164	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
165	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
166	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
167	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
168	C₄⋊C₈	—	2	4	no	—
169	#97	—	4	2	no	—
170	C₄×C₈	—	178	4	no	—
171	#95	—	178	2	no	—
172	#95	—	178	2	no	—
173	C₂²×C₄	—	178	2	no	—
174	C₂²×C₄	—	178	2	no	—
175	C₂²×C₄	#13083551	1	1	yes	characteristic
176	C₂²×C₄	#13083551	1	1	yes	characteristic
177	C₂²×C₄	#13083551	1	1	yes	characteristic
178	C₂²×C₄	#13083551	1	1	yes	characteristic
179	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
180	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
181	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	—	4	2	no	—
184	C₂×C₈	—	4	2	no	—
185	C₂×C₈	—	356	2	no	—
186	C₂×C₈	—	356	2	no	—
187	C₂×C₈	—	178	4	no	—
188	C₄⋊C₄	—	356	2	no	—
189	C₄⋊C₄	—	356	4	no	—
190	C₄⋊C₄	—	356	2	no	—
191	C₄⋊C₄	—	356	2	no	—
192	C₄⋊C₄	—	356	4	no	—
193	C₄⋊C₄	—	356	2	no	—
194	C₄⋊C₄	—	356	2	no	—
195	C₄⋊C₄	—	178	4	no	—
196	C₄²	—	2	2	no	—
197	C₄²	—	2	2	no	—
198	C₄²	—	2	2	no	—
199	C₄²	—	2	2	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	4	1	no	—
202	C₄²	#13083562	1	2	yes	—
203	C₄²	#13083564	1	2	yes	—
204	C₂³	C₂×#13083551	1	1	yes	characteristic
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	356	2	no	—
219	C₂×C₄	—	356	2	no	—
220	C₂×C₄	—	356	2	no	—
221	C₂×C₄	—	356	2	no	—
222	C₂×C₄	—	356	2	no	—
223	C₂×C₄	—	356	2	no	—
224	C₂×C₄	#15667930	1	2	yes	—
225	C₂×C₄	#15667935	1	2	yes	—
226	C₂×C₄	#15667935	1	2	yes	—
227	C₂×C₄	#15667938	1	2	yes	—
228	C₂×C₄	#15667940	1	2	yes	—
229	C₂×C₄	#15668020	1	2	yes	—
230	C₂×C₄	C₂×#13083562	1	1	yes	characteristic
231	C₂×C₄	C₂×#13083564	1	1	yes	characteristic
232	C₂×C₄	#15668040	1	2	yes	—
233	C₈	—	4	4	no	—
234	C₈	—	356	4	no	—
235	C₂²	#19151566	1	1	yes	characteristic
236	C₂²	#19151571	1	1	yes	characteristic
237	C₂²	C₂×#15667930	1	1	yes	characteristic
238	C₂²	C₂×#15667935	1	1	yes	characteristic
239	C₂²	#19151604	1	1	yes	characteristic
240	C₂²	C₂×#15667938	1	1	yes	characteristic
241	C₂²	C₂×#15667940	1	1	yes	characteristic
242	C₄	—	2	2	no	—
243	C₄	—	2	2	no	—
244	C₄	—	2	2	no	—
245	C₄	—	2	1	no	—
246	C₄	—	2	1	no	—
247	C₄	—	2	2	no	—
248	C₄	—	4	1	no	—
249	C₄	—	712	2	no	—
250	C₄	—	712	2	no	—
251	C₄	#19151339	1	4	yes	—
252	C₂	#26018609	1	1	yes	characteristic
253	C₂	#26018670	1	1	yes	characteristic
254	C₂	#26018706	1	1	yes	characteristic
255	C₂	#26018750	1	1	yes	characteristic
256	C₂	#26018761	1	1	yes	characteristic
257	C₂	C₂×#19151339	1	1	yes	characteristic
258	C₂	#26018938	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#13083551
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1767	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1767	D₈₉
hypocenter	C₈₉	#7683
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1767

Complements

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

N	H ≅ G/N	type
134. C₈₉	#7683	semidirect

Maximal subgroups

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

subgroup	index
#26028615	2
#26019998	2
#26018074	2
C₈₉×#1767	2
#7683	89