Subgroups — #36525182

Order 22784.

726 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	#26030831	C₂	1	1	yes	maximal, characteristic
2	#26028965	C₂	1	2	yes	maximal, complemented
3	#26019118	C₂	1	8	yes	maximal, complemented
4	#26019008	C₂	1	1	yes	maximal, characteristic, complemented
5	#26018704	C₂	1	2	yes	maximal, complemented
6	C₂×#322×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#15668018	—	2	2	no	—
8	#19152148	C₂²	1	2	yes	—
9	#19152003	C₄	1	1	yes	characteristic
10	#19151975	C₄	1	1	yes	characteristic
11	#19151945	C₄	1	2	yes	—
12	#19151834	C₄	1	4	yes	—
13	C₄×D₇₁₂	C₂²	1	4	yes	—
14	#19151677	—	2	2	no	—
15	#19151677	C₂²	1	4	yes	—
16	C₈×D₃₅₆	—	2	2	no	—
17	C₈×D₃₅₆	C₂²	1	4	yes	—
18	#19151622	C₂²	1	2	yes	—
19	#19151564	—	2	2	no	—
20	#19151349	C₂²	1	4	yes	—
21	#19151327	C₂²	1	8	yes	—
22	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
25	#322×C₈₉	C₂²	1	4	yes	—
26	C₂×#13083574	—	2	2	no	—
27	C₂×D₇₁₂	C₂×C₄	1	4	yes	—
28	#15668018	—	4	8	no	—
29	C₂×#13083558	—	2	2	no	—
30	C₂×#13083557	—	2	4	no	—
31	C₂×C₈×D₈₉	—	2	4	no	—
32	#15668012	C₂×C₄	1	2	yes	—
33	#15668009	C₂×C₄	1	2	yes	—
34	#15668000	—	2	2	no	—
35	#15667986	—	2	2	no	—
36	#15667986	C₂×C₄	1	4	yes	—
37	C₄×D₃₅₆	—	2	2	no	—
38	C₄×D₃₅₆	C₂³	1	4	yes	—
39	#15667946	C₂×C₄	1	4	yes	—
40	C₄⋊#10623494	—	2	2	no	—
41	C₄⋊#10623494	C₂³	1	4	yes	—
42	#15667934	C₂×C₄	1	8	yes	—
43	#15667924	—	2	4	no	—
44	#15667924	—	2	4	no	—
45	C₂²×C₈×C₈₉	—	2	1	no	—
46	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
47	C₂²×C₈×C₈₉	D₄	1	2	yes	—
48	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
49	C₄×C₈×C₈₉	—	2	1	no	—
50	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
53	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
54	C₄×C₈×C₈₉	C₂³	1	2	yes	—
55	C₄×C₈×C₈₉	C₂³	1	2	yes	—
56	C₄×C₈×C₈₉	C₂³	1	2	yes	—
57	#13083574	—	4	4	no	—
58	#13083574	—	4	4	no	—
59	C₂×Dic₁₇₈	—	2	2	no	—
60	C₂×D₃₅₆	—	2	2	no	—
61	C₂×D₃₅₆	—	2	2	no	—
62	C₂×D₃₅₆	C₂²×C₄	1	4	yes	—
63	C₂×C₂²⋊D₈₉	—	2	4	no	—
64	C₂×C₄×D₈₉	—	2	4	no	—
65	D₇₁₂	—	4	4	no	—
66	#13083558	—	4	4	no	—
67	#13083557	—	4	8	no	—
68	C₂×#10623494	—	2	4	no	—
69	C₈×D₈₉	—	4	8	no	—
70	C₄⋊Dic₈₉	—	2	2	no	—
71	C₄⋊Dic₈₉	C₂²×C₄	1	4	yes	—
72	#13083551	—	2	4	no	—
73	#13083551	—	2	4	no	—
74	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
77	C₂×C₈×C₈₉	—	2	4	no	—
78	C₂×C₈×C₈₉	—	2	4	no	—
79	C₂×C₈×C₈₉	—	2	2	no	—
80	C₂×C₈×C₈₉	—	4	2	no	—
81	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
85	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₈₉	#41	1	1	yes	characteristic
88	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
89	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
90	C₂²×D₈₉	—	2	4	no	—
91	C₂×C₄×C₈₉	—	2	2	no	—
92	C₂×C₄×C₈₉	—	2	2	no	—
93	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
94	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	4	yes	—
97	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
99	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
100	C₂²⋊D₈₉	—	4	8	no	—
101	C₂×Dic₈₉	—	2	4	no	—
102	D₃₅₆	—	2	8	no	—
103	D₃₅₆	—	4	4	no	—
104	D₃₅₆	—	4	4	no	—
105	C₄×D₈₉	—	4	4	no	—
106	C₄×D₈₉	—	4	4	no	—
107	Dic₁₇₈	—	4	4	no	—
108	C₈×C₈₉	—	2	4	no	—
109	C₈×C₈₉	—	2	2	no	—
110	C₈×C₈₉	—	2	2	no	—
111	C₈×C₈₉	—	2	2	no	—
112	C₈×C₈₉	—	2	2	no	—
113	#10623494	—	4	4	no	—
114	C₂²×C₈₉	—	2	1	no	—
115	C₂²×C₈₉	—	2	1	no	—
116	C₂²×C₈₉	#442	1	2	yes	—
117	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
118	C₂×D₈₉	—	4	4	no	—
119	C₂×D₈₉	—	4	4	no	—
120	C₂×D₈₉	—	4	4	no	—
121	C₄×C₈₉	—	2	1	no	—
122	C₄×C₈₉	—	2	1	no	—
123	C₄×C₈₉	#442	1	2	yes	—
124	C₄×C₈₉	#443	1	2	yes	—
125	C₄×C₈₉	#568	1	2	yes	—
126	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
127	Dic₈₉	—	4	4	no	—
128	#17981	—	89	1	no	maximal, Sylow
129	C₂×C₈₉	—	2	2	no	—
130	C₂×C₈₉	#2932	1	1	yes	characteristic
131	C₂×C₈₉	C₂×#442	1	1	yes	characteristic
132	C₂×C₈₉	C₂×#443	1	1	yes	characteristic
133	D₈₉	—	8	4	no	—
134	#3169	—	89	1	no	—
135	#2931	—	89	2	no	—
136	#1586	—	89	8	no	—
137	#1566	—	89	1	no	—
138	#1492	—	89	2	no	—
139	C₂×#322	D₈₉	1	1	yes	characteristic, complemented
140	C₈₉	#17981	1	1	yes	characteristic, complemented, Sylow
141	C₂×#131	—	178	2	no	—
142	#532	—	89	2	no	—
143	#500	—	89	1	no	—
144	#493	—	89	1	no	—
145	#486	—	89	2	no	—
146	#466	—	89	4	no	—
147	C₄×D₈	—	89	4	no	—
148	#436	—	89	4	no	—
149	#436	—	178	2	no	—
150	C₈×D₄	—	89	4	no	—
151	C₈×D₄	—	178	2	no	—
152	#423	—	89	2	no	—
153	#408	—	178	2	no	—
154	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
155	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
156	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
157	#334	—	89	4	no	—
158	D₄⋊C₈	—	89	8	no	—
159	#322	C₂×D₈₉	1	4	yes	—
160	C₂×#41	—	178	2	no	—
161	C₂×D₈	—	89	4	no	—
162	#131	—	356	8	no	—
163	C₂×M₁₆	—	178	2	no	—
164	C₂×M₁₆	—	178	4	no	—
165	C₂²×C₈	—	2	1	no	—
166	C₂²×C₈	—	178	4	no	—
167	C₂²×C₈	C₄×D₈₉	1	2	yes	—
168	C₂²×C₈	D₃₅₆	1	2	yes	—
169	C₄⋊Q₈	—	89	2	no	—
170	#127	—	89	2	no	—
171	#124	—	178	2	no	—
172	#121	—	89	4	no	—
173	#121	—	178	2	no	—
174	C₄×D₄	—	89	4	no	—
175	C₄×D₄	—	178	2	no	—
176	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
177	#107	—	89	4	no	—
178	C₄⋊C₈	—	89	4	no	—
179	C₄⋊C₈	—	178	2	no	—
180	#102	—	89	8	no	—
181	C₂²⋊C₈	—	178	4	no	—
182	C₂²⋊C₈	—	178	4	no	—
183	C₄×C₈	—	2	1	no	—
184	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
185	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
186	C₄×C₈	D₃₅₆	1	1	yes	characteristic
187	C₄×C₈	D₃₅₆	1	1	yes	characteristic
188	C₄×C₈	C₂²×D₈₉	1	2	yes	—
189	C₄×C₈	C₂²×D₈₉	1	2	yes	—
190	C₄×C₈	C₂²×D₈₉	1	2	yes	—
191	#41	—	356	4	no	—
192	#41	—	356	4	no	—
193	C₂×Q₈	—	178	2	no	—
194	C₂×D₄	—	89	4	no	—
195	C₂×D₄	—	178	4	no	—
196	C₂×D₄	—	178	2	no	—
197	C₂×D₄	—	178	2	no	—
198	C₂²×C₄	—	178	4	no	—
199	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
200	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
201	C₂²×C₄	#13083574	1	1	yes	characteristic
202	D₈	—	356	4	no	—
203	M₁₆	—	356	8	no	—
204	M₁₆	—	356	4	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	4	2	no	—
209	C₂×C₈	—	356	8	no	—
210	C₂×C₈	—	178	4	no	—
211	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
212	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
213	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
214	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
215	C₄⋊C₄	—	89	4	no	—
216	C₄⋊C₄	—	178	2	no	—
217	C₂²⋊C₄	—	178	4	no	—
218	C₂²⋊C₄	—	178	4	no	—
219	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
220	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
221	C₄²	#13083574	1	1	yes	characteristic
222	C₄²	C₂³×D₈₉	1	1	yes	characteristic
223	C₂³	—	178	4	no	—
224	C₂³	C₄×D₃₅₆	1	1	yes	characteristic
225	Q₈	—	356	4	no	—
226	D₄	—	356	8	no	—
227	D₄	—	356	4	no	—
228	D₄	—	356	4	no	—
229	D₄	—	178	8	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	356	4	no	—
233	C₂×C₄	—	356	4	no	—
234	C₂×C₄	—	178	4	no	—
235	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
236	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
237	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
238	C₂×C₄	#15668040	1	4	yes	—
239	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
240	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
241	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
242	C₈	—	2	4	no	—
243	C₈	—	2	2	no	—
244	C₈	—	2	2	no	—
245	C₈	—	2	2	no	—
246	C₈	—	2	2	no	—
247	C₈	—	356	4	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	356	4	no	—
251	C₂²	—	356	4	no	—
252	C₂²	—	356	4	no	—
253	C₂²	#19151711	1	2	yes	—
254	C₂²	C₂×C₄×D₃₅₆	1	1	yes	characteristic
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	356	4	no	—
258	C₄	#19151711	1	2	yes	—
259	C₄	#19151716	1	2	yes	—
260	C₄	#19152364	1	2	yes	—
261	C₄	C₂×#15668040	1	2	yes	—
262	C₂	—	2	2	no	—
263	C₂	—	712	4	no	—
264	C₂	#26028971	1	1	yes	characteristic
265	C₂	C₂×#19151711	1	1	yes	characteristic
266	C₂	C₂×#19151716	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂×#322×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	#26030831	C₂
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	C₂×#322	D₈₉
hypocenter	C₈₉	#17981
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
2. #26028965	C₂	semidirect
3. #26019118	C₂	semidirect
4. #26019008	C₂	semidirect
5. #26018704	C₂	semidirect
6. C₂×#322×C₈₉	C₂	semidirect
139. C₂×#322	D₈₉	semidirect
140. C₈₉	#17981	semidirect

Maximal subgroups

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

subgroup	index
#26030831	2
#26028965	2
#26019118	2
#26019008	2
#26018704	2
C₂×#322×C₈₉	2
#17981	89