Subgroups — #36707668

Order 22784.

646 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19152016	C₂	1	2	yes	maximal, complemented
2	#26029156	C₂	1	1	yes	maximal, characteristic
3	#26022093	C₂	1	4	yes	maximal, complemented
4	#26022088	C₂	1	4	yes	maximal, complemented
5	#26021881	C₂	1	2	yes	maximal, complemented
6	#26021849	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#2153	C₂	1	1	yes	maximal, characteristic, complemented
8	#19152398	C₄	1	1	yes	characteristic
9	C₂×#15668018	C₂²	1	2	yes	—
10	#19152016	—	2	2	no	—
11	#19152016	—	2	2	no	—
12	#19152016	C₂²	1	4	yes	complemented
13	#19152016	C₂²	1	4	yes	complemented
14	C₂×#15667952	C₂²	1	1	yes	characteristic
15	C₂×#15667951	—	2	2	no	—
16	C₂²×#13083549	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×D₈₉	—	2	2	no	—
18	#19151721	C₄	1	4	yes	—
19	#19151716	C₂²	1	4	yes	—
20	#19151647	C₄	1	1	yes	characteristic
21	#19151616	C₄	1	2	yes	—
22	#19151383	C₂²	1	8	yes	complemented
23	#19151374	C₂²	1	2	yes	complemented
24	C₄×#13083549	C₂²	1	2	yes	complemented
25	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
26	#405×C₈₉	C₂²	1	1	yes	characteristic
27	#364×C₈₉	C₂²	1	4	yes	complemented
28	C₂×#13083574	C₂×C₄	1	2	yes	—
29	#15668045	C₈	1	4	yes	—
30	#15668040	C₈	1	4	yes	—
31	#15668034	C₂×C₄	1	4	yes	—
32	#15668018	—	2	2	no	—
33	#15668018	—	2	2	no	—
34	#15668018	C₂×C₄	1	4	yes	—
35	#15668018	C₂³	1	4	yes	—
36	C₂×#13083558	C₂×C₄	1	2	yes	—
37	C₂×#13083557	—	2	2	no	—
38	C₂×C₈×D₈₉	—	2	2	no	—
39	#15667952	—	2	2	no	—
40	#15667951	—	2	4	no	—
41	#15667951	—	2	4	no	—
42	C₂×#13083549	D₄	1	2	yes	—
43	C₂×#13083549	D₄	1	2	yes	—
44	C₂×#13083549	C₂³	1	2	yes	—
45	C₂×#13083549	C₂³	1	1	yes	characteristic
46	C₂×#13083549	C₂³	1	1	yes	characteristic
47	C₁₆×D₈₉	—	2	4	no	—
48	C₁₆×D₈₉	—	2	4	no	—
49	#15667948	C₂×C₄	1	4	yes	—
50	#15667940	C₂×C₄	1	8	yes	—
51	C₂×M₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
53	#117×C₈₉	C₂×C₄	1	1	yes	characteristic
54	M₃₂×C₈₉	—	2	2	no	—
55	M₃₂×C₈₉	C₂³	1	4	yes	—
56	C₂×C₁₆×C₈₉	—	2	2	no	—
57	#97×C₈₉	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₈₉	C₂³	1	2	yes	—
59	#13083574	—	2	2	no	—
60	#13083574	—	2	2	no	—
61	#13083574	C₂×C₈	1	4	yes	—
62	#13083574	C₂²×C₄	1	4	yes	—
63	C₂×Dic₁₇₈	C₂×C₈	1	2	yes	—
64	C₂×D₃₅₆	C₂×C₈	1	2	yes	—
65	C₂×C₂²⋊D₈₉	—	2	2	no	—
66	C₂×C₄×D₈₉	—	2	2	no	—
67	Dic₃₅₆	C₂×C₈	1	4	yes	—
68	#13083563	C₂×C₈	1	8	yes	—
69	D₇₁₂	C₂×C₈	1	4	yes	—
70	#13083558	—	2	2	no	—
71	#13083558	C₂²×C₄	1	4	yes	—
72	#13083557	—	2	4	no	—
73	#13083557	—	2	4	no	—
74	C₂×#10623494	—	2	2	no	—
75	C₈×D₈₉	—	2	4	no	—
76	C₈×D₈₉	—	2	4	no	—
77	#13083549	—	2	2	no	—
78	#13083549	C₂×D₄	1	1	yes	characteristic
79	#13083549	C₂×D₄	1	2	yes	—
80	#13083549	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
82	M₁₆×C₈₉	C₂×C₈	1	4	yes	—
83	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₈₉	#41	1	2	yes	—
87	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
88	C₄⋊C₄×C₈₉	—	2	1	no	—
89	C₂²⋊C₄×C₈₉	—	2	1	no	—
90	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
91	C₁₆×C₈₉	—	2	4	no	—
92	C₂³×C₈₉	#131	1	1	yes	characteristic
93	C₂²×D₈₉	—	2	2	no	—
94	C₂×C₄×C₈₉	—	2	2	no	—
95	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
96	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
97	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
98	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
99	C₂²⋊D₈₉	—	2	4	no	—
100	C₂²⋊D₈₉	—	2	4	no	—
101	C₂×Dic₈₉	—	2	2	no	—
102	D₃₅₆	—	2	2	no	—
103	D₃₅₆	C₂²×C₈	1	4	yes	—
104	C₄×D₈₉	—	2	4	no	—
105	C₄×D₈₉	—	2	4	no	—
106	Dic₁₇₈	—	2	2	no	—
107	Dic₁₇₈	C₂²×C₈	1	4	yes	—
108	C₈×C₈₉	C₄×D₄	1	2	yes	—
109	C₈×C₈₉	C₂²×C₈	1	4	yes	—
110	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
111	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
112	#10623494	—	2	4	no	—
113	C₂²×C₈₉	—	4	1	no	—
114	C₂²×C₈₉	C₈×D₄	1	2	yes	—
115	C₂²×C₈₉	C₂×#131	1	1	yes	characteristic
116	C₂×D₈₉	—	2	4	no	—
117	C₂×D₈₉	—	4	4	no	—
118	C₄×C₈₉	—	4	2	no	—
119	C₄×C₈₉	C₈×D₄	1	2	yes	—
120	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
121	C₄×C₈₉	C₂³×C₈	1	1	yes	characteristic
122	Dic₈₉	—	2	4	no	—
123	#33441	—	89	1	no	maximal, Sylow
124	C₂×C₈₉	—	2	1	no	—
125	C₂×C₈₉	—	2	2	no	—
126	C₂×C₈₉	C₂×C₈×D₄	1	1	yes	characteristic
127	D₈₉	—	4	4	no	—
128	C₂×#504	—	89	2	no	—
129	#2955	—	89	1	no	—
130	#2171	—	89	4	no	—
131	#2170	—	89	4	no	—
132	#2153	D₈₉	1	1	yes	characteristic, complemented
133	#2117	—	89	2	no	—
134	#2108	—	89	1	no	—
135	C₈₉	#33441	1	1	yes	characteristic, complemented, Sylow
136	#575	—	89	1	no	—
137	C₂×#131	—	89	2	no	—
138	#504	—	89	4	no	—
139	#504	—	89	4	no	—
140	#504	—	178	2	no	—
141	#504	—	178	2	no	—
142	C₂×M₃₂	—	89	1	no	—
143	C₂×M₃₂	—	178	2	no	—
144	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
145	C₂²×C₁₆	—	89	1	no	—
146	C₂²×C₁₆	—	178	2	no	—
147	#444	—	89	4	no	—
148	#443	—	89	4	no	—
149	#430	—	89	1	no	—
150	#421	—	89	2	no	—
151	#405	C₂×D₈₉	1	1	yes	characteristic
152	#364	C₂×D₈₉	1	4	yes	complemented
153	#350	—	89	8	no	—
154	#346	—	89	2	no	—
155	C₄×C₁₆	—	89	2	no	—
156	C₂×#41	—	89	2	no	—
157	#137	—	89	4	no	—
158	AGL(1,ℤ/8)	—	89	4	no	—
159	#135	—	89	4	no	—
160	#131	—	89	4	no	—
161	#131	—	89	4	no	—
162	#131	—	178	2	no	—
163	#131	—	178	2	no	—
164	C₂×M₁₆	—	89	2	no	—
165	C₂×M₁₆	—	178	2	no	—
166	C₂×M₁₆	C₄×D₈₉	1	1	yes	characteristic
167	C₂²×C₈	—	178	2	no	—
168	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
169	#117	C₄×D₈₉	1	1	yes	characteristic
170	M₃₂	—	2	2	no	—
171	M₃₂	—	178	4	no	—
172	M₃₂	—	178	4	no	—
173	M₃₂	—	178	2	no	—
174	M₃₂	C₂²×D₈₉	1	4	yes	—
175	C₂×C₁₆	—	2	2	no	—
176	C₂×C₁₆	—	89	2	no	—
177	C₂×C₁₆	—	89	2	no	—
178	C₂×C₁₆	—	89	1	no	—
179	C₂×C₁₆	—	89	1	no	—
180	C₂×C₁₆	—	89	2	no	—
181	C₂×C₁₆	—	178	4	no	—
182	C₂×C₁₆	—	178	4	no	—
183	#108	—	89	4	no	—
184	#104	—	89	8	no	—
185	#97	C₄×D₈₉	1	2	yes	—
186	C₄×C₈	C₂²×D₈₉	1	2	yes	—
187	#41	—	89	4	no	—
188	#41	—	89	4	no	—
189	#41	—	178	2	no	—
190	#41	—	178	2	no	—
191	C₂×Q₈	—	89	2	no	—
192	C₂×D₄	—	89	2	no	—
193	C₂×D₄	—	178	2	no	—
194	C₂²×C₄	—	178	2	no	—
195	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
196	Q₁₆	—	89	4	no	—
197	SD₁₆	—	89	8	no	—
198	D₈	—	89	4	no	—
199	M₁₆	—	89	4	no	—
200	M₁₆	—	178	2	no	—
201	M₁₆	—	178	4	no	—
202	M₁₆	—	178	4	no	—
203	M₁₆	C₈×D₈₉	1	4	yes	—
204	C₂×C₈	—	178	2	no	—
205	C₂×C₈	—	178	4	no	—
206	C₂×C₈	—	178	4	no	—
207	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
208	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
209	C₂×C₈	D₄×D₈₉	1	2	yes	—
210	C₂×C₈	#13083575	1	2	yes	—
211	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
212	C₄⋊C₄	—	2	1	no	—
213	C₂²⋊C₄	—	2	1	no	—
214	C₄²	C₂×C₄×D₈₉	1	2	yes	—
215	C₁₆	—	2	4	no	—
216	C₁₆	—	89	1	no	—
217	C₁₆	—	89	2	no	—
218	C₁₆	—	89	1	no	—
219	C₁₆	—	178	2	no	—
220	C₂³	—	178	2	no	—
221	C₂³	#15668020	1	1	yes	characteristic
222	Q₈	—	89	4	no	—
223	Q₈	—	178	2	no	—
224	D₄	—	89	4	no	—
225	D₄	—	178	4	no	—
226	D₄	—	178	4	no	—
227	D₄	—	178	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	178	2	no	—
230	C₂×C₄	—	178	4	no	—
231	C₂×C₄	—	178	4	no	—
232	C₂×C₄	#15667974	1	2	yes	—
233	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
234	C₂×C₄	#15668020	1	1	yes	characteristic
235	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
236	C₈	—	178	4	no	—
237	C₈	#15667974	1	2	yes	—
238	C₈	C₂×C₈×D₈₉	1	4	yes	—
239	C₈	C₂×D₄×D₈₉	1	1	yes	characteristic
240	C₈	C₂×#13083575	1	1	yes	characteristic
241	C₂²	—	4	1	no	—
242	C₂²	—	356	4	no	—
243	C₂²	—	178	4	no	—
244	C₂²	#19151663	1	2	yes	—
245	C₂²	C₂×#15668020	1	1	yes	characteristic
246	C₄	—	4	2	no	—
247	C₄	—	178	4	no	—
248	C₄	#19151663	1	2	yes	—
249	C₄	C₂×#15667974	1	1	yes	characteristic
250	C₄	C₂²×C₈×D₈₉	1	1	yes	characteristic
251	C₂	—	2	1	no	—
252	C₂	—	2	2	no	—
253	C₂	—	356	4	no	—
254	C₂	C₂×#19151663	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×D₄×D₈₉
commutator	C₄×C₈₉	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2153	C₂
socle	C₂×C₈₉	C₂×C₈×D₄
prime radical	#19152398	C₄
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2153	D₈₉
hypocenter	C₈₉	#33441
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2153

Complements

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

N	H ≅ G/N	type
1. C₂×#19152016	C₂	semidirect
3. #26022093	C₂	semidirect
4. #26022088	C₂	semidirect
5. #26021881	C₂	semidirect
6. #26021849	C₂	semidirect
7. C₈₉×#2153	C₂	semidirect
12. #19152016	C₂²	semidirect
13. #19152016	C₂²	semidirect
22. #19151383	C₂²	semidirect
23. #19151374	C₂²	semidirect
24. C₄×#13083549	C₂²	semidirect
27. #364×C₈₉	C₂²	semidirect
132. #2153	D₈₉	semidirect
135. C₈₉	#33441	semidirect
152. #364	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19152016	2
#26029156	2
#26022093	2
#26022088	2
#26021881	2
#26021849	2
C₈₉×#2153	2
#33441	89