Subgroups — #36454240

Order 22784.

694 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	C₂×#19151975	C₂	1	1	yes	maximal, characteristic
2	#26021058	C₂	1	2	yes	maximal, complemented
3	#26021058	C₂	1	2	yes	maximal, complemented
4	#26020380	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#322×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₇₁₂	—	2	2	no	—
7	C₂×#15668009	C₂²	1	2	yes	—
8	#19151975	—	2	2	no	—
9	#19151975	C₄	1	4	yes	complemented
10	C₂×#15667934	—	2	4	no	—
11	C₂×#15667934	—	2	4	no	—
12	#19151468	C₂²	1	2	yes	—
13	#19151361	—	2	2	no	—
14	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	#322×C₈₉	—	2	2	no	—
18	C₂²×D₃₅₆	—	2	2	no	—
19	C₂²×D₃₅₆	—	2	2	no	—
20	C₂²×D₃₅₆	—	2	2	no	—
21	C₂×D₇₁₂	—	2	8	no	—
22	C₂×D₇₁₂	—	4	8	no	—
23	#15668009	—	2	2	no	—
24	#15668009	—	2	2	no	—
25	#15668009	—	2	2	no	—
26	#15668009	C₂×C₄	1	4	yes	—
27	C₂×C₄⋊Dic₈₉	—	2	2	no	—
28	C₂×C₄⋊Dic₈₉	—	2	2	no	—
29	C₂×C₄⋊Dic₈₉	—	2	2	no	—
30	#15667934	—	4	8	no	—
31	#15667934	—	4	8	no	—
32	C₂²×C₈×C₈₉	D₄	1	2	yes	—
33	C₂²×C₈×C₈₉	D₄	1	2	yes	—
34	C₂²×C₈×C₈₉	D₄	1	2	yes	—
35	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
36	C₄×C₈×C₈₉	—	2	1	no	—
37	C₄×C₈×C₈₉	—	2	1	no	—
38	C₄×C₈×C₈₉	—	2	1	no	—
39	C₄×C₈×C₈₉	—	2	1	no	—
40	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₈₉	D₄	1	2	yes	—
42	C₂³×D₈₉	—	4	2	no	—
43	C₂×D₃₅₆	—	2	4	no	—
44	C₂×D₃₅₆	—	2	4	no	—
45	C₂×D₃₅₆	—	2	4	no	—
46	C₂×D₃₅₆	—	4	4	no	—
47	C₂×D₃₅₆	—	4	4	no	—
48	C₂×D₃₅₆	—	4	4	no	—
49	C₂×D₃₅₆	—	4	4	no	—
50	C₂×D₃₅₆	—	4	4	no	—
51	C₂×D₃₅₆	—	4	4	no	—
52	C₂×D₃₅₆	—	4	2	no	—
53	C₂×D₃₅₆	—	4	2	no	—
54	C₂×D₃₅₆	—	4	2	no	—
55	C₂²×Dic₈₉	—	4	2	no	—
56	D₇₁₂	—	4	16	no	—
57	C₄⋊Dic₈₉	—	4	4	no	—
58	C₄⋊Dic₈₉	—	4	4	no	—
59	C₄⋊Dic₈₉	—	4	4	no	—
60	C₂²×C₄×C₈₉	C₂×D₄	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	2	no	—
64	C₂×C₈×C₈₉	—	2	2	no	—
65	C₂×C₈×C₈₉	—	2	4	no	—
66	C₂×C₈×C₈₉	—	2	2	no	—
67	C₂×C₈×C₈₉	—	2	4	no	—
68	C₂×C₈×C₈₉	C₂²⋊C₄	1	4	yes	—
69	C₂×C₈×C₈₉	D₈	1	4	yes	complemented
70	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
71	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₈₉	#41	1	1	yes	characteristic
73	C₄²×C₈₉	#41	1	1	yes	characteristic
74	C₂³×C₈₉	#127	1	1	yes	characteristic
75	C₂²×D₈₉	—	4	4	no	—
76	C₂²×D₈₉	—	8	2	no	—
77	C₂²×D₈₉	—	8	2	no	—
78	C₂²×D₈₉	—	8	2	no	—
79	C₂²×D₈₉	—	8	2	no	—
80	C₂²×D₈₉	—	8	2	no	—
81	C₂²×D₈₉	—	8	2	no	—
82	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
84	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
85	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
86	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
87	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
88	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
89	C₂×C₄×C₈₉	#135	1	2	yes	—
90	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	4	yes	—
91	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	4	yes	—
92	C₂×Dic₈₉	—	8	2	no	—
93	C₂×Dic₈₉	—	8	2	no	—
94	C₂×Dic₈₉	—	8	2	no	—
95	D₃₅₆	—	4	8	no	—
96	D₃₅₆	—	4	8	no	—
97	D₃₅₆	—	4	8	no	—
98	D₃₅₆	—	8	4	no	—
99	D₃₅₆	—	8	4	no	—
100	D₃₅₆	—	8	4	no	—
101	C₈×C₈₉	—	2	4	no	—
102	C₈×C₈₉	—	2	4	no	—
103	C₈×C₈₉	—	4	2	no	—
104	C₂²×C₈₉	#390	1	1	yes	characteristic
105	C₂²×C₈₉	#493	1	1	yes	characteristic
106	C₂²×C₈₉	#495	1	1	yes	characteristic
107	C₂²×C₈₉	#496	1	2	yes	—
108	C₂²×C₈₉	#496	1	2	yes	—
109	C₂×D₈₉	—	8	4	no	—
110	C₂×D₈₉	—	8	4	no	—
111	C₂×D₈₉	—	8	4	no	—
112	C₂×D₈₉	—	16	2	no	—
113	C₂×D₈₉	—	16	2	no	—
114	C₂×D₈₉	—	16	2	no	—
115	C₂×D₈₉	—	16	2	no	—
116	C₄×C₈₉	#418	1	2	yes	—
117	C₄×C₈₉	C₄×D₈	1	2	yes	—
118	C₄×C₈₉	#442	1	2	yes	—
119	C₄×C₈₉	#466	1	2	yes	—
120	C₄×C₈₉	#469	1	2	yes	—
121	C₄×C₈₉	#490	1	2	yes	—
122	Dic₈₉	—	16	2	no	—
123	#9832	—	89	1	no	maximal, Sylow
124	C₂×C₈₉	#1717	1	2	yes	—
125	C₂×C₈₉	#1721	1	2	yes	—
126	C₂×C₈₉	#1961	1	1	yes	characteristic
127	C₂×C₈₉	#1970	1	1	yes	characteristic
128	C₂×C₈₉	#1972	1	1	yes	characteristic
129	D₈₉	—	16	4	no	—
130	C₂×#493	—	89	1	no	—
131	#1957	—	89	2	no	—
132	#1957	—	89	2	no	—
133	#1834	—	89	1	no	—
134	C₂×#322	D₈₉	1	1	yes	characteristic, complemented
135	C₈₉	#9832	1	1	yes	characteristic, complemented, Sylow
136	C₂²×D₈	—	178	2	no	—
137	C₂×#127	—	89	2	no	—
138	#493	—	89	4	no	—
139	#493	—	178	2	no	—
140	C₂×#102	—	178	4	no	—
141	C₂×#102	—	178	4	no	—
142	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
143	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
144	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
145	#384	—	89	2	no	—
146	#340	—	178	2	no	—
147	#322	—	2	2	no	—
148	C₂²×D₄	—	178	2	no	—
149	C₂²×D₄	—	178	2	no	—
150	C₂²×D₄	—	178	2	no	—
151	C₂×D₈	—	356	8	no	—
152	C₂×D₈	—	178	8	no	—
153	C₂²×C₈	D₃₅₆	1	2	yes	—
154	C₂²×C₈	D₃₅₆	1	2	yes	—
155	C₂²×C₈	D₃₅₆	1	2	yes	—
156	#127	—	89	4	no	—
157	#127	—	178	2	no	—
158	#127	—	178	2	no	—
159	#127	—	178	2	no	—
160	C₂×C₄⋊C₄	—	178	2	no	—
161	C₂×C₄⋊C₄	—	178	2	no	—
162	C₂×C₄⋊C₄	—	178	2	no	—
163	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
164	#102	—	356	8	no	—
165	#102	—	356	8	no	—
166	C₄×C₈	—	2	1	no	—
167	C₄×C₈	—	2	1	no	—
168	C₄×C₈	—	2	1	no	—
169	C₄×C₈	—	2	1	no	—
170	C₄×C₈	C₄×D₈₉	1	2	yes	—
171	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
172	C₂⁴	—	356	2	no	—
173	C₂×D₄	—	356	2	no	—
174	C₂×D₄	—	356	2	no	—
175	C₂×D₄	—	356	2	no	—
176	C₂×D₄	—	356	4	no	—
177	C₂×D₄	—	356	4	no	—
178	C₂×D₄	—	356	4	no	—
179	C₂×D₄	—	356	4	no	—
180	C₂×D₄	—	356	4	no	—
181	C₂×D₄	—	356	4	no	—
182	C₂×D₄	—	178	4	no	—
183	C₂×D₄	—	178	4	no	—
184	C₂×D₄	—	178	4	no	—
185	C₂²×C₄	—	356	2	no	—
186	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
187	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
188	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
189	D₈	—	356	16	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	4	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	#13083551	1	4	yes	—
196	C₂×C₈	D₇₁₂	1	4	yes	complemented
197	C₄⋊C₄	—	356	4	no	—
198	C₄⋊C₄	—	356	4	no	—
199	C₄⋊C₄	—	356	4	no	—
200	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
201	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
202	C₄²	#13083574	1	1	yes	characteristic
203	C₄²	#13083574	1	1	yes	characteristic
204	C₂³	—	356	4	no	—
205	C₂³	—	712	2	no	—
206	C₂³	—	712	2	no	—
207	C₂³	—	712	2	no	—
208	C₂³	—	712	2	no	—
209	C₂³	—	712	2	no	—
210	C₂³	—	712	2	no	—
211	C₂³	#15668009	1	1	yes	characteristic
212	D₄	—	356	8	no	—
213	D₄	—	356	8	no	—
214	D₄	—	356	8	no	—
215	D₄	—	712	4	no	—
216	D₄	—	712	4	no	—
217	D₄	—	712	4	no	—
218	C₂×C₄	—	712	2	no	—
219	C₂×C₄	—	712	2	no	—
220	C₂×C₄	—	712	2	no	—
221	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
222	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
223	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
224	C₂×C₄	#15667986	1	1	yes	characteristic
225	C₂×C₄	#15667986	1	1	yes	characteristic
226	C₂×C₄	#15668000	1	1	yes	characteristic
227	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
228	C₂×C₄	#15668034	1	2	yes	—
229	C₂×C₄	#15668040	1	4	yes	—
230	C₂×C₄	#15668040	1	4	yes	—
231	C₈	—	2	4	no	—
232	C₈	—	2	4	no	—
233	C₈	—	4	2	no	—
234	C₂²	—	1424	2	no	—
235	C₂²	—	1424	2	no	—
236	C₂²	—	1424	2	no	—
237	C₂²	—	1424	2	no	—
238	C₂²	—	712	4	no	—
239	C₂²	—	712	4	no	—
240	C₂²	—	712	4	no	—
241	C₂²	#19151500	1	1	yes	characteristic
242	C₂²	#19151975	1	1	yes	characteristic
243	C₂²	#19151982	1	1	yes	characteristic
244	C₂²	#19151987	1	2	yes	—
245	C₂²	#19151987	1	2	yes	—
246	C₄	—	1424	2	no	—
247	C₄	#19151604	1	2	yes	—
248	C₄	C₄×D₇₁₂	1	2	yes	—
249	C₄	#19151711	1	2	yes	—
250	C₄	#19151834	1	2	yes	—
251	C₄	#19151851	1	2	yes	—
252	C₄	#19151963	1	2	yes	—
253	C₂	—	1424	4	no	—
254	C₂	#26019891	1	2	yes	—
255	C₂	#26019918	1	2	yes	—
256	C₂	#26021076	1	1	yes	characteristic
257	C₂	#26021118	1	1	yes	characteristic
258	C₂	#26021128	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15668009
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₂×#322×C₈₉	C₂
socle	C₂³×C₈₉	#127
prime radical	C₂×#19151975	C₂
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂×#322	D₈₉
hypocenter	C₈₉	#9832
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
2. #26021058	C₂	semidirect
3. #26021058	C₂	semidirect
4. #26020380	C₂	semidirect
5. C₂×#322×C₈₉	C₂	semidirect
9. #19151975	C₄	semidirect
69. C₂×C₈×C₈₉	D₈	semidirect
134. C₂×#322	D₈₉	semidirect
135. C₈₉	#9832	semidirect
196. C₂×C₈	D₇₁₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151975	2
#26021058	2
#26020380	2
C₂×#322×C₈₉	2
#9832	89