Subgroups — C₂×C₄×#15667971 #36712613

Order 22784.

4462 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15667971	C₂	1	2	yes	maximal
2	C₄×#15667971	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄×C₄⋊Dic₈₉	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×C₂²⋊Dic₈₉	C₂	1	4	yes	maximal
5	C₂×#19151444	C₂	1	2	yes	maximal, complemented
6	C₂×C₄²×Dic₈₉	C₂	1	2	yes	maximal, complemented
7	C₂³×C₄²×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#15667971	C₄	1	32	yes	direct product factor
9	C₂×#15667971	C₂²	1	16	yes	—
10	C₂²×C₄⋊Dic₈₉	C₂²	1	4	yes	—
11	C₂²×C₂²⋊Dic₈₉	C₂²	1	4	yes	—
12	C₂²×C₄×Dic₈₉	C₂²	1	8	yes	—
13	C₂²×C₄×Dic₈₉	C₂²	1	4	yes	—
14	C₄×C₄⋊Dic₈₉	C₂²	1	32	yes	complemented
15	C₄×C₂²⋊Dic₈₉	C₂²	1	32	yes	—
16	#19151444	C₂²	1	16	yes	complemented
17	C₂×#15667919	C₂²	1	8	yes	—
18	C₄²×Dic₈₉	C₂²	1	16	yes	complemented
19	C₂⁴×C₄×C₈₉	C₂²	1	2	yes	—
20	C₂⁴×C₄×C₈₉	C₂²	1	1	yes	characteristic
21	C₂²×C₄²×C₈₉	C₄	1	16	yes	complemented
22	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
23	C₂²×C₄²×C₈₉	C₂²	1	2	yes	—
24	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
25	C₂²×C₄²×C₈₉	C₂²	1	8	yes	—
26	C₂³×Dic₈₉	C₂³	1	8	yes	—
27	#15667971	C₂×C₄	1	128	yes	direct product factor
28	C₂×C₄⋊Dic₈₉	C₂×C₄	1	32	yes	complemented
29	C₂×C₄⋊Dic₈₉	C₂³	1	16	yes	—
30	C₂×C₂²⋊Dic₈₉	C₂×C₄	1	32	yes	—
31	C₂×C₂²⋊Dic₈₉	C₂³	1	16	yes	—
32	C₂×C₄×Dic₈₉	—	2	32	no	—
33	C₂×C₄×Dic₈₉	C₂×C₄	1	32	yes	complemented
34	C₂×C₄×Dic₈₉	C₂³	1	32	yes	—
35	C₂×C₄×Dic₈₉	C₂³	1	16	yes	—
36	#15667919	C₂³	1	32	yes	—
37	C₂⁵×C₈₉	C₂³	1	1	yes	characteristic
38	C₂³×C₄×C₈₉	—	2	16	no	—
39	C₂³×C₄×C₈₉	C₂×C₄	1	16	yes	—
40	C₂³×C₄×C₈₉	C₂×C₄	1	16	yes	—
41	C₂³×C₄×C₈₉	C₂×C₄	1	8	yes	—
42	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
43	C₂³×C₄×C₈₉	C₂³	1	8	yes	—
44	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
45	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
47	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
48	C₂×C₄²×C₈₉	C₂×C₄	1	8	yes	—
49	C₂×C₄²×C₈₉	C₂×C₄	1	16	yes	—
50	C₂×C₄²×C₈₉	C₂×C₄	1	8	yes	—
51	C₂×C₄²×C₈₉	C₂×C₄	1	64	yes	complemented
52	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
53	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
54	C₂×C₄²×C₈₉	C₂³	1	8	yes	—
55	C₂²×Dic₈₉	—	2	32	no	—
56	C₂²×Dic₈₉	C₂²×C₄	1	32	yes	—
57	C₂²×Dic₈₉	C₂⁴	1	16	yes	—
58	C₄⋊Dic₈₉	C₂²×C₄	1	64	yes	complemented
59	C₂²⋊Dic₈₉	C₂²×C₄	1	64	yes	—
60	C₄×Dic₈₉	—	2	64	no	—
61	C₄×Dic₈₉	C₂²×C₄	1	64	yes	complemented
62	C₂⁴×C₈₉	—	2	8	no	—
63	C₂⁴×C₈₉	C₂²×C₄	1	8	yes	—
64	C₂⁴×C₈₉	C₂²×C₄	1	4	yes	—
65	C₂⁴×C₈₉	C₂⁴	1	2	yes	—
66	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
67	C₂²×C₄×C₈₉	—	2	8	no	—
68	C₂²×C₄×C₈₉	—	2	64	no	—
69	C₂²×C₄×C₈₉	—	2	32	no	—
70	C₂²×C₄×C₈₉	C₄²	1	64	yes	complemented
71	C₂²×C₄×C₈₉	C₂²×C₄	1	8	yes	—
72	C₂²×C₄×C₈₉	C₂²×C₄	1	32	yes	—
73	C₂²×C₄×C₈₉	C₂²×C₄	1	8	yes	—
74	C₂²×C₄×C₈₉	C₂²×C₄	1	8	yes	—
75	C₂²×C₄×C₈₉	C₂²×C₄	1	4	yes	—
76	C₂²×C₄×C₈₉	C₂²×C₄	1	4	yes	—
77	C₂²×C₄×C₈₉	C₂²×C₄	1	32	yes	—
78	C₂²×C₄×C₈₉	C₂²×C₄	1	16	yes	—
79	C₂²×C₄×C₈₉	C₂²×C₄	1	8	yes	—
80	C₂²×C₄×C₈₉	#41	1	16	yes	—
81	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
82	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
83	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
84	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
85	C₄²×C₈₉	C₂²×C₄	1	16	yes	—
86	C₄²×C₈₉	C₂²×C₄	1	32	yes	—
87	C₄²×C₈₉	C₂²×C₄	1	16	yes	—
88	C₂³×C₈₉	—	2	16	no	—
89	C₂³×C₈₉	—	2	32	no	—
90	C₂³×C₈₉	—	2	8	no	—
91	C₂³×C₈₉	C₂×C₄²	1	16	yes	—
92	C₂³×C₈₉	C₂³×C₄	1	8	yes	—
93	C₂³×C₈₉	C₂³×C₄	1	4	yes	—
94	C₂³×C₈₉	C₂³×C₄	1	4	yes	—
95	C₂³×C₈₉	C₂³×C₄	1	2	yes	—
96	C₂³×C₈₉	C₂×#41	1	8	yes	—
97	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
98	C₂×C₄×C₈₉	—	2	16	no	—
99	C₂×C₄×C₈₉	—	2	8	no	—
100	C₂×C₄×C₈₉	—	2	64	no	—
101	C₂×C₄×C₈₉	C₂×C₄²	1	16	yes	—
102	C₂×C₄×C₈₉	C₂×C₄²	1	16	yes	—
103	C₂×C₄×C₈₉	C₂×C₄²	1	64	yes	complemented
104	C₂×C₄×C₈₉	#117	1	32	yes	complemented
105	C₂×C₄×C₈₉	C₂³×C₄	1	8	yes	—
106	C₂×C₄×C₈₉	C₂³×C₄	1	8	yes	—
107	C₂×C₄×C₈₉	C₂³×C₄	1	8	yes	—
108	C₂×C₄×C₈₉	C₂³×C₄	1	8	yes	—
109	C₂×C₄×C₈₉	C₂³×C₄	1	4	yes	—
110	C₂×C₄×C₈₉	C₂³×C₄	1	4	yes	—
111	C₂×C₄×C₈₉	C₂×#41	1	16	yes	—
112	C₂×Dic₈₉	—	2	64	no	—
113	C₂×Dic₈₉	—	2	32	no	—
114	C₂×Dic₈₉	C₂³×C₄	1	32	yes	—
115	C₂²×C₈₉	—	2	8	no	—
116	C₂²×C₈₉	—	2	16	no	—
117	C₂²×C₈₉	—	2	32	no	—
118	C₂²×C₈₉	C₂²×C₄²	1	8	yes	—
119	C₂²×C₈₉	C₂²×C₄²	1	4	yes	—
120	C₂²×C₈₉	C₂×#117	1	8	yes	—
121	C₂²×C₈₉	C₄×#41	1	16	yes	—
122	C₂²×C₈₉	C₂⁴×C₄	1	1	yes	characteristic
123	C₂²×C₈₉	C₂⁴×C₄	1	2	yes	—
124	C₂²×C₈₉	C₂²×#41	1	4	yes	—
125	C₄×C₈₉	—	2	8	no	—
126	C₄×C₈₉	C₂²×C₄²	1	8	yes	—
127	C₄×C₈₉	C₂²×C₄²	1	8	yes	—
128	C₄×C₈₉	C₂×#117	1	16	yes	complemented
129	Dic₈₉	—	2	32	no	—
130	C₂×C₄×#117	—	89	1	no	maximal, Sylow
131	C₂×C₈₉	—	2	8	no	—
132	C₂×C₈₉	C₄×#117	1	8	yes	complemented
133	C₂×C₈₉	C₂³×C₄²	1	1	yes	characteristic
134	C₂×C₈₉	C₂²×#117	1	2	yes	—
135	C₂×C₈₉	C₂×C₄×#41	1	4	yes	—
136	C₂²×#117	—	89	2	no	—
137	C₂³×C₄²	D₈₉	1	1	yes	characteristic
138	C₄×#117	—	89	16	no	—
139	C₂×C₄×C₄⋊C₄	—	89	4	no	—
140	C₂×C₄×C₂²⋊C₄	—	89	4	no	—
141	C₂×#376	—	89	2	no	—
142	C₂×C₄³	—	89	2	no	—
143	C₈₉	C₂×C₄×#117	1	1	yes	characteristic, complemented, Sylow
144	C₂⁴×C₄	C₂×D₈₉	1	2	yes	—
145	C₂⁴×C₄	C₂×D₈₉	1	1	yes	characteristic
146	C₂×#117	—	89	32	no	—
147	C₂×#117	—	89	16	no	—
148	C₂²×C₄⋊C₄	—	89	4	no	—
149	C₂²×C₂²⋊C₄	—	89	4	no	—
150	C₂²×C₄²	—	89	4	no	—
151	C₂²×C₄²	—	89	8	no	—
152	C₂²×C₄²	Dic₈₉	1	16	yes	complemented
153	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
154	C₂²×C₄²	C₂×D₈₉	1	2	yes	—
155	C₂²×C₄²	C₂×D₈₉	1	8	yes	—
156	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
157	C₄×C₄⋊C₄	—	89	32	no	—
158	C₄×C₂²⋊C₄	—	89	32	no	—
159	#376	—	89	16	no	—
160	C₂×#95	—	89	8	no	—
161	C₄³	—	89	16	no	—
162	C₂⁵	C₂²×D₈₉	1	1	yes	characteristic
163	C₂³×C₄	—	2	16	no	—
164	C₂³×C₄	—	89	8	no	—
165	C₂³×C₄	C₄×D₈₉	1	16	yes	—
166	C₂³×C₄	C₂×Dic₈₉	1	16	yes	—
167	C₂³×C₄	C₂×Dic₈₉	1	8	yes	—
168	C₂³×C₄	C₂²×D₈₉	1	2	yes	—
169	C₂³×C₄	C₂²×D₈₉	1	8	yes	—
170	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
171	C₂³×C₄	C₂²×D₈₉	1	4	yes	—
172	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
173	C₂³×C₄	C₂²×D₈₉	1	2	yes	—
174	#117	—	89	128	no	—
175	C₂×C₄⋊C₄	—	89	32	no	—
176	C₂×C₄⋊C₄	—	89	16	no	—
177	C₂×C₂²⋊C₄	—	89	32	no	—
178	C₂×C₂²⋊C₄	—	89	16	no	—
179	C₂×C₄²	—	89	32	no	—
180	C₂×C₄²	—	89	16	no	—
181	C₂×C₄²	—	89	32	no	—
182	C₂×C₄²	—	178	32	no	—
183	C₂×C₄²	C₂×Dic₈₉	1	8	yes	—
184	C₂×C₄²	C₂×Dic₈₉	1	16	yes	—
185	C₂×C₄²	C₂×Dic₈₉	1	8	yes	—
186	C₂×C₄²	C₂×Dic₈₉	1	64	yes	complemented
187	C₂×C₄²	C₂²×D₈₉	1	4	yes	—
188	C₂×C₄²	C₂²×D₈₉	1	4	yes	—
189	C₂×C₄²	C₂²×D₈₉	1	8	yes	—
190	#95	—	89	32	no	—
191	C₂⁴	—	2	8	no	—
192	C₂⁴	C₂×C₄×D₈₉	1	8	yes	—
193	C₂⁴	C₂²×Dic₈₉	1	4	yes	—
194	C₂⁴	C₂³×D₈₉	1	2	yes	—
195	C₂⁴	C₂³×D₈₉	1	1	yes	characteristic
196	C₂²×C₄	—	2	8	no	—
197	C₂²×C₄	—	2	64	no	—
198	C₂²×C₄	—	2	32	no	—
199	C₂²×C₄	—	89	32	no	—
200	C₂²×C₄	—	89	16	no	—
201	C₂²×C₄	—	178	32	no	—
202	C₂²×C₄	C₄×Dic₈₉	1	64	yes	complemented
203	C₂²×C₄	C₂×C₄×D₈₉	1	8	yes	—
204	C₂²×C₄	C₂×C₄×D₈₉	1	8	yes	—
205	C₂²×C₄	C₂×C₄×D₈₉	1	32	yes	—
206	C₂²×C₄	C₂²×Dic₈₉	1	8	yes	—
207	C₂²×C₄	C₂²×Dic₈₉	1	32	yes	—
208	C₂²×C₄	C₂²×Dic₈₉	1	8	yes	—
209	C₂²×C₄	C₂²×Dic₈₉	1	4	yes	—
210	C₂²×C₄	C₂²×Dic₈₉	1	4	yes	—
211	C₂²×C₄	C₂²×Dic₈₉	1	16	yes	—
212	C₂²×C₄	#13083574	1	16	yes	—
213	C₂²×C₄	C₂³×D₈₉	1	4	yes	—
214	C₂²×C₄	C₂³×D₈₉	1	2	yes	—
215	C₂²×C₄	C₂³×D₈₉	1	2	yes	—
216	C₂²×C₄	C₂³×D₈₉	1	4	yes	—
217	C₄⋊C₄	—	89	64	no	—
218	C₂²⋊C₄	—	89	64	no	—
219	C₄²	—	89	64	no	—
220	C₄²	—	178	64	no	—
221	C₄²	C₂²×Dic₈₉	1	16	yes	—
222	C₄²	C₂²×Dic₈₉	1	32	yes	—
223	C₄²	C₂²×Dic₈₉	1	16	yes	—
224	C₂³	—	2	8	no	—
225	C₂³	—	2	16	no	—
226	C₂³	—	2	32	no	—
227	C₂³	C₂×C₄×Dic₈₉	1	16	yes	—
228	C₂³	C₂²×C₄×D₈₉	1	8	yes	—
229	C₂³	C₂²×C₄×D₈₉	1	4	yes	—
230	C₂³	C₂³×Dic₈₉	1	4	yes	—
231	C₂³	C₂³×Dic₈₉	1	2	yes	—
232	C₂³	C₂×#13083574	1	8	yes	—
233	C₂³	C₂⁴×D₈₉	1	1	yes	characteristic
234	C₂×C₄	—	2	8	no	—
235	C₂×C₄	—	2	16	no	—
236	C₂×C₄	—	2	64	no	—
237	C₂×C₄	—	89	32	no	—
238	C₂×C₄	—	178	32	no	—
239	C₂×C₄	—	178	64	no	—
240	C₂×C₄	C₂×C₄×Dic₈₉	1	16	yes	—
241	C₂×C₄	C₂×C₄×Dic₈₉	1	16	yes	—
242	C₂×C₄	C₂×C₄×Dic₈₉	1	64	yes	complemented
243	C₂×C₄	#15667971	1	32	yes	direct product factor
244	C₂×C₄	C₂²×C₄×D₈₉	1	8	yes	—
245	C₂×C₄	C₂²×C₄×D₈₉	1	8	yes	—
246	C₂×C₄	C₂³×Dic₈₉	1	8	yes	—
247	C₂×C₄	C₂³×Dic₈₉	1	4	yes	—
248	C₂×C₄	C₂³×Dic₈₉	1	4	yes	—
249	C₂×C₄	C₂³×Dic₈₉	1	8	yes	—
250	C₂×C₄	C₂×#13083574	1	16	yes	—
251	C₂²	—	2	8	no	—
252	C₂²	—	2	16	no	—
253	C₂²	—	2	32	no	—
254	C₂²	C₂²×C₄×Dic₈₉	1	8	yes	—
255	C₂²	C₂²×C₄×Dic₈₉	1	4	yes	—
256	C₂²	C₂×#15667971	1	8	yes	—
257	C₂²	C₄×#13083574	1	16	yes	—
258	C₂²	C₂³×C₄×D₈₉	1	2	yes	—
259	C₂²	C₂⁴×Dic₈₉	1	1	yes	characteristic
260	C₂²	C₂²×#13083574	1	4	yes	—
261	C₄	—	2	8	no	—
262	C₄	—	178	32	no	—
263	C₄	C₂²×C₄×Dic₈₉	1	8	yes	—
264	C₄	C₂²×C₄×Dic₈₉	1	8	yes	—
265	C₄	C₂×#15667971	1	16	yes	direct product factor
266	C₂	—	2	8	no	—
267	C₂	C₄×#15667971	1	8	yes	direct product factor
268	C₂	C₂³×C₄×Dic₈₉	1	1	yes	characteristic
269	C₂	C₂²×#15667971	1	2	yes	—
270	C₂	C₂×C₄×#13083574	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄²	C₂×D₈₉
commutator	C₂×C₈₉	C₂³×C₄²
Frattini	C₂³	C₂⁴×D₈₉
Fitting	C₂³×C₄²×C₈₉	C₂
socle	C₂⁴×C₈₉	C₂⁴
prime radical	C₂⁵×C₈₉	C₂³
cosocle	C₂³×C₈₉	C₂⁵
hypercenter	C₂³×C₄²	D₈₉
hypocenter	C₈₉	C₂×C₄×#117
LCS	G ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄² ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
2. C₄×#15667971	C₂	direct (N × H)
3. C₂×C₄×C₄⋊Dic₈₉	C₂	semidirect
5. C₂×#19151444	C₂	semidirect
6. C₂×C₄²×Dic₈₉	C₂	semidirect
8. C₂×#15667971	C₄	direct (N × H)
14. C₄×C₄⋊Dic₈₉	C₂²	semidirect
16. #19151444	C₂²	semidirect
18. C₄²×Dic₈₉	C₂²	semidirect
21. C₂²×C₄²×C₈₉	C₄	semidirect
27. #15667971	C₂×C₄	direct (N × H)
28. C₂×C₄⋊Dic₈₉	C₂×C₄	semidirect
33. C₂×C₄×Dic₈₉	C₂×C₄	semidirect
51. C₂×C₄²×C₈₉	C₂×C₄	semidirect
58. C₄⋊Dic₈₉	C₂²×C₄	semidirect
61. C₄×Dic₈₉	C₂²×C₄	semidirect
70. C₂²×C₄×C₈₉	C₄²	semidirect
103. C₂×C₄×C₈₉	C₂×C₄²	semidirect
104. C₂×C₄×C₈₉	#117	semidirect
128. C₄×C₈₉	C₂×#117	semidirect
132. C₂×C₈₉	C₄×#117	semidirect
143. C₈₉	C₂×C₄×#117	semidirect
152. C₂²×C₄²	Dic₈₉	semidirect
186. C₂×C₄²	C₂×Dic₈₉	semidirect
202. C₂²×C₄	C₄×Dic₈₉	semidirect
242. C₂×C₄	C₂×C₄×Dic₈₉	semidirect
243. C₂×C₄	#15667971	direct (N × H)
265. C₄	C₂×#15667971	direct (N × H)
267. C₂	C₄×#15667971	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#15667971	2
C₄×#15667971	2
C₂×C₄×C₄⋊Dic₈₉	2
C₂×C₄×C₂²⋊Dic₈₉	2
C₂×#19151444	2
C₂×C₄²×Dic₈₉	2
C₂³×C₄²×C₈₉	2
C₂×C₄×#117	89

Subgroups — C2×C4×#15667971 #36712613

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#15667971 #36712613