Subgroups — C₂×D₄×#13083549 #36707578

Order 22784.

806 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#13083549	C₂	1	2	yes	maximal, complemented
2	D₄×#13083549	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊#13083549	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂²⋊#13083549	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×#13083549	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×D₄×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×#13083549	—	2	8	no	—
8	C₂²×#13083549	C₂²	1	2	yes	—
9	C₂²×#13083549	C₂²	1	1	yes	characteristic
10	C₂²×#13083549	C₂²	1	1	yes	characteristic
11	C₂²×#13083549	C₂²	1	8	yes	complemented
12	C₄⋊#13083549	C₂²	1	4	yes	complemented
13	C₂²⋊#13083549	C₂²	1	8	yes	complemented
14	C₄×#13083549	C₂²	1	4	yes	complemented
15	C₂³×C₈×C₈₉	C₂²	1	2	yes	—
16	C₂×C₄×D₄×C₈₉	C₄	1	1	yes	characteristic
17	C₈×D₄×C₈₉	C₄	1	4	yes	—
18	C₈×D₄×C₈₉	C₂²	1	4	yes	—
19	C₂×C₄⋊C₈×C₈₉	C₄	1	1	yes	characteristic
20	C₂×C₂²⋊C₈×C₈₉	C₄	1	2	yes	—
21	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×#13083549	—	2	16	no	—
23	C₂×#13083549	—	2	4	no	—
24	C₂×#13083549	—	2	2	no	—
25	C₂×#13083549	D₄	1	4	yes	direct product factor
26	C₂×#13083549	C₂³	1	2	yes	—
27	C₂×#13083549	C₂³	1	4	yes	—
28	C₂×#13083549	C₂³	1	2	yes	—
29	C₂²×D₄×C₈₉	C₈	1	1	yes	characteristic
30	C₂³×C₄×C₈₉	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₈₉	—	2	4	no	—
32	C₂²×C₈×C₈₉	—	2	4	no	—
33	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₈₉	C₂×C₄	1	4	yes	—
36	C₂²×C₈×C₈₉	C₂³	1	4	yes	—
37	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
38	C₄×D₄×C₈₉	C₈	1	4	yes	—
39	C₄×D₄×C₈₉	C₂×C₄	1	4	yes	—
40	C₂×C₄⋊C₄×C₈₉	C₈	1	1	yes	characteristic
41	C₂×C₂²⋊C₄×C₈₉	C₈	1	2	yes	—
42	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
43	C₄⋊C₈×C₈₉	C₂×C₄	1	4	yes	—
44	C₂²⋊C₈×C₈₉	C₂×C₄	1	8	yes	—
45	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₈₉	C₂³	1	2	yes	—
47	#13083549	—	2	4	no	—
48	#13083549	—	2	2	no	—
49	#13083549	C₂×D₄	1	4	yes	direct product factor
50	C₂⁴×C₈₉	C₂×C₈	1	2	yes	—
51	C₂×D₄×C₈₉	C₁₆	1	8	yes	complemented
52	C₂×D₄×C₈₉	C₂×C₈	1	4	yes	—
53	C₂²×C₄×C₈₉	—	2	4	no	—
54	C₂²×C₄×C₈₉	—	2	4	no	—
55	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₈₉	C₂×C₈	1	2	yes	—
57	C₂²×C₄×C₈₉	C₂×C₈	1	4	yes	—
58	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄×C₈₉	C₂²×C₄	1	4	yes	—
60	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₈₉	—	2	4	no	—
62	C₂×C₈×C₈₉	—	2	2	no	—
63	C₂×C₈×C₈₉	—	2	8	no	—
64	C₂×C₈×C₈₉	—	2	4	no	—
65	C₂×C₈×C₈₉	—	2	4	no	—
66	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
68	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₈₉	#41	1	2	yes	—
71	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
72	C₄⋊C₄×C₈₉	C₂×C₈	1	4	yes	—
73	C₂²⋊C₄×C₈₉	C₂×C₈	1	8	yes	—
74	C₄²×C₈₉	C₂×C₈	1	2	yes	—
75	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
76	C₂³×C₈₉	—	2	4	no	—
77	C₂³×C₈₉	—	2	4	no	—
78	C₂³×C₈₉	C₂×C₁₆	1	8	yes	complemented
79	C₂³×C₈₉	C₂²×C₈	1	4	yes	—
80	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
81	D₄×C₈₉	C₂×C₁₆	1	16	yes	complemented
82	C₂×C₄×C₈₉	—	2	4	no	—
83	C₂×C₄×C₈₉	—	2	2	no	—
84	C₂×C₄×C₈₉	—	2	4	no	—
85	C₂×C₄×C₈₉	—	2	4	no	—
86	C₂×C₄×C₈₉	—	2	8	no	—
87	C₂×C₄×C₈₉	C₂×C₁₆	1	4	yes	complemented
88	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
89	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
90	C₂×C₄×C₈₉	C₂²×C₈	1	4	yes	—
91	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
92	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
93	C₂×C₄×C₈₉	#131	1	2	yes	—
94	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
95	C₈×C₈₉	—	2	4	no	—
96	C₈×C₈₉	—	2	2	no	—
97	C₈×C₈₉	C₄×D₄	1	2	yes	—
98	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
99	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
100	C₂²×C₈₉	—	2	4	no	—
101	C₂²×C₈₉	—	2	4	no	—
102	C₂²×C₈₉	—	2	16	no	—
103	C₂²×C₈₉	C₈×D₄	1	2	yes	—
104	C₂²×C₈₉	C₂²×C₁₆	1	2	yes	—
105	C₂²×C₈₉	C₂²×C₁₆	1	8	yes	complemented
106	C₂²×C₈₉	#504	1	2	yes	—
107	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
108	C₄×C₈₉	—	2	2	no	—
109	C₄×C₈₉	—	2	4	no	—
110	C₄×C₈₉	C₈×D₄	1	2	yes	—
111	C₄×C₈₉	C₂²×C₁₆	1	4	yes	complemented
112	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
113	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
114	C₂×D₄×C₁₆	—	89	1	no	maximal, Sylow
115	C₂×C₈₉	—	2	8	no	—
116	C₂×C₈₉	D₄×C₁₆	1	4	yes	complemented
117	C₂×C₈₉	C₂×C₈×D₄	1	1	yes	characteristic
118	C₂×C₈₉	C₂³×C₁₆	1	1	yes	characteristic
119	C₂×C₈₉	C₂×#504	1	1	yes	characteristic
120	C₂³×C₁₆	—	89	2	no	—
121	C₂×C₈×D₄	D₈₉	1	1	yes	characteristic
122	D₄×C₁₆	—	89	8	no	—
123	C₂×C₄⋊C₁₆	—	89	1	no	—
124	C₂×C₂²⋊C₁₆	—	89	2	no	—
125	C₂×C₄×C₁₆	—	89	1	no	—
126	C₈₉	C₂×D₄×C₁₆	1	1	yes	characteristic, complemented, Sylow
127	C₂³×C₈	C₂×D₈₉	1	2	yes	—
128	C₂×C₄×D₄	Dic₈₉	1	1	yes	characteristic
129	C₂²×C₁₆	—	89	1	no	—
130	C₂²×C₁₆	—	89	8	no	—
131	C₂²×C₁₆	—	89	1	no	—
132	C₂²×C₁₆	—	89	2	no	—
133	C₂²×C₁₆	—	178	8	no	—
134	C₈×D₄	Dic₈₉	1	4	yes	—
135	C₈×D₄	C₂×D₈₉	1	4	yes	—
136	C₂×C₄⋊C₈	Dic₈₉	1	1	yes	characteristic
137	C₂×C₂²⋊C₈	Dic₈₉	1	2	yes	—
138	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
139	C₄⋊C₁₆	—	89	4	no	—
140	C₂²⋊C₁₆	—	89	8	no	—
141	C₄×C₁₆	—	89	4	no	—
142	C₂²×D₄	#10623494	1	1	yes	characteristic
143	C₂³×C₄	C₂×Dic₈₉	1	2	yes	—
144	C₂²×C₈	—	2	4	no	—
145	C₂²×C₈	—	2	4	no	—
146	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
147	C₂²×C₈	C₂×Dic₈₉	1	4	yes	—
148	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
149	C₂²×C₈	C₂²×D₈₉	1	4	yes	—
150	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
151	C₄×D₄	#10623494	1	4	yes	—
152	C₄×D₄	C₂×Dic₈₉	1	4	yes	—
153	C₂×C₄⋊C₄	#10623494	1	1	yes	characteristic
154	C₂×C₂²⋊C₄	#10623494	1	2	yes	—
155	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
156	C₂×C₁₆	—	89	4	no	—
157	C₂×C₁₆	—	89	2	no	—
158	C₂×C₁₆	—	89	2	no	—
159	C₂×C₁₆	—	89	4	no	—
160	C₂×C₁₆	—	178	16	no	—
161	C₂×C₁₆	—	178	4	no	—
162	C₂×C₁₆	—	178	2	no	—
163	C₄⋊C₈	C₂×Dic₈₉	1	4	yes	—
164	C₂²⋊C₈	C₂×Dic₈₉	1	8	yes	—
165	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
166	C₄×C₈	C₂²×D₈₉	1	2	yes	—
167	C₂⁴	C₂×#10623494	1	2	yes	—
168	C₂×D₄	#13083549	1	8	yes	direct product factor
169	C₂×D₄	C₂×#10623494	1	4	yes	—
170	C₂²×C₄	—	2	4	no	—
171	C₂²×C₄	—	2	4	no	—
172	C₂²×C₄	C₂×#10623494	1	1	yes	characteristic
173	C₂²×C₄	C₂×#10623494	1	2	yes	—
174	C₂²×C₄	C₂×#10623494	1	4	yes	—
175	C₂²×C₄	C₂×#10623494	1	1	yes	characteristic
176	C₂²×C₄	C₂²×Dic₈₉	1	4	yes	—
177	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	8	no	—
181	C₂×C₈	—	2	4	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
184	C₂×C₈	C₂²×Dic₈₉	1	4	yes	—
185	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
186	C₂×C₈	D₄×D₈₉	1	2	yes	—
187	C₂×C₈	#13083573	1	2	yes	—
188	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
189	C₄⋊C₄	C₂×#10623494	1	4	yes	—
190	C₂²⋊C₄	C₂×#10623494	1	8	yes	—
191	C₄²	C₂×#10623494	1	2	yes	—
192	C₄²	C₂²×Dic₈₉	1	2	yes	—
193	C₁₆	—	89	4	no	—
194	C₁₆	—	178	4	no	—
195	C₁₆	—	178	2	no	—
196	C₂³	—	2	4	no	—
197	C₂³	—	2	4	no	—
198	C₂³	C₂×#13083549	1	8	yes	complemented
199	C₂³	C₂²×#10623494	1	4	yes	—
200	C₂³	C₂²×#10623494	1	1	yes	characteristic
201	D₄	C₂×#13083549	1	16	yes	direct product factor
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	8	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	C₂×#13083549	1	4	yes	complemented
208	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
209	C₂×C₄	C₂²×#10623494	1	2	yes	—
210	C₂×C₄	C₂²×#10623494	1	4	yes	—
211	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
212	C₂×C₄	C₂²×#10623494	1	2	yes	—
213	C₂×C₄	#15668019	1	2	yes	—
214	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
215	C₈	—	2	4	no	—
216	C₈	—	2	2	no	—
217	C₈	D₄×Dic₈₉	1	2	yes	—
218	C₈	C₂×D₄×D₈₉	1	1	yes	characteristic
219	C₈	C₂×#13083573	1	1	yes	characteristic
220	C₂²	—	2	4	no	—
221	C₂²	—	2	16	no	—
222	C₂²	—	2	4	no	—
223	C₂²	D₄×#10623494	1	2	yes	—
224	C₂²	C₂²×#13083549	1	2	yes	—
225	C₂²	C₂²×#13083549	1	8	yes	complemented
226	C₂²	#19152015	1	2	yes	—
227	C₂²	C₂³×#10623494	1	1	yes	characteristic
228	C₄	—	2	4	no	—
229	C₄	—	2	2	no	—
230	C₄	D₄×#10623494	1	2	yes	—
231	C₄	C₂²×#13083549	1	4	yes	complemented
232	C₄	C₂×D₄×Dic₈₉	1	1	yes	characteristic
233	C₄	C₂×#15668019	1	1	yes	characteristic
234	C₂	—	2	8	no	—
235	C₂	D₄×#13083549	1	4	yes	direct product factor
236	C₂	C₂×D₄×#10623494	1	1	yes	characteristic
237	C₂	C₂³×#13083549	1	1	yes	characteristic
238	C₂	C₂×#19152015	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²×D₈₉
commutator	C₂×C₈₉	C₂³×C₁₆
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×C₈×D₄×C₈₉	C₂
socle	C₂³×C₈₉	C₂²×C₈
prime radical	C₂²×D₄×C₈₉	C₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₂×C₈×D₄	D₈₉
hypocenter	C₈₉	C₂×D₄×C₁₆
LCS	G ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. C₂³×#13083549	C₂	semidirect
2. D₄×#13083549	C₂	direct (N × H)
3. C₂×C₄⋊#13083549	C₂	semidirect
4. C₂×C₂²⋊#13083549	C₂	semidirect
5. C₂×C₄×#13083549	C₂	semidirect
11. C₂²×#13083549	C₂²	semidirect
12. C₄⋊#13083549	C₂²	semidirect
13. C₂²⋊#13083549	C₂²	semidirect
14. C₄×#13083549	C₂²	semidirect
25. C₂×#13083549	D₄	direct (N × H)
49. #13083549	C₂×D₄	direct (N × H)
51. C₂×D₄×C₈₉	C₁₆	semidirect
78. C₂³×C₈₉	C₂×C₁₆	semidirect
81. D₄×C₈₉	C₂×C₁₆	semidirect
87. C₂×C₄×C₈₉	C₂×C₁₆	semidirect
105. C₂²×C₈₉	C₂²×C₁₆	semidirect
111. C₄×C₈₉	C₂²×C₁₆	semidirect
116. C₂×C₈₉	D₄×C₁₆	semidirect
126. C₈₉	C₂×D₄×C₁₆	semidirect
168. C₂×D₄	#13083549	direct (N × H)
198. C₂³	C₂×#13083549	semidirect
201. D₄	C₂×#13083549	direct (N × H)
207. C₂×C₄	C₂×#13083549	semidirect
225. C₂²	C₂²×#13083549	semidirect
231. C₄	C₂²×#13083549	semidirect
235. C₂	D₄×#13083549	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#13083549	2
D₄×#13083549	2
C₂×C₄⋊#13083549	2
C₂×C₂²⋊#13083549	2
C₂×C₄×#13083549	2
C₂×C₈×D₄×C₈₉	2
C₂×D₄×C₁₆	89

Subgroups — C2×D4×#13083549 #36707578

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₄×#13083549 #36707578