Subgroups — C₂×C₄×#15668009 #36712942

Order 22784.

5694 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15668009	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×D₃₅₆	C₂	1	6	yes	maximal, complemented
3	C₄×#15668009	C₂	1	16	yes	maximal, direct product factor
4	C₂×#19151500	C₂	1	6	yes	maximal, complemented
5	C₂×#19151468	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄³×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₃₅₆	C₂²	1	6	yes	—
8	C₂³×C₄×D₈₉	C₂²	1	4	yes	—
9	C₂×#15668009	C₄	1	16	yes	direct product factor
10	C₂×#15668009	C₂²	1	8	yes	—
11	C₂×C₄×D₃₅₆	—	2	48	no	—
12	C₂×C₄×D₃₅₆	C₂²	1	48	yes	complemented
13	C₂²×C₄⋊Dic₈₉	C₂²	1	6	yes	—
14	C₂²×#13083551	C₂²	1	12	yes	—
15	#19151500	C₂²	1	48	yes	complemented
16	#19151468	C₂²	1	8	yes	complemented
17	C₂²×C₄²×C₈₉	C₂²	1	3	yes	—
18	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
19	C₂²×C₄²×C₈₉	C₂²	1	3	yes	—
20	C₄³×C₈₉	C₂²	1	8	yes	complemented
21	C₂⁴×D₈₉	C₂³	1	4	yes	—
22	C₂²×D₃₅₆	—	2	24	no	—
23	C₂²×D₃₅₆	—	2	24	no	—
24	C₂²×D₃₅₆	C₂×C₄	1	48	yes	complemented
25	C₂²×D₃₅₆	C₂³	1	24	yes	—
26	C₂³×Dic₈₉	C₂³	1	4	yes	—
27	C₂²×C₄×D₈₉	—	2	48	no	—
28	C₂²×C₄×D₈₉	C₂³	1	16	yes	—
29	#15668009	C₂×C₄	1	64	yes	direct product factor
30	C₄×D₃₅₆	—	2	192	no	—
31	C₂×C₄⋊Dic₈₉	—	2	24	no	—
32	C₂×C₄⋊Dic₈₉	C₂³	1	24	yes	—
33	C₂×#13083551	—	2	48	no	—
34	C₂×#13083551	C₂³	1	48	yes	—
35	C₂³×C₄×C₈₉	C₂³	1	3	yes	—
36	C₂³×C₄×C₈₉	C₂³	1	3	yes	—
37	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₈₉	—	2	12	no	—
39	C₂×C₄²×C₈₉	C₂×C₄	1	8	yes	complemented
40	C₂×C₄²×C₈₉	D₄	1	24	yes	complemented
41	C₂×C₄²×C₈₉	C₂³	1	12	yes	—
42	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
43	C₂×C₄²×C₈₉	C₂³	1	12	yes	—
44	C₂³×D₈₉	—	2	24	no	—
45	C₂³×D₈₉	—	2	24	no	—
46	C₂³×D₈₉	C₂²×C₄	1	16	yes	—
47	C₂³×D₈₉	C₂⁴	1	8	yes	—
48	C₂×D₃₅₆	—	2	192	no	—
49	C₂×D₃₅₆	—	2	48	no	—
50	C₂×D₃₅₆	—	2	48	no	—
51	C₂×D₃₅₆	C₂²×C₄	1	96	yes	complemented
52	C₂²×Dic₈₉	—	2	24	no	—
53	C₂²×Dic₈₉	C₂⁴	1	8	yes	—
54	C₂×C₄×D₈₉	—	2	96	no	—
55	C₂×C₄×D₈₉	—	4	64	no	—
56	C₄⋊Dic₈₉	—	2	48	no	—
57	#13083551	—	2	96	no	—
58	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
59	C₂²×C₄×C₈₉	—	2	12	no	—
60	C₂²×C₄×C₈₉	C₂²×C₄	1	12	yes	—
61	C₂²×C₄×C₈₉	C₂×D₄	1	12	yes	—
62	C₂²×C₄×C₈₉	C₂×D₄	1	12	yes	—
63	C₂²×C₄×C₈₉	C₂×D₄	1	12	yes	—
64	C₂²×C₄×C₈₉	#41	1	12	yes	—
65	C₂²×C₄×C₈₉	C₂⁴	1	6	yes	—
66	C₂²×C₄×C₈₉	C₂⁴	1	6	yes	—
67	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
68	C₄²×C₈₉	—	2	24	no	—
69	C₄²×C₈₉	C₂²×C₄	1	16	yes	complemented
70	C₄²×C₈₉	C₂×D₄	1	48	yes	complemented
71	C₂³×C₈₉	C₂³×C₄	1	2	yes	—
72	C₂³×C₈₉	C₂²×D₄	1	6	yes	—
73	C₂³×C₈₉	C₂×#41	1	6	yes	—
74	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
75	C₂²×D₈₉	—	2	96	no	—
76	C₂²×D₈₉	—	2	24	no	—
77	C₂²×D₈₉	—	2	24	no	—
78	C₂²×D₈₉	—	4	16	no	—
79	C₂²×D₈₉	—	4	48	no	—
80	C₂²×D₈₉	C₂³×C₄	1	16	yes	—
81	C₂×C₄×C₈₉	—	2	24	no	—
82	C₂×C₄×C₈₉	—	2	12	no	—
83	C₂×C₄×C₈₉	C₄×D₄	1	48	yes	complemented
84	C₂×C₄×C₈₉	#127	1	16	yes	complemented
85	C₂×C₄×C₈₉	C₂³×C₄	1	12	yes	—
86	C₂×C₄×C₈₉	C₂²×D₄	1	12	yes	—
87	C₂×C₄×C₈₉	C₂²×D₄	1	12	yes	—
88	C₂×C₄×C₈₉	C₂²×D₄	1	12	yes	—
89	C₂×C₄×C₈₉	C₂×#41	1	12	yes	—
90	C₂×Dic₈₉	—	2	24	no	—
91	C₂×Dic₈₉	—	4	16	no	—
92	D₃₅₆	—	2	192	no	—
93	C₄×D₈₉	—	4	64	no	—
94	C₂²×C₈₉	C₂×C₄×D₄	1	12	yes	—
95	C₂²×C₈₉	C₂×#127	1	4	yes	—
96	C₂²×C₈₉	#532	1	12	yes	—
97	C₂²×C₈₉	C₂⁴×C₄	1	1	yes	characteristic
98	C₂²×C₈₉	C₂³×D₄	1	3	yes	—
99	C₂²×C₈₉	C₂²×#41	1	3	yes	—
100	C₂×D₈₉	—	2	48	no	—
101	C₂×D₈₉	—	4	64	no	—
102	C₂×D₈₉	—	4	8	no	—
103	C₂×D₈₉	—	4	24	no	—
104	C₄×C₈₉	—	2	12	no	—
105	C₄×C₈₉	C₂×C₄×D₄	1	24	yes	complemented
106	C₄×C₈₉	C₂×#127	1	8	yes	complemented
107	Dic₈₉	—	4	8	no	—
108	C₂×C₄×#127	—	89	1	no	maximal, Sylow
109	C₂×C₈₉	C₄×#127	1	8	yes	complemented
110	C₂×C₈₉	C₂²×C₄×D₄	1	3	yes	—
111	C₂×C₈₉	C₂²×#127	1	1	yes	characteristic
112	C₂×C₈₉	C₂×#532	1	3	yes	—
113	D₈₉	—	4	16	no	—
114	C₂²×#127	—	89	1	no	—
115	C₂²×C₄×D₄	—	89	6	no	—
116	C₄×#127	—	89	16	no	—
117	C₂×#390	—	89	6	no	—
118	C₂×#384	—	89	1	no	—
119	C₂×C₄³	D₈₉	1	1	yes	characteristic, complemented
120	C₈₉	C₂×C₄×#127	1	1	yes	characteristic, complemented, Sylow
121	C₂³×D₄	—	89	6	no	—
122	C₂⁴×C₄	—	89	4	no	—
123	C₂×#127	—	89	16	no	—
124	C₂×#127	—	89	8	no	—
125	C₂×C₄×D₄	—	89	48	no	—
126	C₂×C₄×D₄	—	178	48	no	—
127	C₂²×C₄⋊C₄	—	89	6	no	—
128	C₂²×C₂²⋊C₄	—	89	12	no	—
129	C₂²×C₄²	C₂×D₈₉	1	3	yes	—
130	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
131	C₂²×C₄²	C₂×D₈₉	1	3	yes	—
132	#390	—	89	48	no	—
133	#384	—	89	8	no	—
134	C₄³	C₂×D₈₉	1	8	yes	complemented
135	C₂⁵	—	89	4	no	—
136	C₂²×D₄	—	89	48	no	—
137	C₂²×D₄	—	89	24	no	—
138	C₂²×D₄	—	178	24	no	—
139	C₂²×D₄	—	178	24	no	—
140	C₂³×C₄	—	89	4	no	—
141	C₂³×C₄	—	89	16	no	—
142	C₂³×C₄	—	178	48	no	—
143	C₂³×C₄	C₂²×D₈₉	1	3	yes	—
144	C₂³×C₄	C₂²×D₈₉	1	3	yes	—
145	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
146	#127	—	89	64	no	—
147	C₄×D₄	—	178	192	no	—
148	C₂×C₄⋊C₄	—	89	24	no	—
149	C₂×C₄⋊C₄	—	178	24	no	—
150	C₂×C₂²⋊C₄	—	89	48	no	—
151	C₂×C₂²⋊C₄	—	178	48	no	—
152	C₂×C₄²	—	2	12	no	—
153	C₂×C₄²	C₄×D₈₉	1	8	yes	complemented
154	C₂×C₄²	D₃₅₆	1	24	yes	complemented
155	C₂×C₄²	C₂²×D₈₉	1	12	yes	—
156	C₂×C₄²	C₂²×D₈₉	1	4	yes	—
157	C₂×C₄²	C₂²×D₈₉	1	12	yes	—
158	C₂⁴	—	89	16	no	—
159	C₂⁴	—	89	8	no	—
160	C₂⁴	—	178	24	no	—
161	C₂⁴	—	178	24	no	—
162	C₂⁴	C₂³×D₈₉	1	1	yes	characteristic
163	C₂×D₄	—	89	96	no	—
164	C₂×D₄	—	178	192	no	—
165	C₂×D₄	—	178	48	no	—
166	C₂×D₄	—	178	48	no	—
167	C₂²×C₄	—	2	12	no	—
168	C₂²×C₄	—	89	8	no	—
169	C₂²×C₄	—	356	64	no	—
170	C₂²×C₄	—	178	24	no	—
171	C₂²×C₄	—	178	96	no	—
172	C₂²×C₄	C₂×C₄×D₈₉	1	12	yes	—
173	C₂²×C₄	C₂×D₃₅₆	1	12	yes	—
174	C₂²×C₄	C₂×D₃₅₆	1	12	yes	—
175	C₂²×C₄	C₂×D₃₅₆	1	12	yes	—
176	C₂²×C₄	#13083574	1	12	yes	—
177	C₂²×C₄	C₂³×D₈₉	1	6	yes	—
178	C₂²×C₄	C₂³×D₈₉	1	6	yes	—
179	C₂²×C₄	C₂³×D₈₉	1	2	yes	—
180	C₄⋊C₄	—	178	48	no	—
181	C₂²⋊C₄	—	178	96	no	—
182	C₄²	—	2	24	no	—
183	C₄²	C₂×C₄×D₈₉	1	16	yes	complemented
184	C₄²	C₂×D₃₅₆	1	48	yes	complemented
185	C₂³	—	89	16	no	—
186	C₂³	—	356	16	no	—
187	C₂³	—	356	48	no	—
188	C₂³	—	178	24	no	—
189	C₂³	—	178	24	no	—
190	C₂³	—	178	96	no	—
191	C₂³	C₂²×C₄×D₈₉	1	2	yes	—
192	C₂³	C₂²×D₃₅₆	1	6	yes	—
193	C₂³	C₂×#13083574	1	6	yes	—
194	C₂³	C₂⁴×D₈₉	1	1	yes	characteristic
195	D₄	—	178	192	no	—
196	C₂×C₄	—	2	24	no	—
197	C₂×C₄	—	2	12	no	—
198	C₂×C₄	—	356	16	no	—
199	C₂×C₄	—	356	64	no	—
200	C₂×C₄	—	178	24	no	—
201	C₂×C₄	C₄×D₃₅₆	1	48	yes	complemented
202	C₂×C₄	#15668009	1	16	yes	direct product factor
203	C₂×C₄	C₂²×C₄×D₈₉	1	12	yes	—
204	C₂×C₄	C₂²×D₃₅₆	1	12	yes	—
205	C₂×C₄	C₂²×D₃₅₆	1	12	yes	—
206	C₂×C₄	C₂²×D₃₅₆	1	12	yes	—
207	C₂×C₄	C₂×#13083574	1	12	yes	—
208	C₂²	—	356	8	no	—
209	C₂²	—	356	24	no	—
210	C₂²	—	356	64	no	—
211	C₂²	—	178	48	no	—
212	C₂²	C₂×C₄×D₃₅₆	1	12	yes	—
213	C₂²	C₂×#15668009	1	4	yes	—
214	C₂²	#19152148	1	12	yes	—
215	C₂²	C₂³×C₄×D₈₉	1	1	yes	characteristic
216	C₂²	C₂³×D₃₅₆	1	3	yes	—
217	C₂²	C₂²×#13083574	1	3	yes	—
218	C₄	—	2	12	no	—
219	C₄	—	356	8	no	—
220	C₄	C₂×C₄×D₃₅₆	1	24	yes	complemented
221	C₄	C₂×#15668009	1	8	yes	direct product factor
222	C₂	—	356	16	no	—
223	C₂	C₄×#15668009	1	8	yes	direct product factor
224	C₂	C₂²×C₄×D₃₅₆	1	3	yes	—
225	C₂	C₂²×#15668009	1	1	yes	characteristic
226	C₂	C₂×#19152148	1	3	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₈₉
commutator	C₂²×C₈₉	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₈₉
Fitting	C₂×C₄³×C₈₉	C₂
socle	C₂⁴×C₈₉	C₂⁴
prime radical	C₂²×#15668009	C₂
cosocle	C₂³×C₈₉	C₂⁵
hypercenter	C₂×C₄³	D₈₉
hypocenter	C₈₉	C₂×C₄×#127
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×D₃₅₆	C₂	semidirect
3. C₄×#15668009	C₂	direct (N × H)
4. C₂×#19151500	C₂	semidirect
5. C₂×#19151468	C₂	semidirect
6. C₂×C₄³×C₈₉	C₂	semidirect
9. C₂×#15668009	C₄	direct (N × H)
12. C₂×C₄×D₃₅₆	C₂²	semidirect
15. #19151500	C₂²	semidirect
16. #19151468	C₂²	semidirect
20. C₄³×C₈₉	C₂²	semidirect
24. C₂²×D₃₅₆	C₂×C₄	semidirect
29. #15668009	C₂×C₄	direct (N × H)
39. C₂×C₄²×C₈₉	C₂×C₄	semidirect
40. C₂×C₄²×C₈₉	D₄	semidirect
51. C₂×D₃₅₆	C₂²×C₄	semidirect
69. C₄²×C₈₉	C₂²×C₄	semidirect
70. C₄²×C₈₉	C₂×D₄	semidirect
83. C₂×C₄×C₈₉	C₄×D₄	semidirect
84. C₂×C₄×C₈₉	#127	semidirect
105. C₄×C₈₉	C₂×C₄×D₄	semidirect
106. C₄×C₈₉	C₂×#127	semidirect
109. C₂×C₈₉	C₄×#127	semidirect
119. C₂×C₄³	D₈₉	semidirect
120. C₈₉	C₂×C₄×#127	semidirect
134. C₄³	C₂×D₈₉	semidirect
153. C₂×C₄²	C₄×D₈₉	semidirect
154. C₂×C₄²	D₃₅₆	semidirect
183. C₄²	C₂×C₄×D₈₉	semidirect
184. C₄²	C₂×D₃₅₆	semidirect
201. C₂×C₄	C₄×D₃₅₆	semidirect
202. C₂×C₄	#15668009	direct (N × H)
220. C₄	C₂×C₄×D₃₅₆	semidirect
221. C₄	C₂×#15668009	direct (N × H)
223. C₂	C₄×#15668009	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×#15668009	2
C₂²×C₄×D₃₅₆	2
C₄×#15668009	2
C₂×#19151500	2
C₂×#19151468	2
C₂×C₄³×C₈₉	2
C₂×C₄×#127	89

Subgroups — C2×C4×#15668009 #36712942

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#15668009 #36712942