Subgroups — C₄×#63488 #11800760

Order 1152.

1128 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄²×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#63488	C₂	1	2	yes	maximal
3	C₄×#3676	C₂	1	4	yes	maximal
4	C₄×C₄⋊Dic₉	C₂	1	4	yes	maximal, complemented
5	#10588612	C₂	1	2	yes	maximal, complemented
6	C₄²×Dic₉	C₂	1	2	yes	maximal, complemented
7	C₄×#926	—	3	1	no	maximal
8	C₂³×C₄×C₉	C₂²	1	2	yes	—
9	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
10	C₂×C₄²×C₉	C₄	1	8	yes	complemented
11	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₉	C₂²	1	2	yes	—
14	C₂×#3676	C₂²	1	4	yes	—
15	#63488	C₄	1	16	yes	direct product factor
16	C₂×C₄⋊Dic₉	C₂²	1	4	yes	—
17	C₂×C₄×Dic₉	C₂²	1	8	yes	—
18	C₂×C₄×Dic₉	C₂²	1	4	yes	—
19	C₉⋊#95	C₂²	1	8	yes	—
20	C₂²×C₃×C₄²	D₃	1	1	yes	characteristic
21	C₂×#926	—	3	2	no	—
22	C₄×#217	—	3	4	no	—
23	C₄×C₄⋊Dic₃	—	3	4	no	—
24	#4932	—	3	2	no	—
25	C₄²×Dic₃	—	3	2	no	—
26	C₂⁴×C₉	C₂³	1	1	yes	characteristic
27	C₂²×C₄×C₉	—	2	8	no	—
28	C₂²×C₄×C₉	C₂×C₄	1	8	yes	—
29	C₂²×C₄×C₉	C₂×C₄	1	8	yes	—
30	C₂²×C₄×C₉	C₂×C₄	1	4	yes	—
31	C₂²×C₄×C₉	C₂³	1	2	yes	—
32	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₉	C₂³	1	2	yes	—
35	C₂²×Dic₉	C₂³	1	8	yes	—
36	C₄²×C₉	C₂×C₄	1	4	yes	—
37	C₄²×C₉	C₂×C₄	1	8	yes	—
38	C₄²×C₉	C₂×C₄	1	4	yes	—
39	#3676	C₂×C₄	1	16	yes	—
40	C₄⋊Dic₉	C₂×C₄	1	16	yes	complemented
41	C₄×Dic₉	—	2	16	no	—
42	C₄×Dic₉	C₂×C₄	1	16	yes	complemented
43	C₄×#117	—	9	1	no	Sylow
44	C₂³×C₃×C₄	C₂×D₃	1	2	yes	—
45	C₂³×C₃×C₄	C₂×D₃	1	1	yes	characteristic
46	C₂×C₃×C₄²	Dic₃	1	8	yes	—
47	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
48	C₂×C₃×C₄²	C₂×D₃	1	2	yes	—
49	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
50	C₂×#217	—	3	4	no	—
51	#926	—	3	16	no	—
52	C₂×C₄⋊Dic₃	—	3	4	no	—
53	C₂×C₄×Dic₃	—	3	4	no	—
54	C₂×C₄×Dic₃	—	3	8	no	—
55	C₃⋊#95	—	3	8	no	—
56	C₂³×C₉	—	2	4	no	—
57	C₂³×C₉	C₂²×C₄	1	4	yes	—
58	C₂³×C₉	C₂²×C₄	1	2	yes	—
59	C₂³×C₉	C₂⁴	1	1	yes	characteristic
60	C₂×C₄×C₉	—	2	4	no	—
61	C₂×C₄×C₉	—	2	16	no	—
62	C₂×C₄×C₉	C₄²	1	16	yes	complemented
63	C₂×C₄×C₉	C₂²×C₄	1	4	yes	—
64	C₂×C₄×C₉	C₂²×C₄	1	4	yes	—
65	C₂×C₄×C₉	C₂²×C₄	1	4	yes	—
66	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
67	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
68	C₂×C₄×C₉	C₂²×C₄	1	4	yes	—
69	C₂×C₄×C₉	#41	1	8	yes	—
70	C₂×Dic₉	—	2	16	no	—
71	C₂×Dic₉	C₂²×C₄	1	16	yes	—
72	C₂×#117	—	9	2	no	—
73	C₂²×C₄²	D₉	1	1	yes	characteristic
74	C₄×C₄⋊C₄	—	9	4	no	—
75	C₄×C₂²⋊C₄	—	9	4	no	—
76	#376	—	9	2	no	—
77	C₄³	—	9	2	no	—
78	C₂⁴×C₃	C₂²×D₃	1	1	yes	characteristic
79	C₂²×C₃×C₄	—	2	8	no	—
80	C₂²×C₃×C₄	C₄×D₃	1	8	yes	—
81	C₂²×C₃×C₄	C₂×Dic₃	1	8	yes	—
82	C₂²×C₃×C₄	C₂×Dic₃	1	4	yes	—
83	C₂²×C₃×C₄	C₂²×D₃	1	2	yes	—
84	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
85	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
86	C₂²×C₃×C₄	C₂²×D₃	1	2	yes	—
87	C₂²×Dic₃	—	3	8	no	—
88	C₃×C₄²	C₂×Dic₃	1	4	yes	—
89	C₃×C₄²	C₂×Dic₃	1	8	yes	—
90	C₃×C₄²	C₂×Dic₃	1	4	yes	—
91	#217	—	3	16	no	—
92	C₄⋊Dic₃	—	3	16	no	—
93	C₄×Dic₃	—	3	16	no	—
94	C₄×Dic₃	—	6	16	no	—
95	C₂²×C₉	—	2	4	no	—
96	C₂²×C₉	—	2	8	no	—
97	C₂²×C₉	C₂×C₄²	1	4	yes	—
98	C₂²×C₉	C₂³×C₄	1	2	yes	—
99	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
100	C₂²×C₉	C₂×#41	1	4	yes	—
101	C₄×C₉	—	2	4	no	—
102	C₄×C₉	C₂×C₄²	1	4	yes	—
103	C₄×C₉	C₂×C₄²	1	4	yes	—
104	C₄×C₉	#117	1	8	yes	complemented
105	Dic₉	—	2	16	no	—
106	C₂³×C₄	C₂×D₉	1	2	yes	—
107	C₂³×C₄	C₂×D₉	1	1	yes	characteristic
108	#117	—	9	16	no	—
109	C₂×C₄⋊C₄	—	9	4	no	—
110	C₂×C₂²⋊C₄	—	9	4	no	—
111	C₂×C₄²	—	9	4	no	—
112	C₂×C₄²	—	9	8	no	—
113	C₂×C₄²	Dic₉	1	8	yes	complemented
114	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
115	C₂×C₄²	C₂×D₉	1	2	yes	—
116	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
117	#95	—	9	8	no	—
118	C₂³×C₃	—	2	4	no	—
119	C₂³×C₃	C₂×C₄×D₃	1	4	yes	—
120	C₂³×C₃	C₂²×Dic₃	1	2	yes	—
121	C₂³×C₃	C₂³×D₃	1	1	yes	characteristic
122	C₂×C₃×C₄	—	2	4	no	—
123	C₂×C₃×C₄	—	2	16	no	—
124	C₂×C₃×C₄	C₄×Dic₃	1	16	yes	—
125	C₂×C₃×C₄	C₂×C₄×D₃	1	4	yes	—
126	C₂×C₃×C₄	C₂×C₄×D₃	1	4	yes	—
127	C₂×C₃×C₄	#235	1	8	yes	—
128	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
129	C₂×C₃×C₄	C₂²×Dic₃	1	2	yes	—
130	C₂×C₃×C₄	C₂²×Dic₃	1	2	yes	—
131	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
132	C₂×Dic₃	—	3	16	no	—
133	C₂×Dic₃	—	6	16	no	—
134	C₂×C₉	—	2	4	no	—
135	C₂×C₉	C₂²×C₄²	1	1	yes	characteristic
136	C₂×C₉	C₂×#117	1	2	yes	—
137	C₂×C₉	C₄×#41	1	4	yes	—
138	C₂⁴	C₂²×D₉	1	1	yes	characteristic
139	C₂²×C₄	—	2	8	no	—
140	C₂²×C₄	—	9	8	no	—
141	C₂²×C₄	C₄×D₉	1	8	yes	—
142	C₂²×C₄	C₂×Dic₉	1	8	yes	—
143	C₂²×C₄	C₂×Dic₉	1	4	yes	—
144	C₂²×C₄	C₂²×D₉	1	2	yes	—
145	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
146	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
147	C₂²×C₄	C₂²×D₉	1	2	yes	—
148	C₄⋊C₄	—	9	16	no	—
149	C₂²⋊C₄	—	9	16	no	—
150	C₄²	—	9	16	no	—
151	C₄²	—	18	16	no	—
152	C₄²	C₂×Dic₉	1	4	yes	—
153	C₄²	C₂×Dic₉	1	8	yes	—
154	C₄²	C₂×Dic₉	1	4	yes	—
155	C₂²×C₃	—	2	8	no	—
156	C₂²×C₃	—	2	4	no	—
157	C₂²×C₃	C₂×C₄×Dic₃	1	4	yes	—
158	C₂²×C₃	C₂²×C₄×D₃	1	2	yes	—
159	C₂²×C₃	C₂×#235	1	4	yes	—
160	C₂²×C₃	C₂³×Dic₃	1	1	yes	characteristic
161	C₃×C₄	—	2	4	no	—
162	C₃×C₄	C₂×C₄×Dic₃	1	4	yes	—
163	C₃×C₄	C₂×C₄×Dic₃	1	4	yes	—
164	C₃×C₄	#926	1	8	yes	—
165	Dic₃	—	6	16	no	—
166	C₉	C₄×#117	1	1	yes	characteristic, complemented, Sylow
167	C₂³	—	2	4	no	—
168	C₂³	C₂×C₄×D₉	1	4	yes	—
169	C₂³	C₂²×Dic₉	1	2	yes	—
170	C₂³	C₂³×D₉	1	1	yes	characteristic
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	16	no	—
173	C₂×C₄	—	9	16	no	—
174	C₂×C₄	—	18	16	no	—
175	C₂×C₄	C₄×Dic₉	1	16	yes	complemented
176	C₂×C₄	C₂×C₄×D₉	1	4	yes	—
177	C₂×C₄	C₂×C₄×D₉	1	4	yes	—
178	C₂×C₄	#3697	1	8	yes	—
179	C₂×C₄	C₂²×Dic₉	1	4	yes	—
180	C₂×C₄	C₂²×Dic₉	1	4	yes	—
181	C₂×C₄	C₂²×Dic₉	1	2	yes	—
182	C₂×C₄	C₂²×Dic₉	1	2	yes	—
183	C₂×C₃	—	2	4	no	—
184	C₂×C₃	C₄×#235	1	4	yes	—
185	C₂×C₃	C₂²×C₄×Dic₃	1	1	yes	characteristic
186	C₂×C₃	C₂×#926	1	2	yes	—
187	C₂²	—	2	8	no	—
188	C₂²	—	2	4	no	—
189	C₂²	C₂×C₄×Dic₉	1	4	yes	—
190	C₂²	C₂²×C₄×D₉	1	2	yes	—
191	C₂²	C₂×#3697	1	4	yes	—
192	C₂²	C₂³×Dic₉	1	1	yes	characteristic
193	C₄	—	2	4	no	—
194	C₄	—	18	16	no	—
195	C₄	C₂×C₄×Dic₉	1	4	yes	—
196	C₄	C₂×C₄×Dic₉	1	4	yes	—
197	C₄	#63488	1	8	yes	direct product factor
198	C₃	C₄×#926	1	1	yes	characteristic
199	C₂	—	2	4	no	—
200	C₂	C₄×#3697	1	4	yes	—
201	C₂	C₂²×C₄×Dic₉	1	1	yes	characteristic
202	C₂	C₂×#63488	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂×D₉
commutator	C₂×C₉	C₂²×C₄²
Frattini, socle	C₂³×C₃	C₂³×D₃
Fitting	C₂²×C₄²×C₉	C₂
prime radical	C₂⁴×C₃	C₂²×D₃
cosocle	C₂³×C₉	C₂⁴
hypercenter	C₂²×C₄²	D₉
hypocenter	C₉	C₄×#117
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
4. C₄×C₄⋊Dic₉	C₂	semidirect
5. #10588612	C₂	semidirect
6. C₄²×Dic₉	C₂	semidirect
10. C₂×C₄²×C₉	C₄	semidirect
15. #63488	C₄	direct (N × H)
40. C₄⋊Dic₉	C₂×C₄	semidirect
42. C₄×Dic₉	C₂×C₄	semidirect
62. C₂×C₄×C₉	C₄²	semidirect
104. C₄×C₉	#117	semidirect
113. C₂×C₄²	Dic₉	semidirect
166. C₉	C₄×#117	semidirect
175. C₂×C₄	C₄×Dic₉	semidirect
197. C₄	#63488	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×C₄²×C₉	2
C₂×#63488	2
C₄×#3676	2
C₄×C₄⋊Dic₉	2
#10588612	2
C₄²×Dic₉	2
C₄×#926	3

Subgroups — C4×#63488 #11800760

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#63488 #11800760