Subgroups — C₂×#10588743 #11875023

Order 1152.

576 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	C₂×C₉×#125	C₂	1	1	yes	maximal, characteristic
2	C₂×#63431	C₂	1	1	yes	maximal, characteristic
3	#10588743	C₂	1	8	yes	maximal, direct product factor
4	C₂×#63367	C₂	1	2	yes	maximal
5	C₂×#63366	C₂	1	2	yes	maximal
6	C₂×#63362	C₂	1	1	yes	maximal, characteristic
7	C₂×#5064	—	3	1	no	maximal
8	C₂²×Dic₁₈	—	2	1	no	—
9	C₉×#125	—	2	2	no	—
10	C₉×#125	C₂²	1	4	yes	—
11	C₂×C₉×C₄⋊C₄	—	2	1	no	—
12	C₂×C₉×C₄⋊C₄	—	2	1	no	—
13	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₉	—	2	1	no	—
17	C₂×C₄⋊Dic₉	C₂²	1	2	yes	—
18	C₂²×C₉⋊C₈	C₂²	1	2	yes	—
19	#63431	—	2	2	no	—
20	#63431	C₂²	1	4	yes	—
21	#63367	C₂²	1	8	yes	—
22	#63366	C₂²	1	8	yes	—
23	#63362	C₂²	1	4	yes	—
24	C₂×C₃×#125	D₃	1	1	yes	characteristic
25	C₂×#869	—	3	1	no	—
26	#5064	—	3	8	no	—
27	C₂×#808	—	3	2	no	—
28	C₂×#807	—	3	2	no	—
29	C₂×#803	—	3	1	no	—
30	C₂²×C₄×C₉	—	2	1	no	—
31	C₂²×C₄×C₉	—	2	1	no	—
32	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
35	C₂²×Dic₉	—	2	2	no	—
36	C₂×Dic₁₈	—	2	4	no	—
37	C₂×Dic₁₈	—	4	4	no	—
38	C₉×C₄⋊C₄	—	2	4	no	—
39	C₉×C₄⋊C₄	—	2	4	no	—
40	C₉×C₄⋊C₄	—	2	1	no	—
41	C₉×C₄⋊C₄	—	2	1	no	—
42	C₉×C₄⋊C₄	C₂³	1	2	yes	—
43	C₉×C₄⋊C₄	C₂³	1	2	yes	—
44	C₄²×C₉	D₄	1	2	yes	—
45	C₄²×C₉	C₂³	1	2	yes	—
46	C₄⋊Dic₉	—	2	2	no	—
47	C₄⋊Dic₉	—	2	4	no	—
48	C₄⋊Dic₉	C₂³	1	4	yes	—
49	C₂×C₉⋊C₈	Q₈	1	8	yes	complemented
50	C₂×C₉⋊C₈	C₂³	1	4	yes	—
51	C₂×#501	—	9	1	no	Sylow
52	C₂²×Dic₆	—	6	1	no	—
53	C₃×#125	—	2	2	no	—
54	C₃×#125	C₂×D₃	1	4	yes	—
55	C₂×C₃×C₄⋊C₄	—	2	1	no	—
56	C₂×C₃×C₄⋊C₄	—	2	1	no	—
57	C₂×C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
58	C₂×C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
59	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
60	C₂×C₄⋊Dic₃	—	3	2	no	—
61	C₂×C₄⋊Dic₃	—	6	1	no	—
62	C₂²×C₃⋊C₈	—	3	2	no	—
63	#869	—	3	4	no	—
64	#869	—	6	2	no	—
65	#808	—	3	8	no	—
66	#807	—	3	8	no	—
67	#803	—	3	4	no	—
68	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₉	—	2	2	no	—
70	C₂×C₄×C₉	—	2	2	no	—
71	C₂×C₄×C₉	—	2	2	no	—
72	C₂×C₄×C₉	—	2	2	no	—
73	C₂×C₄×C₉	—	4	2	no	—
74	C₂×C₄×C₉	—	4	2	no	—
75	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
76	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
77	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×C₉	C₂×Q₈	1	4	yes	—
79	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
80	C₂×Dic₉	—	2	4	no	—
81	C₂×Dic₉	—	4	4	no	—
82	Dic₁₈	—	4	8	no	—
83	C₉⋊C₈	C₂×Q₈	1	8	yes	complemented
84	C₂×C₄⋊Q₈	—	9	1	no	—
85	C₂×#125	D₉	1	1	yes	characteristic
86	#501	—	9	8	no	—
87	C₂×#107	—	9	2	no	—
88	C₂×#106	—	9	2	no	—
89	C₂×#97	—	9	1	no	—
90	C₂²×C₃×C₄	—	2	1	no	—
91	C₂²×C₃×C₄	—	2	1	no	—
92	C₂²×C₃×C₄	#67	1	1	yes	characteristic
93	C₂²×C₃×C₄	#67	1	1	yes	characteristic
94	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
95	C₂²×Dic₃	—	6	2	no	—
96	C₂×Dic₆	—	6	4	no	—
97	C₂×Dic₆	—	12	4	no	—
98	C₃×C₄⋊C₄	—	2	4	no	—
99	C₃×C₄⋊C₄	—	2	4	no	—
100	C₃×C₄⋊C₄	—	2	1	no	—
101	C₃×C₄⋊C₄	—	2	1	no	—
102	C₃×C₄⋊C₄	C₂²×D₃	1	2	yes	—
103	C₃×C₄⋊C₄	C₂²×D₃	1	2	yes	—
104	C₃×C₄²	#67	1	2	yes	—
105	C₃×C₄²	C₂²×D₃	1	2	yes	—
106	C₄⋊Dic₃	—	3	4	no	—
107	C₄⋊Dic₃	—	6	4	no	—
108	C₄⋊Dic₃	—	6	2	no	—
109	C₂×C₃⋊C₈	—	3	8	no	—
110	C₂×C₃⋊C₈	—	3	4	no	—
111	C₂²×C₉	C₄⋊Q₈	1	2	yes	—
112	C₂²×C₉	AGL(1,ℤ/8)	1	2	yes	—
113	C₂²×C₉	#137	1	2	yes	—
114	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
115	C₄×C₉	—	2	2	no	—
116	C₄×C₉	—	2	2	no	—
117	C₄×C₉	—	4	2	no	—
118	C₄×C₉	—	4	2	no	—
119	C₄×C₉	C₄⋊Q₈	1	2	yes	—
120	C₄×C₉	C₂²×Q₈	1	2	yes	—
121	Dic₉	—	4	4	no	—
122	C₂²×Q₈	—	18	1	no	—
123	C₂²×C₈	—	9	2	no	—
124	C₄⋊Q₈	—	9	4	no	—
125	C₄⋊Q₈	—	18	2	no	—
126	#125	—	2	2	no	—
127	#125	C₂×D₉	1	4	yes	—
128	C₂×C₄⋊C₄	—	2	1	no	—
129	C₂×C₄⋊C₄	—	2	1	no	—
130	C₂×C₄⋊C₄	—	9	2	no	—
131	C₂×C₄⋊C₄	—	18	1	no	—
132	C₂×C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
133	C₂×C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
134	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
135	#107	—	9	8	no	—
136	#106	—	9	8	no	—
137	#97	—	9	4	no	—
138	C₂³×C₃	C₂×#67	1	1	yes	characteristic
139	C₂×C₃×C₄	—	2	2	no	—
140	C₂×C₃×C₄	—	2	2	no	—
141	C₂×C₃×C₄	—	2	2	no	—
142	C₂×C₃×C₄	—	2	2	no	—
143	C₂×C₃×C₄	—	4	2	no	—
144	C₂×C₃×C₄	—	4	2	no	—
145	C₂×C₃×C₄	D₃×Q₈	1	4	yes	—
146	C₂×C₃×C₄	C₂×#67	1	2	yes	—
147	C₂×C₃×C₄	C₂×#67	1	2	yes	—
148	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
149	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
150	C₂×Dic₃	—	6	4	no	—
151	C₂×Dic₃	—	12	4	no	—
152	Dic₆	—	12	8	no	—
153	C₃⋊C₈	—	3	8	no	—
154	C₂×C₉	#501	1	4	yes	complemented
155	C₂×C₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
156	C₂×C₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
157	C₂×C₉	C₂×#137	1	1	yes	characteristic
158	C₂×Q₈	—	18	4	no	—
159	C₂×Q₈	—	36	4	no	—
160	C₂²×C₄	—	2	1	no	—
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	—	18	2	no	—
163	C₂²×C₄	#611	1	1	yes	characteristic
164	C₂²×C₄	#611	1	1	yes	characteristic
165	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
166	C₂×C₈	—	9	8	no	—
167	C₂×C₈	—	9	4	no	—
168	C₄⋊C₄	—	2	4	no	—
169	C₄⋊C₄	—	2	4	no	—
170	C₄⋊C₄	—	2	1	no	—
171	C₄⋊C₄	—	2	1	no	—
172	C₄⋊C₄	—	9	4	no	—
173	C₄⋊C₄	—	18	2	no	—
174	C₄⋊C₄	—	18	4	no	—
175	C₄⋊C₄	C₂²×D₉	1	2	yes	—
176	C₄⋊C₄	C₂²×D₉	1	2	yes	—
177	C₄²	#611	1	2	yes	—
178	C₄²	C₂²×D₉	1	2	yes	—
179	C₂²×C₃	#944	1	2	yes	—
180	C₂²×C₃	#949	1	2	yes	—
181	C₂²×C₃	#951	1	2	yes	—
182	C₂²×C₃	C₂²×#67	1	1	yes	characteristic
183	C₃×C₄	—	2	2	no	—
184	C₃×C₄	—	2	2	no	—
185	C₃×C₄	—	4	2	no	—
186	C₃×C₄	—	4	2	no	—
187	C₃×C₄	#944	1	2	yes	—
188	C₃×C₄	C₂×D₃×Q₈	1	2	yes	—
189	Dic₃	—	12	4	no	—
190	C₉	C₂×#501	1	1	yes	characteristic, complemented, Sylow
191	C₂³	C₂×#611	1	1	yes	characteristic
192	Q₈	—	36	8	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	18	4	no	—
200	C₂×C₄	—	36	4	no	—
201	C₂×C₄	Q₈×D₉	1	4	yes	—
202	C₂×C₄	C₂×#611	1	2	yes	—
203	C₂×C₄	C₂×#611	1	2	yes	—
204	C₂×C₄	C₂×#611	1	1	yes	characteristic
205	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
206	C₈	—	9	8	no	—
207	C₂×C₃	#5064	1	4	yes	—
208	C₂×C₃	C₂×#944	1	1	yes	characteristic
209	C₂×C₃	C₂×#949	1	1	yes	characteristic
210	C₂×C₃	C₂×#951	1	1	yes	characteristic
211	C₂²	#63506	1	2	yes	—
212	C₂²	#63511	1	2	yes	—
213	C₂²	#63512	1	2	yes	—
214	C₂²	C₂²×#611	1	1	yes	characteristic
215	C₄	—	2	2	no	—
216	C₄	—	2	2	no	—
217	C₄	—	4	2	no	—
218	C₄	—	4	2	no	—
219	C₄	—	36	4	no	—
220	C₄	#63506	1	2	yes	—
221	C₄	C₂×Q₈×D₉	1	2	yes	—
222	C₃	C₂×#5064	1	1	yes	characteristic
223	C₂	#10588743	1	4	yes	direct product factor
224	C₂	C₂×#63506	1	1	yes	characteristic
225	C₂	C₂×#63511	1	1	yes	characteristic
226	C₂	C₂×#63512	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#611
commutator, cosocle	C₂×C₄×C₉	C₂⁴
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	C₂×C₉×#125	C₂
socle, prime radical	C₂³×C₃	C₂×#67
hypercenter	C₂×#125	D₉
hypocenter	C₉	C₂×#501
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

Complements

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

N	H ≅ G/N	type
3. #10588743	C₂	direct (N × H)
49. C₂×C₉⋊C₈	Q₈	semidirect
83. C₉⋊C₈	C₂×Q₈	semidirect
154. C₂×C₉	#501	semidirect
190. C₉	C₂×#501	semidirect
223. C₂	#10588743	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₉×#125	2
C₂×#63431	2
#10588743	2
C₂×#63367	2
C₂×#63366	2
C₂×#63362	2
C₂×#5064	3

Subgroups — C2×#10588743 #11875023

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10588743 #11875023