Subgroups — #11919274

Order 1152.

8988 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
2	#10593141	C₂	1	16	yes	maximal, complemented
3	D₉×H(5,𝔽₂)	C₂	1	16	yes	maximal, complemented
4	#10593139	C₂	1	12	yes	maximal, complemented
5	C₂×#63710	C₂	1	18	yes	maximal, complemented
6	#87568	—	3	1	no	maximal
7	C₉×H(5,𝔽₂)	C₂²	1	16	yes	complemented
8	C₂×C₉×#41	C₂²	1	6	yes	complemented
9	C₂²×D₄×C₉	C₂²	1	9	yes	complemented
10	C₂²×#611	C₂²	1	24	yes	complemented
11	#63716	C₂²	1	48	yes	complemented
12	#63715	C₂²	1	48	yes	complemented
13	#41×D₉	C₂²	1	96	yes	complemented
14	#63713	C₂²	1	8	yes	complemented
15	#63710	C₂²	1	72	yes	complemented
16	#63710	C₂²	1	144	yes	complemented
17	C₂×#3699	C₂²	1	72	yes	complemented
18	C₂×D₄×D₉	C₂²	1	72	yes	complemented
19	C₂×#3697	C₂²	1	36	yes	complemented
20	C₂×C₃×H(5,𝔽₂)	D₃	1	1	yes	characteristic
21	#5966	—	3	16	no	—
22	D₃×H(5,𝔽₂)	—	3	16	no	—
23	#5964	—	3	12	no	—
24	C₂×#1004	—	3	18	no	—
25	C₂⁴×C₉	C₂³	1	6	yes	complemented
26	C₂³×D₉	C₂³	1	24	yes	complemented
27	C₉×#41	C₂³	1	48	yes	complemented
28	C₂×Q₈×C₉	C₂³	1	2	yes	complemented
29	C₂×D₄×C₉	C₂³	1	72	yes	complemented
30	C₂×D₄×C₉	C₂³	1	18	yes	complemented
31	C₂²×C₄×C₉	C₂³	1	9	yes	complemented
32	C₂×#611	C₂³	1	72	yes	complemented
33	C₂×#611	C₂³	1	144	yes	complemented
34	C₂²×Dic₉	C₂³	1	24	yes	complemented
35	#3701	C₂³	1	32	yes	complemented
36	Q₈×D₉	C₂³	1	32	yes	complemented
37	#3699	C₂³	1	288	yes	complemented
38	D₄×D₉	C₂³	1	288	yes	complemented
39	#3697	C₂³	1	48	yes	complemented
40	#3697	C₂³	1	144	yes	complemented
41	C₂×D₃₆	C₂³	1	36	yes	complemented
42	C₂×C₄×D₉	C₂³	1	72	yes	complemented
43	C₂×Dic₁₈	C₂³	1	36	yes	complemented
44	H(7,𝔽₂)	—	9	1	no	Sylow
45	C₃×H(5,𝔽₂)	C₂×D₃	1	16	yes	—
46	C₂×C₃×#41	C₂×D₃	1	6	yes	—
47	C₂²×C₃×D₄	C₂×D₃	1	9	yes	—
48	C₂²×#67	—	3	24	no	—
49	#1010	—	3	48	no	—
50	#1009	—	3	48	no	—
51	D₃×#41	—	3	96	no	—
52	#1007	—	3	8	no	—
53	#1004	—	3	72	no	—
54	#1004	—	3	144	no	—
55	C₂×#237	—	3	72	no	—
56	C₂×D₃×D₄	—	3	72	no	—
57	C₂×#235	—	3	36	no	—
58	C₂³×C₉	—	8	6	no	—
59	C₂³×C₉	C₂⁴	1	24	yes	—
60	C₂³×C₉	C₂⁴	1	9	yes	—
61	C₂²×D₉	—	8	24	no	—
62	C₂²×D₉	C₂⁴	1	72	yes	—
63	Q₈×C₉	C₂⁴	1	8	yes	—
64	D₄×C₉	C₂⁴	1	72	yes	—
65	C₂×C₄×C₉	C₂⁴	1	36	yes	—
66	C₂×C₄×C₉	C₂⁴	1	6	yes	—
67	#611	C₂⁴	1	16	yes	—
68	#611	C₂⁴	1	144	yes	—
69	C₂×Dic₉	C₂⁴	1	72	yes	—
70	D₃₆	C₂⁴	1	48	yes	—
71	C₄×D₉	C₂⁴	1	96	yes	—
72	Dic₁₈	C₂⁴	1	48	yes	—
73	#585	—	9	16	no	—
74	#585	—	9	12	no	—
75	C₂×H(5,𝔽₂)	—	9	16	no	—
76	C₂×H(5,𝔽₂)	—	9	18	no	—
77	C₂×H(5,𝔽₂)	D₉	1	1	yes	characteristic, complemented
78	C₂⁴×C₃	C₂²×D₃	1	6	yes	—
79	C₂³×D₃	—	3	24	no	—
80	C₃×#41	C₂²×D₃	1	48	yes	—
81	C₂×C₃×Q₈	C₂²×D₃	1	2	yes	—
82	C₂×C₃×D₄	C₂²×D₃	1	72	yes	—
83	C₂×C₃×D₄	C₂²×D₃	1	18	yes	—
84	C₂²×C₃×C₄	C₂²×D₃	1	9	yes	—
85	C₂×#67	—	3	72	no	—
86	C₂×#67	—	3	144	no	—
87	C₂²×Dic₃	—	3	24	no	—
88	#239	—	3	32	no	—
89	D₃×Q₈	—	3	32	no	—
90	#237	—	3	288	no	—
91	D₃×D₄	—	3	288	no	—
92	#235	—	3	48	no	—
93	#235	—	3	144	no	—
94	C₂×D₁₂	—	3	36	no	—
95	C₂×C₄×D₃	—	3	72	no	—
96	C₂×Dic₆	—	3	36	no	—
97	C₂²×C₉	—	4	24	no	—
98	C₂²×C₉	—	4	9	no	—
99	C₂²×C₉	C₂⁵	1	18	yes	—
100	C₂²×C₉	C₂⁵	1	1	yes	characteristic
101	C₂×D₉	—	4	72	no	—
102	C₂×D₉	C₂⁵	1	16	yes	—
103	C₄×C₉	C₂⁵	1	12	yes	—
104	Dic₉	C₂⁵	1	16	yes	—
105	2₋¹⁺⁴	—	9	48	no	—
106	2₋¹⁺⁴	—	9	8	no	—
107	H(5,𝔽₂)	—	9	48	no	—
108	H(5,𝔽₂)	—	9	144	no	—
109	H(5,𝔽₂)	—	9	72	no	—
110	H(5,𝔽₂)	C₂×D₉	1	16	yes	complemented
111	C₂×#41	—	9	96	no	—
112	C₂×#41	—	9	72	no	—
113	C₂×#41	—	9	36	no	—
114	C₂×#41	C₂×D₉	1	6	yes	complemented
115	C₂²×D₄	—	9	24	no	—
116	C₂²×D₄	—	9	72	no	—
117	C₂²×D₄	C₂×D₉	1	9	yes	complemented
118	C₂³×C₃	—	8	6	no	—
119	C₂³×C₃	C₂³×D₃	1	24	yes	—
120	C₂³×C₃	C₂³×D₃	1	9	yes	—
121	C₂²×D₃	—	3	72	no	—
122	C₂²×D₃	—	24	24	no	—
123	C₃×Q₈	C₂³×D₃	1	8	yes	—
124	C₃×D₄	C₂³×D₃	1	72	yes	—
125	C₂×C₃×C₄	C₂³×D₃	1	36	yes	—
126	C₂×C₃×C₄	C₂³×D₃	1	6	yes	—
127	#67	—	3	16	no	—
128	#67	—	3	144	no	—
129	C₂×Dic₃	—	3	72	no	—
130	D₁₂	—	3	48	no	—
131	C₄×D₃	—	3	96	no	—
132	Dic₆	—	3	48	no	—
133	C₂×C₉	—	2	18	no	—
134	C₂×C₉	—	2	1	no	—
135	C₂×C₉	C₂⁶	1	1	yes	characteristic
136	D₉	—	2	16	no	—
137	C₂⁴	—	9	24	no	—
138	C₂⁴	C₂²×D₉	1	6	yes	complemented
139	#41	—	9	32	no	—
140	#41	—	9	288	no	—
141	#41	—	9	144	no	—
142	#41	—	9	48	no	—
143	#41	C₂²×D₉	1	48	yes	complemented
144	C₂×Q₈	—	9	32	no	—
145	C₂×Q₈	—	9	36	no	—
146	C₂×Q₈	C₂²×D₉	1	2	yes	complemented
147	C₂×D₄	—	9	144	no	—
148	C₂×D₄	—	9	72	no	—
149	C₂×D₄	—	9	288	no	—
150	C₂×D₄	—	9	36	no	—
151	C₂×D₄	C₂²×D₉	1	72	yes	complemented
152	C₂×D₄	C₂²×D₉	1	18	yes	complemented
153	C₂²×C₄	—	9	24	no	—
154	C₂²×C₄	—	9	72	no	—
155	C₂²×C₄	C₂²×D₉	1	9	yes	complemented
156	C₂²×C₃	—	4	24	no	—
157	C₂²×C₃	—	4	9	no	—
158	C₂²×C₃	C₂⁴×D₃	1	18	yes	—
159	C₂²×C₃	C₂⁴×D₃	1	1	yes	characteristic
160	C₂×D₃	—	3	16	no	—
161	C₂×D₃	—	12	72	no	—
162	C₃×C₄	C₂⁴×D₃	1	12	yes	—
163	Dic₃	—	3	16	no	—
164	C₉	H(7,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
165	C₂³	—	8	6	no	—
166	C₂³	—	9	72	no	—
167	C₂³	—	72	24	no	—
168	C₂³	C₂³×D₉	1	24	yes	—
169	C₂³	C₂³×D₉	1	9	yes	—
170	Q₈	—	9	48	no	—
171	Q₈	C₂³×D₉	1	8	yes	—
172	D₄	—	9	144	no	—
173	D₄	—	9	16	no	—
174	D₄	—	9	48	no	—
175	D₄	C₂³×D₉	1	72	yes	—
176	C₂×C₄	—	9	72	no	—
177	C₂×C₄	—	9	96	no	—
178	C₂×C₄	C₂³×D₉	1	36	yes	—
179	C₂×C₄	C₂³×D₉	1	6	yes	—
180	C₂×C₃	—	2	18	no	—
181	C₂×C₃	—	2	1	no	—
182	C₂×C₃	C₂⁵×D₃	1	1	yes	characteristic
183	D₃	—	6	16	no	—
184	C₂²	—	4	9	no	—
185	C₂²	—	4	24	no	—
186	C₂²	—	9	16	no	—
187	C₂²	—	36	72	no	—
188	C₂²	C₂⁴×D₉	1	18	yes	—
189	C₂²	C₂⁴×D₉	1	1	yes	characteristic
190	C₄	—	9	16	no	—
191	C₄	C₂⁴×D₉	1	12	yes	—
192	C₃	#87568	1	1	yes	characteristic
193	C₂	—	2	1	no	—
194	C₂	—	2	18	no	—
195	C₂	—	18	16	no	—
196	C₂	C₂⁵×D₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂⁵×D₉
commutator, cosocle	C₂×C₉	C₂⁶
Frattini, socle	C₂×C₃	C₂⁵×D₃
Fitting	C₂×C₉×H(5,𝔽₂)	C₂
hypercenter	C₂×H(5,𝔽₂)	D₉
hypocenter	C₉	H(7,𝔽₂)
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂ ◁ C₂×H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. C₂×C₉×H(5,𝔽₂)	C₂	semidirect
2. #10593141	C₂	semidirect
3. D₉×H(5,𝔽₂)	C₂	semidirect
4. #10593139	C₂	semidirect
5. C₂×#63710	C₂	semidirect
7. C₉×H(5,𝔽₂)	C₂²	semidirect
8. C₂×C₉×#41	C₂²	semidirect
9. C₂²×D₄×C₉	C₂²	semidirect
10. C₂²×#611	C₂²	semidirect
11. #63716	C₂²	semidirect
12. #63715	C₂²	semidirect
13. #41×D₉	C₂²	semidirect
14. #63713	C₂²	semidirect
15. #63710	C₂²	semidirect
16. #63710	C₂²	semidirect
17. C₂×#3699	C₂²	semidirect
18. C₂×D₄×D₉	C₂²	semidirect
19. C₂×#3697	C₂²	semidirect
25. C₂⁴×C₉	C₂³	semidirect
26. C₂³×D₉	C₂³	semidirect
27. C₉×#41	C₂³	semidirect
28. C₂×Q₈×C₉	C₂³	semidirect
29. C₂×D₄×C₉	C₂³	semidirect
30. C₂×D₄×C₉	C₂³	semidirect
31. C₂²×C₄×C₉	C₂³	semidirect
32. C₂×#611	C₂³	semidirect
33. C₂×#611	C₂³	semidirect
34. C₂²×Dic₉	C₂³	semidirect
35. #3701	C₂³	semidirect
36. Q₈×D₉	C₂³	semidirect
37. #3699	C₂³	semidirect
38. D₄×D₉	C₂³	semidirect
39. #3697	C₂³	semidirect
40. #3697	C₂³	semidirect
41. C₂×D₃₆	C₂³	semidirect
42. C₂×C₄×D₉	C₂³	semidirect
43. C₂×Dic₁₈	C₂³	semidirect
77. C₂×H(5,𝔽₂)	D₉	semidirect
110. H(5,𝔽₂)	C₂×D₉	semidirect
114. C₂×#41	C₂×D₉	semidirect
117. C₂²×D₄	C₂×D₉	semidirect
138. C₂⁴	C₂²×D₉	semidirect
143. #41	C₂²×D₉	semidirect
146. C₂×Q₈	C₂²×D₉	semidirect
151. C₂×D₄	C₂²×D₉	semidirect
152. C₂×D₄	C₂²×D₉	semidirect
155. C₂²×C₄	C₂²×D₉	semidirect
164. C₉	H(7,𝔽₂)	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×H(5,𝔽₂)	2
#10593141	2
D₉×H(5,𝔽₂)	2
#10593139	2
C₂×#63710	2
#87568	3