Subgroups — #11919275

Order 1152.

17976 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×C₃²×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
2	#10596758	C₂	1	16	yes	maximal, complemented
3	C₃⋊D₃×H(5,𝔽₂)	C₂	1	16	yes	maximal, complemented
4	#10596756	C₂	1	12	yes	maximal, complemented
5	C₂×#64361	C₂	1	18	yes	maximal, complemented
6	#87568	—	3	4	no	maximal
7	C₃²×H(5,𝔽₂)	C₂²	1	16	yes	complemented
8	C₂×C₃²×#41	C₂²	1	6	yes	complemented
9	C₂²×C₃²×D₄	C₂²	1	9	yes	complemented
10	C₂²×#638	C₂²	1	24	yes	complemented
11	#64367	C₂²	1	48	yes	complemented
12	#64366	C₂²	1	48	yes	complemented
13	#41×C₃⋊D₃	C₂²	1	96	yes	complemented
14	#64364	C₂²	1	8	yes	complemented
15	#64361	C₂²	1	72	yes	complemented
16	#64361	C₂²	1	144	yes	complemented
17	C₂×#3830	C₂²	1	72	yes	complemented
18	C₂×D₄×C₃⋊D₃	C₂²	1	72	yes	complemented
19	C₂×#3828	C₂²	1	36	yes	complemented
20	C₂×C₃×H(5,𝔽₂)	D₃	1	4	yes	complemented
21	#5966	—	3	64	no	—
22	D₃×H(5,𝔽₂)	—	3	64	no	—
23	#5964	—	3	48	no	—
24	C₂×#1004	—	3	72	no	—
25	C₂⁴×C₃²	C₂³	1	6	yes	complemented
26	C₂³×C₃⋊D₃	C₂³	1	24	yes	complemented
27	C₃²×#41	C₂³	1	48	yes	complemented
28	C₂×C₃²×Q₈	C₂³	1	2	yes	complemented
29	C₂×C₃²×D₄	C₂³	1	72	yes	complemented
30	C₂×C₃²×D₄	C₂³	1	18	yes	complemented
31	C₂²×C₃²×C₄	C₂³	1	9	yes	complemented
32	C₂×#638	C₂³	1	72	yes	complemented
33	C₂×#638	C₂³	1	144	yes	complemented
34	C₂²×C₃⋊Dic₃	C₂³	1	24	yes	complemented
35	#3832	C₂³	1	32	yes	complemented
36	Q₈×C₃⋊D₃	C₂³	1	32	yes	complemented
37	#3830	C₂³	1	288	yes	complemented
38	D₄×C₃⋊D₃	C₂³	1	288	yes	complemented
39	#3828	C₂³	1	48	yes	complemented
40	#3828	C₂³	1	144	yes	complemented
41	C₂×C₄⋊(C₃⋊D₃)	C₂³	1	36	yes	complemented
42	C₂×C₄×C₃⋊D₃	C₂³	1	72	yes	complemented
43	C₂×#634	C₂³	1	36	yes	complemented
44	H(7,𝔽₂)	—	9	1	no	Sylow
45	C₃×H(5,𝔽₂)	C₂×D₃	1	64	yes	complemented
46	C₂×C₃×#41	C₂×D₃	1	24	yes	complemented
47	C₂²×C₃×D₄	C₂×D₃	1	36	yes	complemented
48	C₂²×#67	—	3	96	no	—
49	#1010	—	3	192	no	—
50	#1009	—	3	192	no	—
51	D₃×#41	—	3	384	no	—
52	#1007	—	3	32	no	—
53	#1004	—	3	288	no	—
54	#1004	—	3	576	no	—
55	C₂×#237	—	3	288	no	—
56	C₂×D₃×D₄	—	3	288	no	—
57	C₂×#235	—	3	144	no	—
58	C₂³×C₃²	—	8	6	no	—
59	C₂³×C₃²	C₂⁴	1	24	yes	—
60	C₂³×C₃²	C₂⁴	1	9	yes	—
61	C₂²×C₃⋊D₃	—	8	24	no	—
62	C₂²×C₃⋊D₃	C₂⁴	1	72	yes	—
63	C₃²×Q₈	C₂⁴	1	8	yes	—
64	C₃²×D₄	C₂⁴	1	72	yes	—
65	C₂×C₃²×C₄	C₂⁴	1	36	yes	—
66	C₂×C₃²×C₄	C₂⁴	1	6	yes	—
67	#638	C₂⁴	1	16	yes	—
68	#638	C₂⁴	1	144	yes	—
69	C₂×C₃⋊Dic₃	C₂⁴	1	72	yes	—
70	C₄⋊(C₃⋊D₃)	C₂⁴	1	48	yes	—
71	C₄×C₃⋊D₃	C₂⁴	1	96	yes	—
72	#634	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)	C₃⋊D₃	1	1	yes	characteristic, complemented
78	C₂⁴×C₃	C₂²×D₃	1	24	yes	complemented
79	C₂³×D₃	—	3	96	no	—
80	C₃×#41	C₂²×D₃	1	192	yes	complemented
81	C₂×C₃×Q₈	C₂²×D₃	1	8	yes	complemented
82	C₂×C₃×D₄	C₂²×D₃	1	288	yes	complemented
83	C₂×C₃×D₄	C₂²×D₃	1	72	yes	complemented
84	C₂²×C₃×C₄	C₂²×D₃	1	36	yes	complemented
85	C₂×#67	—	3	288	no	—
86	C₂×#67	—	3	576	no	—
87	C₂²×Dic₃	—	3	96	no	—
88	#239	—	3	128	no	—
89	D₃×Q₈	—	3	128	no	—
90	#237	—	3	1152	no	—
91	D₃×D₄	—	3	1152	no	—
92	#235	—	3	192	no	—
93	#235	—	3	576	no	—
94	C₂×D₁₂	—	3	144	no	—
95	C₂×C₄×D₃	—	3	288	no	—
96	C₂×Dic₆	—	3	144	no	—
97	C₂²×C₃²	—	4	24	no	—
98	C₂²×C₃²	—	4	9	no	—
99	C₂²×C₃²	C₂⁵	1	18	yes	—
100	C₂²×C₃²	C₂⁵	1	1	yes	characteristic
101	C₂×C₃⋊D₃	—	4	72	no	—
102	C₂×C₃⋊D₃	C₂⁵	1	16	yes	—
103	C₃²×C₄	C₂⁵	1	12	yes	—
104	C₃⋊Dic₃	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×C₃⋊D₃	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×C₃⋊D₃	1	6	yes	complemented
115	C₂²×D₄	—	9	24	no	—
116	C₂²×D₄	—	9	72	no	—
117	C₂²×D₄	C₂×C₃⋊D₃	1	9	yes	complemented
118	C₂³×C₃	—	8	24	no	—
119	C₂³×C₃	C₂³×D₃	1	96	yes	—
120	C₂³×C₃	C₂³×D₃	1	36	yes	—
121	C₂²×D₃	—	3	288	no	—
122	C₂²×D₃	—	24	96	no	—
123	C₃×Q₈	C₂³×D₃	1	32	yes	—
124	C₃×D₄	C₂³×D₃	1	288	yes	—
125	C₂×C₃×C₄	C₂³×D₃	1	144	yes	—
126	C₂×C₃×C₄	C₂³×D₃	1	24	yes	—
127	#67	—	3	64	no	—
128	#67	—	3	576	no	—
129	C₂×Dic₃	—	3	288	no	—
130	D₁₂	—	3	192	no	—
131	C₄×D₃	—	3	384	no	—
132	Dic₆	—	3	192	no	—
133	C₂×C₃²	—	2	18	no	—
134	C₂×C₃²	—	2	1	no	—
135	C₂×C₃²	C₂⁶	1	1	yes	characteristic
136	C₃⋊D₃	—	2	16	no	—
137	C₂⁴	—	9	24	no	—
138	C₂⁴	C₂²×C₃⋊D₃	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×C₃⋊D₃	1	48	yes	complemented
144	C₂×Q₈	—	9	32	no	—
145	C₂×Q₈	—	9	36	no	—
146	C₂×Q₈	C₂²×C₃⋊D₃	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×C₃⋊D₃	1	72	yes	complemented
152	C₂×D₄	C₂²×C₃⋊D₃	1	18	yes	complemented
153	C₂²×C₄	—	9	24	no	—
154	C₂²×C₄	—	9	72	no	—
155	C₂²×C₄	C₂²×C₃⋊D₃	1	9	yes	complemented
156	C₂²×C₃	—	4	96	no	—
157	C₂²×C₃	—	4	36	no	—
158	C₂²×C₃	C₂⁴×D₃	1	72	yes	—
159	C₂²×C₃	C₂⁴×D₃	1	4	yes	—
160	C₂×D₃	—	3	64	no	—
161	C₂×D₃	—	12	288	no	—
162	C₃×C₄	C₂⁴×D₃	1	48	yes	—
163	Dic₃	—	3	64	no	—
164	C₃²	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×C₃⋊D₃	1	24	yes	—
169	C₂³	C₂³×C₃⋊D₃	1	9	yes	—
170	Q₈	—	9	48	no	—
171	Q₈	C₂³×C₃⋊D₃	1	8	yes	—
172	D₄	—	9	144	no	—
173	D₄	—	9	16	no	—
174	D₄	—	9	48	no	—
175	D₄	C₂³×C₃⋊D₃	1	72	yes	—
176	C₂×C₄	—	9	72	no	—
177	C₂×C₄	—	9	96	no	—
178	C₂×C₄	C₂³×C₃⋊D₃	1	36	yes	—
179	C₂×C₄	C₂³×C₃⋊D₃	1	6	yes	—
180	C₂×C₃	—	2	72	no	—
181	C₂×C₃	—	2	4	no	—
182	C₂×C₃	C₂⁵×D₃	1	4	yes	—
183	D₃	—	6	64	no	—
184	C₂²	—	4	9	no	—
185	C₂²	—	4	24	no	—
186	C₂²	—	9	16	no	—
187	C₂²	—	36	72	no	—
188	C₂²	C₂⁴×C₃⋊D₃	1	18	yes	—
189	C₂²	C₂⁴×C₃⋊D₃	1	1	yes	characteristic
190	C₄	—	9	16	no	—
191	C₄	C₂⁴×C₃⋊D₃	1	12	yes	—
192	C₃	#87568	1	4	yes	complemented
193	C₂	—	2	1	no	—
194	C₂	—	2	18	no	—
195	C₂	—	18	16	no	—
196	C₂	C₂⁵×C₃⋊D₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁵×C₃⋊D₃
commutator, socle, cosocle	C₂×C₃²	C₂⁶
Fitting	C₂×C₃²×H(5,𝔽₂)	C₂
hypercenter	C₂×H(5,𝔽₂)	C₃⋊D₃
hypocenter	C₃²	H(7,𝔽₂)
LCS	G ▷ C₂×C₃² ▷ C₃²
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×C₃²×H(5,𝔽₂)	C₂	semidirect
2. #10596758	C₂	semidirect
3. C₃⋊D₃×H(5,𝔽₂)	C₂	semidirect
4. #10596756	C₂	semidirect
5. C₂×#64361	C₂	semidirect
7. C₃²×H(5,𝔽₂)	C₂²	semidirect
8. C₂×C₃²×#41	C₂²	semidirect
9. C₂²×C₃²×D₄	C₂²	semidirect
10. C₂²×#638	C₂²	semidirect
11. #64367	C₂²	semidirect
12. #64366	C₂²	semidirect
13. #41×C₃⋊D₃	C₂²	semidirect
14. #64364	C₂²	semidirect
15. #64361	C₂²	semidirect
16. #64361	C₂²	semidirect
17. C₂×#3830	C₂²	semidirect
18. C₂×D₄×C₃⋊D₃	C₂²	semidirect
19. C₂×#3828	C₂²	semidirect
20. C₂×C₃×H(5,𝔽₂)	D₃	semidirect
25. C₂⁴×C₃²	C₂³	semidirect
26. C₂³×C₃⋊D₃	C₂³	semidirect
27. C₃²×#41	C₂³	semidirect
28. C₂×C₃²×Q₈	C₂³	semidirect
29. C₂×C₃²×D₄	C₂³	semidirect
30. C₂×C₃²×D₄	C₂³	semidirect
31. C₂²×C₃²×C₄	C₂³	semidirect
32. C₂×#638	C₂³	semidirect
33. C₂×#638	C₂³	semidirect
34. C₂²×C₃⋊Dic₃	C₂³	semidirect
35. #3832	C₂³	semidirect
36. Q₈×C₃⋊D₃	C₂³	semidirect
37. #3830	C₂³	semidirect
38. D₄×C₃⋊D₃	C₂³	semidirect
39. #3828	C₂³	semidirect
40. #3828	C₂³	semidirect
41. C₂×C₄⋊(C₃⋊D₃)	C₂³	semidirect
42. C₂×C₄×C₃⋊D₃	C₂³	semidirect
43. C₂×#634	C₂³	semidirect
45. C₃×H(5,𝔽₂)	C₂×D₃	semidirect
46. C₂×C₃×#41	C₂×D₃	semidirect
47. C₂²×C₃×D₄	C₂×D₃	semidirect
77. C₂×H(5,𝔽₂)	C₃⋊D₃	semidirect
78. C₂⁴×C₃	C₂²×D₃	semidirect
80. C₃×#41	C₂²×D₃	semidirect
81. C₂×C₃×Q₈	C₂²×D₃	semidirect
82. C₂×C₃×D₄	C₂²×D₃	semidirect
83. C₂×C₃×D₄	C₂²×D₃	semidirect
84. C₂²×C₃×C₄	C₂²×D₃	semidirect
110. H(5,𝔽₂)	C₂×C₃⋊D₃	semidirect
114. C₂×#41	C₂×C₃⋊D₃	semidirect
117. C₂²×D₄	C₂×C₃⋊D₃	semidirect
138. C₂⁴	C₂²×C₃⋊D₃	semidirect
143. #41	C₂²×C₃⋊D₃	semidirect
146. C₂×Q₈	C₂²×C₃⋊D₃	semidirect
151. C₂×D₄	C₂²×C₃⋊D₃	semidirect
152. C₂×D₄	C₂²×C₃⋊D₃	semidirect
155. C₂²×C₄	C₂²×C₃⋊D₃	semidirect
164. C₃²	H(7,𝔽₂)	semidirect
192. C₃	#87568	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×H(5,𝔽₂)	2
#10596758	2
C₃⋊D₃×H(5,𝔽₂)	2
#10596756	2
C₂×#64361	2
#87568	3