Subgroups — C₃×C₄×#713 #11755584

Order 960.

620 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#713	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₅×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×Q₈×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4039	C₂	1	3	yes	maximal, complemented
5	C₃×#4038	C₂	1	3	yes	maximal, complemented
6	C₃×C₄×D₂₀	C₂	1	3	yes	maximal, complemented
7	C₃×C₄²×D₅	C₂	1	3	yes	maximal, complemented
8	C₄×#713	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
10	C₃×#713	C₄	1	8	yes	direct product factor
11	C₂×C₃×D₂₀	C₂²	1	3	yes	—
12	C₂×C₃×C₄×D₅	C₂²	1	3	yes	—
13	C₂×C₃×C₄×D₅	C₂²	1	3	yes	—
14	C₂×C₃×C₄×D₅	C₂²	1	3	yes	—
15	C₃×C₅×C₄⋊C₄	C₂²	1	3	yes	—
16	C₃×C₄²×C₅	C₂²	1	3	yes	—
17	C₃×#685	C₂²	1	6	yes	—
18	C₃×C₄⋊Dic₅	C₂²	1	3	yes	—
19	C₃×C₄×Dic₅	C₂²	1	3	yes	—
20	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
21	C₃×C₄×Dic₅	C₂²	1	3	yes	—
22	C₃×C₄×#41	—	5	1	no	maximal, Hall
23	C₂×#713	C₂×C₃	1	1	yes	characteristic
24	C₄×C₅×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
25	Q₈×Dic₅	C₂×C₃	1	1	yes	characteristic, complemented
26	#4039	C₂×C₃	1	3	yes	complemented
27	#4038	C₂×C₃	1	3	yes	complemented
28	C₄×D₂₀	C₂×C₃	1	3	yes	complemented
29	C₄²×D₅	C₂×C₃	1	3	yes	complemented
30	C₂²×C₃×D₅	C₂³	1	3	yes	—
31	C₃×C₅×Q₈	C₂×C₄	1	4	yes	complemented
32	C₂×C₃×C₄×C₅	C₂³	1	3	yes	—
33	C₂×C₃×C₄×C₅	C₂³	1	3	yes	—
34	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
35	C₂×C₃×Dic₅	C₂³	1	3	yes	—
36	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
37	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
38	C₃×D₂₀	C₂×C₄	1	12	yes	complemented
39	C₃×C₄×D₅	—	2	6	no	—
40	C₃×C₄×D₅	—	2	6	no	—
41	C₃×C₄×D₅	C₂×C₄	1	12	yes	complemented
42	C₂×C₃×#41	—	5	1	no	—
43	C₃×C₄×Q₈	—	5	1	no	—
44	C₃×C₄×Q₈	D₅	1	1	yes	characteristic, complemented
45	C₃×C₄×D₄	—	5	3	no	—
46	C₃×C₄×D₄	—	5	3	no	—
47	C₃×#117	—	5	3	no	—
48	C₂×C₃×C₄²	—	5	3	no	—
49	C₂×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
50	#713	C₃×C₄	1	8	yes	direct product factor
51	C₂×D₂₀	C₂²×C₃	1	3	yes	—
52	C₂×C₄×D₅	C₂²×C₃	1	3	yes	—
53	C₂×C₄×D₅	C₂²×C₃	1	3	yes	—
54	C₂×C₄×D₅	C₂²×C₃	1	3	yes	—
55	C₅×C₄⋊C₄	C₂²×C₃	1	3	yes	—
56	C₄²×C₅	C₂²×C₃	1	3	yes	—
57	#685	C₂²×C₃	1	6	yes	—
58	C₄⋊Dic₅	C₂²×C₃	1	3	yes	—
59	C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
60	C₄×Dic₅	C₂²×C₃	1	3	yes	—
61	C₄×Dic₅	C₂²×C₃	1	3	yes	—
62	C₄×#41	—	5	1	no	Sylow
63	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
64	C₂×C₃×D₅	—	2	3	no	—
65	C₂×C₃×D₅	—	2	3	no	—
66	C₂×C₃×D₅	C₂²×C₄	1	6	yes	—
67	C₃×C₄×C₅	—	2	3	no	—
68	C₃×C₄×C₅	C₂²×C₄	1	6	yes	—
69	C₃×C₄×C₅	#41	1	2	yes	complemented
70	C₃×Dic₅	—	2	3	no	—
71	C₃×Dic₅	C₂²×C₄	1	2	yes	—
72	C₃×Dic₅	#41	1	2	yes	complemented
73	C₃×#41	—	5	8	no	—
74	C₂×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
75	C₂×C₃×D₄	—	5	3	no	—
76	C₂²×C₃×C₄	—	5	3	no	—
77	C₂²×C₃×C₄	—	5	3	no	—
78	C₂²×C₃×C₄	—	5	3	no	—
79	C₃×C₄⋊C₄	—	5	3	no	—
80	C₃×C₄⋊C₄	C₂×D₅	1	3	yes	—
81	C₃×C₂²⋊C₄	—	5	6	no	—
82	C₃×C₄²	—	5	3	no	—
83	C₃×C₄²	—	5	3	no	—
84	C₃×C₄²	—	5	1	no	—
85	C₃×C₄²	C₂×D₅	1	3	yes	—
86	C₂²×D₅	C₂³×C₃	1	3	yes	—
87	C₅×Q₈	C₂×C₃×C₄	1	4	yes	complemented
88	C₂×C₄×C₅	C₂³×C₃	1	3	yes	—
89	C₂×C₄×C₅	C₂³×C₃	1	3	yes	—
90	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
91	C₂×Dic₅	C₂³×C₃	1	3	yes	—
92	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
93	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
94	D₂₀	C₂×C₃×C₄	1	12	yes	complemented
95	C₄×D₅	—	2	6	no	—
96	C₄×D₅	—	2	6	no	—
97	C₄×D₅	C₂×C₃×C₄	1	12	yes	complemented
98	C₂×#41	—	5	1	no	—
99	C₄×Q₈	—	5	1	no	—
100	C₄×Q₈	C₃×D₅	1	1	yes	characteristic, complemented
101	C₄×D₄	—	5	3	no	—
102	C₄×D₄	—	5	3	no	—
103	#117	—	5	3	no	—
104	C₂×C₄²	—	5	3	no	—
105	C₂×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
107	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
108	C₃×D₅	—	2	6	no	—
109	C₂³×C₃	—	5	3	no	—
110	C₃×Q₈	C₄×D₅	1	4	yes	complemented
111	C₃×D₄	—	5	12	no	—
112	C₂×C₃×C₄	—	5	1	no	—
113	C₂×C₃×C₄	—	5	1	no	—
114	C₂×C₃×C₄	—	5	12	no	—
115	C₂×C₃×C₄	—	5	3	no	—
116	C₂×C₃×C₄	—	10	6	no	—
117	C₂×C₃×C₄	—	10	6	no	—
118	C₂×C₃×C₄	C₂²×D₅	1	3	yes	—
119	C₂×C₃×C₄	C₂²×D₅	1	3	yes	—
120	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
121	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
122	C₂×D₅	—	2	3	no	—
123	C₂×D₅	—	2	3	no	—
124	C₂×D₅	C₂²×C₃×C₄	1	6	yes	—
125	C₄×C₅	—	2	3	no	—
126	C₄×C₅	C₂²×C₃×C₄	1	6	yes	—
127	C₄×C₅	C₃×#41	1	2	yes	complemented
128	Dic₅	—	2	3	no	—
129	Dic₅	C₂²×C₃×C₄	1	2	yes	—
130	Dic₅	C₃×#41	1	2	yes	complemented
131	#41	—	5	8	no	—
132	C₂×Q₈	C₂×C₃×D₅	1	1	yes	characteristic
133	C₂×D₄	—	5	3	no	—
134	C₂²×C₄	—	5	3	no	—
135	C₂²×C₄	—	5	3	no	—
136	C₂²×C₄	—	5	3	no	—
137	C₄⋊C₄	—	5	3	no	—
138	C₄⋊C₄	C₂×C₃×D₅	1	3	yes	—
139	C₂²⋊C₄	—	5	6	no	—
140	C₄²	—	5	1	no	—
141	C₄²	—	5	3	no	—
142	C₄²	—	5	3	no	—
143	C₄²	C₂×C₃×D₅	1	3	yes	—
144	C₃×C₅	C₄×#41	1	1	yes	characteristic, complemented, Hall
145	C₂²×C₃	—	5	6	no	—
146	C₂²×C₃	—	10	3	no	—
147	C₂²×C₃	—	10	3	no	—
148	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
149	C₃×C₄	—	2	3	no	—
150	C₃×C₄	—	5	2	no	—
151	C₃×C₄	—	5	2	no	—
152	C₃×C₄	—	10	3	no	—
153	C₃×C₄	C₂×C₄×D₅	1	6	yes	—
154	C₃×C₄	#713	1	2	yes	direct product factor
155	C₂×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
156	C₂×C₅	C₂×C₃×#41	1	1	yes	characteristic
157	C₂×C₅	C₂×C₃×#41	1	1	yes	characteristic
158	D₅	—	2	6	no	—
159	C₂³	—	5	3	no	—
160	Q₈	C₃×C₄×D₅	1	4	yes	complemented
161	D₄	—	5	12	no	—
162	C₂×C₄	—	5	1	no	—
163	C₂×C₄	—	5	1	no	—
164	C₂×C₄	—	5	12	no	—
165	C₂×C₄	—	5	3	no	—
166	C₂×C₄	—	10	6	no	—
167	C₂×C₄	—	10	6	no	—
168	C₂×C₄	C₂²×C₃×D₅	1	3	yes	—
169	C₂×C₄	C₂²×C₃×D₅	1	3	yes	—
170	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
171	C₂×C₃	—	10	6	no	—
172	C₂×C₃	C₂²×C₄×D₅	1	1	yes	characteristic
173	C₂×C₃	C₂×#713	1	1	yes	characteristic
174	C₂×C₃	D₅×#41	1	1	yes	characteristic
175	C₅	C₃×C₄×#41	1	1	yes	characteristic, complemented, Sylow
176	C₂²	—	5	6	no	—
177	C₂²	—	10	3	no	—
178	C₂²	—	10	3	no	—
179	C₂²	C₂³×C₃×D₅	1	1	yes	characteristic
180	C₄	—	2	3	no	—
181	C₄	—	5	2	no	—
182	C₄	—	5	2	no	—
183	C₄	—	10	3	no	—
184	C₄	C₂×C₃×C₄×D₅	1	6	yes	—
185	C₄	C₃×#713	1	2	yes	direct product factor
186	C₃	C₄×#713	1	1	yes	characteristic, direct product factor, Sylow
187	C₂	—	10	6	no	—
188	C₂	C₂²×C₃×C₄×D₅	1	1	yes	characteristic
189	C₂	C₂×C₃×#713	1	1	yes	characteristic
190	C₂	C₃×D₅×#41	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×D₅
commutator	C₂×C₅	C₂³×C₃×C₄
Frattini	C₂²	C₂³×C₃×D₅
Fitting	C₃×C₄×C₅×Q₈	C₂
socle	C₂²×C₃×C₅	C₂⁴
prime radical	C₂×C₃×#713	C₂
cosocle	C₂²×C₅	C₂⁴×C₃
hypercenter	C₃×C₄×Q₈	D₅
hypocenter	C₅	C₃×C₄×#41
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×C₅×Q₈	C₂	semidirect
3. C₃×Q₈×Dic₅	C₂	semidirect
4. C₃×#4039	C₂	semidirect
5. C₃×#4038	C₂	semidirect
6. C₃×C₄×D₂₀	C₂	semidirect
7. C₃×C₄²×D₅	C₂	semidirect
8. C₄×#713	C₃	direct (N × H)
10. C₃×#713	C₄	direct (N × H)
24. C₄×C₅×Q₈	C₂×C₃	semidirect
25. Q₈×Dic₅	C₂×C₃	semidirect
26. #4039	C₂×C₃	semidirect
27. #4038	C₂×C₃	semidirect
28. C₄×D₂₀	C₂×C₃	semidirect
29. C₄²×D₅	C₂×C₃	semidirect
31. C₃×C₅×Q₈	C₂×C₄	semidirect
38. C₃×D₂₀	C₂×C₄	semidirect
41. C₃×C₄×D₅	C₂×C₄	semidirect
44. C₃×C₄×Q₈	D₅	semidirect
50. #713	C₃×C₄	direct (N × H)
69. C₃×C₄×C₅	#41	semidirect
72. C₃×Dic₅	#41	semidirect
87. C₅×Q₈	C₂×C₃×C₄	semidirect
94. D₂₀	C₂×C₃×C₄	semidirect
97. C₄×D₅	C₂×C₃×C₄	semidirect
100. C₄×Q₈	C₃×D₅	semidirect
110. C₃×Q₈	C₄×D₅	semidirect
127. C₄×C₅	C₃×#41	semidirect
130. Dic₅	C₃×#41	semidirect
144. C₃×C₅	C₄×#41	semidirect
154. C₃×C₄	#713	direct (N × H)
160. Q₈	C₃×C₄×D₅	semidirect
175. C₅	C₃×C₄×#41	semidirect
185. C₄	C₃×#713	direct (N × H)
186. C₃	C₄×#713	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×C₃×#713	2
C₃×C₄×C₅×Q₈	2
C₃×Q₈×Dic₅	2
C₃×#4039	2
C₃×#4038	2
C₃×C₄×D₂₀	2
C₃×C₄²×D₅	2
C₄×#713	3
C₃×C₄×#41	5

Subgroups — C3×C4×#713 #11755584

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×#713 #11755584