Subgroups — C₂×C₃×#4097 #11755834

Order 960.

516 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×C₃×C₅×#41	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#4097	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃×#689	C₂	1	2	yes	maximal, complemented
5	C₂×C₃×#687	C₂	1	2	yes	maximal, complemented
6	C₂×C₃×#681	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#4097	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₃×C₅×#41	—	2	2	no	—
9	C₃×C₅×#41	C₂²	1	4	yes	—
10	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₅×D₄	—	2	1	no	—
12	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
13	C₂²×C₃×C₄×C₅	—	2	1	no	—
14	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
15	C₂²×C₃×Dic₅	—	2	1	no	—
16	C₂×C₃×Dic₁₀	—	2	2	no	—
17	C₂×C₃×Dic₁₀	—	2	1	no	—
18	C₂×C₃×Dic₁₀	C₂²	1	2	yes	—
19	C₂×C₃×Dic₁₀	C₂²	1	4	yes	complemented
20	C₃×#689	C₂²	1	8	yes	complemented
21	C₃×#687	C₂²	1	8	yes	complemented
22	C₃×#681	C₂²	1	4	yes	complemented
23	C₂×C₃×#168	C₂²	1	2	yes	—
24	C₂×C₃×#137	—	5	1	no	maximal, Hall
25	C₂×C₅×#41	C₂×C₃	1	1	yes	characteristic
26	C₂²×Dic₁₀	C₂×C₃	1	1	yes	characteristic, complemented
27	#4097	C₂×C₃	1	8	yes	direct product factor
28	C₂×#689	C₂×C₃	1	2	yes	complemented
29	C₂×#687	C₂×C₃	1	2	yes	complemented
30	C₂×#681	C₂×C₃	1	1	yes	characteristic, complemented
31	C₂³×C₃×C₅	—	2	1	no	—
32	C₂³×C₃×C₅	D₄	1	1	yes	characteristic
33	C₃×C₅×Q₈	—	2	1	no	—
34	C₃×C₅×Q₈	C₂³	1	2	yes	—
35	C₃×C₅×D₄	—	2	4	no	—
36	C₃×C₅×D₄	—	2	1	no	—
37	C₃×C₅×D₄	C₂³	1	2	yes	—
38	C₂×C₃×C₄×C₅	—	2	1	no	—
39	C₂×C₃×C₄×C₅	—	2	4	no	—
40	C₂×C₃×C₄×C₅	D₄	1	1	yes	characteristic
41	C₂×C₃×C₄×C₅	D₄	1	2	yes	—
42	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
43	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
44	C₂×C₃×Dic₅	—	2	2	no	—
45	C₂×C₃×Dic₅	—	2	4	no	—
46	C₃×Dic₁₀	—	2	2	no	—
47	C₃×Dic₁₀	—	2	4	no	—
48	C₃×Dic₁₀	C₂³	1	4	yes	—
49	C₃×#168	C₂³	1	4	yes	—
50	C₂×C₃×#41	D₅	1	1	yes	characteristic
51	C₂²×C₃×Q₈	—	5	1	no	—
52	C₃×#137	—	5	8	no	—
53	C₂×C₃×Q₁₆	—	5	2	no	—
54	C₂×C₃×SD₁₆	—	5	2	no	—
55	C₂×C₃×M₁₆	—	5	1	no	—
56	C₅×#41	—	2	2	no	—
57	C₅×#41	C₂²×C₃	1	4	yes	—
58	C₂×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
59	C₂×C₅×D₄	—	2	1	no	—
60	C₂×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
61	C₂²×C₄×C₅	—	2	1	no	—
62	C₂²×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
63	C₂²×Dic₅	—	2	1	no	—
64	C₂×Dic₁₀	—	2	2	no	—
65	C₂×Dic₁₀	—	2	1	no	—
66	C₂×Dic₁₀	C₂²×C₃	1	2	yes	—
67	C₂×Dic₁₀	C₂²×C₃	1	4	yes	complemented
68	#689	C₂²×C₃	1	8	yes	complemented
69	#687	C₂²×C₃	1	8	yes	complemented
70	#681	C₂²×C₃	1	4	yes	complemented
71	C₂×#168	C₂²×C₃	1	2	yes	—
72	C₂×#137	—	5	1	no	Sylow
73	C₂²×C₃×C₅	—	2	2	no	—
74	C₂²×C₃×C₅	—	2	2	no	—
75	C₂²×C₃×C₅	—	4	2	no	—
76	C₂²×C₃×C₅	C₂×D₄	1	2	yes	—
77	C₂²×C₃×C₅	C₂×D₄	1	1	yes	characteristic
78	C₃×C₄×C₅	—	2	2	no	—
79	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
80	C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
82	C₃×Dic₅	—	2	4	no	—
83	C₃×#41	—	2	2	no	—
84	C₃×#41	C₂×D₅	1	4	yes	—
85	C₂×C₃×Q₈	—	5	2	no	—
86	C₂×C₃×Q₈	—	5	4	no	—
87	C₂×C₃×Q₈	—	10	2	no	—
88	C₂×C₃×Q₈	—	10	1	no	—
89	C₂×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
90	C₂×C₃×D₄	—	2	1	no	—
91	C₂×C₃×D₄	C₂×D₅	1	1	yes	characteristic
92	C₂²×C₃×C₄	—	2	1	no	—
93	C₂²×C₃×C₄	—	10	1	no	—
94	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic
95	C₃×Q₁₆	—	5	8	no	—
96	C₃×SD₁₆	—	5	8	no	—
97	C₃×M₁₆	—	5	4	no	—
98	C₂×C₃×C₈	—	5	2	no	—
99	C₂³×C₅	—	2	1	no	—
100	C₂³×C₅	C₃×D₄	1	1	yes	characteristic
101	C₅×Q₈	—	2	1	no	—
102	C₅×Q₈	C₂³×C₃	1	2	yes	—
103	C₅×D₄	—	2	4	no	—
104	C₅×D₄	—	2	1	no	—
105	C₅×D₄	C₂³×C₃	1	2	yes	—
106	C₂×C₄×C₅	—	2	1	no	—
107	C₂×C₄×C₅	—	2	4	no	—
108	C₂×C₄×C₅	C₃×D₄	1	1	yes	characteristic
109	C₂×C₄×C₅	C₃×D₄	1	2	yes	—
110	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
111	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
112	C₂×Dic₅	—	2	2	no	—
113	C₂×Dic₅	—	2	4	no	—
114	Dic₁₀	—	2	2	no	—
115	Dic₁₀	—	2	4	no	—
116	Dic₁₀	C₂³×C₃	1	4	yes	—
117	#168	C₂³×C₃	1	4	yes	—
118	C₂×#41	C₃×D₅	1	1	yes	characteristic
119	C₂²×Q₈	—	5	1	no	—
120	#137	—	5	8	no	—
121	C₂×Q₁₆	—	5	2	no	—
122	C₂×SD₁₆	—	5	2	no	—
123	C₂×M₁₆	—	5	1	no	—
124	C₂×C₃×C₅	—	2	2	no	—
125	C₂×C₃×C₅	—	4	2	no	—
126	C₂×C₃×C₅	#137	1	2	yes	complemented
127	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
128	C₂³×C₃	—	2	1	no	—
129	C₂³×C₃	C₂²⋊D₅	1	1	yes	characteristic
130	C₃×Q₈	—	2	1	no	—
131	C₃×Q₈	—	5	4	no	—
132	C₃×Q₈	—	10	4	no	—
133	C₃×Q₈	—	10	2	no	—
134	C₃×Q₈	C₂²×D₅	1	2	yes	—
135	C₃×D₄	—	2	4	no	—
136	C₃×D₄	—	2	1	no	—
137	C₃×D₄	C₂²×D₅	1	2	yes	—
138	C₂×C₃×C₄	—	2	1	no	—
139	C₂×C₃×C₄	—	2	4	no	—
140	C₂×C₃×C₄	—	10	4	no	—
141	C₂×C₃×C₄	—	10	2	no	—
142	C₂×C₃×C₄	C₂²⋊D₅	1	2	yes	—
143	C₂×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
144	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
145	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
146	C₃×C₈	—	5	4	no	—
147	C₂²×C₅	—	2	2	no	—
148	C₂²×C₅	—	2	2	no	—
149	C₂²×C₅	—	4	2	no	—
150	C₂²×C₅	C₂×C₃×D₄	1	2	yes	—
151	C₂²×C₅	C₂×C₃×D₄	1	1	yes	characteristic
152	C₄×C₅	—	2	2	no	—
153	C₄×C₅	C₂×C₃×D₄	1	2	yes	—
154	C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
155	C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
156	Dic₅	—	2	4	no	—
157	#41	—	2	2	no	—
158	#41	C₂×C₃×D₅	1	4	yes	—
159	C₂×Q₈	—	5	2	no	—
160	C₂×Q₈	—	5	4	no	—
161	C₂×Q₈	—	10	1	no	—
162	C₂×Q₈	—	10	2	no	—
163	C₂×Q₈	C₂×C₃×D₅	1	1	yes	characteristic
164	C₂×D₄	—	2	1	no	—
165	C₂×D₄	C₂×C₃×D₅	1	1	yes	characteristic
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	—	10	1	no	—
168	C₂²×C₄	C₂×C₃×D₅	1	1	yes	characteristic
169	Q₁₆	—	5	8	no	—
170	SD₁₆	—	5	8	no	—
171	M₁₆	—	5	4	no	—
172	C₂×C₈	—	5	2	no	—
173	C₃×C₅	C₂×#137	1	1	yes	characteristic, complemented, Hall
174	C₂²×C₃	—	2	2	no	—
175	C₂²×C₃	—	2	2	no	—
176	C₂²×C₃	—	4	2	no	—
177	C₂²×C₃	C₂×C₂²⋊D₅	1	2	yes	—
178	C₂²×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
179	C₃×C₄	—	2	2	no	—
180	C₃×C₄	—	10	4	no	—
181	C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
182	C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
183	C₃×C₄	C₂³×D₅	1	1	yes	characteristic
184	C₂×C₅	—	2	2	no	—
185	C₂×C₅	—	4	2	no	—
186	C₂×C₅	C₃×#137	1	2	yes	complemented
187	C₂×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
188	C₂³	—	2	1	no	—
189	C₂³	C₃×C₂²⋊D₅	1	1	yes	characteristic
190	Q₈	—	2	1	no	—
191	Q₈	—	5	4	no	—
192	Q₈	—	10	4	no	—
193	Q₈	—	10	2	no	—
194	Q₈	C₂²×C₃×D₅	1	2	yes	—
195	D₄	—	2	4	no	—
196	D₄	—	2	1	no	—
197	D₄	C₂²×C₃×D₅	1	2	yes	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	10	4	no	—
201	C₂×C₄	—	10	2	no	—
202	C₂×C₄	C₃×C₂²⋊D₅	1	1	yes	characteristic
203	C₂×C₄	C₃×C₂²⋊D₅	1	2	yes	—
204	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
205	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
206	C₈	—	5	4	no	—
207	C₂×C₃	—	2	2	no	—
208	C₂×C₃	—	4	2	no	—
209	C₂×C₃	#4097	1	2	yes	direct product factor
210	C₂×C₃	C₂²×C₂²⋊D₅	1	1	yes	characteristic
211	C₅	C₂×C₃×#137	1	1	yes	characteristic, complemented, Sylow
212	C₂²	—	2	2	no	—
213	C₂²	—	2	2	no	—
214	C₂²	—	4	2	no	—
215	C₂²	C₂×C₃×C₂²⋊D₅	1	2	yes	—
216	C₂²	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
217	C₄	—	2	2	no	—
218	C₄	—	10	4	no	—
219	C₄	C₂×C₃×C₂²⋊D₅	1	2	yes	—
220	C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
221	C₄	C₂³×C₃×D₅	1	1	yes	characteristic
222	C₃	C₂×#4097	1	1	yes	characteristic, direct product factor, Sylow
223	C₂	—	2	2	no	—
224	C₂	—	4	2	no	—
225	C₂	C₃×#4097	1	2	yes	direct product factor
226	C₂	C₂²×C₃×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×C₂²⋊D₅
commutator, cosocle	C₄×C₅	C₂⁴×C₃
Frattini	C₄	C₂³×C₃×D₅
Fitting, prime radical	C₂×C₃×C₅×#41	C₂
socle	C₂²×C₃×C₅	C₂×D₄
hypercenter	C₂×C₃×#41	D₅
hypocenter	C₅	C₂×C₃×#137
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×#41

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃×Dic₁₀	C₂	semidirect
3. C₃×#4097	C₂	direct (N × H)
4. C₂×C₃×#689	C₂	semidirect
5. C₂×C₃×#687	C₂	semidirect
6. C₂×C₃×#681	C₂	semidirect
7. C₂×#4097	C₃	direct (N × H)
19. C₂×C₃×Dic₁₀	C₂²	semidirect
20. C₃×#689	C₂²	semidirect
21. C₃×#687	C₂²	semidirect
22. C₃×#681	C₂²	semidirect
26. C₂²×Dic₁₀	C₂×C₃	semidirect
27. #4097	C₂×C₃	direct (N × H)
28. C₂×#689	C₂×C₃	semidirect
29. C₂×#687	C₂×C₃	semidirect
30. C₂×#681	C₂×C₃	semidirect
67. C₂×Dic₁₀	C₂²×C₃	semidirect
68. #689	C₂²×C₃	semidirect
69. #687	C₂²×C₃	semidirect
70. #681	C₂²×C₃	semidirect
126. C₂×C₃×C₅	#137	semidirect
173. C₃×C₅	C₂×#137	semidirect
186. C₂×C₅	C₃×#137	semidirect
209. C₂×C₃	#4097	direct (N × H)
211. C₅	C₂×C₃×#137	semidirect
222. C₃	C₂×#4097	direct (N × H)
225. C₂	C₃×#4097	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×C₅×#41	2
C₂²×C₃×Dic₁₀	2
C₃×#4097	2
C₂×C₃×#689	2
C₂×C₃×#687	2
C₂×C₃×#681	2
C₂×#4097	3
C₂×C₃×#137	5

Subgroups — C2×C3×#4097 #11755834

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#4097 #11755834