Subgroups — C₂×#91288 #11754124

Order 960.

532 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#235	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#681	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#6650	C₂	1	2	yes	maximal, complemented
5	#91288	C₂	1	8	yes	maximal, direct product factor
6	C₂×D₃×#168	C₂	1	2	yes	maximal, complemented
7	C₂×#4094	—	3	1	no	maximal, Hall
8	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
9	C₂×C₅×#67	C₄	1	2	yes	—
10	C₅×#235	C₄	1	4	yes	—
11	C₅×#235	C₂²	1	4	yes	—
12	C₂×C₅×D₁₂	C₄	1	1	yes	characteristic
13	C₂×C₄×C₅×D₃	C₂²	1	2	yes	—
14	C₂×C₅×Dic₆	C₄	1	1	yes	characteristic
15	C₂×C₃⋊#168	C₂²	1	1	yes	characteristic
16	C₂×C₃⋊#168	C₂²	1	1	yes	characteristic
17	C₂×C₃⋊#168	C₂²	1	4	yes	complemented
18	C₃×#681	C₂²	1	4	yes	complemented
19	C₂×C₃×#168	C₂²	1	2	yes	—
20	#6650	C₂²	1	8	yes	complemented
21	D₃×#168	C₂²	1	8	yes	complemented
22	C₂×#907	—	5	1	no	maximal, Hall
23	C₂×C₅×#41	—	3	1	no	—
24	#4094	—	3	8	no	—
25	C₂×#681	—	3	2	no	—
26	C₂×#681	D₃	1	1	yes	characteristic, complemented
27	C₂²×#168	—	3	1	no	—
28	C₂²×#168	—	3	2	no	—
29	C₂³×C₃×C₅	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₅×D₃	C₂×C₄	1	2	yes	—
31	C₂×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
33	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
34	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
35	C₅×#67	C₂×C₄	1	8	yes	—
36	C₂×C₅×Dic₃	C₂×C₄	1	2	yes	—
37	C₅×D₁₂	C₂×C₄	1	4	yes	—
38	C₄×C₅×D₃	C₂×C₄	1	4	yes	—
39	C₄×C₅×D₃	C₂³	1	4	yes	—
40	C₅×Dic₆	C₂×C₄	1	4	yes	—
41	C₃⋊#168	C₂³	1	2	yes	—
42	C₃⋊#168	C₂³	1	2	yes	—
43	C₃×#168	C₂³	1	4	yes	—
44	C₂×#235	D₅	1	1	yes	characteristic
45	C₂×C₃×M₁₆	—	5	1	no	—
46	C₂²×C₃⋊C₈	—	5	1	no	—
47	#907	—	5	8	no	—
48	C₂×ΓL(1,𝔽₂₅)	—	5	2	no	—
49	C₂×D₃×C₈	—	5	2	no	—
50	C₅×#41	—	3	4	no	—
51	C₅×#41	—	3	4	no	—
52	C₂×C₅×Q₈	—	3	1	no	—
53	C₂×C₅×D₄	—	3	2	no	—
54	C₂×C₅×D₄	—	3	1	no	—
55	C₂²×C₄×C₅	—	3	2	no	—
56	C₂²×C₄×C₅	C₂×D₃	1	1	yes	characteristic
57	#681	—	3	8	no	—
58	#681	C₂×D₃	1	4	yes	complemented
59	C₂×#168	—	3	1	no	—
60	C₂×#168	—	3	1	no	—
61	C₂×#168	—	3	4	no	—
62	C₂×#168	—	3	8	no	—
63	C₂×#168	C₂×D₃	1	2	yes	—
64	C₂×#131	—	15	1	no	Sylow
65	C₂²×C₃×C₅	—	2	2	no	—
66	C₂²×C₃×C₅	C₂²×C₄	1	2	yes	—
67	C₂²×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₅×D₃	—	2	4	no	—
69	C₂×C₅×D₃	C₂²×C₄	1	4	yes	—
70	C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
71	C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
72	C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
73	C₅×Dic₃	C₂²×C₄	1	4	yes	—
74	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic
75	C₂×#67	Dic₅	1	2	yes	—
76	#235	Dic₅	1	4	yes	—
77	#235	C₂×D₅	1	4	yes	—
78	C₂×D₁₂	Dic₅	1	1	yes	characteristic
79	C₂×C₄×D₃	C₂×D₅	1	2	yes	—
80	C₂×Dic₆	Dic₅	1	1	yes	characteristic
81	C₃×M₁₆	—	5	4	no	—
82	C₂×C₃×C₈	—	5	2	no	—
83	C₂×C₃⋊C₈	—	5	1	no	—
84	C₂×C₃⋊C₈	—	5	1	no	—
85	C₂×C₃⋊C₈	—	5	4	no	—
86	ΓL(1,𝔽₂₅)	—	5	8	no	—
87	D₃×C₈	—	5	8	no	—
88	C₂³×C₅	—	3	2	no	—
89	C₂³×C₅	C₄×D₃	1	1	yes	characteristic
90	C₅×Q₈	—	3	4	no	—
91	C₅×D₄	—	3	8	no	—
92	C₅×D₄	—	3	4	no	—
93	C₂×C₄×C₅	—	3	2	no	—
94	C₂×C₄×C₅	—	3	4	no	—
95	C₂×C₄×C₅	—	3	4	no	—
96	C₂×C₄×C₅	C₄×D₃	1	1	yes	characteristic
97	C₂×C₄×C₅	C₄×D₃	1	2	yes	—
98	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
99	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
100	#168	—	3	2	no	—
101	#168	—	3	2	no	—
102	#168	C₂²×D₃	1	4	yes	—
103	C₂×#41	—	3	1	no	—
104	#131	—	15	8	no	—
105	C₂×M₁₆	—	5	1	no	—
106	C₂×M₁₆	—	15	2	no	—
107	C₂²×C₈	—	15	2	no	—
108	C₂²×C₈	—	15	1	no	—
109	C₂×C₃×C₅	—	2	2	no	—
110	C₂×C₃×C₅	#131	1	2	yes	complemented
111	C₂×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
112	C₅×D₃	—	2	4	no	—
113	C₂³×C₃	C₂×Dic₅	1	1	yes	characteristic
114	C₂²×D₃	C₂×Dic₅	1	2	yes	—
115	C₂×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
116	C₂×C₃×C₄	C₂×Dic₅	1	2	yes	—
117	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
118	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
119	#67	C₂×Dic₅	1	8	yes	—
120	C₂×Dic₃	C₂×Dic₅	1	2	yes	—
121	D₁₂	C₂×Dic₅	1	4	yes	—
122	C₄×D₃	C₂×Dic₅	1	4	yes	—
123	C₄×D₃	C₂²×D₅	1	4	yes	—
124	Dic₆	C₂×Dic₅	1	4	yes	—
125	C₃×C₈	—	5	4	no	—
126	C₃⋊C₈	—	5	2	no	—
127	C₃⋊C₈	—	5	2	no	—
128	C₂²×C₅	—	2	2	no	—
129	C₂²×C₅	—	3	4	no	—
130	C₂²×C₅	—	6	4	no	—
131	C₂²×C₅	C₂×C₄×D₃	1	2	yes	—
132	C₂²×C₅	C₂×C₄×D₃	1	1	yes	characteristic
133	C₄×C₅	—	3	4	no	—
134	C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
135	C₄×C₅	C₂×C₄×D₃	1	2	yes	—
136	C₄×C₅	C₂³×D₃	1	1	yes	characteristic
137	#41	—	3	4	no	—
138	#41	—	3	4	no	—
139	C₂×Q₈	—	3	1	no	—
140	C₂×D₄	—	3	2	no	—
141	C₂×D₄	—	3	1	no	—
142	C₂²×C₄	—	3	2	no	—
143	C₂²×C₄	D₃×D₅	1	1	yes	characteristic
144	M₁₆	—	5	4	no	—
145	M₁₆	—	15	8	no	—
146	C₂×C₈	—	5	2	no	—
147	C₂×C₈	—	15	8	no	—
148	C₂×C₈	—	15	4	no	—
149	C₂×C₈	—	15	1	no	—
150	C₂×C₈	—	15	1	no	—
151	C₃×C₅	C₂×#131	1	1	yes	characteristic, complemented, Hall
152	C₂²×C₃	—	2	2	no	—
153	C₂²×C₃	C₂²×Dic₅	1	2	yes	—
154	C₂²×C₃	C₂²×Dic₅	1	1	yes	characteristic
155	C₂×D₃	—	2	4	no	—
156	C₂×D₃	C₂²×Dic₅	1	4	yes	—
157	C₃×C₄	C₂²×Dic₅	1	2	yes	—
158	C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
159	C₃×C₄	C₂³×D₅	1	1	yes	characteristic
160	Dic₃	C₂²×Dic₅	1	4	yes	—
161	C₂×C₅	—	2	2	no	—
162	C₂×C₅	—	6	4	no	—
163	C₂×C₅	#907	1	2	yes	complemented
164	C₂×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
165	C₂³	—	3	2	no	—
166	C₂³	D₃×Dic₅	1	1	yes	characteristic
167	Q₈	—	3	4	no	—
168	D₄	—	3	8	no	—
169	D₄	—	3	4	no	—
170	C₂×C₄	—	3	2	no	—
171	C₂×C₄	—	3	4	no	—
172	C₂×C₄	—	3	4	no	—
173	C₂×C₄	D₃×Dic₅	1	1	yes	characteristic
174	C₂×C₄	D₃×Dic₅	1	2	yes	—
175	C₂×C₄	C₂×D₃×D₅	1	2	yes	—
176	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
177	C₈	—	5	4	no	—
178	C₈	—	15	2	no	—
179	C₈	—	15	2	no	—
180	C₂×C₃	—	2	2	no	—
181	C₂×C₃	#4094	1	2	yes	complemented
182	C₂×C₃	C₂³×Dic₅	1	1	yes	characteristic
183	D₃	—	2	4	no	—
184	C₅	C₂×#907	1	1	yes	characteristic, complemented, Sylow
185	C₂²	—	2	2	no	—
186	C₂²	—	3	4	no	—
187	C₂²	—	6	4	no	—
188	C₂²	C₂×D₃×Dic₅	1	2	yes	—
189	C₂²	C₂×D₃×Dic₅	1	1	yes	characteristic
190	C₄	—	3	4	no	—
191	C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
192	C₄	C₂×D₃×Dic₅	1	2	yes	—
193	C₄	C₂²×D₃×D₅	1	1	yes	characteristic
194	C₃	C₂×#4094	1	1	yes	characteristic, complemented, Sylow
195	C₂	—	2	2	no	—
196	C₂	—	6	4	no	—
197	C₂	#91288	1	2	yes	direct product factor
198	C₂	C₂²×D₃×Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₃×D₅
commutator	C₂×C₃×C₅	C₂³×C₄
Frattini	C₄	C₂²×D₃×D₅
Fitting	C₂²×C₃×C₄×C₅	C₂²
socle	C₂²×C₃×C₅	C₂²×C₄
prime radical	C₂×C₅×#235	C₂
cosocle	C₃×C₄×C₅	C₂⁴
hypercenter	C₂²×C₄	D₃×D₅
hypocenter	C₃×C₅	C₂×#131
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃⋊#168	C₂	semidirect
3. C₂×C₃×#681	C₂	semidirect
4. C₂×#6650	C₂	semidirect
5. #91288	C₂	direct (N × H)
6. C₂×D₃×#168	C₂	semidirect
17. C₂×C₃⋊#168	C₂²	semidirect
18. C₃×#681	C₂²	semidirect
20. #6650	C₂²	semidirect
21. D₃×#168	C₂²	semidirect
26. C₂×#681	D₃	semidirect
58. #681	C₂×D₃	semidirect
110. C₂×C₃×C₅	#131	semidirect
151. C₃×C₅	C₂×#131	semidirect
163. C₂×C₅	#907	semidirect
181. C₂×C₃	#4094	semidirect
184. C₅	C₂×#907	semidirect
194. C₃	C₂×#4094	semidirect
197. C₂	#91288	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×C₅×#235	2
C₂²×C₃⋊#168	2
C₂×C₃×#681	2
C₂×#6650	2
#91288	2
C₂×D₃×#168	2
C₂×#4094	3
C₂×#907	5

Subgroups — C2×#91288 #11754124

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#91288 #11754124