Finite Group Databasebeta

← C₂×#11716773 #13103964

Subgroups — C₂×#11716773 #13103964

Order 1600.

280 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 71 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#87863	C₂	1	1	yes	maximal, characteristic
2	C₂×#87856	C₂	1	1	yes	maximal, characteristic
3	C₂×#87767	C₂	1	1	yes	maximal, characteristic
4	#11716773	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
6	C₂×#6034	—	2	1	no	—
7	#87863	—	2	1	no	—
8	#87863	C₂²	1	2	yes	—
9	C₂×#6033	—	2	1	no	—
10	#87856	—	2	1	no	—
11	#87856	C₂²	1	2	yes	—
12	C₂×C₅×#168	C₄	1	1	yes	characteristic
13	C₂×C₅×#168	C₄	1	1	yes	characteristic
14	#87767	C₄	1	2	yes	—
15	#87767	C₂²	1	2	yes	—
16	C₂×#3996	—	5	1	no	maximal
17	C₂×#3995	—	5	1	no	maximal
18	C₂²×C₅⋊D₅	Q₈	1	1	yes	characteristic
19	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
20	C₂×#1039	D₄	1	1	yes	characteristic
21	C₄×C₅⋊D₅	C₂×C₄	1	1	yes	characteristic
22	C₄×C₅⋊D₅	D₄	1	1	yes	characteristic
23	C₄×C₅⋊D₅	Q₈	1	1	yes	characteristic
24	C₄×C₅⋊D₅	C₂³	1	1	yes	characteristic
25	#6034	—	2	2	no	—
26	#6033	—	2	2	no	—
27	C₅×#168	C₂×C₄	1	2	yes	—
28	C₅×#168	C₂×C₄	1	2	yes	—
29	C₂×#700	—	5	1	no	—
30	C₂×#700	—	5	1	no	—
31	C₂×#700	—	5	1	no	—
32	C₂×#700	—	5	1	no	—
33	C₂×#700	—	10	2	no	—
34	C₂×C₈×D₅	—	5	1	no	—
35	C₂×C₈×D₅	—	5	1	no	—
36	#3996	—	5	4	no	—
37	#3995	—	5	4	no	—
38	C₂²×C₅²	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₅⋊D₅	—	2	2	no	—
40	C₂×C₅⋊D₅	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₅⋊D₅	C₂×Q₈	1	1	yes	characteristic
42	C₄×C₅²	C₄⋊C₄	1	1	yes	characteristic
43	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
44	#1039	C₄⋊C₄	1	1	yes	characteristic
45	#1039	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×D₅	—	5	1	no	—
47	C₂×C₄×D₅	—	5	1	no	—
48	C₂×C₄×D₅	—	10	2	no	—
49	C₂×#170	—	10	1	no	—
50	C₂×#170	—	10	1	no	—
51	C₂×#170	—	10	1	no	—
52	C₂×#170	—	10	1	no	—
53	C₂×#170	—	10	4	no	—
54	#700	—	5	2	no	—
55	#700	—	5	2	no	—
56	#700	—	5	2	no	—
57	#700	—	5	2	no	—
58	#700	—	10	4	no	—
59	#700	—	10	1	no	—
60	#700	—	10	1	no	—
61	#700	—	10	1	no	—
62	#700	—	10	1	no	—
63	#700	—	10	4	no	—
64	C₂×C₅×C₈	—	5	1	no	—
65	C₂×C₅×C₈	—	5	1	no	—
66	C₂×#168	AGL(1,𝔽₅)	1	1	yes	characteristic
67	C₂×#168	AGL(1,𝔽₅)	1	1	yes	characteristic
68	C₈×D₅	—	5	2	no	—
69	C₈×D₅	—	5	2	no	—
70	C₈×D₅	—	5	2	no	—
71	C₈×D₅	—	5	2	no	—
72	C₂×#108	—	25	1	no	Sylow
73	C₂×C₅²	#108	1	2	yes	complemented
74	C₂×C₅²	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₅⋊D₅	—	2	1	no	—
76	C₅⋊D₅	—	2	1	no	—
77	C₂²×D₅	—	5	1	no	—
78	C₂²×D₅	—	5	1	no	—
79	C₂²×D₅	—	10	2	no	—
80	C₂×C₄×C₅	—	2	2	no	—
81	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
82	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
83	C₂×Dic₅	—	5	1	no	—
84	C₂×Dic₅	—	5	1	no	—
85	C₂×Dic₅	—	10	2	no	—
86	C₄×D₅	—	5	1	no	—
87	C₄×D₅	—	5	1	no	—
88	C₄×D₅	—	5	1	no	—
89	C₄×D₅	—	5	1	no	—
90	C₄×D₅	—	5	1	no	—
91	C₄×D₅	—	5	1	no	—
92	C₄×D₅	—	5	1	no	—
93	C₄×D₅	—	5	1	no	—
94	C₄×D₅	—	10	2	no	—
95	C₄×D₅	—	10	2	no	—
96	C₄×D₅	—	10	2	no	—
97	C₄×D₅	—	10	2	no	—
98	#170	—	10	2	no	—
99	#170	—	10	2	no	—
100	#170	—	10	2	no	—
101	#170	—	10	2	no	—
102	#170	—	10	8	no	—
103	C₅×C₈	—	5	2	no	—
104	C₅×C₈	—	5	2	no	—
105	#168	C₂×AGL(1,𝔽₅)	1	2	yes	—
106	#168	C₂×AGL(1,𝔽₅)	1	2	yes	—
107	C₂×M₁₆	—	25	1	no	—
108	C₂×M₁₆	—	25	1	no	—
109	C₂²×C₈	—	25	1	no	—
110	#108	—	25	4	no	—
111	C₅²	C₂×#108	1	1	yes	characteristic, complemented, Sylow
112	C₂²×C₅	—	2	2	no	—
113	C₂²×C₅	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
114	C₂²×C₅	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
115	C₂×D₅	—	5	1	no	—
116	C₂×D₅	—	5	1	no	—
117	C₂×D₅	—	5	1	no	—
118	C₂×D₅	—	5	1	no	—
119	C₂×D₅	—	10	2	no	—
120	C₂×D₅	—	10	2	no	—
121	C₂×D₅	—	10	2	no	—
122	C₂×D₅	—	10	2	no	—
123	C₂×D₅	—	20	4	no	—
124	C₄×C₅	—	2	2	no	—
125	C₄×C₅	—	2	2	no	—
126	C₄×C₅	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
127	C₄×C₅	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
128	C₄×C₅	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
129	C₄×C₅	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
130	Dic₅	—	5	1	no	—
131	Dic₅	—	5	1	no	—
132	Dic₅	—	5	1	no	—
133	Dic₅	—	5	1	no	—
134	Dic₅	—	10	2	no	—
135	Dic₅	—	10	2	no	—
136	C₂²×C₄	—	25	1	no	—
137	M₁₆	—	25	2	no	—
138	M₁₆	—	25	2	no	—
139	M₁₆	—	50	1	no	—
140	M₁₆	—	50	1	no	—
141	C₂×C₈	—	5	1	no	—
142	C₂×C₈	—	5	1	no	—
143	C₂×C₈	—	25	2	no	—
144	C₂×C₈	—	25	2	no	—
145	C₂×C₈	—	50	1	no	—
146	C₂×C₈	—	50	1	no	—
147	C₂×C₅	—	2	4	no	—
148	C₂×C₅	—	2	2	no	—
149	C₂×C₅	#3995	1	2	yes	complemented
150	C₂×C₅	#3996	1	2	yes	complemented
151	C₂×C₅	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
152	C₂×C₅	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
153	D₅	—	10	1	no	—
154	D₅	—	10	1	no	—
155	D₅	—	10	1	no	—
156	D₅	—	10	1	no	—
157	D₅	—	20	2	no	—
158	D₅	—	20	2	no	—
159	C₂³	—	25	1	no	—
160	C₂×C₄	—	25	1	no	—
161	C₂×C₄	—	25	1	no	—
162	C₂×C₄	—	25	1	no	—
163	C₂×C₄	—	25	1	no	—
164	C₂×C₄	—	25	1	no	—
165	C₂×C₄	D₅⋊AGL(1,𝔽₅)	1	1	yes	characteristic
166	C₈	—	5	2	no	—
167	C₈	—	5	2	no	—
168	C₈	—	50	2	no	—
169	C₈	—	50	2	no	—
170	C₅	—	2	2	no	—
171	C₅	C₂×#3995	1	1	yes	characteristic, complemented
172	C₅	C₂×#3996	1	1	yes	characteristic, complemented
173	C₂²	—	25	1	no	—
174	C₂²	—	25	1	no	—
175	C₂²	—	50	2	no	—
176	C₂²	#87832	1	1	yes	characteristic
177	C₄	—	25	1	no	—
178	C₄	—	25	1	no	—
179	C₄	#87832	1	1	yes	characteristic
180	C₄	C₂×D₅⋊AGL(1,𝔽₅)	1	1	yes	characteristic
181	C₂	—	50	1	no	—
182	C₂	—	50	1	no	—
183	C₂	#11716773	1	2	yes	direct product factor
184	C₂	C₂×#87832	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#87832
commutator	C₄×C₅²	C₂²×C₄
Frattini	C₄	C₂×D₅⋊AGL(1,𝔽₅)
Fitting	C₂×C₄×C₅²	C₂×C₄
socle	C₂²×C₅²	C₄⋊C₄
prime radical	C₂²×C₅⋊D₅	Q₈
cosocle	C₄×C₅⋊D₅	C₂³
hypercenter	C₂×C₄	D₅⋊AGL(1,𝔽₅)
hypocenter	C₅²	C₂×#108
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
4. #11716773	C₂	direct (N × H)
73. C₂×C₅²	#108	semidirect
111. C₅²	C₂×#108	semidirect
149. C₂×C₅	#3995	semidirect
150. C₂×C₅	#3996	semidirect
171. C₅	C₂×#3995	semidirect
172. C₅	C₂×#3996	semidirect
183. C₂	#11716773	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×#87863	2
C₂×#87856	2
C₂×#87767	2
#11716773	2
C₂×#3996	5
C₂×#3995	5