Finite Group Databasebeta

Subgroups — #66112

Order 320.

258 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 99 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	#4149	C₂	1	1	yes	maximal, characteristic, complemented
2	#4147	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#137	C₂	1	1	yes	maximal, characteristic, complemented
4	#4096	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#688	C₂	1	1	yes	maximal, characteristic, complemented
6	#4065	C₂	1	2	yes	maximal, complemented
7	#4064	C₂	1	2	yes	maximal, complemented
8	#4060	C₂	1	2	yes	maximal, complemented
9	D₅×SD₁₆	C₂	1	2	yes	maximal, complemented
10	#4054	C₂	1	1	yes	maximal, characteristic, complemented
11	#4053	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#41	C₂²	1	1	yes	characteristic
13	C₂×C₅×Q₈	C₂²	1	1	yes	characteristic, complemented
14	#713	—	2	1	no	—
15	#713	C₂²	1	2	yes	—
16	#713	C₂²	1	2	yes	complemented
17	Q₈×D₅	—	2	1	no	—
18	Q₈×D₅	C₂²	1	2	yes	complemented
19	D₄×D₅	—	2	2	no	—
20	D₄×D₅	C₂²	1	2	yes	—
21	#709	—	2	2	no	—
22	#709	—	2	1	no	—
23	#709	C₂²	1	1	yes	characteristic
24	#709	C₂²	1	1	yes	characteristic, complemented
25	C₂×D₂₀	—	2	1	no	—
26	C₂×D₂₀	C₂²	1	1	yes	characteristic
27	C₅×Q₁₆	C₂²	1	2	yes	complemented
28	C₅×SD₁₆	C₂²	1	2	yes	complemented
29	C₅×M₁₆	C₂²	1	1	yes	characteristic, complemented
30	#689	C₂²	1	1	yes	characteristic, complemented
31	#688	C₂²	1	1	yes	characteristic, complemented
32	#688	C₂²	1	2	yes	complemented
33	#688	C₂²	1	2	yes	complemented
34	#687	C₂²	1	1	yes	characteristic, complemented
35	#686	C₂²	1	1	yes	characteristic, complemented
36	C₂×#168	C₂²	1	1	yes	characteristic, complemented
37	D₄₀	C₂²	1	2	yes	complemented
38	#677	C₂²	1	2	yes	complemented
39	#676	C₂²	1	2	yes	complemented
40	C₈×D₅	C₂²	1	2	yes	complemented
41	#577	—	5	1	no	maximal, Sylow
42	C₂²×D₅	—	2	1	no	—
43	C₂²×D₅	—	2	2	no	—
44	C₅×Q₈	—	2	1	no	—
45	C₅×Q₈	C₂³	1	2	yes	—
46	C₅×Q₈	C₂³	1	1	yes	characteristic
47	C₅×D₄	—	2	1	no	—
48	C₅×D₄	C₂³	1	1	yes	characteristic
49	C₂×C₄×C₅	—	2	1	no	—
50	C₂×C₄×C₅	—	2	1	no	—
51	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
52	C₂²⋊D₅	—	2	1	no	—
53	C₂²⋊D₅	—	2	2	no	—
54	C₂²⋊D₅	D₄	1	2	yes	—
55	D₂₀	—	2	2	no	—
56	D₂₀	—	2	2	no	—
57	D₂₀	—	2	1	no	—
58	D₂₀	D₄	1	1	yes	characteristic
59	D₂₀	C₂³	1	2	yes	—
60	D₂₀	C₂³	1	1	yes	characteristic
61	C₄×D₅	—	2	2	no	—
62	C₄×D₅	—	2	1	no	—
63	C₄×D₅	—	2	2	no	—
64	C₄×D₅	—	2	2	no	—
65	C₄×D₅	C₂³	1	2	yes	—
66	Dic₁₀	—	2	2	no	—
67	Dic₁₀	D₄	1	1	yes	characteristic
68	Dic₁₀	C₂³	1	1	yes	characteristic
69	C₅×C₈	C₂³	1	2	yes	—
70	#168	C₂³	1	1	yes	characteristic
71	#168	C₂³	1	1	yes	characteristic
72	2₋¹⁺⁴	—	5	1	no	—
73	H(5,𝔽₂)	—	5	1	no	—
74	#137	—	5	2	no	—
75	#137	D₅	1	1	yes	characteristic, complemented
76	AGL(1,ℤ/8)	—	5	2	no	—
77	AGL(1,ℤ/8)	—	5	1	no	—
78	#135	—	5	1	no	—
79	#135	—	5	2	no	—
80	C₂×SD₁₆	—	5	1	no	—
81	C₂×SD₁₆	—	5	2	no	—
82	#131	—	5	1	no	—
83	C₂²×C₅	—	2	1	no	—
84	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
85	C₂×D₅	—	2	1	no	—
86	C₂×D₅	—	2	2	no	—
87	C₂×D₅	—	8	2	no	—
88	C₂×D₅	—	8	1	no	—
89	C₂×D₅	C₂×D₄	1	2	yes	—
90	C₄×C₅	—	2	2	no	—
91	C₄×C₅	—	2	1	no	—
92	C₄×C₅	C₂×D₄	1	1	yes	characteristic
93	C₄×C₅	C₂⁴	1	1	yes	characteristic
94	Dic₅	—	2	1	no	—
95	Dic₅	C₂×D₄	1	2	yes	—
96	#41	—	5	1	no	—
97	#41	—	5	2	no	—
98	#41	—	5	2	no	—
99	#41	—	5	1	no	—
100	#41	—	10	2	no	—
101	#41	—	10	1	no	—
102	#41	—	10	1	no	—
103	#41	C₂×D₅	1	1	yes	characteristic
104	C₂×Q₈	—	5	2	no	—
105	C₂×Q₈	—	10	1	no	—
106	C₂×Q₈	C₂×D₅	1	1	yes	characteristic, complemented
107	C₂×D₄	—	5	2	no	—
108	C₂×D₄	—	5	1	no	—
109	C₂×D₄	—	10	1	no	—
110	C₂×D₄	—	10	2	no	—
111	Q₁₆	—	5	1	no	—
112	Q₁₆	C₂×D₅	1	2	yes	complemented
113	SD₁₆	—	5	2	no	—
114	SD₁₆	—	5	2	no	—
115	SD₁₆	—	5	1	no	—
116	SD₁₆	—	5	1	no	—
117	SD₁₆	—	5	2	no	—
118	SD₁₆	C₂×D₅	1	2	yes	complemented
119	D₈	—	5	1	no	—
120	D₈	—	5	2	no	—
121	M₁₆	—	5	2	no	—
122	M₁₆	C₂×D₅	1	1	yes	characteristic, complemented
123	C₂×C₈	—	5	1	no	—
124	C₂×C₈	—	5	2	no	—
125	C₂×C₅	—	2	1	no	—
126	C₂×C₅	—	4	1	no	—
127	C₂×C₅	C₂²×D₄	1	1	yes	characteristic
128	D₅	—	2	2	no	—
129	D₅	—	4	1	no	—
130	D₅	—	4	2	no	—
131	C₂³	—	10	2	no	—
132	C₂³	—	10	1	no	—
133	Q₈	—	2	1	no	—
134	Q₈	—	5	1	no	—
135	Q₈	—	5	1	no	—
136	Q₈	—	10	2	no	—
137	Q₈	C₂²×D₅	1	2	yes	—
138	Q₈	C₂²×D₅	1	1	yes	characteristic
139	D₄	—	2	1	no	—
140	D₄	—	5	2	no	—
141	D₄	—	5	1	no	—
142	D₄	—	5	1	no	—
143	D₄	—	5	2	no	—
144	D₄	—	10	2	no	—
145	D₄	—	10	2	no	—
146	D₄	—	10	2	no	—
147	D₄	—	10	1	no	—
148	D₄	—	10	1	no	—
149	D₄	C₂²×D₅	1	1	yes	characteristic
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	5	2	no	—
153	C₂×C₄	—	10	2	no	—
154	C₂×C₄	—	10	2	no	—
155	C₂×C₄	—	10	2	no	—
156	C₂×C₄	—	10	1	no	—
157	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
158	C₈	—	5	1	no	—
159	C₈	—	5	1	no	—
160	C₈	C₂²×D₅	1	2	yes	—
161	C₅	#577	1	1	yes	characteristic, complemented, Sylow
162	C₂²	—	2	1	no	—
163	C₂²	—	5	2	no	—
164	C₂²	—	10	1	no	—
165	C₂²	—	10	2	no	—
166	C₂²	—	40	1	no	—
167	C₂²	—	40	2	no	—
168	C₂²	D₄×D₅	1	1	yes	characteristic
169	C₄	—	2	1	no	—
170	C₄	—	2	2	no	—
171	C₄	—	5	2	no	—
172	C₄	—	10	1	no	—
173	C₄	D₄×D₅	1	1	yes	characteristic
174	C₄	C₂³×D₅	1	1	yes	characteristic
175	C₂	—	2	1	no	—
176	C₂	—	4	1	no	—
177	C₂	—	10	2	no	—
178	C₂	—	20	2	no	—
179	C₂	—	20	1	no	—
180	C₂	C₂×D₄×D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×D₅
commutator, cosocle	C₄×C₅	C₂⁴
Frattini	C₄	C₂³×D₅
Fitting	C₅×#137	C₂
socle	C₂×C₅	C₂²×D₄
hypercenter	#137	D₅
hypocenter	C₅	#577
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #137

Complements

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

N	H ≅ G/N	type
1. #4149	C₂	semidirect
2. #4147	C₂	semidirect
3. C₅×#137	C₂	semidirect
4. #4096	C₂	semidirect
5. C₂×#688	C₂	semidirect
6. #4065	C₂	semidirect
7. #4064	C₂	semidirect
8. #4060	C₂	semidirect
9. D₅×SD₁₆	C₂	semidirect
10. #4054	C₂	semidirect
11. #4053	C₂	semidirect
13. C₂×C₅×Q₈	C₂²	semidirect
16. #713	C₂²	semidirect
18. Q₈×D₅	C₂²	semidirect
24. #709	C₂²	semidirect
27. C₅×Q₁₆	C₂²	semidirect
28. C₅×SD₁₆	C₂²	semidirect
29. C₅×M₁₆	C₂²	semidirect
30. #689	C₂²	semidirect
31. #688	C₂²	semidirect
32. #688	C₂²	semidirect
33. #688	C₂²	semidirect
34. #687	C₂²	semidirect
35. #686	C₂²	semidirect
36. C₂×#168	C₂²	semidirect
37. D₄₀	C₂²	semidirect
38. #677	C₂²	semidirect
39. #676	C₂²	semidirect
40. C₈×D₅	C₂²	semidirect
75. #137	D₅	semidirect
106. C₂×Q₈	C₂×D₅	semidirect
112. Q₁₆	C₂×D₅	semidirect
118. SD₁₆	C₂×D₅	semidirect
122. M₁₆	C₂×D₅	semidirect
161. C₅	#577	semidirect

Maximal subgroups

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

subgroup	index
#4149	2
#4147	2
C₅×#137	2
#4096	2
C₂×#688	2
#4065	2
#4064	2
#4060	2
D₅×SD₁₆	2
#4054	2
#4053	2
#577	5